Dissertation Wenbin He - OPUS 4

GHz-rate harmonically mode-locked fibre laser using optoacoustic effects in photonic crystal fibre Harmonisch modengekoppelter Faserlaser mit GHz Wiederholungsrate unter Verwendung optoakustischer Effekte in photonischen Kristallfasern

Der Naturwissenschaftlichen Fakultät der Friedrich-Alexander-Universität Erlangen-Nürnberg zur Erlangung des Doktorgrades Dr. rer. nat.

vorgelegt von Wenbin He aus Zhenjiang (China)

Als Dissertation genehmigt von der Naturwissenschaftlichen Fakultät der Friedrich-Alexander-Universität Erlangen-Nürnberg

Tag der mündlichen Prüfung:

Vorsitzender des Promotionsorgans:

Gutachter:

28.02.2018

Prof. Dr. Georg Kreimer

Prof. Dr. Philip St.J. Russell Prof. Dr. Roy Taylor

ABSTRACT

This work is concerned with the use of enhanced optoacoustic interactions in the solid-core photonic crystal fibre (PCF) for achieving stable passive high-harmonic mode-locked (HHML) fibre lasers that can generate GHz-rate pulse trains and other types of long-range bound-states of solitons. This specially-designed solid-core PCF, with hollow-channels that surround the μm-sized core, is able to confine both the acoustic mode and the optical mode tightly within the fibre core, leading to a GHzrate optoacoustic interaction that is about two orders-of-magnitude stronger than in the conventional all-solid fibres. By inserting a short piece of such PCF into a conventional passive mode-locked ring-fibre laser, the pulse repetition rate can be locked to the acoustic resonance, while being largely decoupled from the meters-long cavity length, resulting in hundreds of evenly-spaced pulses in the cavity. Such a pulse sequence can coherently drive an acoustic wave in the PCF, while the acoustic wave can act back on the pulse sequence through the opto-elastic effect. The acoustic wave then effectively forms an optoacoustic lattice which divides the cavity round-trip time into hundreds of time-slots. In each time-slot, a single-soliton can be trapped. The temporal trapping potentials, which are formed by the cooperation between the acoustic wave and the cavity group velocity dispersion, ensured the self-stabilized pulse spacings and highly suppressed the relative timing-jitter. Based on this optoacoustic mode-locking mechanism, a stable soliton fibre laser at GHz-rate has been achieved with a wideband tunability and low noise level. In addition, by combining the optoacoustic mode-locking scheme with a newly discovered stretched-soliton effect, a fibre laser with simultaneous GHz-repetition-rate and sub-100-fs pulse duration was realized at a moderate pump power. The multi-pulses that are locked through the optoacoustic interactions in the

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PCF are uncorrelated in phase relations due to the non-interferometric nature of optoacoustic mode-locking. Each pulse can be regarded as an independent sub-lase pulse and thus can be manipulated individually. Using a unique addressing-pulse technique, the intra-cavity pulses can be selectively erased. The remaining pulses that are invariantly trapped in the optoacoustic lattice form a stable optomechanical bound-state which be preserved in the cavity for indefinitely long time. The arbitrarily-controlled on-and off-states of the pulses in different time-slots of the optoacoustic lattice have led to the realization of an all-optical bit-storage in the fibre laser cavity. The interplay between the optoacoustic effects and the dispersive wave perturbations, both being weak long-range interactions between intra-cavity solitons, has led to the generation of stable supramolecular assemblies of optical solitons in the fibre laser. The balance between the carefully-tailored long-range forces of attraction and repulsion, induced by the optoacoustic effects and the dispersive wave perturbations respectively, has enabled stable binding of multi-solitons within each time-slot, while their internal spacings can be over a hundred times of their individual durations. A diversity of long-range bound-states of solitons trapped within different lattice periods have constituted a highly-ordered structure that is distributed throughout the entire laser cavity, with features that resemble the chemical supramolecules, including configurational diversity, reversibility, structural flexibility and dynamic stability.

ii

ZUSAMMENFASSUNG

Diese Arbeit beschäftigt sich mit starken optoakustischen Wechselwirkungen, die in photonischen Kristallfasern (PKF) mit Glaskern auftreten und es ermöglichen, einen passiv modengekoppelten Faserlaser stabil bei hohen Harmonischen (HHML) der fundamentalen Wiederholrate zu betreiben. Dies kann verwendet werden, um Pulszüge mit Wiederholraten von einigen GHz zu erzeugen und andere Arten von langreichweitigen gebundenden Zuständen von Solitonen zu realisieren. Der Querschnitt solcher photonischer Kristallfasern weist eine regelmäßige Anordnung von Hohlkanälen um den Glaskern mit einem Durchmesser von einigen Mikrometern auf. Diese Mikrostruktur dient dazu, sowohl die optische Mode des geführten Lichts als auch die akustische Mode bei ihrer Grenzfrequenz auf engstem Raum zu lokalisieren. Die daraus resultierenden optoakustischen Wechselwirkungen bei Frequenzen von einigen GHz sind etwa zwei Größenordnungen stärker als in gewöhnlichen Glasfasern ohne mikrostrukturierten Mantel. Das Einsetzen eines kurzen PKF-Stücks in einen konventionellen passiv modengelockten Faser-Ringlaser erlaubt es, durch die akustische Resonanz bei einigen GHz die Pulswiederholrate festzulegen. Diese wird dadurch größtenteils von der Länge des Resonators ( einige Meter) unabhängig, so dass darin Hunderte von Pulsen mit festem Zeitabstand zirkulieren. Diese Pulssequenz regt wiederum die akustische Welle in der PKF kohärent an, wobei diese durch optoelastische Effekte auf die einzelnen Pulse rückwirkt. Dadurch kann die akustische Welle als eine Art optoakustisches Gitter aufgefasst werden, das den Resonator in Hunderte Zeitfenster einteilt, in denen sich jeweils ein einzelnes Soliton befindet. Dieser zeitliche Potentialtopf, welcher durch das Zusammenspiel der akustischen Welle und der optischen Dispersion im Resonator gebildet wird, führt zu selbst-stabilisierten Pulsabständen und ermöglicht es daher, deren Schwankungen

iii

effizient zu unterdrücken. Hier wird dieser Mechanismus der optoakustischen Modenkopplung dazu verwendet, um einen stabilen Solitonen-Faserlaser mit GHz-Wiederholrate, breitbandiger Abstimmbarkeit und niedrigem Rauschpegel aufzubauen. Wenn dieser Mechanismus zusätzlich mit dem neu entdeckten Effekt der gedehnten Solitonen kombiniert wird, kann ein Faserlaser realisiert werden, der gleichzeitig eine GHz-Wiederholrate und eine Pulsdauer von unter 100 Femtosekunden bietet. Desweiteren sind die Phasen der zahlreichen Pulse, die durch die optoakustischen Wechselwirkungen in der PKF miteinander verbunden sind, wegen der nichtinterferometrischen Natur der optoakustischen Modenkopplung unkorreliert. Daher lässt sich jeder Puls als eine Art unabhängiger Einzel-Laser auffassen, der individuell manipuliert werden kann. Mit Hilfe einer einzigartigen Technik zur deren Ansteuerung können die Pulse im Resonator selektiv abgeschaltet werden. Die übrigen Pulse, die unverändert im optoakustischen Gitter gefangen sind, bilden einen stabilen gebundenen optomechanischen Zustand, der für eine unbegrenzte Zeit aufrechterhalten werden kann. Die beliebig an- oder ausschaltbaren Zustände der Pulse in verschiedenen Zeitfenstern des optoakustischen Gitters ermöglicht es daher, den Resonator des Faserlasers als rein-optischen Bit-Speicher zu verwenden. Das Zusammenspiel der optoakustischen Effekte und der Störungen durch dispersive Wellen, beides schwache Wechselwirkungen der Laser-Solitonen mit langer Reichweite, kann dazu verwendent werden, supramolekulare Strukturen der optischen Solitonen im Resonator des HHML-Lasers zu erzeugen. Das Gleichgewicht der sorgfältig maßgeschneiderten langreichweitigen Anziehungs- und Abstoßungskräfte, die jeweils durch die optoakustischen Wechselwirkungen und die Störungen durch die dispersiven Wellen hervorgerufen werden, führt innerhalb eines jeden Zeitfensters zu stabilen gebunden Zuständen mehrerer Solitonen, deren Zeitabstände über einhundert mal größer als deren Pulsdauern sein können. Die verschiedenen langreichweitigen gebunden Zustände der Solitonen im optoakustischen Gitter weisen daher eine höchst geordnete Struktur auf, welche in ihren Eigenschaften den Supramolekülen in der Chemie ähnelt. Dies schließt Vielfalt der Konfigurationsmöglichkeiten, Umkehrbarkeit, strukturelle Flexibilität und dynamische Stabilität mit ein.

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CONTENTS

1 Introduction 2 Passive Mode-locked Fibre Lasers 2.1

2.2

2.3

2.4

11

Principle of passive mode-locking . . . . . . . . . . . . . . . . . . . . 11 2.1.1

Saturable absorber: Fast and slow . . . . . . . . . . . . . . . . 12

2.1.2

Artificial saturable absorbers: Kerr mode-locking . . . . . . . 15

2.1.3

Haus master equation: Time-domain description . . . . . . . . 22

2.1.4

Starting of passive mode-locking . . . . . . . . . . . . . . . . . 24

Soliton fibre lasers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.2.1

Soliton-like balance in mode-locked lasers . . . . . . . . . . . . 26

2.2.2

Average soliton model . . . . . . . . . . . . . . . . . . . . . . 28

2.2.3

Periodic perturbations of solitons: Kelly sidebands

. . . . . . 29

Ultrafast mode-locked fibre lasers . . . . . . . . . . . . . . . . . . . . 32 2.3.1

Limitations of soliton fibre lasers . . . . . . . . . . . . . . . . 32

2.3.2

Stretched-pulse fibre lasers . . . . . . . . . . . . . . . . . . . . 33

2.3.3

Dissipative-soliton fibre lasers . . . . . . . . . . . . . . . . . . 34

Multi-pulse states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.4.1

Passive harmonic mode-locking . . . . . . . . . . . . . . . . . 37

2.4.2

Soliton molecules in passive mode-locked fibre lasers . . . . . . 39

3 Optoacoustic Mode-locking 3.1

1

41

SRLS in solid-core PCF . . . . . . . . . . . . . . . . . . . . . . . . . 41 3.1.1

Stimulated Brillouin scattering: Backward and forward . . . . 41

3.1.2

SRLS with CW pump: Frequency conversion . . . . . . . . . . 45

v

Contents 3.1.3 3.2

SRLS with pulsed-light pump: Time-domain description . . . 47

Optoacoustic mode-locking using SRLS . . . . . . . . . . . . . . . . . 52 3.2.1

Temporal trapping potential theory . . . . . . . . . . . . . . . 52

3.2.2

Average soliton model with acoustic index modulation . . . . . 61

3.2.3

The non-interferometric HML . . . . . . . . . . . . . . . . . . 65

4 GHz-rate Soliton Fiber Lasers 4.1

4.2

4.3

Implementation of PCF . . . . . . . . . . . . . . . . . . . . . . . . . 69 4.1.1

Fabrication of PCF . . . . . . . . . . . . . . . . . . . . . . . . 69

4.1.2

The acoustic resonance of PCF . . . . . . . . . . . . . . . . . 71

4.1.3

The GVD of PCF . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.1.4

Splicing PCF with conventional fibres . . . . . . . . . . . . . . 79

4.1.5

Polarization control in PCF . . . . . . . . . . . . . . . . . . . 81

Wideband-tunable GHz-rate Er-fibre soliton laser . . . . . . . . . . . 82 4.2.1

Experimental set-up . . . . . . . . . . . . . . . . . . . . . . . 82

4.2.2

Results: Soliton train and wavelength tuning . . . . . . . . . . 84

4.2.3

Laser noise and stability . . . . . . . . . . . . . . . . . . . . . 87

4.2.4

Optical comb structure and inter-pulse phase-relationship . . . 89

Tm-fibre soliton laser at GHz-rate . . . . . . . . . . . . . . . . . . . . 91 4.3.1

Experimental set-up . . . . . . . . . . . . . . . . . . . . . . . 92

4.3.2

Laser output

. . . . . . . . . . . . . . . . . . . . . . . . . . . 94

5 Stretched-soliton Fibre Laser 5.1

5.2

5.3

vi

69

97

Stretched-soliton laser: Concept . . . . . . . . . . . . . . . . . . . . . 97 5.1.1

The energy dilemma for broadband pulses generation . . . . . 97

5.1.2

A different intra-cavity pulse self-consistency . . . . . . . . . . 98

Stretched-soliton laser: Experiments

. . . . . . . . . . . . . . . . . . 99

5.2.1

Experimental set-up . . . . . . . . . . . . . . . . . . . . . . . 99

5.2.2

GHz-rate sub-100-fs pulses output . . . . . . . . . . . . . . . . 100

5.2.3

Fundamental mode-locking state . . . . . . . . . . . . . . . . . 104

Analysis of intra-cavity pulse evolution . . . . . . . . . . . . . . . . . 105 5.3.1

Numerical simulations and validations . . . . . . . . . . . . . 105

5.3.2

Wavelength-dependent attenuator . . . . . . . . . . . . . . . . 108

5.3.3

The soliton-like balance . . . . . . . . . . . . . . . . . . . . . 109

Contents 6 All-optical Bit-storage 6.1 Working principles . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Optomechanical bound-states . . . . . . . . . . . . . 6.1.2 Writing bit-information in the HHML laser . . . . . . 6.2 Experimental set-up and results . . . . . . . . . . . . . . . . 6.2.1 External addressing pulses and the bit-storage cavity 6.2.2 Selective erasures of intra-cavity pulses . . . . . . . . 6.2.3 Demonstrations of long-term bit-storage . . . . . . .

. . . . . . .

. . . . . . .

7 Supramolecular Assemblies of Optical Solitons 7.1 Balance between long-range interactions . . . . . . . . . . . . . 7.1.1 The observation of balanced interactions . . . . . . . . . 7.1.2 Force of attraction due to optoacoustic effects . . . . . . 7.1.3 Force of repulsion due to dispersive wave perturbations . 7.1.4 Balance between two long-range forces . . . . . . . . . . 7.1.5 The analogy to chemical supramolecular assemblies . . . 7.2 Stable soliton supramolecule: Experiments . . . . . . . . . . . . 7.2.1 Experimental set-up . . . . . . . . . . . . . . . . . . . . 7.2.2 Stable all-double-soliton supramolecule . . . . . . . . . . 7.2.3 Stable all-triple-soliton supramolecule . . . . . . . . . . . 7.2.4 Configurational diversity and dynamic stability . . . . . 7.3 Structural flexibility: Tailoring of long-range interactions . . . . 7.3.1 Internal-spacing tuning: Tailoring of the dispersive waves 7.3.2 Internal-spacing tuning: Tailoring of the acoustic waves . 7.3.3 Directions of repulsive forces . . . . . . . . . . . . . . . .

. . . . . . .

. . . . . . . . . . . . . . .

. . . . . . .

. . . . . . . . . . . . . . .

. . . . . . .

113 113 113 115 118 118 120 124

. . . . . . . . . . . . . . .

129 129 129 133 137 141 143 144 144 146 149 151 155 155 158 160

8 Summary and Outlook

165

List of Figures

169

Bibliography

173

List of Publications

193

Acknowledgements

195

vii

CHAPTER

1 INTRODUCTION

Mode-locked lasers [1–3] are well known for their capability in generating ultra-short pulses, and are important in both practical applications and fundamental researches. On one hand, the format of ultra-short laser pulses confines optical energies into brief time intervals, in addition to the inherent coherence and brightness of lasers, leading to significantly enhanced peak powers, highly localized wave-packets, and broad spectrum bandwidth. For practical applications, ultra-short pulses can be employed for nonlinear light-matter interactions (e.g. material processing [4–6] and supercontinuum generation [7]), bit-information transmission [8–10], and optical frequency metrology [11, 12]. On the other hand, mode-locked lasers have also been widely investigated as platforms for both linear and nonlinear processes, including gain and loss dynamics [13–15], chromatic dispersions [16, 17], nonlinear pulse propagations [18, 19], optoacoustic effects [20], pulse-pulse interactions [21–23], etc. Investigations on these processes have in turn enriched the capabilities of mode-locked lasers in generating pulses with unprecedented parameters and formats. Moreover, mode-locked lasers exhibit profound relevance with many other physical phenomena, e.g. selforganization [24, 25], rogue waves [26, 27], Bose-Einstein condensations [28, 29], and chaos [26, 30, 31], exhibiting opportunities to contribute to a unified picture of various fields of physics. Since the advent of mode-locked lasers around fifty years ago [32], tremendous interests have continued to be vibrant in this field. The basic principle of a mode-locked laser can be interpreted in either the time domain or the frequency domain [33]. The term mode-locking stems from the frequency-domain interpretation, indicating that the axial modes in the laser cavity

1

Ch. 1

INTRODUCTION are mutually locked to a fixed phase relation. Particularly, for narrow pulses generation, the phase profile throughout the consecutive axial modes under the spectral envelope needs to be linear. As a result, these axial modes would periodically turn in phase and add up into a train of narrow pulses. In the time domain, the modelocking process can be understood as an iterative pulse-shaping process induced by an intra-cavity modulation which imposes higher losses for lower intensity light and meanwhile lower losses for higher intensity and has a period that matches the cavity round-trip time. Such modulation can be applied either in an active way through external modulation signal [34] (i.e. active mode-locking) or in a passive way through saturable absorbers [35] (i.e. passive mode-locking). Under the presence of such modulations, the intra-cavity pulse would constantly experience reductions of its tailing part whereas the intense peak part would be largely preserved, leading to ultra-short pulse generation. Fibre lasers are the among most popular platforms to implement mode-locking [2, 36–38]. Due to the geometric advantages of the fibre waveguide, fibre lasers can have compact configurations, efficient heat dissipation, high beam-qualities, easy preparations and maintenances. The advent of rare-earth doped fibres (e.g. with ytterbium, erbium or thulium ions as dopants) [39] has highly enhanced the capability of fibre lasers, which cover a wide range of emission wavelengths and have enabled remarkable power scaling. Mode-locked fibre lasers have given rise to an important category of pulsed lasers: the soliton lasers [40–42]. In such lasers, while the saturable absorbers contribute significantly to initial building-up processes and the suppressions of noise, the anomalous dispersion and Kerr nonlinearity of the optical fibre are able to form a soliton-like balance which dominates the pulse shaping under the steady operation [43]. Conventional soliton lasers consist of fibre sections with anomalous dispersion and the output pulses usually have durations ranging from ps to hundreds of fs with pulse energies below 1 nJ. Variations of the cavity dispersion and gain/loss maps can shift the mode-locked laser away from the “conventional” soliton regime to the more complicated dispersion-managed soliton [44] or dissipative-soliton regimes [18]. which are capable of generating pulses with nJ-level energies and fs durations. In most cases, a mode-locked laser encompasses only a single pulse circulating in the cavity (as shown in Fig.1.1(a) with an exemplary ring-cavity configuration), and thus delivers a periodic pulse sequence at a repetition rate equal to the cavity round-trip frequency fcav , i.e. the axial-mode spacing. This is the so-called funda-

2

mental mode-locking. In some other cases, a mode-locked laser can also encompass multiple evenly-spaced pulses simultaneously in the cavity, corresponding to the harmonic mode-locking (HML) (Fig.1.1(b)), provided that the intra-cavity modulation occurs at an integer multiple of the cavity round-trip frequency, leading to a repetition rate being N th -harmonic of the cavity round-trip frequency, i.e. N · fcav [45–47]. (a)

Single-pulse

Multi-pulse

(b)

Gain

Gain

Laser cavity

Laser cavity

Modulation

Modulation

Fig. 1.1: The schematic illustration of (a) the fundamental and (b) the harmonic mode-locking states in a ring-cavity laser.

The HML is potentially useful for pulse repetition-rate scaling, especially for GHz-rate pulse generations which can find applications in optical communications, high speed sampling, arbitrary waveform generations and precise material processing [6, 36, 38, 48]. The repetition rate of a typical mode-locked fibre laser is generally limited to MHz-range due to the meters-long cavity length [2, 49, 50]. Although scaling of the repetition rate to GHz-range can be achieved using a-few-cm length of optical cavity [51–56], the ultra-short cavity length would impose great difficulties in the practical implementations of functional components in the laser cavity, e.g. the gain fibre, the polarization controller, the couplers, and the isolator. The HML lasers, on the contrary, can have flexible cavity lengths, although high-harmonic mode-locking (HHML) with an order > 100 is usually required for the GHz repetition rate [57–61]. Unlike in the case of fundamental mode-locking, where the stability of repetition rate is largely ensured by the invariant cavity length, the HML demands additional mechanisms to ensure the equal spacings between consecutive pulses so as to have a well-defined repetition rate. Stable HML lasers are usually realized with active mode-locking schemes [33, 62–65], in which this external modulation provides the phase or intensity modulations, the frequency of which matches a certain harmonic of the cavity round-trip

3

Ch. 1

INTRODUCTION frequency. The intra-cavity multi-pulses can be trapped in the periodic potentials created by the external modulation, leading to equal spacings between them [34]. Since the cavity length inevitably suffers random fluctuations due to environmental disturbances, additional stabilizations of cavity length are demanded for longterm operations [62]. Therefore, the active HML usually requires expensive radiofrequency sources and sophisticated feedback electronics, rendering the entire laser configuration rather complicated. Moreover, active mode-locking at GHz repetitionrate can hardly generate ultra-short pulses with a duration below 100 fs [47, 65]. The passive mode-locking scheme has many advantages over the active modelocking. First of all, it is intrinsically capable of generating much shorter pulses [33]. Passive mode-locked lasers usually rely on intra-cavity saturable absorbers made from e.g. semiconductor (SESAM) [66], graphene [67], nano-tube [68], topological insulator [57], or artificial saturable absorber using Kerr-nonlinearity [3, 69–71]. The recovery time of these saturable absorbers could be very fast (e.g. a few fs in the case of artificial saturable absorber), and the induced intensity-modulation depends on the pulse shape itself, leading to a constant or even accelerating pulse-width reduction ratio during the pulse shortening process [3] and consequently, a significantly shorter pulse duration than in the active mode-locking. Secondly, passive mode-locked lasers have simpler configurations, leading to lower costs compared to their active counterparts. Most saturable absorbers have miniaturized sizes and can be easily integrated into all-fibre lasers, rendering the construction of mode-locked fibre laser systems extremely simple [2]. The attractive advantages of passive mode-locked lasers, however, come along with some notable disadvantages. First of all, passive mode-locked lasers are notoriously known for their tricky adjustments and poor stabilities, since the intra-cavity pulses would experience a wide range of self-induced nonlinear effects [33]. In addition, the multi-pulse operation of such lasers can hardly be controlled, making stable HML rather difficult to realize [33, 39]. Unlike in the active mode-locking where the external modulation can force the spacings between the intra-cavity pulses to be equal, the pulses in passive mode-locked lasers do not necessarily have equal spacings and usually experience large temporal fluctuations [72, 73] due to random perturbations. Therefore, multi-pulse states in passive mode-locked lasers are conventionally considered as “unstable and undesired forms of behaviours” [33]. Many efforts have been reported, trying made to realize stable HML in passive mode-locked lasers, for example using gain saturation effects [74, 75], sub-cavity

4

configurations [76–80], and optoacoustic effects [45]. The gain saturation effect is usually ultra-weak due to the long response time of the laser gain and therefore incapable to suppress the timing-jitter of the intra-cavity pulses efficiently. Moreover, the harmonic order, i.e. the pulse number in the laser cavity is erratic [59, 61, 81], especially in the soliton-regime where the pulse number increases accordingly with higher pump power mainly due to the soliton quantization effect [82, 83]. The subcavity configuration employs a short cavity coupled to the main-cavity in order to force a specific harmonic-order in the main cavity [76]. However this scheme is intrinsically interferometric and thus relies on the precise control of cavity lengths against random fluctuations, similar to the case of active mode-locking. The optoacoustic effect, as an inelastic nonlinear effect [84], stands out as a unique mechanism that can be utilized for stable passive HML [45, 60]. The optical fibre, due its geometric profile, can serve as waveguides for both optical and acoustic waves. As the pulses propagate in a fibre cavity, the acoustic wave can be coherently driven in the fibre due to the electrostriction effect [20, 85–87]. The optically-driven acoustic wave acts back on the pulses through the opto-elastic effect, exhibiting itself as a refractive index modulation [20]. The modulation frequency is simply determined by the acoustic resonance frequency, which is usually much higher than the cavity round-trip frequency. With some proper elaborations of cavity parameters, the multiple pulses could eventually have equal temporal spacings due to the longrange interactions induced by the acoustic wave, resulting in the stable HML with a repetition rate locked to the acoustic resonance. The underlying process evolved in this interaction is actually the guided-wave forward stimulated Brillouin scattering (SBS) [88] which stems from the coherent interaction between transverse acoustic resonance of the fibre and the intra-cavity pulse train. The frequency of this acoustic resonance is determined solely by the transverse fibre dimension. Therefore the repetition rate of the laser pulses is not only decoupled from the cavity length, but also invariant at different lasing wavelengths or pump powers. Although the optoacoustic effects have great potentials in passive HML, the reality is that they are usually ultra-weak in the conventional fibre [87], due to the limited overlapping between the optical and the acoustic modes. Such weak optoacoustic effects can hardly be used in practice [20, 46, 49, 56, 81] and were even considered as noise sources in some systems [89]. Actually, although this mechanism has been brought up for over 20 years [39], stable HML lasers based on optoacoustic effects were rarely seen in the literature. As mentioned in the review paper [39],

5

Ch. 1

INTRODUCTION special fibre designs are demanded in order to significantly enhance the optoacoustic effects, otherwise such long-range interactions between intra-cavity pulses would only be too weak to dominate over other noise sources and lead to a steady-state mode-locking. The advent of solid-core photonic crystal fibre (PCF) [90–92] has totally changed the landscape of optoacoustic effects in fibres [93–96] and thus brought exciting opportunities for the renaissance of passive HML [60]. PCFs are well known for their enriched degrees of freedom in waveguide designs, including e.g. mode profiles [97, 98] and dispersion relations [99, 100]. A typical PCF has a structure that is shown in the scanning electron microscopic (SEM) photo in Fig.1.2, which has a solid-core with a diameter of a few μm, surrounded by a triangular array of hollow channels which functions as the microstructured cladding-layer [95]. In general, depending on the structure configuration, such hollow channel array creates either a photonic bandgap [101, 102] that confines the light in the solid-core, or simply an effective step-index profile [100]. The PCFs used for the researches covered by this thesis belong to the latter case, in which the core size and air-filling ratio are the key parameters [100] in the waveguide designs. (a)

10 Pm

(b)

2 Pm

Fig. 1.2: The scanning electron microscopic photo of a typical solid-core PCF. (a) The entire structure. (b) the core and cladding region formed by hollow channels

The tight confinement enabled by the hollow channels in the PCF leads to a small optical mode-area and thus high nonlinearities in the PCF-core [103]. Meanwhile, the discontinuity of the material impedance between the silica and air also tightly confines the transverse acoustic mode in the PCF-core [95]. Consequently, the overlap between the optical mode and the acoustic mode is significantly improved, resulting in enhanced optoacoustic interactions.The forward SBS, which is generally ultra-weak in the conventional fibre becomes remarkably strong in such PCF. Since the PCF-core acts like a μm-sized mechanical oscillator that scatters the

6

pump light in the forward direction, resembling features of the stimulated Raman scattering that originates from the vibrations of molecules, such highly enhanced forward SBS in the PCF is also termed as the stimulated Raman-like scattering (SRLS) [95]. Moreover, due to the small size of the PCF-core, the acoustic resonance frequency can be easily scaled up to GHz-range [96], bringing the possibility of high-harmonic mode-locking (HHML) at GHz-rate [60]. By inserting a short length of solid-core PCF in a conventional passive modelocked fibre laser, the enhanced optoacoustic effect in the PCF can dominate the interactions between the multiple pulses in the cavity [104–106]. The pulse repetition rate of such laser is then mainly determined by the acoustic resonance frequency of the PCF-core instead of the cavity length, given that one specific harmonic of the cavity round-trip frequency lies within the acoustic gain band of the PCF. The index modulation induced by the acoustic wave acts in concert with the cavity dispersion and thus create a series of equally spaced temporal trapping potentials [107], leading to evenly-distributed pulses in the cavity with highly suppressed timing-jitter. The passive HHML laser based on the enhanced optoacoustic effect in the PCF has three key features. Firstly, the emergence of such HHML state is a self-organized process. Not only the individual pulse-shaping is a passive process, but also the arrangement of the hundreds of pulses in the cavity from a disordered pattern to a periodic pulse train is a completely self-organized process [104]. Secondly, this HHML state is self-stabilized due to the presence of temporal trapping potentials against external perturbations. Additional stabilizations are thus not demanded for the long-term operation, rendering the configuration of such HHML laser extremely simple. Thirdly, the HHML state based on the optoacoustic effect is intrinsically non-interferometric, indicating that multiple pulses are uncorrelated in phase relations [105]. Each pulse in the cavity can be regarded as an independent sub-laser pulse that circulates in the cavity at the round-trip frequency [107]. The advantages and exciting opportunities brought by these unique features will be unfolded in later chapters of this thesis. The first phase of this work focused the development of GHz-rate HHML fibre lasers based on the SRLS process in the PCF. We successfully implemented the PCF into a conventional mode-locked fibre laser and made a GHz-rate soliton Erfibre laser at 1.55 μm featuring remarkably small timing-jitter and good long-term stability [104] and additionally continuous wavelength-tuning ability [105]. The same principle was applied to a Tm-fibre laser, and a GHz-rate mode-locked fibre

7

Ch. 1

INTRODUCTION laser around 2 μm was accomplished [106]. The theoretical work of the optoacoustic mode-locking, especially the temporal trapping potential theory, was also completed meanwhile [107]. As a step forward, we developed a unique technique called the stretched-soliton effect and combined it with the optoacoustic mode-locking technique, so that sub-100 fs pulses can be generated at GHz-rate with moderate pump power [108]. The second phase of this work concerned the development of an all-optical bitstorage [107], which was initiated by the fact that the multi-pulses in such HHML laser were independent and could be turned on and off individually. The acoustic core-resonance of the PCF effectively divides the cavity round-trip into hundreds of time-slots, within each of which there could be either one pulse trapped, or no pulse at all, forming a complicated binary pulse sequence that constitutes a optomechanical bound-state. We developed a unique addressing-pulse technique to erase selected pulses in this HHML laser, and the remaining pulse sequence that is left over in the optoacoustic lattice, carrying the bit-information, could be stored in the cavity for indefinitely long time. The third phase of this work stems from an unexpected discovery concerning another long-range interactions between solitons: the dispersive wave perturbations [109, 110], which have long been regarded as noise sources and were far from being controllable [111–113]. We have however observed the unique phenomenon that, by carefully tailoring the dispersive waves and acoustic waves in a HHML soliton fibre laser, the long-range interactions caused by these two different effects can precisely balance each other, leading to the robust binding of multiple solitons within each time-slot of the HHML laser. The various long-range bound-states of solitons that are trapped in the optoacoustic lattice have mutually formed a stable soliton supramolecule that extends to the entire laser cavity and exhibits similar exotic properties of supramolecules in the fields of chemistry and biology [114]. This thesis is structured as follows: In Chapter 2, a brief review of passive mode-locked fibre laser is presented, including general descriptions of passive mode-locked lasers, the conventional soliton lasers, and the dissipative soliton lasers. Then the multi-pulse phenomenon in passive mode-locked lasers is then shortly reviewed, including the HML state and the soliton molecules. In Chapter 3, the optoacoustic mode-locking mechanism is

8

introduced. The principle of the SRLS effect is given at the beginning of this chapter. Then the interactions between the multi-pulses and the acoustic wave in the mode-locked laser are analytically presented. Then the self-stabilization of the optoacoustic mode-locking scheme is interpreted using the temporal trapping potential theory. In Chapter 4, the experimental details of the GHz-rate HHML soliton fibre laser based on the solid-core PCF are presented, including both the Er-fibre and Tm-fibre lasers. In Chapter 5, the experiment details and the underlying mechanism of the stretched-soliton fibre laser are presented. Such laser can deliver highly-stable pulse train with simultaneous GHz repetition-rate and sub-100-fs pulse duration. In Chapter 6, the all-optical bit-storage based on the optomechanical bound-states and an information-addressing technique is presented. In Chapter 7, the supramolecular assemblies of optical solitons originating from the inter-play of two long-range interactions: the optoacoustic effect and the dispersive wave perturbations, are demonstrated with both analytic interpretations and experimental elaborations. At last, a summary of my PhD work and some discussions on the possible future topics are given in Chapter 8.

9

CHAPTER

2

PASSIVE MODE-LOCKED FIBRE LASERS

This chapter provides a brief review of passive mode-locked fibre lasers, concerning their fundamental principles, typical configurations and several working regimes. The review starts with the saturable absorbers, the key element for all the passive mode-locked lasers, with special focus upon artificial saturable absorbers originating from Kerr nonlinearity. In addition, the interplay between cavity group velocity dispersion and self-phase modulation could give rise to soliton-like balance in the mode-locked laser cavity, resulting in soliton-laser regime, while the discrete nature of the fibre sections in the mode-locked soliton fibre laser necessitates the average soliton model. The perturbations upon the solitons caused by discrete fibre parameters would lead to accumulations of dispersive waves at a series of phasedmatched frequencies, which becomes one of the major constraints of the available pulse energies and temporal durations. In order to achieve shorter pulses, different regimes are introduced to the mode-locked lasers, including the stretched-pulse and the dissipative-soliton lasers. At last, multi-pulse phenomena in passive modelocked fibre lasers are briefly discussed, including the harmonic mode-locking and the soliton molecules.

2.1 Principle of passive mode-locking All mode-locked lasers relies on certain forms of intra-cavity modulations in order to favour pulsed waveforms over CW lasing. This is because, otherwise, the gain medium of a laser by itself always provides a higher gain for a narrow-band CW-

11

Ch. 2

PASSIVE MODE-LOCKED FIBRE LASERS waveform than for a broadband pulse mainly due to limited gain bandwidth. The intra-cavity amplitude modulation induces a higher loss for low intensity light while providing transmission window for high intensity light. During the starting process of the mode-locking, a randomly induced long pulse would be reshaped into a sharp pulse during the iterative pulse reshaping, while the weak background would be quickly suppressed. In active mode-locking, this process is accomplished with an external RF signal, and a fixed periodic amplitude modulation is enforced upon the intra-cavity waveforms. A general consequence is that, as the pulse is getting shorter, the pulse width reduction would also become weaker, which would finally be balanced by gain narrowing effect [115]. Therefore, an active mode-locked laser by itself usually results in ps or tens of ps pulses. In contrast, passive mode-locking relies on self-amplitude modulation (SAM), which means that the strength of modulation is determined by the pulse shape itself [3, 33]. Therefore the strength of pulse width reduction would be maintained or even accelerated during the pulse shaping process, resulting in a much shorter pulse duration than in the active mode-locking. The SAM in passive mode-locked laser induced by the saturable absorbers is generally considered as result of nonlinear process, which can be best described as shaping of pulses in the time domain [33]. In practical systems, a wide class of different effects would influence the pulse-shaping process, and they would be gradually introduced in the following sections.

2.1.1 Saturable absorber: Fast and slow Saturable absorbers in passive mode-locked lasers react to incoming light intensity with correspondingly varied transmittance. Specifically, they impose large loss upon weak signal, whereas tend to become transparent for intense signal. Therefore, provided initially with a long pulse, the peak portion of the pulse would experience a lower loss than the tailing part, leading to effective shortening of pulses after iterative round-trips in the cavity. Conventional saturable absorbers are made from real absorptive materials, of which the saturated absorption originates from the depletion of the population difference between the ground state and the upper state [84, 116]. In this sense, the saturable absorption follows the same differential equation as the saturable gain in laser amplifiers, except for a change of sign [33]. The saturable absorber can be characterized with two temporal parameters: the absorption recovery time T1 which represents the energy relaxation, and the de-phasing time T2 which represents the

12

Principle of passive mode-locking

2.1

phase relaxation [33]. In general, given a broadband absorber, the pulses in the mode-locked laser with duration over sub-ps range would have a slow temporal variation compared to T2 . Therefore the saturation process can be largely viewed as solely an energy decay process [33]. This process can be simply modelled by a 2-level system that describes a homogeneous atomic transition. Depending on the relative duration of the recovery time T1 and the modelocked pulse duration τp , the saturable absorbers can be categorized into fast [117] and slow absorbers [116]. In the case of fast absorbers, i.e. with T1  τp , the loss of the absorber reacts instantaneously to the optical intensity of the pulse, which would create a transmission window, the profile of which follows the shape of the pulse. In the case of slow absorbers, i.e. with T1  τp (while T1 is still much shorter than round-trip time TR ), the absorber would saturate with the accumulated energy of the incoming pulse and only recover the loss long after the complete passage of the pulse. The dynamics of population difference density Δn of the absorber provided with the incident pulse can be described as [116, 117] ∂Δn Δn − Δn0 σA |a(t)|2 =− − Δn ∂t T1 ω0

(2.1)

where a(t) is the electric field of the pulse temporal profile normalized to the intensity, i.e. I(t) = |a(t)|2 , Δn0 is the equilibrium population difference density, σA is the optical cross section of the absorbing particle,  the reduced Plank constant, and ω0 the carrier-frequency of the pulse. The absorptive loss q(t), being proportional to Δn, thus follows a similar equation [35]: ∂q q − q0 |a(t)|2 =− − q ∂t T1 Es

(2.2)

where Es is the saturation energy density defined as Es = ω0 /σA , and q0 is the unsaturated small-signal loss. In the limit of fast absorbers with very short T1 , the solution of Eq.(2.2) approaches [35]: q0 q(t) = (2.3) 1 + I(t)/Is where I(t) denotes the pulse intensity profile and Is = ω0 /σA T1 is the saturation intensity. Given that the saturation is relatively weak, then Eq.(2.3) can be

13

Ch. 2

PASSIVE MODE-LOCKED FIBRE LASERS approximated by

q0 |a(t)|2 = q0 − γm |a(t)|2 q(t) = q0 − Is

(2.4)

where γm is the modulation depth of the SAM provided by the fast absorber. On the contrary, for slow absorbers, the absorption depends weakly on the exact pulse shape (mainly the ascending part). In this limit, Eq.(2.2) has the following approximate solution [3, 35, 43]: 

q(t) = q0 exp −

 t −∞

2

|a(t)| dt /Es



(2.5)



t where −∞ |a(t)|2 dt= Ep (t) stands for the accumulated pulse flux density received by the absorber. Obviously, Eq.(2.5) is valid only in the vicinity of the pulse, since the loss would gradually recover over much longer time (T1 ) after the passage of the pulse. The modulation functions Eqs.(2.4) and (2.5) would be used in the Haus master equation (Section 2.1.3) that describes the averaged evolution of the laser pulse in the mode-locked cavity. The instantaneous loss profiles of the fast and slow saturable absorber are compared schematically in Fig.2.1.

(a)

(b)

loss

loss

gain

pulse

pulse

Fig. 2.1: The absorption dynamics of fast and slow saturable absorbers. (a) Fast saturable absorber. (b) Slow saturable absorber, which cooperate with gain saturation dynamics to form a narrow transmission window

The effects of fast and slow absorbers are different mainly in the following features. Firstly, during the starting phase with a long pulse, the pulse-width reduction ratio, defined as rp = τp2 /τp1 (τp1 and τp2 being the pulse-width before and after the absorber), would largely be constant for slow absorbers [33]. For fast absorbers, however, the initial modulation would be rather weak for the long pulse, but the reduction ratio would continuously increase as the pulse becomes narrower. Generically, fast absorbers are capable of forming shorter pulses at the price of the more difficult starting process [3]. Secondly, for slow absorbers, since the transmission window cannot close immediately after the passage of the pulse, the leading

14

Principle of passive mode-locking

2.1

edge of the pulse generally experiences stronger modulation than the tailing edge does, causing asymmetric pulse shapes, while such effect is essentially absent with fast absorbers. In fact, with only the efficacy of the slow absorber, noisy satellite spikes following the main pulse could not be well suppressed, and additional pulseshaping effects are needed. In early realizations of mode-locked lasers, dynamics of gain saturation were arranged to cooperate with the saturable absorption to form a transmission window which was much narrower than the recovery time of the saturable absorber (as illustrated in Fig.2.1(b)) [118]. However, such effect can only occur in gain media with fast relaxation time (e.g. in dye laser). For the rare-earthdoped gain medium typically used in fibre laser, the relaxation time is very long (∼ μs or ms-range) [119], thus for pulse repetition rate exceeding MHz range, the gain is saturated only by the average power of the pulse train. The soliton-like balance due to presence of anomalous dispersion and Kerr nonlinearity can provide additional pulse shortening effect, while the slow absorbers would be responsible mainly for eliminating the noise background under steady operations [43]. A saturable absorber usually has a recovery time that ranges from sub-ps to the ns range. For short enough pulses, all saturable absorbers become slow absorbers. There exists, however, a category of the so-called artificial saturable absorbers based on Kerr nonlinearity which have much shorter recovery time (a few fs). Such ultrafast absorbers are of great importance in many mode-locked fibre lasers delivering fs-pulses, and will be discussed in the next section.

2.1.2 Artificial saturable absorbers: Kerr mode-locking The generic strategy for making artificial saturable absorbers is to transform the SPM-induced nonlinear phase shift ϕNL into the SAM that resembles the conventional saturable absorption. Since the response time of the Kerr-nonlinear process is only of a few fs, the recovery time of such artificial saturable absorber would not impose any essential limit of the resultant pulse duration in practical applications. The most widely used configuration of artificial saturable absorbers is the socall additive pulse mode-locking (APM) [120], which relies on interferometric superposition of two pulses originating from the splitting of the same pulse. Such two pulses are arranged to experience different ϕNL , and so are the different parts of an individual pulse (the pulse peak experiences more ϕNL than the pulse wings). Consequently, given a proper phase bias, the superposition would cause the peak part to have constructive interferences, while the less intense wings to have destructive

15

Ch. 2

PASSIVE MODE-LOCKED FIBRE LASERS interferences, leading to effective shortening of the pulse. The key of APM mechanism lies in the ϕNL experienced by the split pulses, while detailed differences in practical schemes arise in terms of how exactly the pulses are split and interfere with each other. Conventional APM with auxiliary cavity The early realization of APM scheme makes use of an auxiliary cavity that is coupled to the main-cavity via a mutual transmitting mirror (or generally at the output coupler), as shown in Fig.2.2 [40, 121]. The pulse in the main-cavity would be split into two pulses at the mutual mirror. One of them is launched into the auxiliary cavity and experiences SPM in the Kerr medium. When round-trip time of the pulse in the auxiliary cavity matches that of the pulse in the main-cavity, they would interfere at the mutual mirror. Consequently, due to the inhomogeneity of the interference as described above, an effectively shorter pulse would be returned into the main cavity. Usually a linear phase bias was needed in the auxiliary cavity, and it was originally done by stretching the fibre using a piezoelectric transducer. This auxiliary cavity would then effectively act as a terminal with a nonlinear reflectivity, which resembles a saturable absorber mirror with ultra-short response time. Auxillary cavity

Main cavity M1

M2

Kerr medium

Fig. 2.2: Schematic of the conventional APM action using an auxiliary cavity with Kerr medium inside, which is coupled to the main cavity via mutual mirror M1. Interferometric coupling at M1 leads to a shorter pulse to be returned back to the main cavity

Given that the pulse in the auxiliary cavity experiences considerable loss, the interference only leads to a weak modulation on the pulse amplitude. The nonlinear reflectivity at the mutual coupling mirror could then be expressed as: [120, 122] 



R = r + ta 1 − r2 cos (ϕNL + ϕbias )

(2.6)

where r is the amplitude reflectivity of the mirror, ta represents the loss of the

16

Principle of passive mode-locking

2.1

auxiliary cavity (for a large loss ta  1). ϕNL and ϕbias are the nonlinear phase shift and linear phase bias experienced by the pulse in the auxiliary cavity. ϕNL = κ |a(t)|2 , where κ is the effective nonlinear coefficient which is proportional to the Kerr medium length and the Kerr coefficient. The reflectivity actually follows a sinusoidal relation with ϕNL , thus a proper ϕbias is required to make an artificial saturable absorber. Given that ϕNL is relatively small and ϕbias is close to −π/2, the modulation of such APM can be expressed using Eq.(2.4), where the modulation depth γm can be determined as [3] 



γm = −κta 1 − r2 sin (ϕbias )

(2.7)

This artificial saturable absorber would become saturated when ϕNL is on the order of π. In fact nearly all the Kerr-type artificial saturable absorbers would have such feature, as would be discussed later. The first realization of the artificial absorber was achieved by L. F. Mollenauer, et al in 1984 [40], leading to the first soliton laser and was interpreted by E. P. Ippen, et al in 1989 using the APM theory [120]. Haus pointed out later that the soliton-like balanced induced by anomalous GVD of the fibre in this first realization was actually not absolutely necessary [122]. The APM mechanism, which is intrinsically a nonlinear process leading to spectral broadening, can be balanced by gain narrowing effects alone, and even normal dispersion in the auxiliary cavity can still ensure stable operation under the APM scheme. One disadvantage of the conventional APM scheme is that the auxiliary cavity length needs to be held fixed against noisy fluctuations in order to maintained stabilized interferometric superposition at the coupling mirror [2, 122]. Many other APM schemes were later brought up, in which the split pulses propagate in a mutual cavity. As a results the fluctuations are automatically cancelled out between the pulses, and interferometric stabilization was no longer demanded. Two schemes that have widely used in mode-locked fibre lasers are the nonlinear fibre loop mirror and the polarization APM, as described below. Nonlinear fibre loop mirror The nonlinear fibre loop mirror, as a variation of conventional APM scheme, makes use of a 2 × 2 directional fibre coupler for the pulse splitting, with one output port being fed back into the other one [123, 124]. Therefore, the split pulses

17

Ch. 2

PASSIVE MODE-LOCKED FIBRE LASERS would counter-propagate in the same loop cavity, and interfere with each other at the coupler after one round-trip, resembling a Sagnac interferometer. Such interference would re-distribute the power back into the original input port and the other input port. Since the two pulses propagate in the same cavity, the environmental perturbations on the cavity length experienced by the two pulses would be cancelled out. In order to introduce difference in the ϕNL between the two counter propagating pulses, as required in APM scheme, either a non-50/50 coupler is used [123] or an asymmetrically-placed gain is inserted in such external loop [124] so as to make the two pulse to have different nonlinear phase shifts. The ladder case is called the nonlinear amplifying loop mirror (NALM), which has the advantage of having an extra degree of freedom, i.e. the adjustable pump power for this external gain section. The re-distribution of the power back to the two input ports would be power-dependent, and could be arranged so that the peak part of the pulse would be more reflected than the tailing part (or more transmitted into the other input port, depending on desired scheme of the main cavity). Since the interference is polarization-state- and wavelength-dependent, a polarization controller needs to be implemented in the loop. Using polarization-maintaining (PM) fibres to construct such loop is also a feasible option [125]. A schematic of a typical NALM scheme is shown in Fig.2.3. Under the “transmission mode”, i.e. the intense part of the pulse launched to Port-1 tends to be transmitted into Port-2, and the amplifier section in this loop mirror is regarded as a lumped element located close to the 50/50 coupler [124]. Port-2

Port-3

gain

50/50 Port-1

Port-4

Fig. 2.3: Schematic of the NALM operated in transmission mode, with 50/50 coupler and lumped gain section.

The spilt pulses that counter-propagate in the loop are assumed to have the same polarization state. If the launched pulse has an initial intensity of Ii = |a(t)|2 , the nonlinear phase shift experienced by the two counter-propagating pulses would be [3]

18

Principle of passive mode-locking

2.1

Ii Ii and ϕcp = κ (2.8) 2 2 respectively, where g is the power amplification ratio of the gain section. Then the amplitude of the pulse launched to Port-2 would depend sinusoidally upon the difference of the phase shifts in Eq.(2.8), with a necessary addition of phase bias ϕbias . Given weak nonlinear phase shifts, the modulation coefficient γm can be approximated as [3] κ γm = − (g − 1) sin (2ϕbias ) (2.9) 4 The nonlinear fibre loop mirror can be combined with a ring-fibre cavity to form a figure-8 cavity [49, 82, 126, 127] which has been particularly successful in generating fs-pulses [128–130]. When using PM-fibres, such figure-8 mode-locked lasers can be highly stabilized against environmental perturbations [131]. ϕp = κg

Polarization controller

Polarizer Kerr medium (fibre)

NPR elliptically polarized light

Fig. 2.4: Schematic of the polarization APM principle. The input pulse is modified into an elliptical polarization by a polarization controller. Then it is launched into a Kerr-medium (fibre) where the NPR occurs. A polarizer is placed after that, which reshapes the pulse profile accordingly, configuring an artificial fast saturable absorber

Polarization APM Another popular APM scheme for mode-locked fibre lasers is the polarization APM [42, 132, 133], which relies on the nonlinear polarization rotation (NPR)1 that occurs during the propagation of intense pulses along optical fibres [134]. With the polarization discrimination provided by a linear polarizer (or a polarized beamsplitter), the intensity-dependent NPR can be transformed into intensity-dependent loss [135], and given a proper initial polarization-bias, an artificial saturable absorber can be configured. 1

In some literature it is also called the nonlinear polarization evolution (NPE).

19

Ch. 2

PASSIVE MODE-LOCKED FIBRE LASERS Early realizations of the NPR-based APM relied on birefringent fibres. By launching a linear polarized light that is slightly angled with respect to the principle axis of the fibre, two orthogonal linearly-polarization components would experience different phase shift and finally re-combined into a modified polarization state before being discriminated by the polarizer [70, 136]. One limitation of this scheme is that, since the polarization evolution in fibre is wavelength-dependent, such configuration would induce a spectral fringe that limits the final pulse bandwidth. In practice, the length of the birefringent fibre cannot be too long and the pulse should be intense enough for a sufficient SAM. In non-birefringent fibres, NPRs can still occur if the launched pulse has elliptical polarization state [71]. Such pulse can be regarded as a combination of two orthogonal circularly-polarized (or linearly-polarized) pulses with different amplitudes. After propagating in the fibre, the nonlinear phase shifts of the two pulses, contributed by both self- and cross-phase modulation (SPM and XPM), would be different, leading to a variation of final polarization state. Such configuration actually resembles a nonlinear Mach-Zehnder interferometer (MZI), and the key features of APM are preserved: the pulse is effectively split and experiences different nonlinear phase shift. Similarly, a bias is required for a proper working point so that the high intensity pulse would experience a lower loss, forming an artificial saturable absorber. The schematic of a polarization APM configuration is illustrated in Fig.2.4. The nonlinear phase shifts of the two orthogonal circularly-polarized light, denoted as a± (t), can be express as [1] 

ϕN L± = κc |a± (t)|2 + 2 |a∓ (t)|2



(2.10)

in which κc is the nonlinear coefficient of the medium. Obviously only if a+ (t) and a− (t) differ in amplitude (as required for elliptically polarized light), ϕN L would be different for these two circularly polarized pulses. Given the ratio between the two √ pulses being r/ 1 − r2 , the modulation coefficient γm can be approximated as [3] γm = −

 r κc  2 2r − 1 √ sin (2ϕbias ) 4 1 − r2

(2.11)

where the phase bias and the amplitude ratio can be controlled by the polarization controllers in the cavity. The polarization APM has been particularly suitable to fibre laser, since all the relevant components are fibre-compatible and could be configured within a ring-

20

Principle of passive mode-locking

2.1

cavity. This has paved the way for constructing extremely simple and compact all-fibre cavity and in fact has become the most successful configuration for modelocked fibre lasers [2]. All the three schemes of APM based on Kerr-nonlinearity mentioned above tend to saturate with excessive nonlinear phase shift, and the complete response in terms of the cavity loss is a sinusoidal function of the light intensity (see Fig.2.5). The more precise description of these artificial saturable absorptions in terms of the loss function should be [3] 

q(t) = q0 − sin γm |a(t)|2



(2.12)

Gain / Loss

Therefore, the behaviour of such APM action would not always be saturable absorption, favouring more intense pulses. Given some other intensity region, it may become an effective saturable “gain” which imposes a higher loss for more intense pulses.

Working point (gain = loss)

Saturable absorption

Intensity-dependent loss

Gain in EDFA Light intensity in the cavity (a.u.)

Fig. 2.5: Sinusoidal response of the Kerr-based artificial saturable absorber. The intensity-dependent nonlinear phase shift creates a sinusoidal loss profile (blue curve) which encompasses periodic regions of saturable loss and effective saturable “gain” due to the over-driving. In cooperation with the laser gain profile (red line), a stable working point can be formed which can clamp the pulse intensity

This sinusoidal response of artificial saturable absorbers would be very useful in suppressing pulse intensity fluctuations especially under the case of high-harmonic mode-locking. By properly adjusting the modulation depth and the working point, the Kerr-based APM can provide high suppression of weak noise background (with saturable gain), whereas the much more intense multi-pulses would be clamped to a fixed intensity due to the opposite slope of the absorption curve in cooperation with the laser gain, forming a stable “working point” (as shown in Fig.2.5). Moreover, the

21

Ch. 2

PASSIVE MODE-LOCKED FIBRE LASERS working point could be easily adjusted by varying the linear phase bias and making the system extremely flexible for a wide range of pulse parameter. Besides, the sinusoidal response of such APM action would be utilized for a special addressing pulse technique for the HHML fibre soliton laser, which will be described in Section 6.2.2. Spatial beam-width

Dynamic self-focusing

Kerr medium

Fig. 2.6: Schematic of the Kerr-lens mode-locking principle. The input pulse experiences a dynamic self-focusing in the Kerr medium, resulting in narrowed beam widths around the peak region. With an aperture of finite opening, the pulse would be effectively narrowed due to strong trimming of the pulse wing.

Kerr-lens mode-locking Another widely used mode-locking scheme, especially in bulk-solid-state laser, is the Kerr-lens mode-locking (KLM) which relies on the dynamic self-focusing effect [137]. Using an intra-cavity small aperture with a finite opening, the dynamic self-focusing effect, which leads to a smaller beam area for a higher power, can be transformed by an optical aperture into artificial saturable absorption which imposes less loss for higher power, as shown in Fig.2.6. Such transformation can also be accomplished with a “soft-edge” aperture, which is configured by launching the dynamically self-focused beam into a laser gain medium with a spatially-varied gain profile across its transverse section. Usually the self-focusing effect should be avoided in fibres due to the detrimental damage that it might bring along to the fibre structure. In bulk crystals with relatively large mode-area and shorter length, such effects could be better controlled. The KLM scheme is intrinsically suitable for broadband operation and could be applied to a wide range of wavelength, and it has been particularly successful in Ti-sapphire mode-locked lasers [138, 139].

2.1.3 Haus master equation: Time-domain description The saturable absorber is only one of the functional elements, though being the key one, that constitute the entire mode-locked laser. Many other effects could also

22

Principle of passive mode-locking

2.1

provide pulse-shaping effects during the iterative round-trip propagation of pulses in the cavity. H. A. Haus has been well know for summarizing these attributes that could shape pulses in mode-locked lasers into a time-domain master equation [1, 122]. The basic idea of this master equation is to calculate the pulse change per roundtrip induced by various effects in the cavity. To pursuit the steady-state solution of the lasers, we can simply make this change to be zero. The Haus master equation makes a series of simplifications. Firstly, all these shaping effects are assumed to be homogeneously distributed in the cavity, i.e. the variations of pulse parameters within a single round-trip are not considered, although they are inevitable for modelocked lasers. Secondly, the pulse change per round-trip is assumed to be small and could be reasonably regarded as elementary. Thirdly, only one pulse is considered to exist in the cavity. Besides, a moving-frame [134] is used for describing single pulse propagation, with the short-term variable t being the relative time that describes the pulse envelope, and long-term variable T which describes round-trip propagation. The dominant effects that could be incorporated in the master equation include the linear loss, the laser gain (with power saturation and bandwidth narrowing effect), the group velocity dispersion, the saturable absorber and the SPM. Note that in the case of artificial saturable absorbers, the same Kerr nonlinearity contribute to both the saturable absorption action as well as SPM, whereas they are considered separately in the master equation. A typical form of the Haus master equation is as following [122, 140]: 

∂ 1 ∂2 TR a(t) = g 1 − 2 2 ∂T ωc ∂t





∂2 − l0 + γm |a(t)| + iDg 2 − iδ |a(t)|2 − iΨ a(t) ∂t (2.13) 2

On the right-hand side (RHS), TR is the cavity round-trip time, T is the long-term variable. The RHS actually represents the net change over one cavity round-trip, i.e. TR ∂a(t)/∂T = Δa(t). On the left-hand side (LHS), the first term represents the laser gain with a bandwidth limitation. Note that in the time domain, the frequency dependence of gain (ω − ω0 )2 is transformed into the second-order derivative ∂ 2 /∂t2 [122]. ωc is the gain bandwidth and l0 is the linear loss of the cavity. The loss could also become frequency-dependent to represent spectral filtering effects [18]. The γm |a(t)|2 term denotes the fast saturable absorber effect as discussed in the preceding section. The iDg ∂ 2 /∂t2 term represents linear dispersion, in which Dg = 12 β2 L, β2 being the group velocity dispersion and L the propagation length (i.e the cavity

23

Ch. 2

PASSIVE MODE-LOCKED FIBRE LASERS length). The SPM effect is denoted by iδ |a(t)|2 , where δ = (2π/λ) n2 L is the Kerr phase modulation coefficient. At last, the term iΨ is the linear phase shift due to the round-trip pass. There could also be a gain saturation that modifies the value of g. Here we assume that the gain has very long relaxation time and thus becomes saturated only by the average power of the pulse train [122], i.e. g = g0

1 1 1+ TR Is

 TR 0

2

|a(t)| dt

(2.14)

in which g0 is the small signal gain and Is is the saturation intensity. Note that we assume that the pulse spectrum centres at the peak and the local gain profile can be approximated simply with a centred quadratic function. In general, there could also be a linear component which causes an imbalanced gain. Generally this would cause a constant drift of carrier-frequency unless balanced by other effects. We will see later in the Section 3.2.2 that, the optoacoustic effect could lead to continuous frequency conversion, then such gain imbalance would become necessary to fix the carrier-frequency. To obtain the steady-state solution, the RHS is set to zero. Generally, this equation has a solution of hyperbolic-secant pulse profile, together with some frequency chirp across the pulse [122]. However, the pulse shaping mechanism could be very different depending on specific cavity parameters. If the linear dispersion and Kerr nonlinearity are negligible, then the balance would mainly be maintained between the SAM and the gain narrowing effect [122]. If the cavity has additionally normal dispersion, the balanced can still be maintained given that the SPM is not too strong [1, 3]. If, on the contrary, the cavity has an anomalous GVD, then the balanced between the SPM and GVD can lead to soliton-like balance [43, 141, 142]. In many cases, the soliton-like balance would become dominant, while the gain and saturable absorber would only function as weak modulations to stabilize the pulse propagation, leading to soliton laser regime (See Section 2.2)

2.1.4 Starting of passive mode-locking The passive mode-locking usually starts from a noisy background, with a certain random spike being passively selected and favoured by the saturable absorber. During this process, two thresholds are reached successively while increasing the pump

24

Principle of passive mode-locking

2.1

power [33]. The first threshold would be that of the population inversion for a small noisy signal, in which the unsaturated gain could balance the unsaturated absorption. The second threshold would be reached with a higher pump power when a strong noisy spike gains enough power and starts to “burn” itself through the saturable absorber, leading to even higher power compared to the other weaker spikes. Meanwhile the gain saturation would occur due to the increasing pulse energy and finally balances the saturated absorption. On the other hand, other weak spikes and background noises would be quickly eliminated due to the net losses. The process between the first and second threshold is largely a statistic process, unlike in the case of active mode-locking in which the modulation is enforced. The chances that a certain spike can successfully evolve into a stable mode-locked laser pulse depends on the mutual coherence time τc of the axial modes of the cavity [143]. Actually, the initial noisy spikes result from the random mode-beating fluctuations, and such fluctuations would decay within the characteristic time τc which is related to the half-bandwidth of the first beat note of the axial modes (τc = 1/πΔf3dB ) [143]. Stable mode-locked pulses can be created only when the initial fluctuations are strong enough and could be built up within the mutual coherence time (i.e the characteristic decaying time). Therefore, in order to have a higher chance of stable mode-locking, τc should be long enough, or equivalently, the beat note spike should be as narrow as possible. Detrimental broadening of the beat note is mainly caused by two effects. The first effect is the spurious cavity reflections [144] that could occur at the interfaces of different components or sections of the cavity. This could create etalon effect due to the multiple Fabry-Perot cavities and thus unevenly distributed axial modes. The second effect is the spatial hole-burning effect that could induce both gain and refractive index gratings [145]. This would cause frequency-shifts of the cavity axial modes (mode-pulling effect). Beside these deleterious effects, the starting of Kerr mode-locking is especially difficult since the modulation is usually very weak at the beginning, hindering the complete self-starting of the laser [3]. The employment of a unidirectional ring-cavity [133] has been reported to greatly facilitate the self-starting of the mode-locked laser with a low threshold. This is mainly due to the fact that the spurious reflections in the cavity can be highly suppressed by the intra-cavity isolator to the first order (the double-reflected light would become much weaker). In addition, the mode-pulling effects due to spatial hole-burning is weaker in unidirectional ring cavity than in linear cavities [2]. However, due to the finite start-up time of the grating structure of population

25

Ch. 2

PASSIVE MODE-LOCKED FIBRE LASERS inversion inside the gain medium, some external perturbations can help to erase temporally the grating [145] and allow for strong mode-beating within the perturbation time so as to efficiently initiate the self-starting process. In the NPR-based APM ring-fibre laser, such perturbations can be induced simply by rotating the polarization controllers (which would influence the intro-cavity loss) or by changing the pump power under a proper working point. For harmonic mode-locked laser, there exist a third threshold, beyond which the multi-pulses would arise evolve into a regular pattern with equal spacing between them. During the experiments of the HHML fibre laser, this threshold is reached with even higher pump power than required for the previous two thresholds.

2.2 Soliton fibre lasers 2.2.1 Soliton-like balance in mode-locked lasers The Haus master equation (Eq.(2.13)) describes how different effects shape the intra-cavity pulse of a mode-locked laser in the time domain. Particularly, with the presence of negative GVD and SPM, a soliton-like balance could occur and even dominate the pulse shaping process, and such balance is practically most accessible in mode-locked fibre lasers due to the strong Kerr nonlinearity that could arise from the long interaction length in optical fibre. In addition, standard single-mode fibres (SMF-28) usually exhibit anomalous dispersion in the infrared domain (∼ 1.55 μm) which overlaps with the common lasing spectral region of erbium and thulium-doped fibre lasers, making the soliton-like balance easily achieved in such lasers. When the soliton-like balance dominates the intra-cavity pulse shaping, the SAM provided by the saturable absorber would only provide weak modulations to suppress the background noise. Besides, for rare-earth doped fibre lasers, the gain medium can only respond to the average power of the pulse train due to the long relaxation time. With these cavity parameters, this mode-locked fibre laser would become a soliton fibre laser. Soliton fibre lasers could be constructed with either slow or fast saturable absorbers. With slow absorbers, the soliton-like balance can determine the pulse duration which can be significantly shorter than the recovery time of the absorber, while the noisy spikes with lower intensities can be suppressed by the saturable absorber, ensuring stable operation of the mode-locked laser [43]. With fast saturable

26

Soliton fibre lasers

2.2

absorber, especially those based on Kerr nonlinearity, the recovery time of the (artificial) absorber would no longer impose any practical limit upon the pulse duration, thus the soliton-like balance would basically determine the resultant pulse duration. In fact, when the SAM and gain/loss balance are both rather weak, the Haus master equation would degrade to the well-known nonlinear Schrödinger equation (NLSE) that describes the nonlinear pulse evolution in the laser cavity [1] ∂a ∂2 TR = iDg 2 a − iδ |a|2 a ∂T ∂t

(2.15)

Assuming that the pulse propagates homogeneously in the cavity with constant parameters, the equation can be rewritten into a more familiar form [134], with the long time variable T replaced by propagation distance z: i

β2 ∂ 2 ∂a = a − γ |a|2 a 2 ∂z 2 ∂t

(2.16)

in which γ is the nonlinear coefficient defined as γ = ω0 n2 /cAeff , where ω0 is the pulse carrier-frequency, n2 the nonlinear coefficient, and Aeff the effective mode area. Given anomalous dispersion and positive nonlinear coefficient, this equation supports a fundamental “bright” soliton solution with a sech-shaped envelope, i.e. 





t 1 a(z, t) = A0 sech exp i |A0 |2 z τ0 2



(2.17)

where A0 and τ0 are the peak amplitude and pulse duration of the fundamental soliton. An important property of the fundamental soliton solution is that the peak amplitude and pulse duration are related by the soliton-area theorem, which states that 2 |Dg | |β2 | |A0 | · τ0 = = (2.18) δ γ In terms of the peak power P0 = |A0 |2 and the widely used full-width-at-halfmaximum (FWHM) pulse duration τFWHM = 1.76τ0 , the soliton area theorem could be rewritten as: 3.11 |β2 | 2 P0 · τFWHM (2.19) = γ Note that the SAM provided by saturable absorber in soliton fibre lasers, though probably being very weak, is vital for stable operation. After all, pure soliton

27

Ch. 2

PASSIVE MODE-LOCKED FIBRE LASERS without any SAM would never exist in mode-locked lasers, since the narrow-band CW-like light would always experience a higher gain than the relatively broad-band solitons and therefore would dominate [3, 43]. In practice, though, for NPR-based APM laser, a weak CW component could possibly co-propagate with the pulse under steady state, particularly at high pump powers [146]. The appearance of the CW light in the soliton spectrum usually indicates the over-driving of the APM due to the excessive soliton energy [113].

2.2.2 Average soliton model The Haus master equation assumes that all the pulse-shaping effects are homogeneously distributed within one cavity round-trip, and the variations of pulse parameters have been neglected. However, the pulse would never actually propagate in the ideal way that is described by the Eq.(2.16) where GVD is homogeneous and no dissipative effects are involved. In practical mode-locked cavity, periodic gain, loss and dispersion variation along different sections are inevitable. In fact, the periodic gain/loss compensation could be considerable in mode-locked lasers, and the gain fibre usually differs from other sections in GVD, leading to pulse duration variations in the cavity. The Haus master equation could become invalid under such large variations of pulse parameters in a rigid sense. However, if the overall balance between the anomalous GVD and SPM is maintained in the cavity despite those variations, the mode-locked laser could still support the so-called average solitons [147, 148] (or guiding-center solitons [149]), in the sense that the pulse parameters resemble that of pure solitons, provided that they are averaged over one cavity round-trip. It has been proved that stable average soliton propagation in periodically amplified transmission line (i.e. the soliton profile would always return to its initial shape after each period) could be realized only when the amplification period za is much shorter than the soliton period z0 , i.e. [150] za  z0 =

π π τ02 LD = 2 2 |β2 |

(2.20)

The same soliton area theorem that is described by Eq.(2.18) also applies for the average soliton in terms of averaged amplitude A¯0 and duration τ¯0 . The soliton laser cavity can be equivalently viewed as a periodic amplifier sequence to a large extend, with additional SAM provided by the saturable absorber. Therefore this average soliton model can also be applied to the soliton laser cavity, provided that

28

Soliton fibre lasers

2.2

the soliton period is much longer than the cavity length. Note that although the peak amplitude of the intra-cavity soliton could oscillate within a considerable range under the average soliton model, the pulse duration breathing due to the discrete GVD and Kerr nonlinearity of different sections can only be perturbative in order that the averaged parameters remain valid. Under this condition, the intra-cavity pulses then basically experience the averaged cavity dispersion β ave and Kerr nonlinearity γ ave , so that we can estimate the soliton duration from the soliton energy simply by 3.52|β2ave | E¯p · τ¯FWHM ≈ γ ave

(2.21)

in which E¯p and τ¯FWHM denote the averaged soliton energy and duration in the cavity.

2.2.3 Periodic perturbations of solitons: Kelly sidebands Conventional soliton fibre lasers that work in the average soliton domain are usually restricted to pulse durations of many ps in order to fulfill the condition of za  z0 . However, deviations from this restriction can still result in stable soliton lasers, with sub-ps duration, due to the presence of the saturable absorber which provides the stabilizing SAM effect [150]. In this case, the cavity length could be a significant fraction of the soliton period, and considerable perturbations have to be incorporated into the average soliton model. Such perturbations, which physically originate from the discrete nature of cavity parameters, would result in strong coupling of the soliton energy into dispersive waves [110]. In fact, the optical soliton is well known for being a nonlinear attractor which remains robust in terms of shape preservation against perturbations, through shedding of dispersive waves [151]. The dispersive waves shed by the intra-cavity solitons would in general not be able to maintain a fixed phase relation with the soliton itself, due to the fact that the weak dispersive waves experience only linear phase shifts while the solitons would experience additional nonlinear phase shifts due to the SPM effects [150]. However, for some frequency components, the relative phase shifts of the dispersive wave would be integer multiples of 2π after each cavity round-trip. As a result of constructive superposition, the dispersive waves could add upon the solitons and form a series of narrow peaks on the laser spectrum [150, 152–154]. In the time domain, the dispersive waves formed a pedestal under the pulse with exponentially decaying

29

Ch. 2

PASSIVE MODE-LOCKED FIBRE LASERS envelopes that extend much further than the soliton wings. These sidebands in soliton lasers were first interpreted using this phase-matching condition by S. M. Kelly [152] and are usually called as Kelly sidebands. In the frequency domain, the dispersive waves shed by the soliton form a few discrete frequency components, each of which experiences a relative linear phase shift ϕd that scales quadratically with its relative frequency shift Δω from the carrierfrequency of the soliton, i.e. ϕd (za ) =

 za 1 1 (Δω)2 β2 (z)dz ≈ − |β2ave | (Δω)2 za 2 2 0

(2.22)

where za is the cavity length (i.e. the amplification period), β2 (z) describes the dispersion-map of the cavity, and |β2ave | is the average cavity dispersion. On the other hand, the soliton experiences a phase shift ϕs , which is proportional to the soliton peak power and propagation distance, i.e. ϕs (za ) =

1  za 1 γ(z)|A0 (z)|2 dz ≈ γ ave |A¯0 |2 dz 2 0 2

(2.23)

where γ(z) and γ ave are the nonlinear coefficient map and its averaged value over the cavity. Using the soliton-area theorem (Eq.(2.18)), this soliton phase shift can be expressed as: 1 |β2ave |za ϕs (za ) ≈ (2.24) 2 τ02 The phase-matching condition for the constructive superposition of dispersive wave upon the soliton states that ϕs (za ) − ϕd (za ) = 2mπ

(2.25)

in which m is an integer. Then by substituting Eqs.(2.22) and (2.24) into Eq.(2.25), the phase-matched frequencies at which the dispersive wave would be constructively accumulated can be readily obtained to be 1 Δωm = ± τ0

  4πτ 2 m 0 

1 −1=± ave za |β2 | τ0



8mz0 −1 za

(2.26)

or in terms of wavelength differences as: λ20 Δλm = ± 2πcτ0

30



8mz0 −1 za

(2.27)

Soliton fibre lasers

2.2

in which z0 is the soliton period defined in Eq.(2.20) and λ0 is the central wavelength of the soliton. Therefore, with different integer m, there could be a series of sidebands on both sides of the soliton spectral centre. The phase-matching condition of the Kelly sidebands is illustrated in Fig.2.7. phase-shift  Soliton s(za) 4π

2π

2π

4π Frequency ω

Dispersive wave d(za) Soliton spectrum

Fig. 2.7: Illustration of the phase-matching condition of Kelly-sidebands generation.

The peak intensity of a specific Kelly sideband is generally proportional to the intensity of the corresponding frequency component of the soliton, and would be therefore of lower intensity when its location deviates further away from the central frequency of the soliton. However the strength of the sidebands with equal order (on opposite sides) may not necessarily be symmetric [154]. The peak intensities of the Kelly sidebands are determined by the balance between the net loss (due to the spectral narrowing effect and possible wavelength-dependent loss) and the energy coupling from the soliton. Besides, when the cavity net GVD is strong, the location of the Kelly sidebands may also slightly deviate from the symmetric distribution. Higher-order dispersions also play a role in influencing distributions of the Kelly sidebands. Therefore, the spectrum of soliton fibre laser usually has an asymmetric Kelly sidebands distribution in terms of both peak intensities and relative frequencies to the spectral centre. For relatively long pulses, z0 could be very long (e.g. for ∼ 1 ps soliton in SMF-28 z0 ≈ 22 m), the Kelly sidebands would be far from spectral centre and thus have negligible intensities. As τ0 becomes narrower (with higher pulse energy), z0

31

Ch. 2

PASSIVE MODE-LOCKED FIBRE LASERS could drop dramatically (e.g. for ∼ 100 fs soliton,z0 ≈ 22 cm). As we can reveal from Eq.(2.27), when z0  za /8, resonant instability would occur (for first-order sidebands) and stable operation of the soliton laser would essentially collapse. In practice, stable operation requires an even shorter za so that z0 > za /2 [50]. As a result, the presence of Kelly sidebands imposes an upper limit of the nonlinear phase accumulation of the pulse within one cavity round-trip (which should be much smaller than 2π) and ultimately limits the pulse energy (sub-nJ level) as well as the pulse duration (sub-ps level) in soliton fibre lasers. Attempts to generate even shorter pulse duration from mode-locked fibre laser would require dramatic deviations from the conventional soliton regime, as would be discussed in the next section.

2.3 Ultrafast mode-locked fibre lasers 2.3.1 Limitations of soliton fibre lasers Mode-locked fibre lasers that work in the soliton-regime have limited capability to generate ultra-short pulses (< 0.1 ps) due to a couple of reasons. Firstly, the soliton-area theorem indicates that, given a certain mode-locked cavity, the shorter the pulse duration, the higher the individual pulse energy should be, which would consequently lead to a higher peak power and therefore an enhanced SPM effect. However, excessive nonlinear phase shift induced by SPM tends to cause soliton break-up due to the soliton quantization effect [82, 83], leading to unstable multipulse operations. Besides, the Kelly sidebands stemming from the periodical perturbations in the cavity would become detrimental under higher pump power, causing a considerable fraction of the signal power to flow into the dispersive waves, limiting the soliton energy. When the soliton period becomes too short, the soliton-regime mode-locking would simply collapse due to this resonant instability, as elucidated in the foregoing section. Thirdly, for soliton fibre lasers based on APM, the excessive SPM would additionally lead to over-driving of the saturable absorber [50]. In fact, the APM action would clamp the peak intensity of the soliton to only moderate value, which would in turn limit the pulse duration. We can infer from the previous derivation that the soliton energy Es ∝|β2 | while the soliton period z0 ∝ 1/|β2 | which obviously causes difficult dilemma in the cavity design [50]. The real obstacle beneath this dilemma lies in the strength of

32

Ultrafast mode-locked fibre lasers

2.3

SPM effect. Shorter pulse duration relies on stronger SPM effects for broader bandwidth generation, however the excessive nonlinear phase shift induced by SPM would cause instability problems [155]. Therefore, the pulse evolution in the mode-locked laser have to deviate far from such soliton regime in which the anomalous GVD and SPM almost continuously compensate each other, in order that the nonlinear phase shifts can be relieved. Typical strategies employ dispersion management as well as additional gain/loss tailoring and filtering effects. Two salient examples are briefly reviewed below.

2.3.2 Stretched-pulse fibre lasers In 1994, K. Tamura et al from MIT brought up the concept of the stretched-pulse laser as a new design of the mode-locked laser cavity, and it has become a great success experimentally in delivering sub-100 fs pulse directly from a fibre-ring laser ever since [50, 156, 157]. Numerous following works on ultrafast fibre laser have since then striven to emulate this unique configuration. The basic idea is similar to the well-known dispersion-managed soliton laser, which employs fibre sections of both anomalous and normal GVD within the laser cavity therefore the pulse would experience temporal breathing during propagation [44, 158, 159]. The main feature is that in the stretched-pulse laser, both sections would introduce such significant countering group delays that the temporal breathing of the intra-cavity pulse could no longer be regarded as being just perturbative and thus could not be simply approximated by the average soliton model [157]. The typical temporal breathing ratio within one cavity round-trip could be ∼ 10. The considerable pulse breathing could dramatically reduce the nonlinear phase shift per round-trip, and the saturation of APM action is largely relieved as well. Meanwhile, the net cavity group dispersion is usually compensated close to zero so as to defer the locations of the Kelly sidebands far away from the spectral centre and thus eliminate the resonant instability [153]. As a result, the intra-cavity pulse could tolerate more pulse energy and broader bandwidth. In the early realizations of the stretched-pulse Er-fibre lasers, the cavity consists of an Er-doped gain fibre with normal dispersion and another section of standard SMF with anomalous dispersion, compensating the net GVD close to zero [157]. Experiments showed that optimal characteristic of the laser was achieved when then net cavity GVD was slightly normal [156]. The intra-cavity pulse would experience twice breathing cycles for each round-trip, with the narrowest pulse width appearing

33

Ch. 2

PASSIVE MODE-LOCKED FIBRE LASERS in the middle point of both fibre sections with opposite signs of GVD. The pulse experienced significant nonlinear spectral broadening in the normal GVD section and then the bandwidth narrowed back in the anomalous GVD section, before relaunching into the gain fibre. The output pulse of this laser was inserted right after the end of the gain fibre and the output pulse usually has a nJ-level pulse energy together with a large linear chirp. External dispersion compensation is necessary to remove the chirp, and the output transform-limited pulse typically has a Gaussianshaped temporal profile (with exponentially decaying wings). The Gaussian-shaped profile at the pulse peak corresponds to the spectral wings that also follows a Gaussian profile, which features a much fast decaying rate at the tailing regions compared to a sech2 -shaped soliton spectrum. Therefore, the Kelly sidebands that are deferred far off the pulse spectral centre are further suppressed [1]. The high pulse energy allowed by strong dispersion compensation leads to broad bandwidth, whereas large breathing of the pulse width suppresses the excessive nonlinear phase shift, which ensures the stability to a large extend, although in most cases band-pass filtering effects are necessary for stable performance. In fact, given a highly broadened pulse with large chirp, the APM effect that suppresses the tailing part of the pulse works equivalently as a spectral band-pass filter. The nonlinear spectral broadening via SPM, which acts in concert with the pulse width broadening in the normal dispersion section, is mainly responsible for the broad pulse bandwidth. Actually, the Gaussian-shaped output pulse, with a higher time-bandwidth product than that of the conventional soliton, is an indication of strong nonlinear spectral broadening [155]. Under this condition, the gain bandwidth no longer sets the upper limit for pulse bandwidth [160] and only induces the gain filtering effects.

2.3.3 Dissipative-soliton fibre lasers The conventional optical soliton that exists in fibres arises from a mutual balance between the anomalous SPM and GVD, which is essentially a non-dissipative process. Although energy dissipation is inevitable in mode-locked soliton lasers, gain and loss in the cavity can be largely considered as perturbations, against which the soliton-like propagation process stays robust. However, the conventional soliton is only one type of the attractors that can exist in nonlinear systems. The dissipative soliton, which depends crucially upon the energy dissipation process, could also behaviour like an attractor [18]. In general, dissipative solitons arise from mutual balance of nonlinearity, dispersion, gain, and loss. Although the concept of dissi-

34

Ultrafast mode-locked fibre lasers

2.3

pative solitons extends to very wide range of fields and phenomena [161], it was introduced to mode-locked fibre lasers mainly in pursuit of higher pulse energies and broader bandwidth. The stability of the dissipative-soliton lasers is ensured by both the attractor-like property of dissipative-solitons as well as the additional intra-cavity filtering effects [18]. The self-similar pulse [162–164], or the similariton [165, 166] is a well-known type of dissipative soliton that can emerge inside a mode-locked fibre laser, with a much higher tolerance of pulse energies (tens of nJ or higher). The self-similar pulse profile is an asymptotic solution of the generalized NLSE and behaves as a distinct nonlinear attractor which has a parabolic temporal profile and large chirp across the pulse [167]. The self-similar pulse can experience expansions in both time and frequency domain during propagation without wave-breaking [168, 169] while maintaining the parabolic profile in a self-similar fashion. The self-similar pulse achieved in gain fibre with normal dispersion and Kerr-nonlinearity is also called the amplifier similariton. In this case, even with an initial pulse profile that is not parabolic, the pulse would asymptotically evolve into the parabolic shape after sufficient propagation and gradually develop a linear chirp cross the pulse. Moreover, instead of shedding dispersive waves during the propagation, all the energy is maintained inside the pulse during evolution [167]. Self-similar pulses can also be achieved at cavity-scale, in which the gain fibre section is followed by a passive fibre with normal dispersion [164]. Nevertheless, the basic mechanism rooted in the mutual balance is unchanged, and the self-similar pulse still serves as a nonlinear attractor in such mode-locked fibre lasers. Another popular configuration of dissipative soliton lasers is the all-normal-dispersion (ANDi) fibre laser [170–172], in which the intra-cavity pulse maintains a large chirp while its temporal breathing becomes less significant. The nonlinear spectral broadening in gain fibre with normal dispersion is still the dominant factor for the broadband pulse generation. In order to form stable pulse evolutions in the cavity, the self-consistency condition must be fulfilled in both temporal and spectral domains, which means that the intra-cavity pulse must return to its original state after each cavity round-trip. In the stretched-pulse laser, both temporal and spectral self-consistencies are ensured by the corresponding pulse “breathing” in the normal and anomalous dispersion section, additionally with weak filtering effects for stabilization. The dissipativesolitons lasers follow a similar map of the self-consistency within one round-trip, which is summarized in Fig.2.8.

35

Ch. 2

PASSIVE MODE-LOCKED FIBRE LASERS Re-injection

anomalous dispersion passive fibre

normal dispersion gain fibre pulse temporal broadening

pulse temporal narrowing

spectral nonlinear broadening

spectral nonlinear narrowing

Band-pass filter spectral reshaping noise supression Fig. 2.8: Map of self-consistency of typical dissipative soliton lasers with a unidirectional ring-cavity. In the normal dispersion section with gain (red section) the pulse experiences monotonic expansion in both temporal and spectral domain. A band-pass filter is inserted after that for bandwidth reduction and noise suppression. Then the pulse is launched in to anomalous section (blue section), usually with low power, for pulse duration narrowing and possibly nonlinear spectral narrowing. At last the pulse would return to the initial state before re-injected into the normal dispersion section.

As a start, the pulse with a relatively high pulse energy would first experience monotonic expansions in both temporal and spectral domains in the gain fibre with normal dispersion (or a lumped gain fibre followed by a passive fibre with normal dispersion), as shown in red-section of Fig.2.8. Then the pulse needs to narrow back before being re-injected into the normal dispersion section. In order to avoid deleterious effects brought by excessive nonlinear phase shift, the pulse that is launched in the successive anomalous fibre (blue-section in Fig.2.8) should have relatively lower energy than in the normal dispersion section. Generally it is not preferred to narrow the pulse back mainly by passive fibres. Instead, band-pass filters and free-space bulk grating-pairs are often used to force the intra-cavity self-consistency. The band-pass filter is inserted also to suppress the noise and stabilize the operation of the laser, while the bulk-grating pairs can induce anomalous dispersions without any accompanying SPM effects. A question rose up when the author was reviewing the literature on ultrafast

36

Multi-pulse states

2.4

mode-locked fibre lasers. It is that why all these lasers seem to feature simultaneous high-pulse energies and ultra-short pulse durations, which seems to suggest that these two parameters are highly coupled in laser sources [170, 171]. The author suspected that it was due to the fashion of the intra-cavity pulse evolution that relies heavily on nonlinear spectral broadening via SPM, which naturally required high pulse energy. However, the pulse duration should in principle only depends on the spectral bandwidth, while the rare-earth doped gain fibre already provides sufficient bandwidth for ultra-short pulses [173, 174]. In Chapter 5, a novel strategy of mode-locked fibre laser, termed as stretched-soliton laser, will be presented, in which sub-100 fs pulse can be generated with only tens of pJ intra-cavity pulse energy thanks to the efficient utilization of the bandwidth of the gain fibre instead of relying on nonlinear spectral broadening.

2.4 Multi-pulse states 2.4.1 Passive harmonic mode-locking Given sufficiently high pump power and proper cavity parameters, the passive modelocked laser can run under multi-pulse states. In the time domain, the most regular pattern for multi-pulses in a mode-locked lase cavity corresponds to the harmonic mode-locking (HML) state [45, 46, 175], in which all the pulses are evenly spaced while co-propagating in the cavity. One obvious advantage of HML is that the scaling of repetition rate would become rather simple, especially when GHz-repetition rate is desired, which would otherwise demands cm-long cavity for the fundamental mode-locking. Typical fibre lasers with meters-long cavity lengths would only produced tens of MHz-rate pulse trains. Although ideal HML state can theoretically exist, such state cannot stably persist in passive mode-locked lasers, at least not in the long-term, due to lack of dominant mechanisms that maintains the regular pulse pattern against external perturbations (environment disturbances via acoustic noise or temperature drifting, spontaneous emission in gain fibre, and pulse-pulse interactions, weak CW or continuum background, etc.). After all, the saturable absorber in the passive modelocked laser can only respond to individual pulse shape, being essentially insensitive to pulse spacing. Although some weak mechanisms can help to regulate the pulse spacings in passive HML lasers, the timing-jitter of the multi-pulse pattern could

37

Ch. 2

PASSIVE MODE-LOCKED FIBRE LASERS still be severe. Several strategies have been reported to achieve stable HML in passive modelocked laser. For example, by using a coupled sub-cavity which has a round-trip frequency being an integer-multiple of that in the main cavity, the pulse spacings would be determined by the sub-cavity length and therefore the multi-pulses in the main cavity could become evenly spaced. However, since the sub-cavity functions as a Fabry-Perot [77] or Mach-Zehnder filter [79, 80] which relies on the interferometric overlapping between the pulses in sub- and main-cavity, precise control of cavity length would be demanded against environmental perturbations. Therefore, similar feedback electronics as in active HML would be required to lock the sub-cavity length with that of the main-cavity. The gain depletion effect can also lead to equal space between the multiple laser pulses and can in principle lead to stable HML. However, the harmonic order is usually very low [81, 176] and also depends on the pump power [59, 61] while having large timing-jitter [177, 178]. In fact, such effect is practically absent in rare-earth doped fibre lasers due to the long relaxation time. The optoacoustic effect in conventional fibre has also been revealed to be able to induce long-range pulse-pulse interactions and ultimately leads to equal spacing between the multiple pulses in the mode-locked fibre cavity [20]. In practice, however, the strength of the optoacoustic effect in conventional fibres is generally very weak due to limited overlapping between the optical mode and acoustic mode in the transverse cross-section of the fibre. The optical mode profile is usually well confined in the fibre core due to the refractive index step at the core-cladding interface, while the transverse acoustic mode profile could extend throughout the entire cylindrical structure of the fibre due to the trivial difference in acoustic impedance (or sound velocity) between the fibre core and cladding [179]. For standard SMF, the optical mode is confined in the core with a diameter of ∼ 10 μm, while the acoustic mode would extend to the entire fibre structure with a diameter over 100 μm. In addition, the ∼ 100μm fibre diameter corresponds to acoustic resonance frequencies of only tens of MHz to hundreds of MHz), which can hardly enable GHz-rate HML state. Although it has been reported in literature that the HML laser with a repetition rate of a few hundreds of MHz [81] can be achieved with weak optoacoustic interactions, the performance of such laser is still far from being robust, with erratic repetition rate, large timing-jitter and unstable long-term operation. The passive HML is conventionally not considered as a promising strategy for repetition rate scaling [33]. This is however the starting point of the work covered by

38

Multi-pulse states

2.4

this thesis. By using a PCF with a small solid-core surrounded by hollow channels, the optoacoustic interactions can be highly enhanced. The specially-designed PCF introduces a dominant kind of long-range interactions between the pulses, which can extend to the entire optical cavity. Stable HHML laser can be achieved simply by inserting such a short piece of PCF inside a conventional soliton fibre laser [104]. Moreover, the introduction of such enhanced acoustic resonance has highly enriched physical picture of the conventional mode-locked fibre laser, leading to a series of exciting phenomena beyond the original intention of a sheer scaling of the repetition rate.

2.4.2 Soliton molecules in passive mode-locked fibre lasers As discussed above, the interactions between multi-pulses that exist in a mode-locked soliton fibre laser could leads to mutual arrangement of the multi-pulses distributed across the entire cavity. However, when the pulses are closely placed, direct pulsepulse interactions due to partial overlapping could result in localized structures consisting multiple solitons, which are termed as soliton molecules [18, 180, 181]. A widely known type of soliton molecules is the so-called soliton pair, with two solitons narrowly spaced and interact with each other at their overlapping tails through XPM [182–184]. The two solitons would be stably bound at a fixed, narrow inter-spacing which is at most a few of times of their individual durations, and the resultant pair of solitons would have a fixed phase-relation between them (typically being 0 or ±π) [183, 185, 186]. Soliton pairs have been observed both in soliton transmission lines [184] and in mode-locked fibre lasers [187], and such short-range bound-states exhibit good robustness against perturbations. The same interaction mechanism could also bind multiple solitons, leading to a soliton triplet or even a soliton crystal consisting a bunch of solitons, still with fixed phase relations among the bunch of equally-spaced solitons [188]. Soliton crystals do not only exist in fibre lasers, but also have recently been discovered in silicon micro-resonators [189]. Since the soliton molecules have ultra-compact inner structure, with inter-soliton spacings of merely a few ps, the commercially-available photodetectors with resolutions of ∼ 20 ps (∼ 50 GHz bandwidth) can hardly resolve their inner structures. In fact, the soliton molecules are usually characterized through their autocorrelation traces and spectral fringes [187, 188]. The XPM between directly interacting solitons is not the only interaction mechanism between the multiple solitons. Long-range interactions through optoacoustic

39

Ch. 2

PASSIVE MODE-LOCKED FIBRE LASERS effects [20, 85, 87], optothermal effects [190], dispersive waves [110, 111, 191], and CW backgrounds [21, 192] can also arise in mode-locked lasers. However they can hardly form stable bound-states and only lead to disordering of solitons [112, 193]. In Chapter 7, a novel type of long-rang bound-state of solitons will be present, which arises from the stable balance between weak, long-range interactions stemming from optoacoustic effects and dispersive wave perturbations. The internal soliton separation could be ∼ 100 ps, being hundreds of times of the individual soliton durations, and no fixed phase-relation exists among the bound solitons. Such structures offer significant diversity, flexibility,and remarkable robustness, highlighting the supramolecular features of these structures [114].

40

CHAPTER

3 OPTOACOUSTIC MODE-LOCKING

This chapter presents the core principle of optoacoustic mode-locking, which settles the basis for stable HHML fibre laser as well as other types of optomechanical bound-states that can exist in such laser cavity. Due to the special design of PCF, the optoacoustic interaction in form of stimulated Raman-like scattering (SRLS) is greatly enhanced. The basic principle of SRLS is usually interpreted in the frequency domain with CW pump light, while the pulse-driven SRLS is interpreted in the time domain as well. The multi-pulses in the mode-locked cavity coherently drive the acoustic resonance in the PCF, while the acoustic wave, through modulating the refractive index, acts back on the pulses and forces them to evenly distribute in the cavity. The pulse timing stabilization is found to result from the temporal trapping potential created by the cooperation between the acoustic wave and the cavity dispersion. An average soliton model with the optoacoustic index-modulation is developed for demonstrating the influences of the acoustic wave on individual pulse parameters. The non-interferometric nature of the optoacoustic mode-locking, which leads to multi-pulses with uncorrelated carrier-phases, is also briefly discussed.

3.1 SRLS in solid-core PCF 3.1.1 Stimulated Brillouin scattering: Backward and forward Stimulated Brillouin scattering (SBS) in optical fibres is an inelastic χ(3) nonlinear process that involves the interaction between the optical wave and the acoustic

41

Ch. 3

OPTOACOUSTIC MODE-LOCKING

(b)

(a)

Backward optical wave

qa k2

k1 O/2

k2

1st anti-Stokes

Frequency Z

wave (or photons and phonons from the quantum viewpoint) [84]. The acoustic wave is driven by the optical wave via optical forces which tend to compress the fibre material. As a back-action, the acoustic wave would cause material density variations and scatter the optical wave with a frequency-shift. The SBS process could be easily excited in silica optical fibres with very low threshold due to the high optical intensity and long interaction length. In conventional silica fibre, the SBS process can occur in the both backward and forward direction [194, 195]. The principle of the backward SBS is illustrated in Fig.3.1. The acoustic wave involved in this process is a longitudinal quasi-plane wave that propagates in the forward direction with frequency Ωa , as shown in Fig.3.1(a), which leads to scattering of the forward pump light at frequency of ωp into the backward Stokes light at frequency ωs = ωp − Ωa . The interference between the counter-propagating lights with a frequency difference would create to a moving “grating” of the optical densities which would drive the longitudinal acoustic wave, then this acoustic wave would in turn reinforce the backward Stokes light [84]. Forward optical wave

:a

Pump

1st Stokes U:a qa

k1

z

Acoustic wave, moving density “grating”

Backward acoustic wave

Forward acoustic wave kz axial wave-vector

Fig. 3.1: Principle of the backward SBS in conventional fibres. (a) The phasematching condition for the forward pump light, the backward scattered light, and the forward moving density grating due to the longitudinal acoustic wave along the fibre. (b) The phase-matching condition for multi-sidebands generation illustrated in dispersion relation map.

In order to have the phase-matching condition fulfilled, the wave vector of the acoustic wave would be the difference in that of the forward and backward light, i.e. qa = k1 − k2 . Since the frequency difference between pump and stokes light is usually much smaller than the optical frequency, we can approximately have |k1 | ≈ |k2 |, therefore qa ≈ 2k1 , which indicates that the wavelength of the longitudinal acoustic wave would be half of the optical wavelength of the pump light

42

SRLS in solid-core PCF

3.1

(Fig.3.1(a)). Therefore there exists a direct dependency of the acoustic frequency upon the optical frequency. Besides, in order to have further scattering into higherorder Stokes (or anti-Stokes) light, different acoustic waves with opposite propagating directions need to be involved as shown in the dispersion relation map in Fig.3.1(b) [95]. Therefore, unless under very high pump power, the backward SBS can hardly occur in a cascaded fashion, due to the distinct phase-matching conditions for each sidebands. Since the SBS strength scales inversely proportional to the pulse bandwidth, efficient backward SBS would generally require narrow-linewidth pump. In practice, the backward SBS would essentially disappear for pulse durations below 1 ns [134]. In conventional fibres, the SBS process could also occur in the forward direction, in which the pump and scattered light would co-propagate along the fibre. The acoustic wave involved in this scattering process is a longitudinal wave that exists mainly in the transverse cross-section of the fibre (close to cut-off), which is driven by the optical mode with a transverse intensity gradient. Such acoustic wave would have a wave-vector that is almost perpendicular to the fibre axis, with meanwhile a trivial longitudinal component in the same direction of the pump light. The acoustic frequency would be independent of the pump wavelength, while it mainly depends on the transverse dimension of the fibre. The strength of the forward scattering depends on the overlapping between the optical mode and the acoustic mode. The optical mode would be mainly confined in the fibre core due to the difference in refractive index, while the acoustic mode would extend to the entire cross-section of the fibre since the mechanical impedance are almost identical in the core and cladding of the fibre. As a consequence, the overlapping would be so weak, that the forward scattering would largely become negligible perturbations. The PCF with small a solid-core surrounded by hollow channels (as shown in Fig.1.2), unlike the conventional fibres, could confine not only the optical mode in the fibre core, but also the acoustic mode at cut-off in the core. Due to the large optoacoustic overlapping, the interactions between optical and acoustic waves in such solid-core PCF are greatly enhanced. Typically, two-orders of magnitude of enhancement can be achieved using such structure compared to the conventional fibre [95]. The principle of the forward SBS process in PCF is illustrated in Fig.3.2. The acoustic wave is mainly driven by the transverse optical intensity profile which has a gradient electrostrictive force, and the beating pattern between pump and forward scattered light creates a sinusoidal density-variation along the PCF-core that

43

Ch. 3

OPTOACOUSTIC MODE-LOCKING

(a)

(b) k1 k2 S/|qa|=2Svg/:a

qa

Frequency Z

moves along with the beating envelope (Fig.3.2(a)) [196]. Due to the trivial acoustic vector along the fibre axis, the period of this density-variation along the fibre is very long in order to be phase-matched with the co-propagating light.

Forward optical wave 2nd anti-Stokes

k2

1st anti-Stokes

z

Pump

qa

k1

:a

density variation along the fibre axis transverse optical intensity profile

:a

1st Stokes 2nd Stokes

Forward acoustic wave kz axial wave-vector

Fig. 3.2: Principle of the forward SBS in a solid-core PCF. (a) The phasematching condition for the forward pump light, the forward scattered light, and the forward moving density variation of the PCF-core, which moves along with the beat pattern of the pump and the scattered light at the optical group velocity. The acoustic wave only has trivial z-component propagation (b) The phase-matching conditions for cascaded sidebands generations illustrated in the dispersion relation map. The acoustic wave that is involved in this process has cut-off frequency of the suspended solid-core of PCF.

The strong core vibration in the PCF which scatters the light in the forward direction is a reminiscent of another inelastic nonlinear optical phenomenon: the stimulated Raman scattering (SRS) which stems from interactions between light and molecular vibrations. The typical oscillation frequency of Raman scattering is usually in THz-range due to the small size of individual molecules, while the microsized PCF core can be viewed as a “macro” mechanical oscillator which vibrates at frequency of a few GHz. Based on such analogue, this highly enhanced forward SBS in solid-core PCF is also termed as stimulated Raman-like scattering (SRLS). A important feature of SRLS distinct from the backward SBS is that the phase-matching conditions are automatically fulfilled for higher-order Stokes and anti-Stokes sidebands in SRLS, since the same acoustic wave (phonons) mediates all different orders of scattering. The pump and scattered light propagate in the same direction and the acoustic wave involved has nearly the cut-off frequency which depends only on the core-size. Therefore, SRLS could easily happen in a cascaded fashion, generating mutually coherent multiple sidebands. The phase-matching con-

44

SRLS in solid-core PCF

3.1

dition of SRLS for cascaded frequency conversions is illustrated in Fig.3.2(b). Besides, unlike the backward SBS which can occur only with narrow-linewidth pump light, SRLS can be driven by short-pulse trains with individual durations well below ns-range, due to the co-propagation fashion of the pump and scattered light. These properties set the basis for the application of the SRLS process in HML lasers, in which the intra-cavity pulses would be efficiently modulated in the PCF that is inserted in the cavity.

3.1.2 SRLS with CW pump: Frequency conversion The analysis of SRLS process in solid-core PCF pumped by CW light has been thoroughly presented in [95]. Here we only shortly review the basic conclusions. In order to efficiently excite the transverse acoustic mode, dual-frequency CW pump light was used, with frequency components of ω1 and ω2 . The beating pattern of the dual-frequency light would coherent drive the acoustic wave in PCF at the frequency of Ωa = ω2 − ω1 . As a result, the energy distribution of both frequency components would be changed due to the scattering by the acoustic wave. Under relatively low pump power, the dominant SRLS process would be the energy conversion from the higher frequency component to the lower frequency one, corresponding to the coherent generation of acoustic phonons, as shown in Fig.3.3. The dominant optical mode and acoustic mode that are involved in this process are the LP01 -mode and the R01 -mode of the PCF-core, respectively, (a)

(b)

E2

E1

E2

E1

:a

:a

SRLS gain g(:a)

Z

Z

Z

Z

Z

Z

Fig. 3.3: Principle of the SRLS pumped with a dual-frequency CW light. (a) The dual-frequency pump has a frequency-difference of Ωa . The lower frequency component would have gain due to the SRLS, as shown in grey curve, which is a function of Ωa . (b) As a result, there would be an energy conversion from higher frequency component to the lower one.

45

Ch. 3

OPTOACOUSTIC MODE-LOCKING The coupled equations of the optical intensity of the two frequency components, which describe the SRLS process under this dual frequency pump, are as below: ∂I1 = gI1 I2 ∂z ∂I2 = −gI1 I2 ∂z

(3.1a) (3.1b)

in which the intensity Ii , (i = 1, 2) relates to the electric field as Ii = 12 n0 0 c|Ei |2 , and g = g(Ωa ) is the acoustic gain spectrum for the energy conversion from higher frequency component to the lower one, which has the following expression: g(Ωa ) = g0

(ΓB /2)2 (Ωa − Ω01 )2 + (ΓB /2)2

(3.2)

The coefficient g0 is the peak gain in the case of on-resonance driving g0 =

ω1 γe |Q0 Q1 | 2n2eff c2 ρ0 Ω01 ΓB

(3.3)

in which Ω01 is the resonant frequency of the R01 -mode of the PCF-core, ΓB is the Brillouin linewidth, γe is the electrostrictive constant, c is the vacuum light speed, ρ0 is the average density of fibre core (silica), neff is the effective refractive index of the LP01 -mode. Q1 ≡ ∇2⊥ E02 , ρ01 , which describes the strength of electrostrictive force that drives out the acoustic wave. The angled bracket  is defined as f (r, θ) =

 2π  +∞ 0

0

f (r, θ)rdrdθ

(3.4)

Q0 ≡ E02 , ρ01 , which describes the strength of the opto-elastic effect that causes the index modulations experienced by the optical wave. Eqs.(3.1) describe the basic process of the SRLS in terms of frequency conversion, with an acoustic gain spectrum having a Lorentzian profile. When the dual-frequency pump becomes stronger, additional frequency components, i.e. higher-orders of Stokes and anti-Stokes light, would appear, and the energy conversion process would become rather complicated and needs to be described by a set of infinite coupled-equations concerning equidistant frequency components. When the input pump light is in form of pulse train, which consists of numerous frequency components, the SRLS would becomes hard to describe in the frequency domain. Pulsed-light-induced SRLS process would be easier to illustrate in the time domain, as will be covered in the next section.

46

SRLS in solid-core PCF

3.1

3.1.3 SRLS with pulsed-light pump: Time-domain description The practical case of optoacoustic mode-locked laser involves a SRLS process driven by a pulse train. In this case, the electrostrictive force would be in form of periodic impulsion instead of a sinusoidal beat pattern. However, since the silica material has much longer mechanical response time than the practical pulse duration, the PCF can only respond to the fundamental frequency beat note of the pulse train. Therefore, in order to have strong SRLS process, the repetition rate of the pulse train should be tuned close to the core-resonance of the PCF. As a back-action, the acoustic wave driven by the optical pulse train would induce a refractive-index wave that would co-propagate with the pulses, and each pulse would see an index-slope across its pulse duration. As a consequence, the carrier-frequency of the pulses would be continuously shifted during propagation in the PCF, leading to modifications of the pulse group velocity due to the fibre GVD. The index modulation is derived in this section, while the resultant re-timing effect will be covered in Section 3.2.1. We start with describing a pulse train of repetition rate Ω that propagates in the PCF, the electric field of which can be expressed as E(z, t, r, θ) =

 1 +∞ D(nTp ) {E01 (r, θ)a(r, t) exp [−i (ω0 t − β0 z)]} + c.c. 2 n=−∞

(3.5)

in which D is the time-delay operator that describes the periodic pulse pattern, Tp = 2π/Ω is the pulse interval, E01 (r, θ) is the normalized field distribution of LP01 -mode. a(z, t) is the slowly varying envelope of individual pulses, β0 is the propagation constant along z-direction. The fibre loss is neglected in this expression. The electrostrictive force f induced by the optical wave can be expressed as [84]: 1 f = − 0 γe ∇E 2 (3.6) 2 in which 0 is the vacuum permittivity, γe the electrostrictive constant. E 2 stands for the time-averaged intensity over one optical period. The acoustic wave equation in terms of material density variation ρ, with the electrostrictive force as the driving term, can be written as [84, 197]



∂2ρ ∂ 1 − va2 1 + Γ ∇2 ρ = ∇ · f = − 0 γe ∇2 E 2 2 ∂t ∂t 2

(3.7)

in which va is the sound velocity and Γ is the damping factor.

47

Ch. 3

OPTOACOUSTIC MODE-LOCKING Then we can derive the driving term for the acoustic wave equation Eq.(3.7) using the slowly varying envelope assumption, which is +∞  1 1 2 ∇ · f = − 0 γe ∇2 E 2 ≈ − 0 γe ∇2⊥ E01 D(nTp )|a(r, t)|2 2 2 n=−∞

(3.8)

in which n denotes the pulse number. The periodic pulse train that is described by the time-delay operator can be expanded into a Fourier series, then we have +∞  1 2 ∇ · f = − 0 γe ∇2⊥ E01 sl exp [−i (lΩt − ql )] + c.c. 4 l=1

(3.9)

where l is the Fourier series number, ql is the propagation constant of the lth component along the z-direction and is defined as ql = lΩ/vg , vg being the group velocity of the pulse. The Fourier series coefficients sl can be given by sl =

1  Tp /2 |a(0, t)|2 exp (ilΩt ) dt Tp −Tp /2

(3.10)

By substitute Eq.(3.9) into Eq.(3.7) we can obtain density variation equation



+∞  ∂2ρ ∂ 1 2 2 2 2  − v 1 + Γ ∇ ρ = − γ ∇ E sl exp [−i (lΩt − ql )] + c.c. 0 e a ⊥ 01 ∂t2 ∂t 4 l=1

(3.11)

The steady-state solution of Eq.(3.11) can be considered as the overlapping of the responses to each frequency component of the driving term, i.e. ρ(z, t, r, θ) =

+∞ 

ρl (z, t, r, θ)

(3.12)

l=1

ρl is density variation driven by the lth frequency components of the driving force, which can be expressed as the sum of multiple acoustic modes that can be possibly driven by that particular frequency components ρl (z, t, r, θ) =

M 1  bm ρm (r, θ) exp [−i (lΩt − ql z)] + c.c. 2 m=1 l

(3.13)

where m is the mode number (for simplicity we use only m instead of mn here), bm l is the weight factor of the corresponding mode, and ρm (r, θ) is the normalized profile of the mth -order acoustic mode. The acoustic wave has the same frequency and

48

SRLS in solid-core PCF

3.1

propagation constant along the fibre as its driving force. Different acoustic modes supported by the PCF structure can be determined by the modal equation

∇2⊥ ρm (r, θ)



Ω2m 2 + − qm ρm (r, θ) = 0 2 va

(3.14)

where Ωm is the resonance frequency of the mth acoustic mode. Then by substituting Eqs.(3.12)–(3.14) into Eq.(3.11), we can obtain the steady-state equation as M   m=1



m m 2 2 −ibm l ρm (r, θ)lΩΓB − bl ρm (r, θ) (lΩ) − Ωm



1 2 = − 0 γe ∇2⊥ E01 · sl 2

(3.15)

2 th where Γm B = Ωm Γ is the Brillouin bandwidth of the m -order acoustic mode. Then by multiplying both sides of Eq.(3.15) with ρm (r, θ) and integrating over the crosssection, we obtain 0 γe sl Qm bm (3.16) l = − 2 [Ω2m − (lΩ)2 − ilΩΓm B] 2 in which Qm = ∇2⊥ E01 , ρm (r, θ) is the overlap integral of the mth -order acoustic mode and the electrostrictive force field. In practice, the fundamental R01 acoustic mode has the largest overlap integral Q1 with the LP01 optical mode and the repetition rate of the pulse train is close to the resonance frequency of the R01 -mode. Therefore, we only need to include the fundamental frequency component of the driving force (for l = 1), i.e., bm l = 0 only for l = 1 and m = 1. Then the density variation under steady-state can be expressed as

ρ(z, t, r, θ) = −

4 (Ω201

0 γe s1 Q1 ρ01 (r, θ) exp [−i (Ωt − qz)] + c.c. − Ω2 − iΩΓB )

(3.17)

in which Ω01 and ΓB are the resonance frequency and bandwidth of the R01 -mode, q is the propagation constant of the acoustic wave. Since the individual pulse duration is much shorter than the period of phase envelope of the acoustic wave along the fibre (i.e. TP = 2π/Ω), the first-order series coefficient s1 of the driving term can be approximated as 1  TP /2 s1 ≈ |a(0, t )|2 dt (3.18) TP −TP /2 The optical intensity is related to the electric field by the relation I(t) = 1/20 cneff |a(t)|2 , and the total power P (t) ≈ I(t) · Aeff , Aeff being the effective mode area. Then we

49

Ch. 3

OPTOACOUSTIC MODE-LOCKING can rewrite Eq.(3.18) as  +TP /2 2 Ω ΩEP Pav s1 ≈ · P (t )dt = = 2π 0 cneff Aeff −TP /2 π0 cneff Aeff π0 cneff Aeff

(3.19)

where EP is the individual pulse energy, and Pav is the average power of the pulse train. By substituting Eq.(3.19) into Eq.(3.17) and assuming that Ω ≈ Ω01  ΓB , we can obtain ρ(z, t, r, θ) =

γe |Q1 |EP ρ01 (r, θ) exp [−i (Ωt − qz)]+c.c. (3.20) 4πcneff Aeff [2 (Ω201 − Ω2 ) − iΓB ]

Since the denominator of Eq.(3.19) is a complex number, we can rewrite the density variation into the “phasor” form [33], with real amplitude and complex number phase term, which is ρ(z, t, r, θ) =

γe |Q1 |EP 

4πcneff Aeff 4δa2 + Γ2B

ρ01 (r, θ) exp [−i (Ωt − qz − Δϕ)] + c.c. (3.21)

in which δa ≡ Ω − Ω01 , and Δϕ is the phase difference between the acoustic wave and driving pulse train which is Δϕ = cot−1

−2δa , ΓB

Δϕ ∈ (0, π)

(3.22)

Eq.(3.21) describes the action of the optical pulse train upon the PCF-core, which creates an acoustic wave that induces a moving “grating” of density variation along the fibre. The pulse train would ride upon such density variation along the fibre at the group velocity. The density variation can transformed into modulation of the relative permittivity via the following relation Δr = γe Q0

ρ(z, t, r, θ) ρ0

(3.23)

in which Q0 ≡ ρ01 , |E01 |2 , i.e. the overlap integral of optical and acoustic mode. The modulation of Δr is practically very small, thus we could derive the corresponding modulation of refractive index simply using the approximated relation Δn ≈ Δr /2n. Moreover, since the density grating co-propagate with the pulse train along the fibre at the same velocity, the moving frame can also be employed,

50

SRLS in solid-core PCF

3.1

in which the relative time tm can be defined as tm = t −

z q =t− z vg Ω

(3.24)

Then the refractive-index wave due the pulse-driven acoustic vibration can be rewritten as Δn(tm , r, θ) =

γe2 |Q1 |Q0 EP 

4πcn2eff Aeff 4δa2 + Γ2B

ρ01 (r, θ) cos (Ωtm − Δϕ)

(3.25)

With typical fibre parameters (∼ 2 μm-core diameter) and pulse energies (tens of pJ), the modulation depth of the refractive index can be estimated to be in the order of 10−8 [104]. (b) Pulse train

Δϕ=π/2

(a) Acoustic resonance On resonance δa=0

Acoustic gain (a.u.)

Off resonance δa 0

(3.31)

Therefore, in case of a soliton laser where β2av < 0, a negative Δn0 is required, which can be fulfilled when Ω < Ω01 , as could be revealed from Eq.(3.25). This is actually in accordance with experimental observations, where it was, at least apparently, impossible to tune the pulse repetition rate above the acoustic core-resonance frequency without totally destroy the HHML state. The temporal trapping potential due to this optical spring effect is illustrated in Fig.3.7, with an approximate quadratic-profile (solid blue curve) in the vicinity of the balanced position. On the contrary, if Δn0 > 0, then the pulse cannot be stably trapped to the balanced position, due to the reversed potential profile (dashed curve). Temporal trapping potential

Stable trapping Soliton

2av ⋅ Δn0′′ > 0 Δt0

Unstable case

Balanced position

2av ⋅ Δn0′′ < 0 t − z / vg

t0

Fig. 3.7: The temporal trapping potential in the vicinity of the balanced position of the pulse. When β2av ·Δn0 > 0, a stable trapping potential can be formed (solidblue curve). The soliton that is slightly shifted away from the balanced position at t0 would then be shifted back. On the contrary, if β2av · Δn0 < 0, the pulse position would be unstable due to the opposite curvature (dashed curve).

58

Optoacoustic mode-locking using SRLS

3.2

The “depth” of this potential is actually proportional to the product β2av Lc Δn0 , where β2av Lc is the net group delay of the cavity. On one hand, the larger the cavity group delay, the “deeper” the trapping potential would be, i.e. the re-timing effect can be stronger per round-trip. On the other hand, a stronger curvature of the indexmodulation profile where the pulse was located would also make the potential deeper. Since the index modulation follows a sinusoidal profile as described by Eq.(3.25), the second-order derivative |Δn0 | seemingly takes maximum magnitude when the pulse is located at the wave peak point. However, the pulse cannot be actually located there, since it corresponds to the case of far off-resonance driving that leads to trivial acoustic wave amplitude. For the case of exact on-resonance driving (the pulse repetition rate is same as the core-resonance frequency), the acoustic-wave amplitude would be maximum. However, the pulse would then be located at the inflection point of sine function, and the second-order derivative would be zero. In the experiments, we observed that the HML laser became unstable when the pulse repetition rate was too close to the acoustic core-resonance, even though it was the position where the optoacoustic interaction is most intense. The optimized trapping point requires that the repetition rate of the driving pulse is smaller than core-resonance by a certain difference, and it can be derived as following. The indexmodulation given by Eq.(3.25) is under moving frame, and the pulse can be set to centre at tm = 0. Therefore, we only need to expand the index profile at this point, and we can get the value of Δn0 . We rewrite the Eq.(3.25) with a summarized coefficient Bm cos (Ωtm − Δϕ) Δn(tm , r, θ) = Bm ·  (3.32) 4δa2 + Γ2B where Bm > 0. Then Δn0 can be expressed as 

Δn0

∂ 2 Δn(tm )  Ω2 cos (Δϕ)   = = −B · m ∂t2m tm =0 4δa2 + Γ2B

(3.33)

Using the expression of the phase difference Eq.(3.22), we can obtain that Δn0 = Bm ·

2Ω2 δa 4δa2 + Γ2B

(3.34)

where we have δa < 0 since Ω < Ω01 . Then magnitude of Δn0 obviously takes a

59

Ch. 3

OPTOACOUSTIC MODE-LOCKING maximum value when

ΓB (3.35) 2 which means that the repetition rate should be tuned approximately to the position inside the acoustic resonance where the response amplitude drops to half of the maximum value, and the pulse position consequently has a π/4 phase difference relative to the acoustic wave. According to Eq.(3.31), another configuration can be devised for a similar trapping effect: given a net normal cavity GVD and a pulse repetition rate slightly higher than the acoustic core-resonance of the PCF (i.e. an opposite sign in index slope), each pulse can still be trapped at the same position in each acoustic cycle. This question actually has already been mentioned in [81] and the change of sign in the frequency-detuning has been observed under the weak optoacoustic interactions with SMF. In principle this configuration should also work for with GHz-core resonance in PCF, with however a practical difficulty. Mode-locked lasers with net normaldispersion are not intrinsically self-consistent unless aided with additional filtering effect in the cavity, and the individual pulse-energy for such schemes can be quite high as previously discussed in Section 2.3.3. With the HHML fibre lasers at GHzrate, the scaling of average power can be significant and would require high pump power and double-cladding fibre technology [171]. In this thesis we mainly focus on the soliton-regime under a modest pump power, and the other configuration is out of the scope of this thesis. Interestingly, the Gordon-Haus effect is originally used to interpret the timingjitter of pulse propagation due to perturbations stemming from e.g. spontaneous emissions in the amplifier section [140, 201]. Here we use such effect to interpret the stabilized timing of pulses due to the cooperation between the acoustic wave and the cavity dispersion. Gordon-Haus jitter nevertheless still exists in the HML laser cavity, serving as the noise sources in Eq.(3.30). However, with a negative feedback induced by the optoacoustic effect, the timing-jitter can be highly suppressed, as we will see later in the experiment results. Most importantly, such negative feedback is not imposed from external sources, but induced in a completely passive way through the interactions between the pulsed light and the acoustic wave in the PCF. Therefore, the optoacoustic mode-locking can be extremely simple to prepare, while the timing stability can be remarkable. |δa | =

60

Optoacoustic mode-locking using SRLS

3.2

3.2.2 Average soliton model with acoustic index modulation We developed an averaged soliton model that describes the pulse-shaping effects per round-trip in the optoacoustic mode-locking cavity, in order to find out the steady-state solution for individual pulses. Compared to the model that has been mentioned in Section 2.1.3, one additional effect that needs to be included is the phase-shift induced by the acoustic wave. The average soliton model assumes that all the pulse-shaping effects are homogeneously distributed in the laser cavity, and each of these effects can be expressed as a transfer operator upon the input field. Using this model, we can obtain not only the individual pulse shape under the steady state, but also the details of the balanced effects underneath. The electric field of a single pulse inside the cavity under the moving frame can be expressed as E(t) = A(t) exp (iω0 t) = v(t) exp [i (ω0 t + ϕ(t))]

(3.36)

in which ω0 is the carrier-frequency of the pulse. A(t) describes the slowly-varying pulse envelope, v(t) and ϕ(t) are the amplitude and phase of the pulse, respectively. Then we define the normalized energy density u(t) to be [117, 142] : u(t) =

 t −∞

v 2 (t)dt − u0 where u0 =

1  +∞ 2 v (t)dt. 2 −∞

(3.37)

u0 is half of the normalized pulse energy density, i.e. EP = neff 0 cAeff u0 . The resultant pulse shape of the average soliton model should have a sech2 -shape, with a frequency chirp that is proportional to the pulse intensity as dϕ(t)/dt = ξu(t) due to the SPM effect. The pulse parameters are summarized as below 

 

u0 t sech τ τ ϕ(t) = −ξu0 τ ln(v)   t u(t) = u0 tanh τ v(t) =

(3.38a) (3.38b) (3.38c)

where τ is the pulse-width. Then we employ the master equation of a single pulse that propagates in the mode-locked laser and by setting the change per round-trip to be zero, we can obtain the steady-state equation. We first employ simple notations for the coefficients that describe different effects involved in the master equation, and then explain them individually afterwards. The steady-state master equation

61

Ch. 3

OPTOACOUSTIC MODE-LOCKING can be express as [142] 



d d2 g − l + e1 + a1 2 + b1 v 2 dt dt   d d2 2 −i Ψ0 + e2 + a2 2 + b2 v − Φa (t) · v(t) exp (iϕ(t)) = 0 dt dt

(3.39)

where g and l give the optical gain and loss of the pulse at its central frequency, a1 defines the net gain bandwidth of the fibre gain, a2 defines the GVD of the cavity. b1 v 2 describes the SAM given by the intra-cavity fast saturable absorber, while b2 v 2 describes the phase shift due to Kerr-nonlinearity; e2 describes the gain tuning effect, which is caused by the offset of the pulse central frequency from the gain spectral centre. The term with e1 introduces a temporal delay due to the linear dispersion, while Ψ0 is an unimportant phase shift determined by the cavity transit time. The last term Φa describes the phase shift due to the index modulation caused by the optoacoustic effect. The gain spectrum of the laser is assumed to have a Lorentzian shape with a peak gain of g0 at frequency of ωg and a gain bandwidth of ΔΩg . Then we can express the coefficients g, e2 and a1 as below: g0 1 + 4η 2 8g0 η e2 = (1 + 4η 2 )2 ΔΩg 8g0 a1 = (1 + 4η 2 )3 ΔΩ2g g=

(3.40a) (3.40b) (3.40c)

where η is the detuning parameter defined as η = (ω0 − ωg ) /ΔΩ2g . The coefficient a2 and b2 are defined as 1 a2 = β2av 2 1 b2 = − γKerr neff 0 cAeff 2

(3.41a) (3.41b)

where β2av is the average GVD, and γKerr is the average Kerr nonlinear coefficient of the laser cavity. The phase shift term due to acoustic wave Φa , according to the conclusion from Section 3.1.3, can be expressed as Φa (t) =

62

ω0 (LPCF /Lc ) na (t) = Aa cos (Ωt − Δϕ) c

(3.42)

Optoacoustic mode-locking using SRLS

3.2

in which Aa is the amplitude of the average phase modulation, which has been determined in Section 3.1.3 to be Aa =

ω0 γe2 |Q1 |Q0 Ep LPCF 

(3.43)

4πn2eff c2 Aeff ρ0 LR 4δa2 + Γ2B

where LR is the cavity length. In order to derive the steady-state solution of Eq.(3.39), the phase shift term Φa can be expanded into a series of functions of tanh(t/τ ) of different power, similar to that of the Taylor expansion. The reason for such kind of expansion is related to the type of solution of the pulse shape, which is of sech2 -shape. As we will see later, the solution can be determined simply by summarizing the coefficients of independent hyperbolic functions and making them to be zero [142]. The expansion of Φa up to the second-order can be written as   (1) Φa (t) = Φ(0) a + Φa tanh

t τ

  2 + Φ(2) a tanh

t τ

(3.44)

where Φ(l) a , (l = 0, 1, 2) are the zero- to second-order expansion coefficients, which can be determined as ⎧ ⎪ ⎪ Φ(0) ⎪ a ⎪ ⎪ ⎨

= Aa cos (Δϕ)

Φ(1) = Aa Ωτ sin (Δϕ) 1 = − Aa Ω2 τ 2 cos (Δϕ) 2

a ⎪ ⎪ ⎪ ⎪ ⎪ ⎩Φ(2) a

(3.45)

By substituting Eqs.(3.38), (3.44) and (3.45) into Eq.(3.39), and setting the coefficients of the terms with independent hyperbolic functions to zero, we can obtain six equations: e1 = e2 X

(3.46a)

Φ(1) a Y − e1 X u0

(3.46b)

2 Ψ0 τ 2 = Φ(0) a τ − b2 Y + a1 X + a2

(3.46c)

(g − l)τ 2 + b1 Y + a2 X − a1 = 0

(3.46d)

b1 Y + 3a2 X + a1 X 2 − 2a1 = 0

(3.46e)

Φ(2) a Y + b2 Y − 3a1 X − 2a2 = 0 u20

(3.46f)

e2 = −

a2 X 2 +

63

Ch. 3

OPTOACOUSTIC MODE-LOCKING where X = ξu0 τ and Y = u0 τ . The parameters of this model (Ψ0 , e1 , ω0 , u0 , τ and ξ) can be determined from these six equations. The parameter Ψ0 determined by Eqs.(3.46c) gives a phase delay caused by the linear dispersion per cavity round-trip. e1 and e2 can be determined by Eqs.(3.46a) and (3.46b). While the term of e1 gives an unimportant time-delay, e2 defines the offset of the central frequency ω0 of the pulses, which can be rewritten as 8g0 η −Φ(1) a τ = e2 = 2 2 1 + X2 (1 + 4η ) ΔΩg

(3.47)

where we used the expression of e2 given in Eq.(3.40b). Eq.(3.47) describes the relationship between detuning parameter η of the pulse carrier-frequency and the first-order coefficient Φ(1) a of the optoacoustic effect in the PCF. We can reveal from Eqs.(3.44) and (3.22) that the acoustic wave always gives a positive Φ(1) a (dna /dt > 0), which causes a ref-shift of the pulse carrier-frequency. In order to compensate this red-shift, a gain detuning from the centre of the laser gain is required, and thus the η parameter should be negative. Therefore, when the pulse with a finite bandwidth is amplified in the gain section of the laser cavity, its carrier-frequency would be shifted slightly towards the peak frequency of the gain spectrum. Such a frequency-shift would compensate the red-shift that occurred in the PCF. With the practical parameters used in our experiment, the frequency detuning parameter is estimated to be in the order of ∼ −1 × 10−3 , which corresponds to a wavelength offset of ∼ 0.01 nm from the EDF gain centre. This detuning is rather weak and can be hardly revealed from the experiments, and the central wavelength of the resultant pulses would still be dominantly determined by the wavelength-dependent gain/loss profiles of the cavity. The normalized pulse energy u0 , pulse-width τ and the pulse chirp ξ can be determined by Eqs.(3.46d) to (3.46f). By eliminating X 2 terms in Eqs.(3.46e) and (3.46f), the pulse chirp ξ can be expressed as ξ=

−1 a1 b2 − b1 a2 + a1 Φ(2) a τ u0 3 (a21 + a22 )

(3.48)

Since the period of the acoustic wave along the PCF is much longer than the soliton duration, the second-order phase-modulation Φ(2) a induced by the acoustic wave within the duration of the pulse is much weaker than the modulation induced by −7 Kerr-effect. As an estimation, the second order coefficient Φ(2) a is in the order of 10

64

Optoacoustic mode-locking using SRLS

3.2

m−1 , while the typical Kerr-effect coefficient b2 v 2 is estimated to be ∼ 0.1 m−1 . So the term with Φ(2) a can be neglected in Eq.(3.48). Then using this chirp parameter in Eq.(3.46d), we can obtain u0 τ =

3 [a1 − (g − l) τ 2 ] (a21 + a22 ) 2a21 b1 + 2a22 b1 + a1 a2 b2

(3.49)

Several assumptions can be made to simplify this expression. Firstly, (g − l) is approximately zero, which means that the gain balances the linear loss, while the loss induced by SAM can be neglected. Secondly, b2  b1 , meaning that the SPM effect is much stronger than the SAM effect, which is also in accordance with our practical parameters. Thirdly, a2  a1 , indicating that the limited bandwidth of the laser gain dose not impose practical limitations upon the pulse bandwidth (and thus the pulse duration), whereas the cavity GVD and SPM are the dominant parameters that determine the finial soliton duration. Given these assumptions, we can simplify Eq.(3.49) into 3a2 u0 · τ = (3.50) b2 Using the definitions of the parameters u0 , a2 and b2 , we can obtain Ep · τFWHM ≈

3.40|β2av | γKerr

(3.51)

which approximates the soliton-area theory of a fundamental soliton (Eq.(2.21)). In summary, the average soliton model described above gives two main conclusions. Firstly, the red-shift of pulse carrier-frequency induced by the optoacoustic effect needs to be compensated by the gain detuning effect. Secondly, the balance between the averaged cavity GVD and SPM effect gives rise to soliton-like pulse profiles, and the related soliton-area theorem can be used to predict soliton parameters.

3.2.3 The non-interferometric HML We have seen from above that the optoacoustic interaction in the PCF can be used to stabilize the pulse spacing between the multi-pulses in the HML laser thanks to the negative feedback from the index modulation induced by the acoustic wave. An important question is that whether there exists any carrier-phase relationship between the multi-pulses in such HML laser. Equivalently, phase-relationship between

65

Ch. 3

OPTOACOUSTIC MODE-LOCKING the axial-modes that is involved in the HML laser is also undetermined. For fundamentally mode-locked lasers, as we previously discussed, the single intra-cavity pulse simply repeats itself after each round-trip, and within the coherence time of the laser gain medium, the pulse should be able to maintain its phase, leading to a coherent train of output pulses. This is actually the basis of the frequency comb technology based on femtosecond fibre lasers [11]. In the frequency domain, fundamental modelocking requires that the all the axial-modes under the spectral envelope should be in phase, as shown in Fig.3.8(a). In the case of HML, for sure, the axial modes should follow a different profile that supports a regular multi-pulse pattern. We first understand it from the time domain. The pulse-spacing is stabilized due to the presence of the acoustic wave excited in the PCF-core. Since the acoustic wave can only respond to the pulse envelope (more precisely only the lowest frequency beat-note of the pulse train), the optoacoustic effect in the PCF can hardly discriminate the difference of carrier-phase between the multiple pulses in the laser cavity. The artificial saturable absorber based on NPR can clamp the pulse intensity, but it is also insensitive to the pulse carrier-phases. The laser gain saturation played a role in stabilizing the pulse energies, while the long relaxation time make it responsive only to the average-power of the intra-cavity pulses. Therefore, in the time domain, there are no obvious effects from the cavity that can impose any certain phase-relationship between the multiple pulses inside the cavity, although each pulse would be able to maintain its own carrier-phase after each round-trip. As a result, these intra-cavity pulses are largely independent from each other except for the even spacings and equal peak intensities. Actually, since the multi-pulses in the HML cavity arise from a noisy background during the self-starting process, they would actually have random phase-relationship between them, as later confirmed in the experiments. The optoacoustic mode-locking can be interpreted also in the frequency domain, where the SRLS process imposes coherence between the axial modes that are involved in the mode-locked laser. We have seen that in order to efficiently drive the acoustic wave in the PCF-core, the pulse repetition rate fp needs to be close to the acoustic resonance frequency. In addition, the pulses are equally spaced in the laser cavity, indicating that the repetition rate needs to be an integer multiple N of the fundamental cavity round-trip frequency fcav , i.e. fp = N · fcav . Both conditions can be easily achieved simultaneously due to the MHz bandwidth of the acoustic core-resonance, which is close to the typical cavity round-trip frequency of

66

Optoacoustic mode-locking using SRLS

3.2

the mode-locked laser. Only some rough adjustment of cavity length can ensure that on particular harmonic-order of the fundamental mode-locking frequency lies within the gain band of the acoustic resonance. In the frequency domain, mode-locked laser should consist of equally spaced axial modes, and the SRLS process would induce an energy conversion between axial modes that are spaced by N · fcav . The axial modes under the spectral envelope of the mode-locked laser is then effectively separated into N sets of evenly interleaved frequency comb. Each frequency comb has an internal frequency spacing of N · fcav (at GHz) while different sets of combs are spaced by fcav (at MHz) (as schematically plotted in Fig.3.8(b)). Such modelocking configuration leaves the phase relationship between these different sets of comb undetermined, and so far we have not yet able to obtain the complete picture of the comb-structure. Nevertheless, we still attempted to draw some qualitative predictions based on experimental measurement. Firstly, all the axial modes of the cavity should exist, meaning that no specific set of the GHz comb was selected by the mode-locked laser. Within each set of comb, all the axial modes are locked in-phase due to the energy conversion induced by the acoustic wave. Secondly, the N sets of combs should take the random phase relations so as to add up into a sequence of intra-cavity pulses with equal intensities. We will see in Chapter 4 some interferometric measurement that gives partial insights and clues to the actual comb structures of the optoacoustic mode-locked laser, while the detailed properties and potential applications of such “partially coherent” comb structure would require further investigations. Due to the fact that the optoacoustic mode-locking does not impose fixed phaserelation between adjacent intra-cavity pulses, it differs significantly from the HML laser that is based on Mach-Zehnder interferometer [78, 80] or coupled sub-cavity [76] where the intra-cavity pulses would interfere with adjacent ones. As a result, they would be mutually locked in phase. In the frequency domain, such interferometric filter actually selects a specific comb with spacing of the free-spectral range of the filter itself and suppresses all the others (as shown in Fig.3.8(c)). The sub-cavity configuration can keep the coherence between adjacent pulse, with however stability problem due to cavity length fluctuations. Compared to such case, the optoacoustic mode-locking was non-interferometric since it does not rely on the precise interference between the intra-cavity pulses. It is this non-interferometric nature that renders the optoacoustic mode-locking robust against environmental perturbations and meanwhile highly flexible, which gives rise to diverse phenomena beyond the

67

Ch. 3

OPTOACOUSTIC MODE-LOCKING conventional HML state, as we will see later in Chapters 6 and 7. (a)

fcav

axial modes

N*fcav

axial modes

(b)

(c)

fcav

axial modes

Fig. 3.8: The phase-relations between the cavity axial modes in different modelocking regimes. (a) In fundamental mode-locking, all the adjacent axial modes are locked in phase, so as to form a single pulse in the cavity. (b) In the optoacoustic mode-locking, each axial mode is phase-locked to the neighbours being N axial modes away, resulting in N set of evenly interleaved frequency comb, leading to N evenly-spaced and phase-unrelated pulses in the cavity. (c) In HML imposed by sub-cavity or with intra-cavity MZI, only the axial modes that are spaced by N · fcav are selected by the spectral filtering effect (grey curve), leading to N phase-locked phases in the cavity.

68

CHAPTER

4

GHZ-RATE SOLITON FIBER LASERS

This chapter demonstrates the experimental realization of GHz-rate soliton fibre lasers based on the optoacoustic mode-locking scheme using a short length of solidcore PCF. We start with the implementation technique of the solid-core PCF by describing its basic properties and how it was inserted in the mode-locked laser cavity. With a short piece of solid-core PCF inserted in the conventional soliton fibre laser based on the polarization APM, we realized for the first time a stable passive HHML laser at GHz repetition rate, corresponding to a harmonic order of a few hundreds. This HHML laser has remarkably low timing-jitter and good long-term stability. In addition, due to the robust optoacoustic mode-locking strategy, continuous tuning of both operating wavelength and pulse repetition rate are achieved while the laser is under freely running. The same mode-locking strategy also leads to a GHz-rate thulium-doped fibre laser emitting at ∼ 2 μm wavelength.

4.1 Implementation of PCF 4.1.1 Fabrication of PCF The solid-core PCF used in the mode-locked fibre laser is made from silica and its micro-structure is enabled by the standard “stack and draw” procedure [202]. The procedure starts from manual assemblies of glass capillaries (corresponding to hollow channels) and rods (corresponding to the solid-core or “defect” among the hollowchannel array/lattice) into the approximate structure of the PCF. Then the preform

69

Ch. 4

GHZ-RATE SOLITON FIBER LASERS stack is inserted into a glass tube, which would become the outer-cladding around the hollow channels. Next, this preform is fused and drawn into a micro-structured preform or cane. At last, this cane is drawn into the fibre with desired geometrical parameters, including the d/pitch-ratio1 , the core and outer-cladding diameters. The fine tuning of the fibre geometry can achieved during the drawing process by controlling the fusion temperature in the furnace, the feed rate of the preform, the drawing speed of the tractor system, and the air pressure inside the preform. In the final step, the fabricated PCF is coated with a polymer jacket (immediately hardened by exposure to a UV lamp) to improve the mechanical strength. The schematic drawing set-up and procedure is illustrated in Fig.4.1. (a)

Feed screw and holder Preform

(b)

Step 1 Capillaries fabrication

Furnace Diameter Monitor Cane Drawing system

Step 2 Preform stacking

Tractor system cutter

Step 3

Coating cone

Drawing preform into cane

UV lamp Tractor system Spooling wheel

Step 4 Drawing cane into fibre

Fig. 4.1: (a) The sketch of the PCF-fabrication set-up (the fibre drawing tower). (b) The PCF fabrication procedure.

The capillary that is used for stacking process has a diameter of ∼ 1 mm. 18 capillaries and a solid-rod with the same diameter are stacked into symmetrical triangular array with a defect (rod), as the solid-core, in the centre. The stack 1

70

The ratio between the diameter of individual hollow channel and the mutual spacing between the hollow channels

Implementation of PCF

4.1

structure is inserted into a glass tube with thin thickness, and then this preform is drawn into a cane with a diameter of ∼ 1.75 mm. In order to have a small solidcore in the final fibre, the outer cladding of the preform need to be scaled up to a significantly larger diameter. To do so, the cane is inserted into two layers of glass tube (inner/outer diameters are 2 mm/8 mm and 8 mm/15 mm respectively). Then this preform is installed into the holder on top of the feed screw of the fibre drawing tower, with gas pressure controllers connected to it in order to “open up” the hollow channels during the drawing process. The preform feed speed is ∼ 3 mm/min and the tractor drawing speed is about ∼ 60 m/min. The fast drawing speed ensures the homogeneity of PCF diameters, although slow drift of the fibre dimension usually cannot be completely avoided2 . The resultant PCF would have a core diameter around 2 μm, while the outer cladding would have a diameter over 100 μm. The large outer-cladding diameter also facilitate the splicing between PCF and standard SMF, as will be described later. At last the fibre is fed into the coating cone and the polymer layer would have a diameter about 300 μm. The SEM photo of the typical PCF micro-structure is given in Fig.1.2. The solid-core PCF was designed to operate in practical single-mode regime, which means that the fundamental LP01 optical mode would have a much lower loss than the others. The intrinsic loss of the solid-core PCF generally scales up quickly with smaller core-size. However, with a typical core diameter of ∼ 2 μm, the PCF loss could be < 0.02 dB/m (at 1.55 μm), which is quite acceptable for practical use.

4.1.2 The acoustic resonance of PCF The repetition rate of the HHML fibre laser is determined by the acoustic coreresonance of the PCF, which is dominated by the R01 “breathing” mode driven by the LP01 optical mode. Using the simulation tool Comsol, the LP01 mode (with electrostriction force profile) and the R01 acoustic mode profiles can be obtained, as illustrated in Fig.4.2. The optical wavelength used in the simulation is 1.55 μm, while the core-diameter of the PCF is 1.95 μm and the d/pitch ratio is 0.80. The resonance frequency of the R01 mode is ∼ 1.89 GHz, which is determined mainly by the PCF core-size and meanwhile influenced by the d/pitch ratio (i.e. the thickness of the “bridge” between the hollow channels intimately surrounding the core). Simulation results as well as experimental observations revealed that the 2

Empirically, in terms of the core-resonance frequency, several MHz drift would occur over tensof-meter long PCF that was drawn from the same preform.

71

Ch. 4

GHZ-RATE SOLITON FIBER LASERS acoustic resonance frequency would increase with smaller core-size, which was rather intuitive, and meanwhile also with lower d/pitch ratio (or thicker bridge). The simulation results illustrating such dependencies are shown in Fig.4.3. (b)

PCF core structure

(c)

LP01 optical mode

Length (Pm)

1 0 1 2 2

1 0 1 Length (Pm)

2

0

1

2 1 0

0 1 2 2

1 0 1 Length (Pm)

2

Density change (a.u.)

Optical intensity (a.u.)

2

2 Pm

R 01 mechanical mode

1

Length (Pm)

(a)

1

Fig. 4.2: (a) The SEM photo of the solid-core PCF core structure. The two hollow channels marked by the red arrows are intentionally made smaller than the others by gas pressure control during the fabrication process, so as to induce fibre birefringence. (b) The simulated intensity profile of the LP01 optical mode. The corresponding electrostriction force field is given by the black arrows. (c) The R01 acoustic mode driven by the LP01 optical mode. The density change and the deformation (highly exaggerated) are presented.

Frequency (GHz)

2.00 1.96 1.92 d/pitch=0.70 d/pitch=0.75 d/pitch=0.80

1.88 1.5

1.6

1.7

1.8

1.9

2.0

2.1

Core diameter (μm)

Fig. 4.3: The simulated dependencies of the PCF-core resonance frequency (R01 mode upon the PCF core diameter and the d/pitch ratio).

The axially symmetric (breathing) radial R01 mode is not the only acoustic mode that could be effectively driven by the LP01 optical mode in the solid-core PCF. The TR21 torsional-radial mode can also be excited by the LP01 mode, with however lower overlapping by a factor of ∼ 3 compared to R01 mode [95]. Given some proper starting condition, the repetition rate of the HHML laser can also be locked to the resonance frequency of the TR21 mode of the PCF, with although

72

Implementation of PCF

4.1

slightly unstable performance. The acoustic resonance can be experimentally characterized by the pump-probe method which makes use of the SRLS process pump by a dual-frequency CW source [95]. The measurement set-up of the acoustic resonance is sketched in the Fig.4.4 below [107]. 2f0

BC Z

PC-1

CW fibre laser 2 kHz bandwidth

Z

EOM

Z

EDFA

RF source f

FC 90/10

2f0 4 GHz FSR OSC

SI

PCF Z

Z

PC-2

TA

Fig. 4.4: The set-up for measuring the optoacoustic gain spectrum using the SRLS process. A dual-frequency CW light is used as the pump light, which is obtained through modulating a single-frequency CW laser. PC, polarization controller; EOM, electro-optic modulator; BC, bias controller; EDFA, Er-doped fibre amplifier; FC, fibre coupler, TA, tunable attenuator; SI, scanning interferometer; FSR, free spectral range; PD, photodetector; OSC, oscilloscope.

A 2-kHz-bandwidth fibre laser at 1550 nm was used as the light source. The dualfrequency pump light is generated using the electro-optic modulator (EOM). When this single-wavelength light is launched into the EOM with an optimized polarization state (adjusted by PC-1), and a radio-frequency (RF) wave at frequency f0 with an appropriate bias voltage is applied to the EOM, two sidebands with equal amplitude and a frequency difference of 2f0 can be generated (at frequency ω1 and ω2 respectively), while the original frequency component from the fibre laser (at ω0 ) can be highly suppressed. Then this dual-frequency light is amplified using an EDFA to around 25 dB − 30 dB. A 90/10 fibre coupler was inserted afterwards simply to monitor the optical power. With another PC-2 to adjust the polarization state3 , the amplified pump light was launched into the PCF with a length of a few metres. The output signal from the PCF was sent to a scanning interferometer (SI) with reduced power using the tunable attenuator (TA). Then the output signal from the SI was 3

The polarization states of the driving light are found to have slight impact on the optoacoustic interactions, while systematic studies have not been completed.

73

Ch. 4

GHZ-RATE SOLITON FIBER LASERS detected by a photodetector (PD) and recorded using an oscilloscope (OSC). When this beating frequency 2f0 is tuned close to the acoustic resonance of the PCF-core, efficient SRLS process would occur and cause energy conversions from frequency component ω2 to ω1 [95]. The optoacoustic gain coefficient g (Ωa ) at the frequency Ωa = 2f0 can then be calculated and the simplified formula would be [107] g (Ωa ) =

ln(P1 /P2 ) P0 LPCF

(4.1)

Optoacoustic gain (m-1W-1)

where P1 and P2 are the powers of frequency components at ω1 (Stokes light) and ω2 (pump light) respectively, P0 is the total power that is launched into PCF. By scanning the beating frequency 2f0 , the gain spectrum over the entire acoustic resonance can be obtained. A typical measurement result at the R01 resonance is given in Fig.4.5 below. 0.4 0.3 1.95μm

R01-like acoustic resonance

0.2 0.1 0.0

1.80

1.84

1.88 1.92 1.96 Frequency (GHz)

2.00

Fig. 4.5: The optoacoustic gain spectrum measured at the R01 acoustic resonance of the solid-core PCF with a core diameter of 1.95 μm. Inset: the simulated acoustic mode profile at the R01 resonance

The acoustic resonance at the fundamental mode should in principle take a singlepeak Lorentzian profile with a bandwidth that is determined by the silica material itself [95]. In reality, the acoustic resonance takes a rather complicated profile, with multi-peaks over the resonance range and broader bandwidth than the intrinsic bandwidth of the silica. The multi-peak structure probably might be caused by the anisotropic PCF-core structure due to the hexagonal shape and the suspending bridges. The coupling between the longitudinal wave and the sheer wave probably yields a few hybrid acoustic modes around the major R01 mode. The built-in stress in the PCF-core structure during the fabrication process might also contribute to such multi-peak structure. Such multi-peak structure did not cause any repetition-rate

74

Implementation of PCF

4.1

Optoacoustic gain (m-1W-1)

instability of the optoacoustic mode-locking, as we realized later in the experiments. In fact, stable mode-locking would only occur close to the main peak at ∼ 1.89 GHz. The broad bandwidth of the acoustic resonance, however, puzzles us till now. According to the theory in [95], the intrinsic bandwidth of silica at frequency around 2 GHz should be about only ∼ 1 MHz, while the measurement yielded a bandwidth of 10 MHz−20 MHz. This indicates some unknown damping mechanisms other than that from the material itself. We first suspected that the energy leakage through the suspending bridge has induced additional damping. By removing the polymer coating, we found that the acoustic bandwidth only narrowed slightly, as shown in Fig.4.6, suggesting that such leakage might not the be the dominant damping mechanism4 . 0.4 0.3

with coating without coating

0.2 0.1 0.0 1.84

1.86 1.88 1.90 Frequency (GHz)

1.92

Fig. 4.6: The optoacoustic gain spectrum of solid-core PCF measured at R01 resonance, with (blue-line) and without (red-line) the polymer coating,

We then assumed that the air trapped inside the hollow channels (and sealed by the splice at two ends) has induced additional damping. To verify this mechanism, we evacuated the air in the PCF for over 72 hours, then measured the optoacoustic gain spectrum of such PCF. The measured spectrum is compared with that in ordinary ambient (1 bar) as illustrated in Fig.4.7. The bandwidth of the same resonance was also only slightly narrowed after the evacuation, indicating that the trapped air also provided only trivial acoustic damping. The investigation of the acoustic damping mechanism of such solid-core PCF is still on-going. Major reduction of the bandwidth would probably need some special phononic bandgap formed by some specially-designed hollow channels array [203]. 4

This experiment actually only indicates that the leaked acoustic energy that reached to the polymer coating is very weak. It is also suspected that the leaked energy might have already been dissipated in the hollow channels or the thick outer-cladding before reaching the coating.

75

Ch. 4

GHZ-RATE SOLITON FIBER LASERS

Optoacoustic gain (m-1W-1)

Besides, the bandwidth generally scales quadratically with the resonance frequency of the acoustic mode. In order to achieve a higher acoustic frequency, the core-size can be straightforwardly reduced. However, this would influence the confinement factor of the driving optical mode, and beyond a certain size, the fibre loss would become unacceptably high for practically applications. At the moment, the highest R01 resonance frequency in the solid-core PCF we achieved in experiments is ∼ 2.9 GHz. Shorter optical wavelength might push forward this limit (for example using Yb-doped gain fibre), due to the smaller optical mode area. 0.4

1 bar Evacuated 72h

0.3 0.2 0.1 0.0 1.84

1.86 1.88 1.90 Frequency (GHz)

1.92

Fig. 4.7: The optoacoustic gain spectrum of solid-core PCF measured at R01 resonance, in ambient at 1 bar (blue line) and after evacuation of the air in the hollow-channels for 72 hours (red line).

The acoustic resonance frequency also depends on the temperature of the PCF. We used liquid nitrogen as well as a heating oven to create different temperatures of the PCF, ranging from −196 ◦C to 400 ◦C (i.e. 77 K − 673 K), and measured the R01 acoustic resonance at these temperatures. The results are illustrated in Fig.4.8. The resonance frequencies at the three dominant peaks all vary linearly with the temperature change, with a slope of ∼ 100 kHz/K. Over the entire temperature tuning range of 600 K, the dominant resonance frequency at 1.89 GHz (Peak 1) was tuned by ∼ 65 MHz. Such tuning of resonance frequency can be interpreted by the change of acoustic wave velocity in silica which decreases linearly with the increasing temperature [204]. In addition to the peak frequency tuning, the relative strengths of individual peaks are also varied. Note that in Fig.4.8(a) with temperature of 77 K, the bandwidth is significantly broader than that under higher temperatures. This is partially due to the experimental constraints that only 70 % of the test fibre was immersed in the liquid nitrogen, leading to inhomogeneous distribution of the temperature along the fibre.

76

Implementation of PCF 0.4

77 K Liquid N2

0.2

Optoacosutic gain (W-1m-1)

0.0 0.4 0.2

2

peak 3 1

0.0 0.4 0.2

473 K

0.2 0.0 0.4

573 K

0.2

1.92

1.90

1.88

1.86

0.0 0.4

Slope 109 kHz/K 112 kHz/K 88 kHz/K

673 K

0.2 0.0

Peak 1 Peak 2 Peak 3

1.94

288 K Room temp. 373 K

0.0 0.4

(b)

Resonance (GHz)

(a)

4.1

1.84 1.80 1.84 1.88 1.92 1.96 2.00 Frequency (GHz)

0

200 400 600 800 Temperature (K)

Fig. 4.8: (a) The acoustic resonance measured at R01 resonance under different temperatures ranging from 77 K to 673 K. (b) The relations between resonance frequencies of the three main peaks of the R01 mode under different temperatures.

By applying a longitudinal strain upon the PCF, the resonance frequency can also be tuned, however, within a much smaller range before the PCF was stretched to break. In terms of longitudinal deformation (relative change of fibre length), the change of the resonance frequency is ∼ 1 MHz/0.1 %, while the fibre would break at ∼ 0.5 % stretching.

4.1.3 The GVD of PCF For an optical pulse at a central frequency of ω0 with relatively narrow bandwidth, the propagation constant of the pulse in a dispersive medium can be expanded around ω0 as [134]: 1 β (ω) ≈ β0 + β1 (ω − ω0 ) + β2 (ω − ω0 )2 2

(4.2)

where β0 = β (ω0 ) which is the propagation constant for the carrier phase, β1 = (dβ/dω)|ω0 = 1/vg determines the group velocity in the vicinity of ω0 . β2 =

77

Ch. 4

GHZ-RATE SOLITON FIBER LASERS (d2 β/dω 2 )|ω0 is the GVD, which is related to the refractive index n (ω) as

d2 n 1 dn +ω 2 β2 = 2 c dω dω



(4.3)

The typical unit of β2 is ps2 /km, and 1 ps2 /km means that, for a pulse with a bandwidth of 1 ps−1 (i.e. 1 THz), the pulse duration would be broadened by 1 ps after a propagation length of 1 km. An alternative parameter that defines the GVD is the so-called D parameter, which is a function of wavelength λ as [134]: D≡

2πc λ d2 n dβ1 = − 2 β2 = − dλ λ c dλ2

(4.4)

The typical unit for D is ps/km/nm, and D = 1 ps/km/nm means that for a pulse with linewidth of 1 nm, after propagating 1 km, the pulse duration would be broadened by 1 ps. Note that the signs of β2 and D are opposite. For a carrier wavelength of 1.55 μm, D (ps/km/nm) ≈ −0.78β2 (ps2 /km). 200

D (ps/nm/km)

100 0 Core diameter

-100

1.2μm 1.4μm 1.6μm 1.8μm 2.0μm 2.2μm

-200 -300 -400 0.6

0.8

1.0 1.2 Wavelength (μm)

1.4

1.6

Fig. 4.9: The GVD of solid-core PCF (D-parameter) with d/pitch ratio of 0.85 and core diameters ranging from 1.2 μm − 2.2 μm.

In order to obtain the GVD of the solid-core PCF, the dependence of refractive index on the frequency (or wavelength) should be obtained and the refractive index used in Eq.(4.3) should be replaced by the effective refractive index (neff ) of the optical mode (usually the LP01 mode) in the PCF. When the core size is relatively large, neff (ω) of the LP01 mode would be close to index of the material itself, meaning that the atomic resonance of silica contributes dominantly to the dispersion relation.

78

Implementation of PCF

4.1

However, when the core-size becomes smaller, and even become comparable with the optical carrier wavelength, neff (ω) would be strongly influenced by the waveguide structure. In fact, the transverse optical mode confined in the fibre core by the hollow channels introduces an additional optical resonance upon the intrinsic atomic resonance of silica and thus alters the dispersion relation. Therefore, the GVD of the solid-core PCF can be properly controlled via careful core-structure engineering [99, 100]. Using Comsol, we scanned the neff of the LP01 mode of solid-core PCFs with different core sizes, and the dispersion relations in terms of D-parameter obtained using Eqs.(4.3) and (4.4) are illustrated in Fig.4.9. The wavelength range of interests in our case is around 1.55 μm. The core size of the PCF used in later experiments is of ∼ 2 μm with d/pitch ratio ∼ 0.8, and according to the simulation results shown in Fig.4.9, the PCF would have a strong anomalous dispersion. By inserting such a PCF of a few metres long, the mode-locked fibre laser would operate in soliton regime if no other dispersion compensation fibres are used simultaneously. As the core-size becomes smaller, the GVD would tend to shift from being anomalous to normal. The influence of d/pitch ratio upon the GVD is however much weaker. A higher d/pitch ratio (bigger hollow channels) tends to only slightly shift the GVD toward anomalous dispersion, given a core-diameter of ∼ 2 μm. The GVD of the PCF is experimentally measured using the Mach-Zehnder interferometer (MZI) with a broadband pulsed-light source [205]. The two path of the MZI are constituted by free-space path and the solid-core PCF respectively, By comparing the group-delay of these two path over different central wavelength, the group index can be obtained at these wavelengths, from which the GVD can be determined.

4.1.4 Splicing PCF with conventional fibres Mode-locked fibre lasers are desired to have all-fibre configurations which can eliminate free-space optics and lead to robust performance against environmental disturbances. The mode-locked soliton fibre laser at ∼ 1.55 μm we built has a ringcavity and is mostly constructed by standard SMF-28, which has anomalous dispersion (β2 = −22.5 ps2 /km) at this wavelength. Other type fibre components in the ring-cavity include the EDF and the wavelength-division multiplexer (WDM), which are made from conventional fibres with comparable core-size with the SMF-28 (∼ 10 μm). The splicing between such conventional fibres with similar core-size is

79

Ch. 4

GHZ-RATE SOLITON FIBER LASERS already state-of-art technique and the fusion parameters are available in commercial splice machines. The resultant splice losses are usually well below 1 dB. However, the insertion of solid-core PCF would require the splice between the SMF-28 and the PCF with hollow channels and a significantly smaller core. Direct splice between them would induce huge loss (> 5 dB). To reduce the splice loss, a transition fibre section is introduced between the PCF and SMF, which has an intermediate core-size [206]. We use a short transition fibre with high numerical aperture (Nufern UHNA-7) and a core diameter of 2.4 μm between the SMF-28 and the PCF, as illustrated in Fig.4.10. UHNA-7 to SMF-28 Single splice strong arc, seconds

UHNA-7 to PCF multiple splice weak arc, miliseconds Transiton fibre UHNA-7 3~5 mm

Solid-core PCF

Icore 1.9 Pm

SMF-28

2.4 Pm

Icladding 100 Pm

8.2 Pm 125 Pm

125 Pm

Pre-splice gap 50 dB. The two polarization controllers, PC-1 and PC-2, together with the linear polarizer configured the polarization APM (Section 2.1.2). The self-starting of the HHML usually requires some initial trigger using PC-1 and -2 after cranking up the LD pump power. WDM

WDM EDF

50:50 Output

LD-1

LD-2

PC-1

ISO

Delay Line Tunable Filter

Polarizer

PC-3

PC-2 PCF

Fig. 4.11: The schematic of a unidirectional ring-cavity for the HHML soliton fibre laser. A 2-m-long solid-core PCF is inserted to enable the optoacoustic mode-locking. A F-P cavity based tunable filter is inserted for laser wavelength tuning. The tunable delay line is inserted for adjusting the cavity length. EDF, Erbium-doped fibre; LD, laser diode (pump); WDM, wavelength division multiplexer; ISO, isolator; PC, polarization controller.

The solid-core PCF that was inserted in the ring-cavity has a core-diameter of ∼ 1.95 μm and an air-filling ratio of ∼ 0.53 (or a d/pitch ratio of ∼ 0.85). The birefringence of the PCF induced by the specially engineered hollow-channel sizes is roughly 1.6 × 10−4 (group index birefringence). The PCF-core supports a R01 acoustic mode with a resonance frequency of ∼ 1.887 GHz and a bandwidth of ∼ 13 MHz. The GVD of this PCF is measured to be ∼ −157 ps2 /km at 1550 nm, while the Kerr-nonlinearity coefficient is estimated to be ∼ 9.3 W−1 km−1 . An additional PC-3 was inserted between the polarizer and the PCF to launch the linearly polarized light into one principle axis of the PCF. A tunable delay-line was inserted to adjust the cavity length so as to have one harmonic-order of the fundamental cavity roundtrip frequency (fcav ) lying within the resonance bandwidth of the R01 acoustic mode. Moreover, the repetition rate would also become continuously tunable within this bandwidth. In order to achieve the laser wavelength tuning, a F-P cavity based manual tunable-filter was inserted in the cavity. The filter had a tuning range of

83

Ch. 4

GHZ-RATE SOLITON FIBER LASERS ∼ 40 nm (1530 nm − 1570 nm) and a filtering bandwidth of 12 nm. Note that the tunable filter is not a necessary component for a stable operation of this soliton laser, and meanwhile it did not degrade the laser stability. The total ring-cavity length was very flexible and could range from a few metres to near a hundred metres. The exemplary ring-cavity we demonstrate has a cavity length of ∼ 45 m, corresponding to fcav = 4.84 MHz. All the components are connected by the standard SMF-28. The insertion loss of the PCF is < 2 dB, including the intrinsic loss of the PCF (< 0.02 dB). The total cavity loss, excluding the EDFA and the tunable filter, was ∼ 11 dB. The average cavity GVD is strongly anomalous, which is dominated by the GVD of the PCF, resulting in soliton-regime operation. A 50/50 fibre coupler was used as the output port, and the pulse train generated by this HHML laser was detected by a 30-GHz photodetector and then recorded by a 16-GHz oscilloscope and a 26-GHz electric spectral analyser (ESA).

4.2.2 Results: Soliton train and wavelength tuning The lasing threshold for this ring-cavity laser was reached with a pump power of ∼ 100 mW, while a stable HHML state required a higher pump power (∼ 500 mW) together with careful adjustments of the PC-1 and PC-2 in order to initiate the multi-pulse operation and eliminate the CW background (which manifests itself as a narrow peak upon the soliton spectrum). Without the tunable filter, this HHML laser emitted at a central wavelength of ∼ 1560 nm, close to the wavelength at which the EDF has the highest net gain. The emitted solitons had a bandwidth of 4 nm under the highest pump power, with sub-ps duration. With the tunable filter, the pulse bandwidth was narrowed to ∼ 2 nm. The experimental results when the central frequency is tuned to 1550 nm are shown in Fig.4.12. The pulse train recorded by the oscilloscope is shown in Fig.4.12(a), with a zoom-in on ten consecutive pulses in Fig.4.12(b), which gives a regular pulse spacing of ∼ 531 ns. From the power spectrum recorded by the ESA (Fig.4.12(c)), the repetition rate of this HHML laser can be determined to be 1.8826 GHz, and the sharp harmonic peaks indicated the presence of stable HHML state. The harmonic-order was determined to be 389 given fcav = 4.84 MHz. The measured autocorrelation-trace of the pulse train is shown in Fig.4.12(d), which gives a FWHM of 2.5 ps and thus an estimated soliton duration of ∼ 1.6 ps given a sech2 profile. The acoustic gain spectrum of the solid-core PCF is shown in Fig.4.12(e),

84

Wideband-tunable GHz-rate Er-fibre soliton laser

4.2

with the repetition rate of the pulse train marked in it. By tuning the delay-line, the pulse repetition rate can be continuously tuned within a few MHz (1.878 GHz − 1.884 GHz) during the laser free-running (marked as blue shading in Fig.4.12(e)), which remains lower than the acoustic resonance frequency (the reason is explained in Section 3.2.1). Provided that the cavity length did not change over one acoustic period (∼ 10.9 cm), the specific harmonic-order is highly reproducible. Besides, although the acoustic resonance takes a form of multi-peaks as shown in Fig.4.12(e), the pulse repetition rate apparently preferred the strongest central peak and did not hopping into the side-peaks.

0.4 0.0

-0.4

(b)

(d)

0.8

0

10

Voltage (V)

1.2

Intensity (dBm)

40

50

~531 ps

0.8

1.0 0.8

2.5 ps

0.4 0.2 0.0

-5 -4 -3 -2 -1 0 1 2 Time (ps)

3

4

5

(e)

0.0 5

6

7 8 Time (ns)

9

10

0 -25 -50 -75

measured data sech2 fitting

0.6

0.4

-0.4

(c)

20 30 Time (ns)

Normalized intensity

Voltage (V)

1.2

0

5

10 15 20 Frequency (GHz)

25

Gain coefficient (m-1W-1)

(a)

0.4

measured data

0.3 0.2 0.1

repetition rate 1.8826 GHz

R01 resonance 1.887 GHz

0.0 1.85 1.86 1.87 1.88 1.89 1.90 1.91 1.92 Frequency (GHz)

Fig. 4.12: The HHML soliton laser output with the central wavelength at 1550 nm. (a) The output pulse train recorded by the 16 GHz oscilloscope over 50 ns span. (b) Zoom-in on 10 consecutive pulses in (a). (c) The RF-spectrum of the output pulse train recorded by the ESA. (d) The autocorrelation trace of a typical output soliton (the measured data in black square, with a fitting curve assuming a sech2 pulse shape). The 2.5 ps FWHM autocorrelation width corresponds to ∼ 1.6 ps soliton duration. (e) The acoustic gain profile of the R01 resonance of the PCF-core (yellow square). The pulse repetition rate (1.8826 GHz, marked in blue line) is slightly smaller than the resonance peak frequency 1.887 GHz.

While this HHML is laser is freely running, the central lasing wavelength can be continuously tuned using the tunable filter in a range of 1532 nm−1566 nm. Fig.4.13(a)

85

Ch. 4

GHZ-RATE SOLITON FIBER LASERS shows the exemplary output spectra of the laser with 5 nm-step under a pump power of 1.6 W. Most importantly, the harmonic-order of the HHML laser remains constant during the tuning-process. The strong Kelly sidebands distributed upon the otherwise smooth spectral profile indicated soliton-regime operation. In the experiment, we observed that the central wavelength of the laser solitons usually deviated slightly (a few nm) away from the pre-set central wavelength of the filter. This was probably due to the fact that spectral centre was supposed to be at the maximum of the effective gain profile of the laser cavity, which was determined not only by the transmission profile of the filter, but also the gain spectrum of the EDFA and some other wavelength-dependent loss contributed by other components in the cavity. As a consequence, the central wavelength of the soliton laser would very unlikely be exactly at the transmission centre of the tunable filter, although it could be precisely tuned nevertheless by the filter alone. The HHML soliton laser became unstable when the filter was set close to the edge wavelengths. This might be caused by the fact that the EDFA had significantly lower gain at these wavelength provided with only moderate pump power. The soliton energy then became so low that the optoacoustic interaction in the PCF-core is no longer strong enough to stably lock the inter-pulse spacings. Over the entire tuning range (34 nm) the laser performance was maintained. The FWHM pulse duration and the 3-dB spectral bandwidth of the output solitons measured during the wavelength tuning process as shown in Fig.4.13(b) were found to be almost constant over the entire tuning range. The time-bandwidth product (TBP) is around ∼ 0.4, which is close to the Fourier transform limit of the sech2 -shaped pulse. The residual pulse chirp (∼ −0.1 ps2 ) was caused by the uncompensated SMF-28 pigtail fibre (∼ 4 m-long) between the laser output and the autocorrelator. The transform-limited soliton duration was estimated to be ∼ 1.4 ps. This HHML laser also exhibited flexibility in the soliton energy over the entire tuning range at a fixed harmonic-order. The FWHM pulse duration and the 3-dB bandwidth under different soliton energies (determined by the pump power) are plotted in Fig.4.13(c), with the theoretical fitting curves based on the fundamentalsoliton assumption. When the estimated pulse energy in the PCF increased from ∼ 9.5 pJ to ∼ 14 pJ, the pulse duration declined from ∼ 1.8 ps to ∼ 1.45 ps, meanwhile the 3-dB bandwidth increased from ∼ 1.4 nm to ∼ 2.0 nm. The TBP at different pulse energies remained being close to the transform limit, indicating trivial pulse chirp.

86

Wideband-tunable GHz-rate Er-fibre soliton laser (a)

-30 -45 -60 1550 1560 Wavelength (nm)

1.8

2.2

1.7

2.0

1.6

1.8

1.6 1.5 1530 1540 1550 1560 1570 Wavelength (nm)

1570

1580

(c) 1.8

2.0

1.6

1.6

1.4

1.2

3-dB bandwidth (nm)

1540

FWHM duration (ps)

FWHM duration (ps)

1530

3 dB bandwidth (nm)

Intensity (dBm)

-15

-75 1520

(b)

4.2

10 11 12 13 14 Pulse energy (pJ)

Fig. 4.13: (a) The spectrum of the soliton laser with its central wavelength tuned between 1535 nm − 1565 nm, with 5 nm step. (b) The measured pulse durations and 3-dB spectral bandwidths of output solitons with different central wavelengths under a constant pump power of 1.6 W. (c) The measured pulse durations (red-squares) and the 3-dB bandwidths (blue triangles) with different pulse energies in the laser cavity when lasing at 1550 nm. The fitting curves (solid-lines) are based on fundamental soliton-area theorem.

4.2.3 Laser noise and stability The temporal trapping potentials formed by the cooperation between the acoustic wave and the cavity dispersion enables the self-stabilized inter-pulse spacings in the HHML laser, as we have theoretically described in Section 3.2.1. In the experiments, we observed that the HHML laser stably locked to the acoustic resonance of the PCF has exhibited remarkable robustness against external perturbations, with days-long stable operation without additional assistances. The supermode suppression ratio over the 100 MHz span was > 50 dB (as shown in Fig.4.14(a)). Besides, we observed slow drifting of this repetition rate within only few kHz over many hours operation, which we believe was mainly due to environmental temperature fluctuations. A few snap-shorts of the first harmonic peak recorded every 10 min are shown in Fig.4.14(b), exhibiting a drifting range of ∼ 1.5 kHz within one hour. Long-term drifting range (> 10 h) was observed to be within 10 kHz.

87

GHZ-RATE SOLITON FIBER LASERS

Intensity (dBm)

(a)

(b)

0 -20

1.8826 GHz > 50 dB

-40 -60

4.84 MHz

-80 -100

1.84

1.86 1.88 1.90 Frequency (GHz)

1.92

Intensity (dBm)

Ch. 4

0

1.88259736 GHz

-20 -40 -60 -80 -100 -1.5

-1.0 -0.5 0.0 0.5 1.0 Relative Frequency (kHz)

1.5

Fig. 4.14: (a) Zoom-in at the first harmonic peak of the RF-spectrum (100 MHzspan) of the output pulse train, exhibiting a super-mode-noise suppression ratio > 50 dB. (b) Zoom-in at the first harmonic peak of the RF-spectrum (3 kHzspan), with 7 snap-shots recorded in 1 h.

The HHML laser also exhibited remarkably low short-term noise, including the timing-jitter and amplitude (energy) fluctuations of the output GHz-rate pulses. To estimate the short-term pulse amplitude fluctuations, the single-sideband (SSB) noise spectrum around the baseband was measured as a function of the offset frequency ranging from 20 Hz to 1 MHz, and the result is shown in Fig.4.15(a) (compared with the noise floor induced by the photodetector and ESA without any input light) [140]. The sharp peaks at ∼ 1 kHz might be caused by the vibrational perturbation in the laboratory. By integrating the baseband noise (excluding the noise floor contribution), we estimated the relative pulse amplitude fluctuation of this HHML laser to be < 0.2 % (20 Hz − 1 MHz). The relative pulse timing-jitter, which is the key parameter for the HHML laser, was estimated from the higher-order SSB noise spectrum recorded from ESA, using the following expression [207] 1 ΔTp = 2Pnoise Tp 2π

(4.5)

where Tp is the temporal spacing between two pulses in the pulse train, ΔTp is the corresponding relative timing-jitter between them, and Pnoise is the SSB noise power integrated over the selected range. The factor 2 before Pnoise concerns both sidebands of the same order. Note that the SSB of the first harmonic actually includes both the amplitude noise and the timing-jitter between adjacent pulses. Similarly, the SSB of the N th -harmonic band (N > 1) includes both the amplitude noise and the timing-jitter between every N pulses, the ladder being supposed to be N times of the timing-jitter of the adjacent pulses. The SSB noise spectra of the first, fourth, and eighth harmonic-order peaks (at 1.8826 GHz, 7.5304 GHz, 15.0608 GHz) are plotted

88

Wideband-tunable GHz-rate Er-fibre soliton laser

4.2

versus the off-set frequency related to the corresponding peak frequencies in Fig.4.15. The estimated timing-jitter calculated from these sidebands are 84 fs, 51 fs, and 52 fs (first, fourth and eighth harmonics) in the range of 200 Hz − 1 MHz. Therefore, we can estimate that the actual timing-jitter of the output pulse train should be < 60 fs over 200 Hz to 1 MHz. (b) -90

SSB noise (dBc/Hz)

SSB noise (dBc/Hz)

(a) baseband noise noise floor ESA+PD

-105 -120 -135 -150

2

10

3

4

5

10 10 10 Off-set frequency (Hz)

6

10

-60 -80

1st sideband 4th sideband 8th sideband

-100 -120 -140 103 104 105 Off-set frequency (Hz)

106

Fig. 4.15: (a) The baseband SSB noise spectrum of the laser with a noise floor set by the ESA and the PD. (b) The SSB noise spectra at the sidebands of first, fourth, and eighth harmonic-orders in the RF spectrum of the output pulse train.

4.2.4 Optical comb structure and inter-pulse phase-relationship The frequency comb structure of the HHML laser was investigated using the heterodyne method. The experimental set-up is sketched in Fig.4.16(a). A single-frequency laser-source (NKT, 2 kHz bandwidth) was used as the local oscillator to interfere with the pulse train. A 50/50 coupler was used to combine the output from the HHML laser and the local oscillator. Two PCs and tunable attenuators (TAs) were used to adjust the polarization states and relative powers of the two lights so as to reach the maximum contrast out of their interference. When the central wavelength of the HHML laser was tuned to approach that of the local oscillator (the combined spectrum is shown in Fig.4.16(c)), the optical comb structure of the HHML laser was down-shifted to the radio-frequency range (Fig.4.16(b)), which could be measured directly by the ESA. The heterodyne signal recorded by the ESA shown in Fig.4.17(a) over a frequency span of 100 MHz. From the two sets of frequency comb, each with internal spacing of fcav = 4.84 MHz, we can reveal the fact that the spectrum of the laser consists of a frequency comb constituted by all the cavity axial-modes under the pulse spectral envelope, as previously discussed in Section 3.2.3. The two interleaved frequency combs (marked by the two sets of arrows in Fig.4.17(a)) are caused by the two different sets of beating signal (between the local oscillator and the axial-modes

89

Ch. 4

GHZ-RATE SOLITON FIBER LASERS that is higher and lower than the frequency of the oscillator). Due to the drifting of both the HHML laser comb and local oscillator, these two sets of beating single were randomly drifting across each other during the acquisition time of the measurement. The frequency drift of both lasers also contributed to the amplitude fluctuations of the comb lines. By decreasing the sweep time of the ESA, the amplitude fluctuations could be reduced, with however a decreased frequency resolution. Each frequency component in the beating signal actually consists of a side peak which is about 0.2 kHz away from the main comb line, which we believe was an artefacts of the local oscillators [104]. (a) HHML laser

Local oscillator

TA

PC

PC

OSA

50:50 Coupler

TA

OSC 20

(c) Intensity (dBm)

(b) Cavity axial-modes

Local oscillator

...

ESA PD

...

0 -20 -40 1540

Frequency

1545

1550 1555 Wavelength (nm)

1560

Fig. 4.16: (a) Experimental set-up for investigating the optical comb structure and the pulse-to-pulse phase-relation. (b) The conceptual sketch of the comb structure of the HHML laser (blue line) and the local oscillator (red-line).(c) The optical spectrum of the heterodyne signal, including the soliton laser (blueline) and the local oscillator (red-line).

(b)

-50

4.84 MHz

-60

Voltage (mV)

Intensity (dBm)

(a)

-70 -80 -90 0

20

40 60 Frequency (MHz)

80

100

80 60 40 20

0

40

80

120 Time (ns)

160

Fig. 4.17: (a) The RF-spectrum of the heterodyne signal of the pulse train over 100 MHz. Two combs marked by the two sets of the arrows have a comb spacing of 4.84 MHz. (b) The time-domain heterodyne signal recorded over a single round-trip time, which exhibits random modulations of pulse amplitudes.

90

200

Tm-fibre soliton laser at GHz-rate

4.3

30

1.0

25

0.8

20

0.6

15

0.4

10

Intensity

Round-trip number

The heterodyne signal over one cavity round-trip time (206.6 ns) was recorded by the oscilloscope and plotted in Fig.4.17(b). The random interference pattern was translated from the random phase relationship between the 389 uncorrelated sub-laser pulses, each locked at the fundamental frequency. Each of the intra-cavity pulse, however, maintained a fixed phase after each round-trip. To reveal this fact, we continuously recorded the interference signal over many round-trips, and plotted the successive round-trips in parallel to illustrate the evolution of each pulse over many round-trips in Fig.4.18. Due to the limited space, only 26 out of the 389 pulses are plotted over 30 round-trips. A clear sinusoidal pattern along the “vertical line” can be observed, which indicates the persistence of both amplitude and phase of each pulse. The period of this sinusoidal pattern is determined by the frequency detuning between the local oscillator and the carrier-frequency of the pulse, which was the same for all the intra-cavity pulses. Meanwhile, the beating patterns within different time-slots exhibit no obvious timing relationships, indicating a random phase-relationship between the multiple pulses in the cavity. Similar experiments, in which adjacent pulses in the output train were arranged to interfere with each other, also resulted in random modulation of individual pulse peak-amplitude, which was also resulted from the random phase relationship between the pulses.

0.2

5 0

2

4

6 8 Relative intra-cavity time (ns)

10

12

0.0 (a.u.)

Fig. 4.18: The heterodyne signal recorded for 30 round-trips plotted in parallel according to the cavity round-trip time. Only part of the intra-cavity pulses (26 out of 389) are plotted due to the limited space.

4.3 Tm-fibre soliton laser at GHz-rate The optoacoustic mode-locking scheme has also been successfully applied to Tmdoped fibre laser (TDFL) at a longer lasing wavelength (∼ 2 μm) [106]. This particular eye-safe wavelength region has attracted increasing interests, and mode-locked

91

Ch. 4

GHZ-RATE SOLITON FIBER LASERS lasers working in this region have potential applications in biomedicine, spectroscopy, and remote sensing. Previously, Tm-doped fibre lasers were limited to MHz-range repetition rate. With the insertion of a specially-designed small-core PCF, we have generated GHz-rate laser pulses using the optoacoustic mode-locking scheme in this wavelength region.

1 0 1 2

1 1 optoacoustic gain (mW)

2 1 0 1 2 Length (Pm)

0.6

(c) R 01 mechanical mode

0

1

2 1 0 1 2

0

2 1 0 1 2 Length (Pm)

Density change

(d)

Optical intensity

2

2 Pm

1

Length (Pm)

(b) LP01 optical mode Length (Pm)

(a)

1

1.453 GHz

0.5 0.4

1.446 GHz

0.3 0.2 28 MHz

0.1 0 1.40

1.42

1.44 1.46 1.48 Frequency (GHz)

1.50

Fig. 4.19: The nano-bore PCF fabricated for the GHz-rate Tm-doped HHML fibre laser. (a) The SEM-photo of the core-structure of the nanobore PCF. (b) The normalized intensity profile (colour scale) of the LP01 optical mode of the PCF. (c) The normalized density-variation profile of the R01 acoustic mode (colour scale). The corresponding strain (deformation) is highly exaggerated. (d) The measured optoacoustic gain spectrum of the PCF.

4.3.1 Experimental set-up Compared to the HHML soliton fibre laser at 1.55 μm based on EDF, the similar realization at 2 μm using TDF has some practical difficulties. Firstly, the optical mode that is involved in the optoacoustic interactions in the solid-core PCF would become larger, leading to a higher propagation loss as well as splice loss with conventional fibres. The GVD of the PCF and also the SMF-28 at longer wavelength would also tend to become even more strongly anomalous, leading to a cavity with ultra-high net GVD. According to the soliton-area theorem (Eq.(2.19)), given a fixed soliton energy, the resultant soliton duration would be so long that the nonlinear

92

Tm-fibre soliton laser at GHz-rate

4.3

interactions in the PCF would be highly weakened. The soliton bandwidth (also energy) would also be severely limited by the sideband instability imposed by the cavity periodical perturbations (Section 2.2.3). Trying to solve these problems that come along with the longer operating wavelength, we fabricated a special nano-bore PCF, the SEM photo of which is shown in Fig.4.19(a). This nano-bore PCF has a core-diameter of 1.76 μm and an air-filling ratio of 80 % in order to better confine the LP01 optical mode. Meanwhile, a nano-bore with diameter of 560 nm is opened at the center of the PCF-core in order to shift the GVD of the PCF towards normal value. The estimated GVD of the LP01 mode of this PCF at the operating wavelength (1.85 μm) is about +160 ps2 /km. The dominant acoustic resonance frequency (R01 mode) was measured to be at 1.453 GHz with a bandwidth of ∼ 18 MHz. The simulated optical and acoustic modes of this PCF are shown in Fig.4.19(b) and (c), and the measured acoustic gain spectrum is shown in Fig.4.19(d). WDM

WDM TDF

90:10 LD-1

Output

LD-2 ISO

PC-1 Polarizer

PC-2

Nanobore PCF

PC-3

Fig. 4.20: Sketch of the Tm-doped fibre (TDF) laser cavity with the nano-bore PCF. LD, laser diode; WDM, wavelength division multiplexer; PC, polarization controllers.

The unidirectional fibre laser with a ring-cavity is sketched in Fig.4.20. An 1-m-long commercial TDF is used as the gain fibre, which has a peak absorption of 27 dB/m at 790 nm, a core diameter of 9 μm and a GVD of −50 ps2 /km. Two pump laser diodes (LD-1 and LD-2) at 785 nm provided the CW pump light with combined maximum power of 500 mW. Note that due to the low quantum efficiency and low absorption of TDF, the resultant laser power would be much lower than that of the Er-laser. The nano-bore PCF inserted in this cavity has a length of ∼ 0.6 m and an insertion loss of 4.7 dB (4.2 dB at input end and 0.5 dB at the output end

93

Ch. 4

GHZ-RATE SOLITON FIBER LASERS using a transition section as described in Section 4.1.4). Two pressured PCs (PC-1 and PC-3) and a polarizer configured the polarization APM, while the PC-2 was used for polarization alignment in the PCF. Note that due to the high bending loss of SMF-28 at ∼ 2 μm, pressurized PCs were used instead of the previously used fibre-loop based PCs. A 90/10 coupler provided the laser output. The rest of the cavity were made from SMF-28, which has GVD of −70 ps2 /km. The total cavity loss was ∼ 9 dB. The total cavity length was ∼ 7.4 m, corresponding to a round-trip frequency of 27.8 MHz. The mode-locked laser then worked at a harmonic-order of 52.

4.3.2 Laser output The lasing threshold of this TDFL was reached with a pump power of ∼ 360 mW, above which noisy pulsations were generated from the laser output. Stable HHML operation required a higher pump power of ∼ 420 mW, while proper adjustments of the PCs were also demanded. The repetition rate of the output pulse was 1.446 GHz, which was fixed by the core-resonance frequency of the PCF at 1.453 GHz. After tuning off and recovering the pump power, the same HHML laser output can be easily re-established through slight adjustments of the PCs. This HHML laser can stably operate in the laboratory for many hours without any additional re-alignments, exhibiting good robustness against the temperature drifts and weak acoustic vibrations. The output pulse train recorded under infinite persistence-mode using an oscilloscope (33-GHz) is shown in Fig.4.21(a). The RF-spectrum recorded by an ESA is shown in Fig.4.21(b), in which the first peak at 1.446 GHz and its higher harmonics indicate stable HHML state. Fig.4.21(c) shows a zoom-in of the first harmonic peak over a 200-MHz span, which exhibits a supermode suppression ratio of > 50 dB. The estimated amplitude fluctuation is < 0.6 % (50 Hz − 1 MHz), and the timing-jitter is < 120 fs (200 Hz − 1 MHz). The repetition rate also had a long-term drifting range of several kHz due to the slow temperature variations. The average laser power output from the 90/10 coupler was measured to be 1.7 mW, corresponding to an intra-cavity power of ∼ 17 mW and thus an individual pulse energy of ∼ 12 pJ. Due to the low pulse energy, the pulse shape (and thus the duration) could not be measured using the autocorrelator based on second-harmonic generation, while the 12.5-GHz photodetector only has a temporal resolution of 55 ps. The pulse duration could nevertheless be estimated from the output spectrum

94

Tm-fibre soliton laser at GHz-rate

4.3

of the laser shown in Fig.4.21(d). The dominant central part of the spectrum has a sech2 -shape, indicating soliton-like pulse propagation in the fibre laser cavity. The 3dB bandwidth was measured to be ∼ 0.16 nm, from which the pulse duration can be estimated to be ∼ 22 ps using the soliton-area theorem (Eq.(2.19)). The simulation result (based on the experimental parameters and using the conventional split-step Fourier method) gives a pulse duration of ∼ 25 ps and a TBP of 0.35, which agrees quite well with the measured pulse bandwidth of ∼ 0.16 nm.

0.1

691 ps 55 ps

0 0.1

Power (dB)

(b)

(c) Power (dB)

0.2

(d) 0

0

0.5 1.446 GHz

1 1.5 Time (ns)

2

2.5

20 40 60 0 0 20 40 60 80 1.346

5

10 15 Frequency (GHz)

20

Spectral power (dB)

Voltage (V)

(a)

0 0.16 nm

10 20

Measured

30

Simulated

40 50 60 1847.5

1848 1848.5 1849 Wavelength (nm)

1849.5

> 50 dB

1.446 Frequency (GHz)

1.546

Fig. 4.21: The output of the HHML TDFL. (a) A typical pulse train recorded by an oscilloscope under infinite persistence mode over ∼ 10 min. (b) The RF spectrum of the pulse train measured by the ESA. (c) Zoom-in at the first harmonic peak (1.446 GHz) of the RF-spectrum with a 200-MHz span. (d) The optical spectrum of the TDFL, measured (red solid line) and the simulated (grey dashed line) results.

The relatively narrow bandwidth and long pulse duration of this HHML soliton laser were mainly caused by the limited pulse energy and the strong anomalous cavity GVD. Using a longer TDF with a higher pump power, together with proper cavity dispersion compensation, could in principle lead to significantly shorter pulse durations.

95

CHAPTER

5

STRETCHED-SOLITON FIBRE LASER

The precedent chapter demonstrates the application of the PCF-based optoacoustic mode-locking in generating GHz-rate pulses in the soliton regime, where the pulse duration is limited to ps-level. Efforts to achieve significantly shorter pulses usually employ stretched-pulse or dissipative-soliton schemes, which generally require high pulse energies and therefore demand very high pump powers for the laser. In this chapter, a new strategy called stretched-soliton effect is applied to the HHML fibre laser, which allows the generation of fs-pulses with only pJ-level single pulse energy, thanks to a unique elaboration of the self-consistency map of the intra-cavity pulse evolution. When this strategy was combined with the optoacoustic mode-locking scheme, an all-fibre mode-locked laser was developed, being able to deliver, for the first time, a pulse train with GHz-repetition-rate and sub-100 fs pulse duration simultaneously under a moderate pump power [108].

5.1 Stretched-soliton laser: Concept 5.1.1 The energy dilemma for broadband pulses generation In many practical applications of mode-locked lasers, ultra-short pulse durationd and or broad spectral bandwidth are desired features of the output pulses. In the soliton regime, as previously discussed, the pulse shortening is restricted by the soliton area theorem and sideband instability due to intra-cavity nonlinearity. To mitigate these problems, the strategy of stretched-pulse laser and dissipative-soliton

97

Ch. 5

STRETCHED-SOLITON FIBRE LASER laser have been introduced for generating broadband, sub-100-fs pulses directly from the lasers. Despite the many types of detailed configurations, two elements are shared by most of these lasers. Firstly, although the excessive nonlinear phase shift has been significantly reduced due to the large temporal breathing of the intra-cavity pulses, strong nonlinear spectral shaping still exists during the pulse evolution within the cavity due to the nJ-level pulse energy. Actually, the output broadband spectrum relies heavily upon the SPM-induced spectral broadening, which, in many realizations, results in Gaussian-shaped pulses. Secondly, a band-pass filter is usually required to be inserted in the cavity in order to stabilize the intra-cavity pulses and most importantly to force the self-consistency of the laser pulses over each round-trip [18]. These pulse-shortening strategies, though proven to be applicable in the MHzrate laser, would encounter several difficulties at GHz-repetition-rate. Combined with the nJ-level pulse energy, an average power of many watts would be expected, which necessitates high-pump power and probably complex cladding-pumped configurations. However, it is worth noticing that, the gain bandwidth provided by the Er-doped fibre is wide enough to support tens of nm bandwidth, or equivalently a pulse duration of ∼ 100 fs. In principle, a low pulse energy, which causes only weak Kerr-nonlinearity should have no intrinsic obstacle to acquire a broad bandwidth directly from the laser gain medium without the aid of nonlinear spectral broadening. The realization of such laser, however, requires a self-consistency map of the intra-cavity pulses that is distinct from the configuration summarized in Fig.2.8.

5.1.2 A different intra-cavity pulse self-consistency During our development of the dispersion-managed soliton laser, we have realized a novel type of intra-cavity pulse self-consistency map. In one hand, strong dispersionmanagement is introduced into the fibre laser cavity, leading to only weakly anomalous net GVD. The intra-cavity pulses then have a temporal breathing ratio of ∼ 70 together with several periods of breathing over each cavity round-trip. With only tens of pJ intra-cavity pulse energy, the resultant nonlinear phase shift due to SPM becomes negligibly low, leading to only trivial nonlinear spectral shaping. As a result, the pulse actually experiences almost linear pulse breathing in the time domain, distinct from the traditional strategies mentioned above. The over-all balance between the residual anomalous GVD and the weak SPM leads to a soliton-like balance, while the actual pulse evolution deviates far from the soliton-regime. On the

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Stretched-soliton laser: Experiments

5.2

other hand, a lumped wavelength-dependent attenuator caused by fibre birefringence was delicately implemented in the cavity so as to encounter the gain narrowing effect in the EDF section. Such opposite “filtering-effects” dominate the spectral shaping within each cavity round-trip. This pulse self-consistency strategy was named as the stretched-soliton, which is summarized in Fig.5.1. EDF: Gain narrowing effects

Normal dispersion passive fibre

Linear pulse broadening

Linear pulse narrowing

Anomalous dispersion passive fibre

Wavelength-dependent attenuator

Fig. 5.1: The self-consistency map of the stretched-soliton laser. The gainnarrowing effect in EDF is balanced by the wavelength-dependent attenuator which suppresses the central part of the pulse spectrum. The passive fibre sections with large and opposite signs of GVD lead to quasi-linear pulse temporal breathing with large breathing ratio.

5.2 Stretched-soliton laser: Experiments 5.2.1 Experimental set-up The experimental realization of the stretched-soliton laser still employs a unidirectional fibre ring-cavity, similar to that in the previous soliton lasers. The laser gain medium was an 1.05-m-long EDF with 110 dB/m peak absorption at 1530 nm. Two pump laser diodes at 976 nm (LD-1 and LD-2) had a maximum combined power of 1.5 W. The solid-core PCF inserted in the cavity had a core diameter of ∼ 1.95 μm, corresponding a R01 acoustic resonance frequency at ∼ 1.882 GHz. The PCF had a GVD of −157 ps2 /km at 1550 nm. The insertion-loss of the PCF-section was ∼ 1.7 dB, with an intrinsic loss of ∼ 0.05 dB/m. Two pieces of dispersioncompensation fibres (DCF-1),having an identical GVD of +121 ps2 /km at 1550 nm

99

Ch. 5

STRETCHED-SOLITON FIBRE LASER and with lengths of 3.0 m and 0.8 m, were inserted in the cavity, each with an insertion loss of ∼ 2 dB. The DCF has a weak birefringence (∼ 1 × 10−5 ), which can lead to spectral filtering effects [208] when cooperating with a polarizer and proper adjustments of the PCs, as will be described later in Section 5.3.2. WDM Output Port-1

DCF-2

TA

ISO

WDM

Monitor Port-3

EDF

50:50

0.6 m

1m

97:3

3m

LD-1

LD-2 ISO

DCF-1

PC-4

Delay line

HNLF 0.8 m SMF 0.6 m

0.8 m

PC-1 Output Port-2

DCF-1

Polarizer PC-3

PC-2 PCF 2m

Fig. 5.2: Sketch of the experimental set-up of the stretched-soliton fibre laser.

The total cavity group delay was compensated to only slightly anomalous (−0.008±0.003 ps2 ). The implementation of the three PCs and polarizer was similar to that in the soliton laser (See Section 4.2.1). A 50/50 coupler was inserted after the EDFA as the output port where the pulse energy was supposed to be the largest. Another 97:3 coupler was inserted before the EDF to monitor the intra-cavity pulse evolution. The total cavity length was 17.2 m, corresponding to a round-trip frequency of 12.1 MHz, and the total cavity loss was ∼ 13 dB.

5.2.2 GHz-rate sub-100-fs pulses output The CW-lasing threshold of this laser was reached with a pump power of ∼ 100 mW, while the stable HHML operation requires a much higher pump power of ∼ 1 W. The repetition rate was ∼ 1.8726 GHz locked to the PCF core resonance. Similarly, careful alignments of the PC-2 and adjustments of the NPR working point set by PC1 and PC-3 were required for laser starting and stable operation. Compared to the soliton regime, the self-pulsing of this laser and the self-organization of the intracavity pulses for this strongly dispersion-managed cavity required highly precise adjustments of the PCs. The weaker temporal trapping potential, which is directly related to the cavity net group delay, might be responsible for it. Nevertheless,

100

Stretched-soliton laser: Experiments

5.2

once the HHML state was reached, we observed the same level of stability and robustness in this laser, despite the weaker re-timing forces. When the pump power was increased from ∼ 1 W to ∼ 1.5 W, the stable HHML state at 1.8726 GHz was maintained, with an increase of output signal power of ∼ 20 %. The output average signal power from the 50/50 coupler port was measured to be 158 mW, corresponding to a maximum intra-cavity average power of ∼ 340 mW. The pulse train recorded by a 33-GHz PD and 16-GHz OSC is shown in Fig.5.3(a) under an infinite persistence mode for over 300 hours. The RF-spectrum of the output pulse train measured using a 26-GHz ESA is shown in Fig.5.3(b). The first harmonic peak at 1.872625 GHz corresponds to a harmonic-order of 155, which is located close to the acoustic core-resonance of the PCF as shown in Fig.5.3(c). The supermode suppression ratio was measured to be > 50 dB. The repetition rate could be continuously tuned using the delay-line in the range of 1.869 GHz − 1.876 GHz while the HHML state was stably maintained with an invariant harmonic-order. 120

Voltage (mW)

(a)

534 ps

60 0 0.0

Power (dB)

(b)

1.5

2.0

8

12

16

-50 -75 4

0.4 0.3 0.2

Frequency (GHz)

R01 resonance at 1.882 GHz > 50 dB

-40 -60

0.1 0.0 1.80

-20

-80 1.85

1.90 1.95 Frequency (GHz)

Intensity (dBm)

OA-Gain (W-1m-1)

1.0

Time (ns)

1.8726 GHz

-25

0

(c)

0.5

2.00

Fig. 5.3: (a) The output pulse train from the stretched-soliton laser recorded by an oscilloscope under infinite persistence mode for over 300 hours. (b) The RF spectrum of the output pulse train. (c) Zoom-in plot of the RF spectrum at the first harmonic peak (blue curve), plotted together with the optoacoustic gain (OA-gain) spectrum of the solid-core PCF (grey curve)

The short-term pulse amplitude noise was measured to be < 0.1 % (20 Hz − 1 MHz),

101

Ch. 5

STRETCHED-SOLITON FIBRE LASER

(b)

-60

noise floor baseband

-80

-100 -120 -140

102 103 104 105 Offset frequency (Hz)

Power (mW)

(c)

106

SSB noise (dBc/Hz)

(a)

SSB noise (dBc/Hz)

using the baseband SSB noise spectrum shown in Fig.5.4(a), and the pulse timingjitter was measured to be < 50 fs (200 Hz − 1 MHz) using the higher-order harmonic SSB noise spectra (see Fig.5.4). This stable HHML laser could last continuously for a few weeks without external stabilizations or re-adjustments. A 3 % long-term fluctuation of the average power with a period of 24 h was observed, as shown in Fig.5.4(c), which we attributed to the daily temperature cycle in the laboratory. After tuning-off and restarting the pump source, the HHML laser at invariant repetition rate could be easily reproduced with slight assistance of the PC-adjustments. -60

1st harmonic 4th harmonic 8th harmonic

-80 -100 -120 -140

103 104 105 Offset frequency (Hz)

106

124 122 120

0

24

48 72 96 Time (hours)

120

Fig. 5.4: (a) The baseband SSB noise spectrum of the stretched-soliton laser with a noise floor set by the ESA and the PD. (b) the SSB noise spectra at the first, fourth, and eighth harmonic-orders in the RF spectrum. (c) A long-term recording of output average power.

The laser output spectrum directly measured at the 50/50 coupler is shown as the red-curve in Fig.5.5(a) (with relative spectral densities), which gives a 3dB bandwidth of 20.9 nm. The spectrum has a smooth profile without any Kelly sidebands or CW noisy-peaks, indicating a low background noise level. The output pulse train was delivered through an isolator (ISO) and then a tunable attenuator (TA) in order to avoid backward reflections from the diagnostic set-up to enter the laser cavity. To compress the output laser pulse, a short piece of DCF (DCF-2,with a GVD of +45 ps2 /km at 1550 nm) was used to compensate the pulse chirp using the cutback method [157]. The pulse duration variation at different lengths of DCF2 is shown in Fig.5.5(b). With a 0.62-m-long DCF-2 fibre, the shortest pulse was

102

Stretched-soliton laser: Experiments

5.2

obtained with a FWHM-duration of ∼ 117 fs (∼ 180 fs autocorrelation (AC) trace), estimated by fitting the measured data assuming a sech2 -shaped pulse, as shown in Fig.5.5(c). The TBP was calculated to be ∼ 0.31, which was in good agreement with a transform-limited sech2 -shaped pulse.

-30 -45

Port 1 Port 2 Port 3

-60

0.8

AC FWHM (ps)

(b)

0.6

Negative chirp

1475

1500

(c)

Positive chirp

0.4 0.2 0.0 0.2

1.54 ×117 fs 0.4 0.6 0.8 DCF-2 length (m)

1.0

1525

1550 1575 Wavelength (nm) 1.54 ×117 fs

0.5

0.0 1.0

1600

(d) 1.0 Intensity (a.u.)

-75 1450

Intensity (a.u.)

Intensity (dBm)

(a) -15

1625

1650

1.54 × 83 fs

0.5

0.0 -0.4 -0.2 0.0 0.2 Delay (ps)

0.4

-0.4 -0.2 0.0 0.2 Delay (ps)

0.4

Fig. 5.5: (a) The optical spectra measured at Port-1, -2, and -3 as illustrated in Fig.5.2. (b) The pulse chirp compensation using the cutback method. The output pulse durations with different DCF-2 lengths was plotted. (c) and (d) AC traces measured at output Port-1 (red) and Port-2 (blue). Both traces are fitted with curves assuming a sech2 -shaped pulse profile.

The output pulse from the HHML laser cavity was directly launched into a short piece of highly nonlinear fibre (HNLF) for further spectral broadening and pulse compression, as shown in Fig.5.2. The de-chirped pulse train output from Port-1 had an average power of ∼ 112 mW, corresponding to an individual pulse energy of ∼ 60 pJ. The peak power of the output pulse was estimated to be ∼ 500 W. The HNLF used for nonlinear spectral broadening had a length of 0.8 m, a Kerr-nonlinearity coefficient of 10.9 W−1 km−1 and a normal GVD of +16.5 ps2 /km. An additional PC-4 was inserted before the HNLF to compensate the slight fibre birefringence of the HNLF. The spectrum of the pulses output from Port-2, after the nonlinear broadening, is shown in Fig.5.5(a) (blue curve). The broadened spectrum had a 3-dB bandwidth of 35.6 nm. Then a 0.6-m length of SMF-28 was used to further de-chirp the broadened pulses, leading to a pulse duration of ∼ 83 fs. The AC trace measured at Port-2 is shown in Fig.5.5(d). The fitting curve still assumes a sech2 -shaped pulse, compared to which the measured pulse profile had a slightly

103

Ch. 5

STRETCHED-SOLITON FIBRE LASER

Intensity (dBm)

(b)

150 100 50 0 -50 0.0

83 ns

0.1

0.2

(c)

-45

21 nm

-60

0.3

1.0

0.4

0.5

1.54×119 fs

0.5

-75 -90

Time (μs) Intensity (a.u.)

(a)

Voltage (mV)

stronger pedestal. The TBP was calculated to be ∼ 0.37. The increased TBP, as well as the slight deviation of the pulse shape from the sech2 -shape, can be attributed mainly to the highly nonlinear propagation of the output pulses in the HNLF. The average power at Port-2 was measured to be ∼ 76 mW, corresponding to an individual pulse energy of ∼ 41 pJ.

Measured spectrum ASE spectrum

1500

0.0

1550 1600 Wavelength (nm)

-0.4 -0.2 0.0 0.2 Delay (ps)

0.4

Fig. 5.6: (a) The output pulse train from the stretched-soliton laser running under fundamental mode-locked state. (b) The optical spectrum of this pulse train measured at Port-1 (red solid curve), which exhibits considerable ASE background. The ASE spectrum of the EDF measured below lasing threshold is plotted in grey dashed line. (c) The measured AC trace of this pulse train with a fitting curve assuming a sech2 -shaped pulse.

5.2.3 Fundamental mode-locking state The stretched-soliton strategy led to successful ultra-short generation at GHz repetition rate in the HHML laser, with only tens-of-pJ intra-cavity pulse energy. In principle, the stretched-soliton effect is independent of the intra-cavity pulse number, and could also be applied to the fundamental mode-locking state at MHz repetition rate. In experiments, we achieved such state by graduating decreasing the pump power when the laser was stably running at the HHML state. The optoacoustic mode-locking collapsed when the pump power dropped below 1 W. When the pump power was further reduced, the number of intra-cavity pulse continuously decreased. As the pump power reached close to the CW lasing threshold, the fundamental mode-locked state was eventually reached at a repetition rate of 12.1 MHz. The recorded pulse train in shown in Fig.5.6(a). Despite the considerable pulse drop-out

104

Analysis of intra-cavity pulse evolution

5.3

during pump-power decreasing, the spectral bandwidth as well as the pulse duration were basically maintained. The optical spectrum and AC trace of the 12.1 MHz output pulse train measured at Port-1 are shown in Fig.5.6(b) and (c) respectively, exhibiting a nearly invariant bandwidth of ∼ 21 nm and pulse duration of ∼ 119 fs. Since the laser was ruining close to the threshold, there was a strong ASE background beneath the pulse spectrum, as can be easily revealed from Fig.5.6. The average intra-cavity signal power was measured to be only ∼ 1 mW, still corresponding to only tens of pJ individual pulse energy. The extraordinary advantage of the stretched-soliton scheme, which requires only moderate pump power for ultra-short pulse generation with simple cavity-design, can possibly lead to the realization of low-cost, high-quality, ultrafast seed lasers that can run under extremely low power.

5.3 Analysis of intra-cavity pulse evolution 5.3.1 Numerical simulations and validations To uncover the detailed mechanism of the stretched-soliton formation in such speciallydesigned fibre laser cavity, numerical simulations were performed to analyse the evolution of an individual pulse during the propagation in the fibre-cavity. We assume that all the 155 pulses that are propagating in the cavity have identical envelope, which is valid under the steady state. Then we only need to focus on a single pulse for simplicity. The simulation was based on the symmetrized split-step Fourier method [134]. The dispersion map of the cavity loop taken from experimental parameters is illustrated in Fig.5.7(a), which consists of eight fibre sections with alternating normal and anomalous dispersion, leading to significant pulse temporal breathing. The Kerr-nonlinearity coefficients used for SMF, DCF and PCF were 1.1, 2.86, and 28.6 W−1 km−1 respectively. A wavelength-dependent attenuator (WDA), as a lumped component, was included in the cavity to model in-line polarizer, as marked in Fig.5.7(a). The WDA had a central wavelength of 1550 nm, a bandwidth of 40 nm and a peak attenuation factor of ∼ 6 dB. The inclusion of the WDA is based on the fact that the polarizer acts in concert with the DCF fibre birefringence and thus suppresses the central region of the pulse spectrum. A pulse with a duration of ∼ 100 ps was set as the initial input of the pulse propagation simulation. The evolution of this pulse in the fibre loop was followed for over several thousand round-trips until the steady state was reached, i.e. the pulse

105

Ch. 5

STRETCHED-SOLITON FIBRE LASER parameters becomes invariant. In the simulation, a gain saturation element was introduced to stabilize the pulse energy, and a fast saturable absorber was included in order to accelerate the process from initial pulse to the steady-state pulse. Both the gain saturation and the fast saturable absorber only functioned as weak “noise eaters” and influenced trivially the pulse shaping under the steady state. When they were removed from the simulations, the pulse became unstable after several hundreds of round-trips.

(d)

2

DCF 4

6

8 Fiber length (m)

EDF 10

12

14

16

0.2 0.1 0.0 10 5 0 -5 -10 8 4 0

(ii)

0

0.2 0.4 0.6 0.8

1

0

0.2 0.4 0.6 0.8

1

(iii) (iv)

(i)

1.59

λ (μm)

(e)

DCF

-0.2 0

T (ps)

(c)

Output Coupler (50:50)

PCF

0.0

τp (ps)

(b)

WDA (lumped element)

SMF

0.2

Ep (nJ)

β2 (ps2/m)

(a)

1.56 1.53

(f)

Δλp (nm)

1.50

40 30 (i) 20 0

(iv)

(iii) (ii) 2

4

6

8 Fiber length (m)

10

12

14

16

Fig. 5.7: The simulation results of intra-cavity pulse evolution in the stretchedsoliton fibre laser under the steady state. (a) The laser loop and the dispersion map used in the pulse propagation simulation. (b) The steady-state evolution of pulse energy in the laser cavity (c) and (d) The steady-state evolutions of the pulse shape (normalized to peak) and the pulse FWHM duration. (e) and (f) The steady-state evolution of pulse spectrum (normalized to local maximum) and the 3-dB bandwidth.

The steady-state evolution of the pulse energy is shown in Fig.5.7(b). The maximum

106

Analysis of intra-cavity pulse evolution

5.3

pulse energy, directly after EDF, is ∼ 0.1 nJ, while the averaged pulse energy is only ∼ 50 pJ. The total cavity loss is ∼ 19 dB, including ∼ 13 dB component loss and an extra ∼ 6 dB from the WDA effect at the polarizer. The temporal and spectral evolutions of the simulated pulse under steady state is shown in Fig.5.7(c) and (e). Both figures are plotted using normalized intensities at each position in the laser loop. The corresponding evolutions of the FWHM pulse duration and spectral bandwidth in the fibre cavity are plotted in Fig.5.7(d) and (f) in solid curves. As we can reveal from Fig.5.7(d), the pulse experienced a temporal breathing ratio of ∼ 70 within a single cavity round-trip, from ∼ 110 fs to ∼ 8 ps. Such temporal breathing ratio is about 10 times higher than that in typical stretchedpulse [50] or dissipative-soliton lasers [165]. The considerable breathing ensured the low nonlinear spectral shaping experienced by the pulse. As we can see from Fig.5.7(f), the spectral breathing ratio was only ∼ 1.7, ranging from ∼ 20 nm to 35 nm. Most importantly, this spectral “breathing” only occurred at the EDFA and the WDA, in which the WDA counteracted the bandwidth narrowing effect induced by the EDFA. Meanwhile over the rest part of the fibre cavity, the spectrum remained almost unchanged.

0.5

(b) (i) (ii) (iii) (iv)

0.0 -15 -10 -5 0 5 10 15 Delay (ps)

Intensity (dBm)

1.0

Intensity (a.u.)

(a)

-20 -40 -60

(i) (ii) (iii) (iv)

-80 1500

1550 1600 Wavelength (nm)

Fig. 5.8: Experimental validations of the stretched-soliton simulation results(a) AC traces of the pulses and (b) optical spectra measured at different positions as marked in Fig.5.7.

In order to validate the simulation results with the actual pulse evolution in the laser cavity, an external replica-fibre was built to repeat the part of the fibre-cavity starting from the 50/50 coupler to the 97/3 coupler including the SMFs, DCFs, PCF, the polarizer, and three PCs (see Fig.5.2). This replica-fibre was connected to the output port of the 50/50 coupler, so that a replica-pulse starting from this coupler could propagate in the replica-fibre from which we were able to retrieve the pulse evolution in the cavity. Under the steady-state of this HHML laser, the pulse

107

Ch. 5

STRETCHED-SOLITON FIBRE LASER durations and spectral widths were measured at several different positions of the replica fibre, and the results are shown as square dots in Figs.5.7(b), (d), and (f). The measured results agree quite well with the simulated curves. In addition, AC traces and optical spectra measured at four different positions (marked in Fig.5.7(d) and (e)) directly demonstrate the temporal breathing and spectral variations in the laser cavity, as shown in Fig.5.8.

5.3.2 Wavelength-dependent attenuator The enabling factors for generating broadband sech2 -shaped pulses are the relatively low pulse-energy and the huge temporal breathing in the strongly dispersionmanaged fibre cavity, which nearly eliminate the nonlinear spectral shaping upon the pulses. As the pulse bandwidth broadens, the gain narrowing effect in the EDF becomes considerable, which needs to be compensated within each cavity roundtrip in order to ensure self-consistent pulse evolution. In the conventional type of stretched-pulse fibre lasers, this gain filtering effect is compensated by the nonlinear spectral broadening which relies on the cooperation between strong SPM effects and the pulse chirp, leading to strong spectral broadening in the gain fibre with normal dispersion. In this special stretched-soliton laser, however, the gain narrowing effect in EDF is balanced by the WDA formed by the fibre birefringence of the DCF-1 and the in-line polarizer under proper adjustment of the PCs. The birefringence-based WDA has a free spectral range of ∼ 80 nm between its transmission peaks, and the transmission minimum (∼ 6 dB) was able to be arranged to coincide with gain maximum of EDF section. The suppression of the spectral center-region, or the “effective spectral broadening” induced by the WDA, was verified experimentally by measuring the laser spectra before and after the in-line polarizer. The measured spectra are shown in Fig.5.9(a). The spectral bandwidth broadened from 22.6 nm (blue curve) to 35.0 nm (grey curve), while the spectral peak intensity was reduced by ∼ 6 dB. The simulation also exhibited similar effects as shown in Fig.5.9(b). The WDA profile measured from the experimental results is show in Fig.5.9(c) in solid yellow curve, exhibiting a bandwidth of ∼ 40 nm and modulation depth of ∼ −6 dB. The simplified WDA profile used in the simulations is shown as dashed line. The wavelength-dependent gain in the EDFA was measured using the spectra of pulses before and after the EDFA, which has a profile that closely resembles the profile of the measured WDA, though with a reversed curvature.

108

Analysis of intra-cavity pulse evolution (b)

-30

Intensity (dBm)

Intensity (dBm)

(a)

-45 -60 -75

before after

-30 -45 -60 -75

1500 1550 1600 Wavelength (nm) -3 -4 -5 -6 -7 -8

1500 1550 1600 Wavelength (nm) (d)

measured in simulation

1500 1525 1550 1575 1600 Wavelength (nm)

before after

20

EDF Gain (dB)

Transmission (dB)

(c)

5.3

18 16 14 1500 1525 1550 1575 1600 Wavelength (nm)

Fig. 5.9: Characterization of the wavelength-dependent attenuator. (a) The measured and (b) the simulated pulse spectra before and after the in-line polarizer. (c) The transmission profiles of the intra-cavity WDA calculated using the experimentally measured data (solid yellow curve) and those obtained from simulations (dashed curve). (d) The measured wavelength-dependent gain of the EDF section.

The balance between wavelength-dependent gain and attenuation has permit a spectral domain self-consistency that does not rely on strong Kerr-nonlinear spectral shaping and thus permits its implementation with very low pulse energy. Although this scheme has not been used before in a fibre laser cavity, components with similar concept, known as the gain flattening filters, have been widely used in optical amplifiers [209, 210]. Recently, this concept has also been used in semiconductor [211] and bulk-material mode-locked lasers [212, 213].

5.3.3 The soliton-like balance The strong dispersion-management in the laser cavity led to a huge temporal breathing of the intra-cavity pulses, while these laser pulses experienced soliton-like balances between the weak residual SPM effect and the also weak anomalous dispersion of the cavity. The resultant soliton-like pulse, not surprisingly, features a hyperbolicsecant profile, rather than the commonly seen Gaussian-profile in stretched-pulse or dissipative-soliton lasers with strong nonlinear spectral shaping. Due to the low

109

Ch. 5

STRETCHED-SOLITON FIBRE LASER pulse energy and large temporal breathing ratio, energy re-distributions between different frequency components of the intra-cavity pulses were negligibly small, so that such soliton-like balance would be easier to illustrate in the spectral domain. In the moving frame at the group velocity, the spectral envelope of the pulse is denoted as A (ω, z), which is the Fourier transform of the pulse temporal envelope A (t, z). For a single round-trip, the accumulated linear phase-shift for each frequency component due to the cavity GVD is given by ϕGVD, total

1 2  LR = ω β2 (z)dz, 2 0

(5.1)

where LR is the total cavity length, and β2 (z) describe the cavity dispersion-map. Then we calculate the corresponding nonlinear phase-shift. Only with the SPM effect, the pulse temporal envelope, after an infinitesimal propagation step dz, becomes   ANL (t, z + dz) = A (t, z) exp iγ(z)|A(z, t)|2 dz, (5.2) where γ(z) describes the Kerr-nonlinearity distribution in the laser cavity. The nonlinear phase-shift of the laser pulse, as a function of the frequency, can be calculated using   ANL (ω, z + dz) , (5.3) dϕNL (ω, z) = arg  A(ω, z) where ANL (ω, z + dz) is the Fourier transform of ANL (t, z + dz). The accumulated nonlinear phase shift over each cavity round-trip can then be expressed as ϕNL, total (ω) =

 LR 0

dϕNL (ω, z).

(5.4)

Using Eqs.(5.1) to (5.4), we were able to calculate the accumulated phase-shifts of the intra-cavity pulses, both linear and nonlinear, using the simulated pulse evolution results. The calculated phase-shifts in the frequency domain are plotted in Fig.5.10, together with the spectral envelope plotted as the grey dashed curve. The linear and nonlinear phase-shifts almost completely cancelled each other close to the central part of the spectrum, which implies that different spectral components of the laser pulses had the same net group-delay over each cavity round-trip, ensuring the self-consistency of the pulse evolution. Small discrepancies observed at the edge frequencies were probably due to the saturable absorber and higher-order dispersion in the fibre cavity. The accumulated nonlinear phase-shift over each cavity round-trip,

110

Analysis of intra-cavity pulse evolution

5.3

1.5

1.0

1.0

0.8

0.5

0.6

0.0 0.4

-0.5 linear nonlinear pulse spectrum

-1.0 -1.5 -4

-3

-2

-1 0 1 2 Frequency (THz)

0.2 3

4

normalized intensity (a.u.)

Relative phase shift (rad)

as shown in Fig.5.10, was well below π, resulting from weak nonlinearity experienced by the pulse.

0.0

Fig. 5.10: The calculated phase-shift of the stretched-soliton propagation in one cavity round-trip. The linear phase-shift due to cavity GVD is shown in blue solid-curve, while the nonlinear phase-shift due to SPM effect is shown in red solid-curve. The simulated spectrum of the laser pulse is plotted in grey dashed curve.

111

CHAPTER

6 ALL-OPTICAL BIT-STORAGE

The enhanced optoacoustic interactions in solid-core PCF have enabled stable passiveHHML soliton lasers with typically hundreds of pulses evenly-spaced in the laser cavity. These pulses are uncorrelated with each other due to the non-interferometric nature of the optoacoustic mode-locking. As a result, each pulse can be regarded as an independent sub-laser pulse that can be manipulated individually. In this chapter, a unique technique is demonstrated, which could selectively erase some of the intra-cavity pulses in the HHML fibre laser, while the complicated sequence of pulses that remained in the cavity formed an optomechanical bound-state that could stably persist inside the cavity. These bound-states, in forms of arbitrarilycontrolled presence or absence of soliton within each time-slot, can be used to carry bit-information that could be stably stored in the laser cavity over extraordinary long time.

6.1 Working principles 6.1.1 Optomechanical bound-states In the HHML laser demonstrated in Chapter 4 and 5, the acoustic core-resonance of the solid-core PCF has effectively divided the cavity round-trip time into hundreds of identical time-slots that were homogeneously distributed in the ring-cavity. Each time-slot corresponds to one acoustic cycle of the PCF core vibration, within which one soliton can be trapped, thanks to the re-timing force provided by the temporal

113

Ch. 6

ALL-OPTICAL BIT-STORAGE trapping potentials (Section 3.2.1). Since these pulses are uncorrelated in phase, they can be regarded as independent sub-laser pulses that could be manipulated independently. If, for example, a few of the time-slots were “empty” (i.e. with pulse drop-outs [63, 214]), whereas the cavity still has enough pulses, being invariantly trapped in their corresponding time-slots, that could coherently drive the acoustic resonance of the PCF, then this specific pulse sequence forms a stable optomechanical bound-state that could still persist in the laser cavity [107]. As illustrated in Fig.6.1, the case of HHML without any pulse drop-out is compared with the case with some empty time-slots. The acoustic waves driven by the pulses have no fundamental difference, except that the ladder case might have a slightly lower acoustic amplitude. The pulse sequences in both cases, being two different optomechanical bound-states, are self-stabilized. The enabling mechanism for such bound-states includes two aspects. On one hand, the GHz-acoustic resonance of the PCF-core has a relatively long life-time of tens of nanoseconds (with a mechanical Q-factor of ∼ 150), therefore some empty time-slots in the driving pulse sequence would have negligible influence upon the waveform of the resultant acoustic wave. As long as the fundamental beat note of such pulse sequence is fixed, the acoustic frequency in the PCF-core would also be unchanged, leading to an invariant optoacoustic lattice in the cavity. On the other hand, the trapping of each pulse is largely a localized effect, meaning that the re-timing force is confined only in the vicinity of the balanced position within the relevant time-slot and does not directly depend on the presence of any other particular pulse, as described in Section 3.2.1. The optomechanical bound-states in the HHML laser could take diverse patterns, with arbitrary presence or absence of pulses in different time-slots (shown in Fig.6.1(b)). As a result, binary information can be stored using such boundstates (time-slots with and without pulses represent 1s and 0s respectively) [192, 215]. These long-range, highly flexible, and meanwhile self-stabilized optomechanical bound-states, once formed, can persist for very long time in the laser cavity, which brought us with the idea of realizing an all-optical bit-storage in such optoacoustic mode-locked laser cavity. The technical problem to be solved was how to selectively control the presence or absence of solitons in each time-slot, which is described in the following section.

114

Working principles

0

1

1 1 0 0 0 1 1 1 1 01 01 1 1 1 1 1 1

1

1 1

0

1

1 1 1 1 1 1 1 1 1 1 1 1 1 1

1

Pulse sequence without drop-outs

0 1 1 0 1 1 0 1 0 01 1 1 1

1 1 1 1 1 1 1 1 1 1 1 1 1 1

PCF

1 1 0 1 0 1 1 0 1 1 1 1 1 0

1

EDF

0

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0

1

PCF

1

1

EDF

1 0 1 1 1 1 0 0 1 01 1 0 1 0 1 0

1

(b)

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

0

(a)

Pulse sequence with drop-outs

z-QGt

Index modulation

Time-slot

6.1

z-QGt

Index modulation

Time-slot

Fig. 6.1: Optomechanical bound-states in a HHML fibre laser using the optoacoustic mode-locking scheme. (a) The HHML state without pulse drop-outs(all “1”s). (b) The HHML state with some pulse drop-outs (with “1”s and “0”s simultaneously), which forms a complicated pulse sequence. The upper figures illustrate the key components in the unidirectional ring-fibre laser cavity. the optoacoustic interactions in the PCF provide index modulation. The lower figures illustrate the acoustic waves driven by the pulse sequences, which effectively divide the cavity round-trip into many time-slots.

6.1.2 Writing bit-information in the HHML laser In the HHML laser based on the optoacoustic mode-locking, the NPR-based polarization APM is employed to initiate the self-pulsing process, to suppress the background noise, and to clamp the pulse peak amplitude under steady-state. The intensity-dependent loss transformed from the NPR leads to a stable working point based on the gain/loss balance for each pulse (see Fig.2.5), while the EDFA is saturated by the average intra-cavity power. Our idea was that, by manipulating the gain/loss balance of some selected intra-cavity pulses using externally launched addressing pulses, we could erase them through increasing the cavity loss specifically on them. The addressing pulses, which are supposed to have strong nonlinear propagation in the fibre cavity, were arranged to overlap and co-propagate with selected intra-cavity pulses after being launched externally into the laser. The intensity-dependent loss profile experienced by these pre-existing pulses at the po-

115

Ch. 6

ALL-OPTICAL BIT-STORAGE larizer would be instantaneously modified to XPM effect, leading to an interruption of the gain/loss balance. Given properly-set polarization bias and addressing pulse intensities, the selected pulses could experience net loss in the laser cavity due to this interruption, and would be eventually erased. Once the erasure was completed, the addressing pulses can be turned off. The conceptual set-up of this erasing technique is illustrated in Fig.6.2. To avoid the influence of the addressing pulses (which is relatively strong) on the EDFA gain, the polarization states of these pulses were prepared so that they would be completely eliminated by the polarizer. External addressing (erasing) pulses EDF

Overlap between addressing and selected intra-cavity pulses NPR

PCF

Polarizer

Fig. 6.2: The principle of selective erasure of intra-cavity pulses using external addressing pulses. The addressing pulses (in red) launched externally were arranged to overlap with selected intra-cavity pulses (in blue, enclosed in the grey dashed-box) and modify their NPR through XPM, so as to change the intensitydependent loss at the polarizer. The interruption of gain/loss balance of the selected intra-cavity pulses would eventually lead to the erasure of them.

The step-by-step procedure of the erasing dynamics is illustrated in Fig.6.3, using the balance between EDFA gain and intensity-dependent loss profiles. Before the launch of addressing pulses, all the intra-cavity pulses have the same clampedintensities due to stable gain/loss balance (Step-1). The region in which the pulse intensity could be stabilized is shaded in red, while in the region shade in blue (close to zero light intensity), net cavity loss was obtained to suppress low intensity light, behaving as a saturable absorber. The addressing pulses then entered and perturbed the selected pulses through cross-phase modulation (Step-2). As a consequence, the profile of the intensity-dependent cavity loss is modified (dashed blue curve), leading to the net loss for the perturbed pulses, while the unperturbed pulses maintained the

116

Working principles

6.1

stable balance as before. The perturbed pulses would be attenuated until they were out of the region with stable balance (Step-3). Note that this particular step was not necessarily able to be accomplished within only a single-shot. Empirically, repeated addressing pulses were necessary for sufficient attenuations for the same selected intra-cavity pulse, as will be later revealed in the experiment results. The transition process occurred (Step-4), after the addressing pulses were already off, in which the perturbed pulses would be further attenuated due to the saturable absorption until they were completely eliminated. Meanwhile, since fewer pulses were present after the erasure, the average power would decline, leading to a reduced saturation level of the EDFA and consequently a slightly higher gain for the remaining pulses. Their working points, as a result, would shift a bit accordingly until the new stable gain/loss balance was reached at last (Step-5). Note that for encoding bit-information in the fibre cavity, the erasure of selective pre-existing pulses is in principle equivalent to the writing process, i.e to generate pulses in previously empty time-slots. However, in practice, the erasure process was found to be much more reliable and reproducible. For generating a desired pulse sequence in the fibre laser cavity, the addressing pulses should be the complementary sequence of it. After the erasure of selected pulses, the addressing pulses which are launched externally can be turned off, and the remaining pulses can persist for indefinitely long-time with almost equally high stability as the HHML lasers. The self-stabilization of such optomechanically bound-state in the fibre cavity was challenged, however, if too many pulses were erased and thus the left-over ones would not be able to drive a sufficiently strong acoustic wave in the PCF. In addition, when too few pulses were remained, EDFA gain would increase considerably, leading to excessively high amplitude for the remaining pulses. As a result, the polarization APM would easily be over-driven, and the stable optoacoustic mode-locking would simply collapse. Due to this challenge, the bit-information that could be encoded using all the time-slots within one-round-trip was not absolutely arbitrary. We tackled this problem by preserving a certain block of many consecutive time-slots within one round-trip (typically tens out of hundreds of time-slots). Within this particular block, arbitrary number of pulses could be erased, while the rest of timeslots were all filled with pulses so that the acoustic wave in the PCF could be maintained with a sufficient amplitude.

117

Ch. 6

ALL-OPTICAL BIT-STORAGE

Net-loss for perturbed pulses Gain in EDF Cavity-loss for perturbed pulses Light intensity in the cavity (a.u.)

Working point (gain = loss)

Step 3: Selective pulses attenuated due to net loss

Step 5: New stable state

Intensity-dependent cavity loss

Cavity loss/gain (a.u.)

Gain in EDFA

Light intensity in the cavity (a.u.)

Cavity loss/gain (a.u.)

Cavity-loss for unperturbed pulse

Intensity-dependent cavity loss

Intensity-dependent cavity loss

Unperturbed pulses

Gain in EDFA

Perturbed pulses

New working point (gain = loss)

Light intensity in the cavity (a.u.)

Step 4: Transition: complete erasure

Gain in EDF

Light intensity in the cavity (a.u.)

Cavity loss/gain (a.u.)

Cavity loss/gain (a.u.)

Step 1: HHML state (full of solitons)

Cavity loss/gain (a.u.)

Step 2: Modified gain/loss balance

Intensity-dependent cavity loss

Erased pulses Unperturbed pulses

Shifted gain in EDF

Light intensity in the cavity (a.u.)

Fig. 6.3: The mechanism of intra-cavity pulses erasure using external addressing pulse, which is based on the dynamic modifications of the intensity-dependent cavity loss. See the text above for detailed descriptions.

6.2 Experimental set-up and results 6.2.1 External addressing pulses and the bit-storage cavity The experimental set-up for realizing the all-optical bit-storage, as sketched in Fig.6.4(a), consists of two parts, one for generating and controlling the external addressing pulses, the other being simply a unidirectional ring-cavity HHML soliton fibre laser for storing the bit-information. In the ring-cavity part, a 0.6-m length of EDF provided the optical gain at 1550 nm, pumped by the two laser diodes with a

118

Experimental set-up and results

6.2

combined power of 1 W. A 2.5-m length of solid-core PCF was inserted to enable the optoacoustic mode-locking. This PCF has a core resonance at ∼ 1.881 GHz with bandwidth of 13 MHz. An additional 15-m length of SMF was inserted after the EDFA section in-order to enhance the NPR effect so that the erasing process based on XPM would be more efficient. (a)

Pulse pattern generator

Pulse generator EDFA

BPF

HPOS

EOM Addressing pulse sequence

FPC 4

BC

DCF 0.9 m

BPF PD FC 10/90

FC 50/50

OSC

FPC 5

1545 nm Single-frequency laser

EDFA

Cavity output

Autocorrelator FC 50/50

OSC ESA

PD

FPC 2 WDM 1

FC 50/50 Fibre laser cavity

SMF 15 m Polarizer

FPC 1

(c)

EDF 0.6 m PCF 1 m

WDM 2 Pump LD 1 976 nm

Isolator FPC 3

Spectral power (dB)

(b) Pump LD 2 976 nm

Autocorrelation (a.u.)

OSA 5

760fs×1.54

4 3 2 1 0 0

2

1 0 1 Delay (ps)

2

3.5 nm

20 40 60 1540 1550 1560 1570 1580

Wavelength (nm)

Fig. 6.4: (a)All-optical bit-storage experimental set-up consisting of two parts: the set-up for generating addressing pulse sequence and the fibre laser cavity for bit-information storage. LD, laser diodes; WDM, wavelength division multiplexer; FPC, fibre polarization controller; FC, fibre coupler; EOM, electro-optical modulator; BC, biased controller; EDF, erbium-doped fibre; BPF, band-pass filter (optical); HPOS, high-power optical switch; DCF, dispersion compensation fibre; PD, photodetector; OSC, oscilloscope; ESA, real-time electrical spectrum analyser; OSA, optical spectrum analyser. (b) The AC-trace of the output soliton. (c) The optical spectrum of the output solitons

The total cavity length was 28.3 m, corresponding to a fcav of 7.33 MHz. The pulse repetition rate was adjusted to 1.876 GHz, which was the 256th harmonic

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Ch. 6

ALL-OPTICAL BIT-STORAGE of the fcav . The intra-cavity solitons had a FWHM pulse duration of 760 fs and 3-dB bandwidth of 3.5 nm (Fig.6.4(b) and (c)). The soliton laser operated with low intensity fluctuations and timing-jitter. The addressing pulse sequence, which was used to selectively erase the intra-cavity pulses, had an underlying time-grid that matched the repetition rate of the fibre laser (1.876 GHz). It was generated by modulating a single frequency source at 1550 nm using an EOM. The exact sequence pattern was determined by the electrical pulse-pattern generator. Then the addressing pulse sequence would experience multi-stage amplification and bandpass filtering (to suppress the broadband ASE noise) before it was launched into the fibre laser cavity through the 50/50 coupler. Each addressing pulse had a peak power of ∼ 10 W with a duration of ∼ 100 ps. The addressing pulse sequence was repeated for several times at intervals that matches the cavity round-trip, in order to ensure sufficient attenuations upon the selected pulses. The entire span of the addressing pulse sequence was controlled by a high-power optical switch (HPOS), which had a rising/falling edge of 300 ns and an open-window-time of ∼ 1 μs. Since the addressing pulse had a much longer duration than intra-cavity pulse (sub-ps), while still being sufficiently shorter than the time-slot, the full overlap between addressing pulses and the selected intra-cavity pulses could be ensured without any synchronization even when the repetition rate was constantly drifting during the erasure process due to the slow environmental disturbances.

6.2.2 Selective erasures of intra-cavity pulses The selective erasing process achieved in the experiment was shown in Fig.6.5. Out of the 256 time-slots, a block of consecutive 42 time-slots were reserved, in which the pulses can be arbitrarily erased. Before the erasure, the fibre laser was operated under stable HHML state, as shown in Fig.6.5(a) (Only the reserved block with some neighbouring pulses next to them are plotted). Then the addressing pulse sequence was launched (shown in Fig.6.5(b)), in which the same pulse pattern was repeated for seven times at a period of 136.4 ns (cavity round-trip time). The zoomin of a single-period pattern is shown in Fig.6.5(c). Then those intra-cavity pulses that overlapped with addressing pulses were successfully erased, leaving only the complementary pulse sequence, as show in Fig.6.5(d).

120

(b)

Voltage (V)

(a)

Voltage (V)

Experimental set-up and results

0.2 0.1 0 0.1

0

5

Voltage (V) Voltage (V)

(d)

20

10 15 Time (ns)

25

0.3 0.2

1 Ps

0.1 0 0

(c)

6.2

5

0.3

10 Time (Ps)

20

15

~10 W

0.2 0.1 0 0

5

10 15 Time (ns)

20

25

5

10 15 Time (ns)

20

25

0.2 0.1 0 0.1

0

Fig. 6.5: A experimental demonstration of using an addressing pulse sequence to generate a complementary pulse sequence in the HHML laser cavity. (a) The initial HHML state with all time-slots filled with pulses (only the 47 relevant time-slots were plotted). (b) The addressing pulses sequence, in which the same pulse pattern was repeated for 7 times at a period of 135.4 ns. (c) The detailed addressing pulse pattern within each addressing pulse period, consisting of a pattern of 100-ps pulses with 10-W peak-power. (d) The remaining intra-cavity pulses after the erasure, which form the complementary pattern of the addressing pulse sequence.

The remaining pulses in the block, as well as the ones outside, still followed an invariant time-grid, as we can reveal from the RF spectrum of the resultant pulse sequence after the erasing process (as shown in Fig.6.6(a)). The pulse intensity noise and timing-jitter were measured to be almost invariant before and after the erasing process. The pulse intensity noise was estimated to be ∼ 0.11 % (20 Hz−1 MHz) from the SSB noise spectrum at baseband (Fig.6.6(b)).The pulse timing-jitter, estimated from SSB noise spectra of higher-order harmonic bands (Fig.6.6(c)), was ∼ 74 fs (100 Hz − 1 MHz). The remaining pulses had a slightly shorter pulse duration and broader bandwidth due to the higher EDFA gain, as we have previously explained.

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ALL-OPTICAL BIT-STORAGE

Power (dB)

(a)

20

1.876 GHz

40 60 80

0

10

20

30

Frequency (GHz) (b)

(c) -60

-100

-120 -140

SSB noise (dBc/Hz)

SSB noise (dBc/Hz)

-80 Soliton sequence

HHML 10 2 10 3 10 4 10 5 Off-set frequency (Hz)

10 6

-80 -100

Soliton sequence

-120 -140 10 2

HHML 10 3 10 4 10 5 Off-set frequency (Hz)

10 6

Fig. 6.6: (a) The RF-spectrum of the output pulse sequence of the HHML laser after the erasing process was completed. (b) SSB noise spectra at the baseband measured for the original HHML state output pulse train (black curve) and the pulse sequence after the erasing process, exhibiting an invariant level of intensity noise. (c) The SSB noise spectra measured at higher order harmonic band, before and after the erasure, exhibiting an invariant level of time-jitter.

The erasing process, recorded at the laser 50/50 output port over a span of 100 μs (equivalently 731 round-trips), is shown in Fig.6.7(a), with a zoom-in figure of the first three erased pulses shown in Fig.6.7(b). This dynamic process is plotted so that the change of intra-cavity pulse pattern after each cavity roundtrip are compared in parallel (only the relevant 47 time-slots (25 ns) is plotted). Before Nr.70 round-trip, the laser was under the stable HHML state (Step-1 as explained in Section 6.2.2). Then the addressing pulse (exhibiting themselves as intense red-dots in Fig.6.7) entered and overlapped with selected intra-cavity pulses. This process (corresponding to Step-2 and -3) lasted for around 8 round trips before the addressing pulse was turned off, which had caused significant attenuations of the selected pulses. At this moment, these selected pulses did not completely disappear. They were gradually attenuated by the artificial saturable absorption over the next few hundreds of round-trips (Step-4) as we can reveal from the gradually decaying traces of the selected pulse within their time-slots. Then finally a new stable state was reached (Step-5) after these pulses have completely disappeared.

122

Experimental set-up and results (a)

M

P

L



I

Y

O

(b)

L

11100110110000011001101101001110010111101100011

0.4

700

OSC voltage (V)

Roundtrip number

700 600

600 500 400 300

Step-5

500 400 300

Step-4

200

200 100 0

6.2

0

100

Step-2 & -3

0

0

5

Zoom-in

10 15 Relative intra-cavity time (ns)

20

25

Step-1 2 3 4 Relative intra-cavity time (ns)

Fig. 6.7: (a) A erasing process recorded over 731 round-trips plotted as the dynamic change of the intra-cavity pulses after each round-trip. Only the relevant 47 time-slots out of the entire 256 time-slots are shown. (b) Zoom-in of the erasing process of the first three erased pulses. The 5 steps within this process, as previously described in Fig.6.3 were marked at the corresponding round-trips

We used the encoding strategy listed in Table 6.1 to store a simple string “MPL–IOYL” in the ring-cavity fibre laser (as the abbreviation of the Max Planck institute for the science of Light–International Year Of Light1 ), as shown in Fig.6.7(a). The bit-information started from the first empty time-slot, i.e. an initial 0, then lasted for 8 characters (40-bits) before ending with a terminal 0. Table 6.1: The encoding list for the all-optical bit-storage.

Character A B C D E F G H I

1

Code 00001 00010 00011 00100 00101 00110 00111 01000 01001

Character J K L M N O P Q R

Code 01010 01011 01100 01101 01110 01111 10000 10001 10010

Character S T U V W X Y Z -

Code 10011 10100 10101 10110 10111 11000 11001 11010 11011

This work was mainly done in 2015 when it was the International Year of Light led by UNESCO.

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Ch. 6

ALL-OPTICAL BIT-STORAGE

6.2.3 Demonstrations of long-term bit-storage After the erasing process was completed, the bit-information carried by the remaining pulse sequence, being an optomechanical bound-state, was able to be invariantly preserved in the cavity for many hours. As some examples, some of the stored pulses sequence, recorded over 2 hours at 5 min interval, are plotted in Fig.6.8 which consists of some extreme cases, including, “all 0” (Fig.6.8(a)), “isolated 1” (Fig.6.8(b)), and “isolated 0” (Fig.6.8(c)), which indicated that any arbitrary pulse pattern was able to be stored within this block. Fig.6.8(d) demonstrates a 2-hour storage of the string “ERLANGEN” 2 . The total number of empty time-slots could be greater than 42, however, if only they were not closely next to each other. Fig.6.9 shows the stored pulse sequence after the selective erasing, in which 64 evenly-spaced pulses were erased through the entire cavity round-trip. In principle, given a fixed acoustic frequency (i.e. the time-slot length), the bit-information stored in the cavity can be increased simply by increasing the cavity length, given that the empty slots are not completely consecutive. The influence of excessive empty time-slots on the HHML state was not completely understood, and we only have empirical knowledge of the maximum tolerance of this system, beyond which the HHML state would collapse. Recently, however, we analysed the data shown in Fig.6.8(a) and realized that the remaining pulses close to the two edges of the empty block actually slightly deviated from their original relative positions within their own time-slots. This indicates that these pulses were no longer precisely equally-spaced, although such shift was too small to be revealed from such round-trip plot. This phenomenon could be equivalently understood as that, due to highly-biased distribution of the driving pulse sequence, the optoacoustic lattice has become inhomogeneous. This topic is out of the scope of this thesis, while some experimental progresses have been made.

2

124

The name of the German city where this work was done.

Experimental set-up and results

6.2

(a) 120 Rec ordi ng t ime (min )

100

Voltage (V)

80

60

0.2

40 20

0 0

5

10 15 Relative intra-cavity time (ns)

20

25

0

(b) 120 Rec ordi ng t ime (min )

100

Voltage (V)

80

60

0.2

40 20

0 0

5

10 15 Relative intra-cavity time (ns)

20

25

0

(c) 120 Rec ordi ng t ime (min )

100

Voltage (V)

80

60

0.2

40 20

0 0

5

10 15 Relative intra-cavity time (ns)

20

25

0

(d) 120 Rec ordi ng t ime (min )

100

Voltage (V)

80

60

0.2

40 20

0 0

5 0.2

0

10 15 Relative intra-cavity time (ns)

20

25

0

OSC voltage (V)

Fig. 6.8: Demonstrations of different bit-patterns stored over 2 hours. (a) Full of 0 with 42 consecutive pulses erased. (b) An isolated 1 with the other 41 pulses erased. (c) An isolated 0, in which only one pulse was erased. (d) The pulse sequence carrying bit information of “ERLANGEN” according to the encoding list in Table 6.1.

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Ch. 6

ALL-OPTICAL BIT-STORAGE

120 Reco rding time (min )

100

80

Voltage (V)

60 40

0.2

20

0 0

10

20

30 40 Relative intra-cavity time (ns)

50

60

0

120 Reco rding time (min )

100

80

Voltage (V)

60 40

0.2

20

0 70

80 0

90 0.2

100 110 Relative intra-cavity time (ns)

120

130

0

OSC voltage (V)

Fig. 6.9: A intra-cavity pulse pattern with 64 evenly-spaced pulses erased, which extends to the entire cavity and was stored for over 2 hours.

At last, an 100-hour storage test was performed, the stored string being the same as in Fig.6.7. The read-outs were performed at every 30 min. The intra-cavity pulse sequence was perfectly preserved without any trace of degradation. This was, for the first time, that a complicated long-range soliton bound-state was stored without any external stabilization. The corresponding propagation distance inside the laser cavity was a record 72 trillion kilometres. At the end of the last recording day, after these 100 hours, this pattern finally experienced a slight degradation. A few pulses close to the edge of the reserved block suddenly changed, although the basic time-grid, and most of the bit-information remained intact. We attributed this degradation mainly to some cavity perturbations caused by the accumulated misalignment of the FPCs in the cavity over these many days, which induced some mechanical strains on the fibre sections. It might also be caused some random laser spikes emerged from the ASE noise of the EDFA.

126

Experimental set-up and results

6.2

100 Rec ordi ng t ime (h)

80

0.2

40 20

0 0

5 0

0.2

10

15 Relative intra-cavity time (ns)

20

25

0

Voltage (V)

0.2 Recording at 0 h 0.1 0 0

5

20 10 15 Relative intra-cavity time (ns)

25

0.2 Recording at 70 h 0.1 0 0

5

20 10 15 Relative intra-cavity time (ns)

25

0.2 Recording at 30 h 0.1 0 0

Voltage (V)

Voltage (V)

OSC voltage (V)

Voltage (V)

Voltage (V)

60

5

20 10 15 Relative intra-cavity time (ns)

25

0.2 Recording at 100 h 0.1 0 0

5

20 10 15 Relative intra-cavity time (ns)

25

Fig. 6.10: An 100-hour bit-storage demonstration. Read-out was performed over one cavity round-trip time at every 30 min.

127

CHAPTER

7

SUPRAMOLECULAR ASSEMBLIES OF OPTICAL SOLITONS

This chapter concerns the newly discovered long-range bound-states of optical solitons that originate from two types of long-range interactions between the intra-cavity solitons in the optoacoustic mode-locked laser: the dispersive wave perturbations and the optoacoustic effects. The delicate balance between these two long-range interactions of different origins enables stable binding of multi-solitons within each time-slot of the HHML laser, with their internal spacings being two orders of magnitude longer than the individual soliton duration. Moreover, this giant assembly of solitons throughout the entire HHML cavity, being constituted by a diversity of different long-range bound-state units, resembles in many ways to the supramolecular assemblies of particles, exhibiting evident analogy to many of their key properties. We start with analytical interpretation of the binding mechanism, then the experimental observations and carefully designed validations are demonstrated later.

7.1 Balance between long-range interactions 7.1.1 The observation of balanced interactions The enhanced optoacoustic interaction in the solid-core PCF has been demonstrated to enable stable HHML lasers in the previous chapters. The resonant vibrations in the PCF-core effectively divided the cavity round-trip into hundreds of time-slots. In each of these time-slots, a single soliton can be trapped at a fixed relative position.

129

Ch. 7

SUPRAMOLECULAR ASSEMBLIES OF OPTICAL SOLITONS The trapping potential for this soliton is formed by the cooperation between the index modulation induced by the acoustic wave and the cavity GVD. The acoustic vibration is driven by the soliton sequence, and as a back action the solitons are modulated by the vibration. These solitons formed a long-range optomechanical bound-state, in which the solitons interact with each other through the acoustic wave indirectly. The result is a self-stabilized, equally-spaced soliton sequence that circulated in the fibre cavity. Moreover, as demonstrated in Chapter 6, successful manipulations upon the on- and off-state of each soliton have been achieved, which relied on the independence of each sub-laser pulse, as well as the long life-time of the acoustic vibration. As an unexpected discovery, it was found in subsequent experiments that, more than one (up to four) solitons can be stably trapped within each time-slot. The spacings between the multi-solitons in each time-slot (from now on referred to as the “internal spacings”) were surprisingly found to be around 100 ps, while the duration of the individual soliton was in the sub-ps range. Moreover, all the time-slots with multi-solitons shared the same characteristic internal spacings, even though these time-slots could have different numbers of solitons. The double-soliton, triplesoliton, and quadruple-soliton states, in addition to the non-soliton and single-soliton state as demonstrated in Chapter 6, have all become the configurational units that can be arbitrarily arranged throughout the time-slots in the cavity. Meanwhile, the over-all sequence of the intra-cavity solitons still follows the same time-grid so as to coherently drive the acoustic resonance in the PCF-core. A difficulty was met when we tried to interpret the mechanism of such multisoliton state. As we have described in the Section 3.2.1, the index-slope seen by the pulse at different positions of the acoustic cycle would be different. Provided with a non-zero cavity dispersion, these solitons at such distance should have relative drift in positions during the propagation and would never form any bound-state. In fact, one soliton always tends to be attracted towards the other until they reach the same balanced position which is the cornerstone of the self-stabilized HHML state using the optoacoustic mode-locking scheme. Therefore it was a shock to realized that, there actually exist other stable positions within each time-slot, in which additional solitons can be trapped. The over-all pattern, though still being highly organized, can take far more complicated patterns than simply an evenly-spaced pulse sequence. With careful measurements of the output soliton spectrum, we then notice a weak but rather uncommon spectral fringe that appeared only in the vicinity

130

Balance between long-range interactions

7.1

of dominant Kelly sideband. This specific spectral fringe suggested an unrevealed interaction between dispersive waves and solitons in a laser cavity. This discovery has inspired the following model of balanced long-range interactions, in which these dispersive waves shed by the solitons are assumed to participate in the interaction between the solitons within one time-slot. This working principle is schematically illustrated in Fig.7.1. On one hand, the acoustic wave driven by the overall soliton sequence still divides the round-trip time into many time-slots, which is the same as the previous HHML state.(Fig.7.1(a)). Within each time-slot (with exemplary double-soliton state, as enclosed by the dashed-line box), the different index slope seen by the two solitons (dna /dt) causes one soliton to shift towards the other due to the relative red-shift of its carrierfrequency, yielding an effective force of attraction (Fig.7.1(b)). On the other hand, the dispersive wave shed from one soliton, which corresponds to the dominant Kelly sideband of m = −1 order (Fig.7.1(c) inset)1 , perturbs the other soliton by induce a blue-shift of its carrier-frequency. As a result, one soliton tends to push the other away from it through the dispersive wave which acts as a force of repulsion (Fig.7.1(c)). It is the balance between these two long-range forces that leads to a stabilized internal spacing within each time-slot. Since the acoustic wave magnitude and dispersive wave strength are both the same in all the time-slots (all solitons being identical), these multi-solitons would consequently share the same characteristic internal spacings. The first soliton in each time-slot effectively creates a trapping potential beside it (Fig.7.1(d)). Then with this position filled with a second soliton, a new trapping potential is created beside this one, which can trap a third soliton, and so on (Fig.7.1(e) and (f)). Since both interactions are long-range in nature, the internal spacings are typically much longer than the soliton itself. Meanwhile, the perturbation of dispersive wave shed by each soliton was assumed to be mostly confined within the time-slot, number of solitons in each time-slot is generally independent from others, leading to a high degree of configurational diversity in the soliton sequence throughout the entire cavity laser cavity.

1

Note that, according to Section 2.2.3, for the same order number m, there are two sidebands on opposite sides of the spectrum. In order to better specify the sideband, we used a slightly different notation in this chapter. The first order sideband with higher frequency was denoted to be the m = −1 order sideband, while the lower frequency one to be the m = +1 order sideband, the ± signs only indicate the relative locations of the sidebands.

131

Ch. 7

SUPRAMOLECULAR ASSEMBLIES OF OPTICAL SOLITONS

(a) Acoustic wave in PCF core

z-

Gt

Index modualtion

(dna

Index modulation

(dna

dt ) t =t

(c) 2

Spectral density

(b) dt ) t =t

1

Attraction 2nd soliton

(d)

of e rc o F Forc e of a

t2

ttrac

t-z/

1st Repulsion DW (m= −1)

Frequency

t1

ion uls p re

2nd

DW (m=+1)

1st soliton

t2 Trapping potential

DW (m=−1)

G

DW (m= +1)

t2

(e)

t1

t-z/

G

(f)

tion

t1

t3

t2

t1

t4

t3

t2

t1

Fig. 7.1: The principle of a supramolecular assembly of optical solitons. (a) A sequence of solitons drives an acoustic wave in a PCF-core. The optically-driven acoustic wave, which creates an index modulation na (z − vg ), forms a time-grid based on the mechanical vibrations. (b) Within each time-slot, a long-range force of attraction between the solitons arises due to the different index slopes seen by this two solitons, leading to relative drift of the soliton at t2 towards the one at t1 . (c) The dispersive wave shed by the soliton at t1 perturb the soliton at t2 , resulting in a force of repulsion between the two solitons. The inset figure shows a typical soliton spectrum with asymmetric Kelly sidebands. The dominant sideband (m = −1 order) is the main source of dispersive wave (DW) perturbation. (d) The balance between these two long-range forces of attraction and repulsion forms a temporal trapping potential for the second soliton. (e) and (f) Stable multi-soliton units can form via the build-up of additional trapping potentials.

In many ways, such long-range bound-state of solitons is by itself a unique phenomenon. First of all, multi-solitons in each time-slot are practically contactless and phase unrelated, due to the pulse spacing being two orders of magnitude longer than their individual pulse durations. Therefore, the binding mechanism is distinct from the well-know soliton pairs [187] and soliton crystals [188] in which the solitons are spaced at most a few times of the soliton duration. In that case, the XPM that arises through their tailing fields would result in a “rigid” bound-state with fixed phase-relation between them, leading to strong spectral fringe throughout the entire soliton spectrum. Secondly, long-range interactions between optical

132

Balance between long-range interactions

7.1

solitons, especially through the perturbations of dispersive waves, have long been regarded as being uncontrollable or even as sources of noises leading to disordering of soliton train during propagation [112]. Here, however, the carefully tailored long-range interactions originating from long-lived optical and acoustic waves were able to form stable self-assembled complex soliton structures. The theory of longrange interactions, including the formation of forces of attraction and repulsion, are analytically presented in the following sections. Guided by the theory, a few validation experiments were conducted, with the results that highly match the theoretical predictions, as will be demonstrated in Section 7.2 and 7.3.

7.1.2 Force of attraction due to optoacoustic effects We start from the analysis of the force of attraction that arises from the optoacoustic effect. For the HHML laser under steady-state, the intra-cavity solitons ride on the acoustic wave which is predominantly a transverse vibration with a cut-off frequency and meanwhile a phase velocity along fibre axis that matches the soliton group velocity. This acoustic wave produces a moving sinusoidal refractive index modulation (wave) that modulates the solitons while co-propagating with them (Fig.7.1(a)). The resulting optical field of the pulse can be expressed as Es = as (z − vg t) exp i (β0 z − ω0 [t + (zn1 /c) cos (z − vg t + ϕ)])

(7.1)

where as is the slowly varying soliton envelope, β0 is the propagation constant at the carrier-frequency ω0 , vg is the soliton group velocity, n1 is the amplitude of the travelling refractive index wave, and ϕ is the phase of the acoustic wave relative to the soliton center. By taking the first-order time-derivative of the complex exponential argument, we can obtain the instantaneous frequency ωs (z) during the propagation, which is ωs (z) = ω0 (1 + (n1 vg z/c) sin (ϕ + z − vg t)) (7.2) showing that the soliton would experience a blue- or red-shift in the carrier-frequency at its peak (z = vg t), the exact sign and magnitude depending on the value ϕ, i.e. the relative position of the soliton within that acoustic cycle. Besides, this frequency shift also increases linearly with the propagation distance over which the solitons ride on the acoustic wave in the PCF. If two solitons are in the same time-slot, then their rates of frequency shift will be different, due to the different index-slopes seen

133

Ch. 7

SUPRAMOLECULAR ASSEMBLIES OF OPTICAL SOLITONS by them. If they have the correct relative timing with refractive index wave (Section 3.2.1), their relative frequency shift, acting in concert with the cavity net GVD, will cause them to drift closer to each other as they propagate, which manifests itself as a long-range force of attraction. It can also be equivalently understood that, the two solitons drive the acoustic wave with different powers (which are proportional to instantaneous velocities of the mechanical vibrations at the positions of the driving solitons), leading to different rates of energy conversions from the solitons to the acoustic wave and eventually causing different carrier-frequency shifts. For simplicity, we only calculate this relative frequency shift to represent the strength of this force, which would later be compared with the force of repulsion expressed in the same way. We take one double-soliton unit as example, meanwhile assuming that all the time-slots are filled with the same type of units. The relative positions of the index modulation and the two solitons within one acoustic cycle, all being represented in the time domain under the moving frame, are shown in Fig.7.2. 2nd soliton

dna dt t = t 1

Δt

dna dt t = t 2

-Ta/2

1st soliton +Ta/2

0 t2 te t1

t

Equivalent driving center Index modulation na(t) Fig. 7.2: One double-soliton unit of the all-double-soliton state and the opticallydriven index modulation within one time-slot in the optomechanical lattice plotted under the moving frame. The two solitons are located at t1 and t2 respectively, giving an internal spacing of Δt and an equivalent driving center of te . When propagating in the PCF-core, these two solitons would experience different indexslopes.

Similar to the derivation in Section 3.1.3, the acoustic wave driven by the alldouble-soliton train at a repetition rate of frep = Ωa /2π can be expressed under the moving frame as below: γe |Q1 |Pav



Ωa Δt  sin (Ωa t) cos ρ(t) = 2 2πcn0 Aeff Ωa 4δa2 + Γ2B

134



(7.3)

Balance between long-range interactions

7.1

in which ρ(t) is the material density variation as a function of time, γe is the electrostrictive constant of the silica, c is the speed of light in vacuum, n0 is the refractive index of silica at rest, Aeff is the effective mode area of LP01 optical mode in the PCFcore, δa is the frequency offset defined as δa = Ωa − Ω01 , and Pav is the average power of this driving soliton train. Q1 is the overlap integral defined as ρ01 , Δ2⊥ |E01 |2 , where ρ01 and E01 are the transverse mode profile of the R01 acoustic mode and the LP01 optical mode. The additional factor cos (Ωa Δt/2) appears because the strength of the driving force at the frequency component of Ωa is weakened by this factor due to the finite the spacing Δt between the two solitons in each unit. The coherent driving of the R01 acoustic resonance would be most efficient when Δt = 0, i.e. the all-single-soliton case, while this coherent driving would basically fade away when Δt = Ta /2 where Ta is the acoustic period (i.e. driven by a frequency-doubled soliton train). The refractive index modulation induced by the density variation through the stress-optical effect has been determined to be

ρ(t) Pav Ωa Δt na = γe Q0 = Ca  sin(Ωa t) cos 2n0 ρ0 2 Ωa 4δ 2 + Γ2 a

in which Ca =

B 2 γe |Q1 |Q0 4πcn20 Aeff ρ0



(7.4)

>0

The timing of the double-soliton unit relative to the optically driven acoustic wave in one time-slot can be expressed using the equivalent driving center te , which is defined as

1 2δ a te = tan−1 − (7.5) Ωa ΓB te is simply the middle point of the two solitons and is determined by the frequency offset δa , as previously determined in Eq.(3.22). The locations of the two solitons can therefore be expressed as Δt 2 Δt t2 = te − 2

t1 = te +

(7.6a) (7.6b)

In almost all the derivations and descriptions used in the following sections (except in Section 7.3.3), we always refer to the soliton located at t1 , which arrives at a later time, as the first soliton, while the located at t2 , which arrives earlier, as second

135

Ch. 7

SUPRAMOLECULAR ASSEMBLIES OF OPTICAL SOLITONS soliton (t1 > t2 ). The reason for such notation would become clear later. The back-action of the mechanical vibration results in a carrier-frequency shift of the driving solitons, which is directly related to the refractive index slope experienced by the soliton. The rate of frequency shift during the co-propagation with the index modulation wave can be expressed as [20] dωs ωs dna =− dz c dt

(7.7)

Assuming that the total frequency shift of each solitons after propagation in the PCF with a length of LPCF is much smaller than the carrier-frequency, and the index modulation wave maintains the same amplitude throughout the PCF, the frequency shift of each soliton can then be expressed as ω0 δωsj = − c



dna dt



LPCF j = 1, 2

(7.8)

tj

Using Eqs.(7.4) to (7.6), we can calculate the relative frequency shift between this two solitons after a single-trip propagation in the PCF, which is Δωs = δωs2 − δωs1 =

δa 2Ca ω0 Pav LPCF sin(Ωa Δt) c 4δa2 + Γ2B

(7.9)

Since we usually only have one piece of PCF in the cavity, this is also the relative frequency shift induced by the acoustic wave per cavity round-trip. As previously explained,that for stable trapping of optical solitons, we should have a driving frequency slightly lower than the acoustic resonance frequency, i.e. δa < 0. Besides, the internal spacing was always smaller than half of the time-slot length, therefore sin(Ωa Δt) > 0. As a result, Δωa is always negative, which means that the second soliton has a net red-shift with respect to the first soliton (as the reference soliton),. Given the anomalous GVD of the cavity, the second soliton would tend to be delayed and consequently shift toward the first soliton during the propagation. The acoustic wave therefore exerts effectively a force of attraction between these two solitons. Given an addition soliton that comes earlier than the second soliton, the calculation would be similar, resulting in relative drift of this additional soliton to be attracted towards the second one, as so on2 . This derivation above can also be 2

136

Note that for more than two solitons within each time-slot, the spacing between the two edge solitons should still be smaller than half of the acoustic cycle

Balance between long-range interactions

7.1

regarded as an extension of the trapping potential theory that has been presented in 3.2.1, where the pulse deviation only occurs in the close vicinity of the balance position and thus the index modulation can be approximated by Taylor expansion to the second order. Given a much larger deviation of pulse position, similar effect of temporal trapping still exists.

7.1.3 Force of repulsion due to dispersive wave perturbations A competing long-range interaction between the two solitons in one time-slot arises from the dispersive waves that are shed from one soliton and extend in the time domain to the neighbouring solitons and perturb them through XPM [109, 110, 134]. The dispersive waves in a soliton laser corresponds to the Kelly sidebands which appears in form of narrow peaks at discrete frequencies on the soliton spectrum, as shown in Fig.7.3(a). The Kelly sidebands accumulate at these phase-matched frequencies due to the periodic disturbances arising from the discrete dispersion, nonlinearity, gain/loss distributions in the soliton laser, as previously reviewed in Section 2.2.3. Due to the gain narrowing effects in the EDF, these sidebands, being a few nm away from the spectral centre, experience net loss in the cavity compared to the spectral centre. During the round-trip propagations in the laser cavity, these sidebands acquire energies coupled from solitons due to the periodic disturbances, so as to balances their net loss. In practice, the asymmetric gain profile in the EDFA as well as higher-order dispersion in the fibre cavity can cause asymmetric distribution of these Kelly sidebands, typically with one sideband having a dominant intensity over others. The situation we consider in the following analysis is that the first order sideband with higher frequency (the m = −1 order sideband) has the dominant intensity, while the other sidebands are relatively weak and cannot induce obvious perturbations. This is also the situation which we will illustrate in the experimental results (except in Section 7.3.3). This dominant sideband corresponds, in the time domain, to a packet of dispersive wave shed from the solitons during propagation in the laser cavity. Within one double-soliton unit, the dispersive wave shed from the first soliton travels faster than the solitons, due to its higher carrier-frequency. Thus it would gradually extend to the second soliton located earlier in time. As the dispersive wave drifts away from the soliton, net cavity loss reduces its amplitude. The part of dispersive wave that is located further away from the soliton experiences heavier attenuations due to the higher round-trip times. As a result, the overall envelope of the dispersive

137

Ch. 7

SUPRAMOLECULAR ASSEMBLIES OF OPTICAL SOLITONS wave would approximate an exponentially-decaying pedestal that extends from the soliton which emits it, as illustrated in Fig.7.3(b). (a) Spectrum power (a.u.)

Dominant Kelly sideband

(b) 2nd soliton Us(t)

Δt

1st soliton

Ud(t) s

d

t2=0

Frequency

(c)

2nd soliton (carrier freq. at s)

t2=0

t1

t

Dispersive wave (freq. at d) emitted by 1st soliton

t

Fig. 7.3: The long-range soliton interactions due to the dispersive wave perturbations. (a) A typical soliton spectrum with a dominant (m = −1 order) Kelly sideband at the frequency ωd . (b) This dominant sideband corresponds, in the time domain, to an extending dispersive wave shed from the soliton, which has a faster group velocity than the soliton and would thus extend to the second soliton. (c) This dispersive wave shed from the first soliton interacts with the second soliton by causing a carrier-frequency shift.

In the moving frame (with the origin t2 ), the second soliton can be expressed as Us (t) = As sech (t/τs ) exp [iωs t − iϕs (z)] = As us (t)

(7.10)

where As is the soliton peak amplitude, us (t) is the normalized profile, τs is the soliton duration, and ϕs is the phase of the second soliton varying with the propagation length. For simplicity, we neglect the variation of the soliton envelope within one round-trip, thus the its duration, bandwidth and the peak amplitude are all considered as constant during propagation.

138

Balance between long-range interactions

7.1

The dispersive wave shed from the first soliton can be expressed as Ud (t) = Ad exp [h (t − Δt)] exp [iωd (t − Δt) − iϕd (z)] = As ud ,

(7.11)

where Ad is the dispersive wave amplitude, h is the decay rate of envelope in the time domain, ωd is its carrier-frequency, ϕd is the phase of the dispersive wave, and ud is the waveform of the dispersive wave normalized to the peak amplitude of the soliton. Note that the phase term ϕd (z) includes the spatial propagation term, thus it would vary as the dispersive wave propagates in the cavity. In practice, Ad and ωd could be estimated using peak intensity and central frequency of the dominant sideband, h could be estimated from the bandwidth of the sideband. We employed the conventional theory of soliton perturbation to calculate the carrier-frequency shift the soliton under the perturbations from the dispersive wave (as sketched in Fig.7.3(c)). The rate of carrier-frequency shift can be expressed as [109, 134]  +∞ dωs 1 =  u∗s tanh(t/τs )ud (t)dt (7.12) dz τ s LD −∞ where LD is the dispersion length of the soliton [134], and {...} stands for the imaginary part. By substituting Eqs.(7.10) and (7.11) into Eq.(7.12), we can obtain 

 +∞ Ad dωs = exp (−hΔt)  exp(iΔϕ(z)) sech (t/τs ) tanh (t/τs ) exp (iΔωt) dt dz As τs LD −∞ (7.13) in which Δω = ωd − ωs > 0 is the difference in carrier-frequency, and Δϕ(z) = ϕs (z) − ϕd (z) is the phase difference between the soliton and dispersive wave that perturbs it. In order to evaluate them with the imaginary operator {...}, we should calculate the sum of following two terms: the real part of exp(iΔϕ(z)) times the  +∞ imaginary part of −∞ ...dt term, and the imaginary part of exp(iΔϕ(z)) times the  +∞ real part of the −∞ ...dt term. Since this integral is zero for an odd integrandfunction, we can obtain that  +∞ −∞

sech (t/τs ) tanh (t/τs ) cos (Δωt) dt = 0

(7.14)

139

Ch. 7

SUPRAMOLECULAR ASSEMBLIES OF OPTICAL SOLITONS Then we denote the remaining non-zero part of the integral as B(Δω), which is B(Δω) ≡

 +∞ −∞

sech (t/τs ) tanh (t/τs ) sin (Δωt) dt

= πΔωτs sech (πΔωτs /2)

(7.15a) (7.15b)

In practice B(Δω) ≈ 1 given the experimental parameters and it remains positive given that ωd > ωs . Then we can rewrite Eq.(7.13) as dωs Ad = B(Δω) exp (−hΔt) cos (Δϕ(z)) , dz As τ s L D

(7.16)

the accumulated carrier-frequency shift of the second soliton after each round-trip can be expressed as: ΔωdRT =

 z+LR Ad B(Δω) exp (−hΔt) cos (Δϕ(z)) dz, As τ s L D z0

(7.17)

where z0 is the starting point of the path integral along the cavity, and LR is the cavity-length, assuming the perturbation is homogeneous within one cavity roundtrip3 . The phase-matching condition for the Kelly sidebands, as described in Section 2.2.3, requires that the accumulated phase difference between the soliton and the dispersive waves over one cavity round-trip should be an integer multiple of 2π. For the m = −1 order sideband, this accumulated phase difference should be simply 2π. If we define the initial phase difference at the arbitrarily-defined starting point z0 to be Δϕ0 , we should have, for this particular sideband, that  z0 ! 1

"

1 Δϕ(z) = γKerr (z)|As (z)| − β2 (z)Δω 2 dz + Δϕ0 2 2 z Δϕ(z + LR ) − Δϕ(z) = 2π 2

(7.18a) (7.18b)

in which β2 and γKerr describe the dispersion and nonlinearity map of the laser cavity, respectively. By substituting Eq.(7.18a) into Eq.(7.17), we can obtain ΔωdRT = 3

140

Ad B(Δω) exp (−hΔt) Ψ (Δϕ0 ) As τ s L D

(7.19)

It is suspected that, there exists some nonlinear frequency coupling between the dispersive wave and the perturbed soliton based on XPM or FWM process, which leads to the carrier-frequency shift. The PCF, which has the dominant Kerr-nonlinearity (30-times higher than SMF-28) was actually main section in which the perturbation occurred.

Balance between long-range interactions

7.1

in which Ψ (Δϕ0 ) is defined as Ψ (Δϕ0 ) ≡

 z+LR z0

cos

 z0 ! 1 z

"

1 γKerr (z)|As (z)| − β2 (z)Δω 2 dz + Δϕ0 dz (7.20) 2 2 2

Given fixed dispersion and nonlinearity distributions in the cavity, the integral Ψ would only depend on the initial phase difference Δϕ0 . Since all the other terms in Eq.(7.19) are already known, Δϕ0 would determine the final sign and magnitude of the ΔωdRT . In order to form stable double-soliton state, a fixed Δϕ0 is required to have a fixed ΔωdRT , which is supposed to be a blue-shift, in order to exactly balance the red-shift Δωs induced by the optoacoustic effects. In this case, the dispersive wave exerts an effective force of repulsion that tends to push the second soliton away from the first one. The fixed initial-phase-difference Δϕ0 has been verified by the spectral fringes that was observed in the experiments, as will be presented in next Sections. The spectral fringe appeared only in the vicinity of the dominant sideband and quickly faded away outside that region, while the rest of the soliton spectrum is as smooth as the “normal” HHML state. This indicates a phase-locking between the dispersive wave and the soliton instead of directly between the two solitons. In addition, Δϕ0 is the same in all the time-slots, leading to identical internal spacings within all the time-slots. Curiously enough, it is well known that no fixed phase-relation exists between them, mainly due to the nonlinear phase-shift that is experienced only by the soliton itself. Therefore, it was an unexpected discovery that, although the dispersive wave cannot maintain a fixed phase relation from the emitting soliton, it can do with the other soliton in the same time-slot. Moreover, such bound-states were self-assembled and exhibited excellent robustness over a long-time, behaving as a nonlinear attractor similar to the conventional bound soliton-pairs4 .

7.1.4 Balance between two long-range forces The build-up of the long-range binding of the two solitons in one time-slot is based on the exact balance between the opposite frequency shifts upon the second soliton, i.e. Δωa + ΔωdRT ≡ 0 (7.21)

4

In our theoretical derivation, we simply assumed the fixed phase-relation, while the exact mechanism and origin for such attractor-like behaviour is left as an open question.

141

Ch. 7

SUPRAMOLECULAR ASSEMBLIES OF OPTICAL SOLITONS so that these two solitons can have exactly the same carrier-frequency and thus propagate at the same group velocity, resulting in a fixed internal spacing. Using Eqs.(7.9) and (7.19), we can rewrite Eq.(7.21) as −

2Ca ω0 Pav LPCF δa Ad sin(Ωa Δt) ≡ B(Δω) exp (−hΔt) Ψ (Δϕ0 ) (7.22) 2 2 c 4δa + ΓB As τs LD

in which the LHS and the RHS represent the force of attraction and repulsion respectively. When the soliton laser parameters are fixed, it can be seen that the force of attraction would increase when the internal spacing Δt increases up to a Ta /4, and than decrease back to zero when Δt further increase to Ta /2. The force of repulsion presented by the RHS, on the contrary, would decrease with larger Δt given a fixed Δϕ0 . In the experiments, the internal spacing observed for double-soliton state is usually smaller than Ta /4, which is ∼ 130 ps. Within this range, it is easy to realized that these two distance-dependent forces actually form a stable trapping potential, as described in Fig.7.1(d), which can partially explain the robustness of such long-range bound-states. An interesting consequence of such balance is that, when some of the laser parameters are changed, the balance of forces at the original internal spacing would break, and the solitons would shift to a new internal spacing to re-establish the balance. Therefore, this theory predicts that this internal spacing is continuously tunable, provided that the related laser parameters are carefully tailored. The principle of the internal-spacing tuning for double-soliton state is illustrated in Fig.7.4, where the position of the first soliton in the double-soliton time-slot is set as the reference, and the force of attraction/repulsion are plotted in difference colours. The cross point of the force curves represents the balanced position of the second soliton. The force of repulsion (shown in blue) can be actively tuned by changing the dispersive wave amplitude Ad . A weaker dispersive wave would result in a narrower internal spacing so as to re-establish the balance, as illustrated in Fig.7.4(a)5 . Similarly, the force of attraction can be actively tuned by changing the amplitude of the acoustic wave. This can be achieve with detuning of the repetition rate of the multi-soliton unit simply by actively changing the cavity length (i.e.to change δa ). The decrease in the force of attraction can also lead to a decrease in the internal spacing, as illustrated in Fig.7.4(b). The experimental details of this tuning process 5

142

Note that Fig.7.4(a) is drawn with some simplification. As the internal spacing of all the doublesoliton units decreased, the curve for the force of attraction should also change slightly, as predicted in Eq.(7.9), leading to an even smaller spacing.

Balance between long-range interactions

7.1

are presented in Section 7.3.1.

(a)

(b) Positions of 2nd soliton

t2

st

1 soliton

t1

t

Force of repulsion Force of attraction Positions of 2nd soliton

t2

1st soliton

t1

t

Fig. 7.4: Schematic illustrations of the internal spacing tuning in a double-soliton unit. (a) Tailoring of the dispersive wave can lead to variation of the force of repulsion, and eventually result in shifts of the balanced position for the second soliton relative to the first one. (b) Similarly, tailoring of the mechanical wave amplitude can lead to variations of the force of attraction and therefore changes of the internal spacing.

It is also worth noticing that, the dispersive wave emitted from the second soliton can hardly perturb the first soliton in turn due the weaker intensity of the corresponding Kelly sideband. If, within one specific time-slot, the second soliton becomes absent all of a sudden, the relative position of the first soliton in this time-slot would basically not be influenced. Therefore, additional solitons in each time-slot are practically attached to the previously existing ones without affecting their positions. For this reason, we denoted the soliton that maintained its relative position regardless of the presence of the others to be the first soliton. Each time-slot can have different numbers of attached solitons, corresponding to different long-range bound-states, while the acoustic cycle is only determined by the long-term driving force provided by the entire soliton train.

7.1.5 The analogy to chemical supramolecular assemblies In many ways, the large assembly of optical solitons bound together by the weak, long-range interactions in the HHML laser cavity is analogue to the supramolecule [114, 216–221]—an assembly of pre-existing particles or other elements held together by weak interactions such as Van der Wass forces—which is in contrast to the

143

Ch. 7

SUPRAMOLECULAR ASSEMBLIES OF OPTICAL SOLITONS molecules that formed by atoms bound by strong covalent or ionic bonds. The supramolecular structure exhibits many remarkable features compared to molecules [114], such as flexibility and reversibility, which allows them to vary reversibly in response to environmental perturbations [216–219]. Another remarkable feature is the dynamic stability, which means that the basic elements can be robustly bound together to form a thermodynamically-stable state with a Gibbs free energy lower than that of dissociated elements [217–220]. Therefore, supramolecular assemblies can persist long-term even when their internal structure is re-arranged, or when individual elements are exchanged, added or removed [221]. Moreover, the large number of possible configurational units means that large amounts of information can be stored, retrieved, and processed at the molecular level [218, 221]. The conventional soliton pairs and soliton crystals have long been regarded to be in analogy to chemical molecules form by strong covalent bonds, featuring close proximity and fixed phase-relation. In our work, the weak long-range interactions between the solitons induced by optoacoustic effects and nonlinear perturbations are carefully tailored to achieve stable, highly-ordered, hierarchical structures. The features we observed in such large assemble optical solitons in the optoacoustic mode-locked fibre laser—the widely-tunable internal-spacing, the uncorrelated phase between the solitons, the independent number of bound solitons in each time-slot, the long-term robustness and the self-organized fashion—makes it, so far as we know, the first observation of supramolecular assemblies of optical solitons which are particlelike light that are constantly under propagation instead of real particles. As a result, such soliton supramolecule can have potential applications in storage, replication and manipulation of optical information, by analogy with molecular computing [219, 222] and molecular machines [220, 223] in supramolecular chemistry and biology. Such soliton supramolecules discovered in the soliton laser, which extend throughout the cavity, also opens up the possibility of waveform-programmable lasers [189] or adaptive lasers with responsiveness to the change of environmental parameters.

7.2 Stable soliton supramolecule: Experiments 7.2.1 Experimental set-up The platform used in the experiments was a ring-cavity soliton fibre laser that is similar to the one used in Section 4.2, as shown in Fig.7.5. A 2-m-long solid-core

144

Stable soliton supramolecule: Experiments

7.2

PCF was spliced into the laser cavity with an insertion loss of ∼ 1.7 dB. The PCF has an acoustic core-resonance at 1.887 GHz. The laser gain medium was a 1.2m-long EDF with 110 dB/m peak absorption at 1530 nm. Two pump laser diodes (LD-1 and LD-2) provided a maximum combined pump power of ∼ 1.6 W at 976 nm. A tunable delay (TD) line was used to adjust the cavity length, and a tunable attenuator (TA) to adjust the cavity loss. WDM

WDM EDF

90:10

PD

LD-1

50:50

LD-2 ISO

SMF TD OSC

OSA

PC-1 TA

Polarizer PC-3

PC-2 PCF

Fig. 7.5: Experimental set-up for generating supramolecular assemblies of optical solitons.A unidirectional fibre laser cavity is operated in the soliton regime. The laser output was recorded using a fast oscilloscope (OSC) and a high-resolution (0.01 nm) optical spectrum analyser (OSA). EDF: erbium-doped fibre; WDM: wavelength division multiplexer; LD: laser diode; SMF: single-mode fibre; PC: polarization controller, PCF: photonic crystal fibre; TA: tunable attenuator; TD: tunable delay line; ISO: optical isolator; PD: photodetector.

The total cavity length was ∼ 17 m, corresponding to a free spectral range (FSR) of ∼ 11.7 MHz. This HHML laser was then operated under a harmonic order of 160. The cavity average GVD was calculated to be ∼ −0.046 ps2 /m, ensuring the soliton-regime operation of this laser. The cavity loss could be tuned from ∼ 6 dB to > 30 dB by adjusting the intra-cavity TA with a manual tuning-resolution of ∼ 0.02 dB. The laser cavity length could be varied using an intra-cavity TD with a tuning range of 0.15 m and a resolution of 3 μm (or equivalently 0.001 ps). In the diagnostic set-up, the laser output was detected using a 30-GHz photodetector, and the electrical signal from the detector was recorded using a 33-GHz OSC. The response time of this real-time detection is ∼ 20 ps which sets the pulse width shown in all the plots of the recorded data using the OSC. The timing-jitter of

145

Ch. 7

SUPRAMOLECULAR ASSEMBLIES OF OPTICAL SOLITONS the OSC in sampling is ∼ 2 ps which gives the measurement error in reading the fine structures of the soliton supramolecules. For measuring the actual durations of the laser solitons, a SHG-based autocorrelator was used with a time resolution of 20 fs. Before the autocorrelator, a 0.9-m-long fibre with normal dispersion of +0.045 ps2 /m at 1550 nm was used to compensate for the anomalous dispersion of the 2-m-long SMF-28 fibre patch cord at the laser output port. The optical spectrum of the laser output was measured using an optical spectrum analyser (OSA) with a resolution of 0.01 nm. In the experiments, both the self-starting of the laser and the fine structure of the steady-state soliton supramolecule depended critically on manual adjustments of the PCs. However, after we set a proper working point of the NPR, no further adjustment of the PCs was required for long-term preservation of the soliton supramolecule.

7.2.2 Stable all-double-soliton supramolecule The simplest case in the soliton supramolecule is the all-double-soliton state (ADS state). With ultra-slow PC-travels during the starting process, a homogeneous distribution of the soliton bound-states was achieved throughout all the 160 time-slots within one cavity round-trip, and such supramolecular pattern can persist for hourslong. The typical soliton train recorded by the oscilloscope is shown in Fig.7.6(a), with only the starting and the ending parts of one cavity round-trip record are shown, the intermediate time-slots being filled with the same patterns. A continuous record of ∼ 230 round-trips (20 μs) of the output pulse train were plotted in parallel in Fig.7.6(b). We can readily reveal that the double-soliton units in the optomechanical lattices are perfectly preserved after each round-trip. In order to check the relative positions of each double-soliton unit within each time-slot, we separately plot the time-domain trace within each time-slot (∼ 0.532 ns) along the ordinate, and all the 160 time-slots in parallel along the abscissa, as shown in Fig.7.6(c). From the almost-perfect horizontal “row” of pulse signal (in yellow), we can reveal that there were only trivial drifts of relative positions between different time-slots. Therefore the entire soliton supramolecular under with ADS had a highly-ordered and stable structure.

146

Stable soliton supramolecule: Experiments

Voltage (V)

(a)

7.2

0.3 0.2 0.1 0.0 0

1

2

0

1

2

0

20

40

3

4 81 Time (ns)

82

83

84

85

83

84

85

140

160

(b)

Round-trip number

200

150

100

50

0 3 4 81 82 Relative intra-cavity time (ns)

Relative time (ns)

(c) 0.4

0.2

0.0 60

80 100 Time-slot number

120

Fig. 7.6: Illustrations of the all-double-soliton supramolecule in three difference ways.(a) Time domain trace of the supramolecule recorded using OSC. Due to figure-size limit, only 8 ns out of the entire round-trip time of ∼ 85 ns is plotted. (b) The time-domain trace over 230 consecutive cavity round-trips plotted in parallel (also only partially plotted as in (a)). (c) The time-domain traces of the consecutive 160 time-slots within one round-trip plotted in parallel, so as to see the relative position of each double-soliton unit within the time-slot.

The same pulse train as in Fig.7.6 was recorded under the persistence mode with an amplitude-trigger over a single time-slot span, as shown in Fig.7.7(a). Using this plot, we can measure the characteristic internal spacing to be 90 ps. Even though the OSC only has a response time of ∼ 20 ps, the large spacing between the two solitons still enable satisfactory estimations of the spacings with an error bar of only a few ps (intrinsic timing-jitter). The FFT spectrum of the time-domain trace, as shown in Fig.7.7(b), has sharp peaks at multiplies of 1.882 GHz, indicating the global lattice of the supramolecule structure. The modulation on the spectral envelope (as indicated by the dashed line) has a period of 11.1 GHz which matches

147

Ch. 7

SUPRAMOLECULAR ASSEMBLIES OF OPTICAL SOLITONS the 90-ps internal spacing.

Intensity (a.u.)

(a)

1

90 ps

Index modulation 0 0.0

FFT power (dB)

(b)

0.1

0.2 0.3 Relative time (ns)

0.4

0.5

0

-30

-60 0

10

20 Frequency (GHz)

30

Fig. 7.7: (a) The time-domain trace of the ADS supramolecule recorded under persistence mode (with amplitude trigger), which exhibit a stable 90-ps internal spacing. The dashed curve presents the index modulation na (t) within the timeslot. (b) The FFT power spectrum of the time-domain trace. The sinusoidal envelope agrees well with the 90-ps internal spacing.

The AC-trace of the soliton train measured at the laser output is shown in Fig.7.8, which gives a sech2 -profile and a FWHM-duration of only 670 fs. Therefore, the internal spacing of the double-soliton unit is about two orders of magnitude longer than the individual soliton which is the essential feature of such unique boundstate of solitons. The laser spectrum measured by the OSA is shown in Fig.7.7(b), which exhibits multiple Kelly sidebands on both sides of the spectral peak, indicating the existences of dispersive waves. However, unlike in conventional soliton lasers, a spectral fringe in the vicinity of the dominant Kelly sideband (m = −1 order) can be readily revealed from the zoom-in plot shown in Fig.7.7(c). This spectral fringe is a clear indication of the perturbations of the Kelly sideband upon the soliton with a locked phase-relation. The period of this fringe was measured to be ∼ 0.09 nm (equivalent to ∼ 11.1 GHz around 1.56 μm), which matched the internal spacing between the two solitons. This suggests that, the perturbed soliton actually received the perturbation from the other soliton which was 90-ps away from it. On the other side of the spectrum, the relatively weak sideband of m = +1 order, however, did not exhibit such fringe, indicating that the dispersive wave perturbations were unidirectional. The dominant part of the soliton spectrum, i.e. the central part,

148

Stable soliton supramolecule: Experiments

7.2

is as smooth as a conventional soliton laser, indicating that there exists no fixedphase relation directly between the two solitons within one time-slot.(Other wise the 0.09-nm period would be present throughout the entire spectrum and reach the maximum contrast at the centre, similar in the case of conventional soliton-pairs). These experimental observations are all in accordance with the theoretical model described in Section 7.1.3.

0.5

0.0 -2

Spectral Power (dB)

(c)

-30

-1 0 1 Time (ps)

-40

(d) m=−1

0.09 nm 1557 1558 1559 Wavelength (nm)

m=−1

m=+1

-60 -80

2

-40

-50

(b) Spectral Power (dB)

1.54×670 fs

Spectral Power (dB)

Signal (a.u.)

(a) 1.0

1550 1560 1570 1580 Wavelength (nm) -30

m=+1

-40

-50 1567 1568 1569 Wavelength (nm)

Fig. 7.8: (a) The AC-trace of the output ADS supramolecule train. (b) The optical spectrum measured at the laser output using an OSA with a resolution of 0.01 nm. (c) Zoom-in of the spectrum in (b) at the dominant Kelly sideband (m = −1 order) which exhibits a spectral fringe that matches the internal spacing. (d) Zoom-in of the spectrum in (b) at the other Kelly sideband (m = +1 order) which exhibits no obvious spectral fringe.

7.2.3 Stable all-triple-soliton supramolecule Similar soliton supramolecules under stable all-triple-soliton (ATS) state have also been achieved simply with a different working point and a higher pump power. The recorded soliton train within one cavity round-trip shown in Fig.7.9 with the 160 time-slots plotted in parallel. Similar to the case of ADS state (in Fig.7.6(b)), the ATS state also constituted a highly-ordered structure throughout the entire cavity. The internal spacing between the first and third soliton was measured to be 147 ps

149

Ch. 7

SUPRAMOLECULAR ASSEMBLIES OF OPTICAL SOLITONS as shown in Fig.7.9(b), while the internal spacing between consecutive solitons are 76 ps (first-second) and 70 ps (second-third), respectively. The FFT power spectrum of the recorded soliton train is shown in Fig.7.9(c), where the modulated envelope also reflected the triple-soliton units that repeat at 1.882 GHz.

Relative time (ns)

(a) 0.4

0.2

0.0 0

20

40

60 80 100 Time-slot number

(b) Voltage (V)

160

Index modulation

0.1 0.0 0.0

FFT power (dB)

140

147 ps

0.2

(c)

120

0.1

0.2 0.3 Relative time (ns)

0.4

0.5

0

-30

-60 0

10

20 Frequency (GHz)

30

Fig. 7.9: (a) The time-domain trace within a single round-trip, the 160-time-slots being plotted in parallel. (b) The time-domain trace of the ATS supramolecule recorded under persistence mode (with amplitude trigger), which exhibits a stable 147-ps internal spacing between the first and third soliton. The dashed curve represents the index modulation na (t) within the time-slot. (c) The FFT power spectrum of the ATS time-domain trace.

The AC trace under the ATS state was measured to be 790 fs (as shown in Fig.7.10(a)), which is slightly higher than the ADS case due to the lower individual soliton energy. The optical spectrum measured under such ATS state is shown in Fig.7.10(b), which also exhibits an asymmetric Kelly sidebands distribution. The m = −1 order sideband, as shown in Fig.7.10(c), also exhibits a spectral fringe that corresponds to an internal spacing of ∼ 75 ps. The fringe contrast is slightly lower than the ADS case (Fig.7.8(c)) due to the fact that there are two fringes, which

150

Stable soliton supramolecule: Experiments

7.2

corresponds to the 76-ps and 70-ps internal spacing respectively6 , smearing out each other and leading to the lower contrast. The opposite sideband (m = +1 order) still exhibits no fringe at all, similar to the ADS state.

0.5

0.0 -2

Spectral Power (dB)

(c)

-1 0 1 Time (ps)

-40

0.11 nm 1556 1557 1558 Wavelength (nm)

m=+1

-80 1550 1560 1570 1580 Wavelength (nm)

(d)

-40

m=−1

-60

2

-30 m=−1

-50

(b)

Spectral Power (dB)

1.54×690 fs

Spectral Power (dB)

Signal (a.u.)

(a) 1.0

-30

m=+1

-40 -50 1566

1567 1568 Wavelength (nm)

Fig. 7.10: (a) The AC-trace of the output ATS supramolecule. (b) The optical spectrum measured at the laser output using an OSA with a resolution of 0.01 nm. (c) Zoom-in of the spectrum in (b) at the dominant Kelly sideband (m = −1 order) which exhibits a spectral fringe that matches the internal spacing. (d) Zoom-in of the spectrum in (b) at the other Kelly sideband (m = +1 order) which exhibits no obvious spectral fringe.

7.2.4 Configurational diversity and dynamic stability The numbers of solitons that can be bound within different time-slots were found to be independent from each other. We realized that, if the polarization adjustment through the PC-travel was relatively fast during the self-starting process, the boundstates of solitons would become highly inhomogeneous throughout all the time-slots within one cavity round-trip. This particular state, apart from the ADS and ATS 6

Note that in each triple-soliton unit, the internal spacings between consecutive solitons are not necessarily the same, and could even become highly unequal, as we revealed from some carefully-designed experiments. Such unequal internal spacings could also be interpreted using the balanced force model, while the detailed derivations and experiment results have been omitted in this thesis.

151

Ch. 7

SUPRAMOLECULAR ASSEMBLIES OF OPTICAL SOLITONS state, is named as the hybrid-state. One example at relatively low pump power is a soliton supramolecule that accommodate units of null-soliton (empty time-slots), single-soliton, double-soliton, and triple soliton in different time-slots of one cavity round trip time simultaneously, as shown in Fig.7.11. 1

0 1

0

2

4

6

8

10

0 12

1

14

16

18

20

Signal (a.u.)

0 22

1

24

26

28

30

0 1

32

34

36

38

40

42

0 44

1

46

48

50

52

0 54

1

56

58

60

62

0 1

64

66

68

70

72

74

0 76

78

80

82

84

Relative intra-cavity time (ns)

Fig. 7.11: A soliton supramolecule under the hybrid-state with null-soliton up to triple-soliton states. All the 160 time-slots within one round-trip (85.12 ns) are plotted, being separated in to 8 rows with 20 time-slots along each rows.

Moreover, such hybrid-state supramolecule can be self-stabilized over longterm, just as the ADS and ATS state as demonstrated in previous sections. Due to

152

Stable soliton supramolecule: Experiments

7.2

the limited space, we only demonstrate part of the supramolecule (highlighted in yellow in Fig.7.11) that was recorded at every 5 min, totally over 20 min. The structure was precisely maintained, with invariant soliton numbers and internal spacing(s) in each time-slot, as shown in Fig.7.12.

1 20 min 0

Signal (a.u.)

1 10 min 0 1 5 min 0 1 0 min

0 0

2

4

6 8 Relative intra-cavity time (ns)

10

12

Fig. 7.12: The long-term preservation of a soliton supramolecule consisting of null-, single-, double-, and triple-soliton states. The recording was performed every 5 min over 20 min. Only 25 consecutive time-slots (marked by the dashed line) are plotted, which corresponds to the highlighted part in Fig.7.11.

With a higher pump power, a soliton supramolecule could consist of double- and triple-soliton units, together with a few quadruple-soliton units. The long-term preservation of a typical supramolecule is partially demonstrated in Fig.7.13. 1 20 min 0

Signal (a.u.)

1 10 min 0 1 5 min 0 1 0 min 0 0

2

4

6 8 Relative intra-cavity time (ns)

10

12

Fig. 7.13: The long-term preservation of a soliton supramolecule consisting of double-, triple-, and quadruple-soliton states. The recording was performed every 5 min over 20 min. Only 25 consecutive time-slots (marked by the dashed line) are plotted.

153

Ch. 7

SUPRAMOLECULAR ASSEMBLIES OF OPTICAL SOLITONS By further increasing the pump power, all the time-slots would be filled with only triple- or quadruple-soliton units, as shown in Fig.7.14. Due to the limited pump power, the all-quadruple-soliton (AQS) state was not successfully achieved. The maximum ratio of quadruple-soliton units in the soliton supramolecule was ∼ 70 %. 1 20 min 0

Signal (a.u.)

1 10 min 0 1 5 min 0 1 0 min 0 0

2

4

6 8 Relative intra-cavity time (ns)

10

12

Fig. 7.14: The long-term preservation of a soliton supramolecule consisting of triple-, and quadruple-soliton states. The recording was performed every 5 min over 20 min. Only 25 consecutive time-slots (marked by the dashed line) are plotted.

The ATS and ADS state were found to be highly reproducible, while the hybridstate could only be partially controlled, in terms of the total number of solitons in the cavity, by varying the pump power and the PCs-working point. Independent control of the soliton number in each time-slot possibly be achieved by employing and further perfecting the technique of addressing-pulses which has been demonstrated in Chapter 6.Moreover, the fundamental configurational element at present is still only single-soliton units. In the future, the conventional soliton-pair or triplets with only few-ps internal spacings could also be incorporated as additional elements, which would further enhance the diversity of the soliton supramolecule structure. If each time-slot can accommodate any of the M -state of bound-solitons (at the moment M = 5, from null- to quadruple-soliton), the such supramolecule constitute a modulo-M system. The largest number that this supramolecule can store is then M N , in which N is the total number of time-slots.

154

Structural flexibility: Tailoring of long-range interactions

7.3

7.3 Structural flexibility: Tailoring of long-range interactions In contrast to the strong covalent-like bonds in the conventional soliton molecules, the weak nature of the long-range interactions (via acoustic waves and dispersive waves) allows the soliton supramolecules to re-arrange their internal structures in response to the variations of the external parameters. During the experiments, we have found that the internal-spacing of the double-soliton (under ADS state) can be continuously tuned to-and-fro over a range of tens of ps, while the overall supramolecular structure remained intact. The strengths of the two long-range interactions involved in the supramolecular structures can be actively tailored by tuning the relevant laser parameters. When one of interactions was changed, the other would respond passively and re-establish their balance at a different internal spacing. These two tuning processes, corresponding to the continuously tailoring of the two types of long-range interactions respectively, are presented using the experimental results in the following sections.

7.3.1 Internal-spacing tuning: Tailoring of the dispersive waves The internal spacing of the ADS supramolecule was firstly tuned by tailoring the strength of the dispersive waves corresponding to the dominant Kelly sideband. This was achieved by continuous tuning of the cavity loss with the tunable attenuator (TA) inserted before the EDFA in the soliton laser (see Fig.7.5). The EDF was highly saturated under the pump power of ∼ 1.5 W. When the cavity-loss was gradually changed from ∼ 6 dB to ∼ 11 dB, we observed that the intensity of the dominant Kelly sideband dropped by around 3 times, as shown in Fig.7.15(a). During this process, the average optical power output (after the EDF section) was observed to dropped by only < 5 %, and could be simply compensated by slightly increasing the pump power. Most importantly, the spectral bandwidth and individual soliton duration only experienced minor changes (within 5 %). Therefore, we ensured that the optoacoustic interaction that occurred in the solid-core PCF (before the TA) was basically not influenced. As explained in Fig.7.4(a), that decrease in dispersive wave strength would result in a weaker force of repulsion at the original position, leading to a narrower internal-spacing between the two solitons. In practice, while the basic structure of the ADS supramolecule was stably preserved in the laser

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SUPRAMOLECULAR ASSEMBLIES OF OPTICAL SOLITONS cavity, the internal spacing in the double-soliton unit could be tuned at will within a broad range (from 40 ps to 116 ps), as shown in Fig.7.15(b), with a theoretical fitting curve based on Eq.(7.22). Some of the intermediate ADS states (marked by the number (i)–(iv)) are illustrated in Fig.7.15(b) under persistence-mode recording. Note that the spectral fringe over the dominant sideband as shown Fig.7.15(a) has a varying period during the tuning process, corresponding to the changing internalspacing. Besides, the relative position of the spectral fringe, which can be revealed from the slight “dip” on top of the sideband, were also almost invariant, indicating a practically unchanged phase-relation between the dispersive wave and the second soliton. This unchanged phase-relation leaves the term Ψ (Δϕ0 ) in Eq.(7.19), which represents the force of repulsion, to be practically invariant. Therefore, the change in strength of this force was dominantly due to the variation of the amplitude of the dispersive wave 7 . (b) 120

(a)

Experiment Theory

-30

(i)

(i)

Internal spacing (ps)

Spectral power (dB)

-50

(ii)

0

80 (iii)

1554

1555

1556 1557 1558 Wavelength (nm)

1575

1559

67 ps

1

(iii) 0

60 (iv)

1560

1

(ii)

-60 1545

91 ps

Intensity (a.u.)

-30

1

0

100

-40

116 ps

40 ps

(iv)

40 0.5

1.0

1

0 1.5

Dispersive wave intensity (a.u.)

0.0 0.1 0.2 0.3 0.4 0.5 Relative time (ns)

Fig. 7.15: The continuous tuning of the internal-spacing of double-soliton unit by tailoring of dispersive waves. (a) The dominant Kelly sideband at m = −1 order under the ADS state that is continuously varied in peak intensity, while spectral bandwidths of the solitons is remained invariant (see the inset). (b) Tuning the internal spacing from 40 to 116 ps by varying the intensity of the dispersive wave as shown in (a). The corresponding sidebands for the internal spacings marked by the arrows are shown in (a) with the indicated colours. Four exemplary OSC traces under persistence-mode are shown by figures (i)–(iv) on the right side.

The internal-spacing has a tuning range up to ∼ 200 %, which was much larger than 7

156

A minor change that also appeared during the tuning process is the slight broadening in the bandwidth of the Kelly sideband when the peak intensity was reduced. This factor has also been incorporated into the theoretical fitting.

Structural flexibility: Tailoring of long-range interactions

7.3

the trivial changes of individual soliton parameters. The internal-spacing between the solitons was able to be tuned back and forth without destroying the steady-state of the HHML soliton laser, highlighting the structural flexibility and reversibility of the supramolecular assembly of optical solitons in the cavity. The ATS supramolecule achieved under a higher pump power also had a similar flexibility in terms of the tunability in its internal spacing using the same technique as above. The tuning results are shown in Fig.7.16. We observed that in the experiments that, as we gradually varied the intensity of the Kelly sideband of m = −1 order, the internal spacing of the triple-soliton unit (Δt13 in Fig.7.16(b)) could be tuned within a range of 102 ps to 191 ps, while the spacing between the first and the second soliton (Δt12 ) could be tuned from 51 ps to 97 ps and the spacing between the second and third soliton (Δt23 ) from 50 ps to 94 ps. (a)

(b)

200

Δt23

-30

(i)

Δt12

187ps

1

Δt13

Internal spacing (ps)

-40

-50

-40

0 (i)

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1555

1556 1557 Wavelength (nm)

1575

1558

1

0 (iii)

(iii)

133 ps

1

100 0 Δt12

1560

157 ps

(ii)

(iv) -80 1545

(ii)

50

(iv)

102 ps

Intensity (a.u.)

Spectral power (dB)

t

1

Δt23 Δt13

0

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.8 1.0 1.2 Kelly sideband intensity (a.u.) Relative time (ns)

Fig. 7.16: The continuous tuning of internal-spacing of triple-soliton by tailoring of dispersive waves. (a) Tailoring of the intensity of the m = −1 order dispersive wave without changing spectral widths of the solitons (see the inset). (b) The internal spacings increase as the peak intensity of dominant sideband grows. Δt12 , Δt23 and Δt13 are defined in the inset. Four examples are plotted in figure (i)–(iv) on the right side

For measuring the internal-spacing, long-time averaging was used to obtain its mean value, and the root mean square of the spacing jitter was adopt as the error bars in Figs.7.15(b) and 7.16(b). In the experiments, we observed slightly higher jitter spacing (∼ 10 ps) near the upper and lower edges of the tuning range (as shown in Fig.7.16(b) for example) which could be attributed to the fact that we always optimized our system near the mid-point of the tuning range. Nevertheless, if all

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SUPRAMOLECULAR ASSEMBLIES OF OPTICAL SOLITONS the laser parameters were maintained, no obvious degradation of the spacing jitter could be observed over time.

7.3.2 Internal-spacing tuning: Tailoring of the acoustic waves We were also able to cycle the internal spacing back and forth by actively varying the force of attraction between the solitons. This was achieved by adjusting the cavity length and thus the lattice frequency of the ADS supramolecule relative to the resonant frequency of the core (denoted as δa previously) as shown in Fig.7.17(a). Then the amplitude of the acoustic wave, thus the force of attraction as expressed in Eq.(7.9) would also be changed. Tuning of the lattice frequency from 1.8725 GHz to 1.882 GHz led to a continuous decrease of the internal spacing from 87 ps to 63 ps (Fig.7.17(b)). Further increasing the lattice frequency caused a roll-over in internal spacing, in good agreement with the theoretical predictions by the forcebalance condition Eq.(7.22) with only slight deviations at the roll-over regions. Note that during this process, the time-slot length was varied by only ∼ 0.5 %, while the internal spacing was changed by > 30 %. 0.4

0.4

(b)

1.887 GHz

(i)

90

87 ps

0.2

0.3

Internal spacing (ps)

Optomechanical gain (m-1W-1)

(a)

0.0 1.84 1.86 1.88 1.90 1.92 1.94

0.2

Gain profile 0.1 1.870

1.875 1.880 1.885 Frequency (GHz)

1

0 80

(ii)

(i)

78 ps

(iii) 70

(ii)

0 (iii)

63 ps

1

0

60 1.872

1

1.876 1.880 1.884 0.0 0.1 0.2 0.3 0.4 0.5 Frequency (GHz) Relative time (ns)

Fig. 7.17: The continuous tuning of the internal spacing between double-soliton by tailoring the acoustic waves.(a) Tuning of the acoustic wave frequency of the ADS supramolecule by varying the cavity length (i.e. the repetition rate). The optoacoustic gain spectrum of the solid-core PCF is plotted in the inset. (b) The internal spacing first decreases as the lattice frequency of the supramolecule is tuned toward the resonance frequency, and then rolls over in the vicinity of ∼ 1.882 GHz. The internal spacing could be tuned from 63 to 87 ps, and three oscilloscope traces are shown in the figures (i)–(iii) on the right.

The optical spectrum of the ADS supramolecule was measured during this cavity-length tuning process, as shown in Fig.7.18. The dominant Kelly sideband

158

Structural flexibility: Tailoring of long-range interactions

7.3

exhibits an almost invariant bandwidth and peak level as well (Fig.7.18(a)), indicating an unchanged strength of dispersive waves. However, the spectral fringe on this Kelly sideband did not maintain the same relative position on the sideband during the tuning process. When |δa | gradually increased from the minimum value, the fringe position (indicated by the relative position between the spectral-dip and the sideband centre) shifted towards the left, which indicated a varying initial phasedifference (Δϕ0 in Eq.(7.19)). When |δa | exceeded 2π · 10 MHz, the position shift of the fringe became trivial, suggesting an invariant Δϕ0 . Therefore, the value of Ψ (ΔΦ0 ) which determines the strength of repulsive force (Eq.(7.19)) had some nontrivial change over the region close to the acoustic resonance, which was likely to be responsible to the deviation of experimental data at the roll-over region as shown in Fig.7.17(b) compared to the theoretical fitting curve. Therefore, during this tuning range, the phase-relation between the dispersive wave and the soliton was also varied, in addition to the internal spacing, as a response to the active tuning of cavity length. This phenomenon suggests that the current theory of the binding mechanism is still not complete. The thorough understanding of this attractor-like long-range bound-state demands more in-depth studies. (a)

(b)

Spectral power (dBm)

-30

-28

-35 -29

-40 -45

-40

|Ω|/2π (MHz) 3.31 5.62 7.92 10.21 12.5 14.36

-30

-60

-50

-80 1540

1560

1580

1555 1556 1557 Wavelength (nm)

-31 1555.9

1556.0 1556.1 Wavelength (nm)

1556.2

Fig. 7.18: The spectrum of the ADS state soliton supramolecule during the repetition-rate tuning. (a) inset: the exemplary 6 spectra at different detuned repetition rate, with enlarged details shown at the dominant Kelly sideband. The large overlapping between these spectra indicates almost invariant sideband profiles. (b) Zoom-in of the fringe on the dominant sideband during the tuning process.

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SUPRAMOLECULAR ASSEMBLIES OF OPTICAL SOLITONS

7.3.3 Directions of repulsive forces In all the experimental demonstrated above, the force of repulsion was exerted by the soliton that was located later in the time domain upon the soliton that was located earlier, through the dispersive wave that had a carrier-frequency to be higher than that of the soliton itself. This dispersive wave corresponded to the Kelly sideband at m = −1 order with a much higher intensity than the others. In the following experiments we were able to adjust the relative intensities of the two sidebands at m = ±1 orders by carefully adjusting the gain value in the EDF and the dispersionmap in the cavity. Variations of relative intensities of the two sidebands could change the configuration of the force of repulsion due to dispersive wave perturbations, which can be revealed by plotting the soliton supramolecules in the same way as in Fig.7.6(c). Such time-slots-plotting can reveal the the relative positions of the double-soliton units in the optomechanical lattice. in the ADS state, the information about the directions of the forces can hardly be revealed due to the homogeneous distribution of multi-soliton units. Using the experimental results of the hybridstate soliton supramolecules, with for example single-soliton or double-soliton units trapped in different time-slots simultaneously, we can reveal direction-information of the repulsive force between the two solitons originated from the dispersive wave perturbations by comparing the relative positions of the single-soliton and multisoliton units within the time-slots. In another word, we can reveal in which direction (or from which side) that the second soliton was attached to the first one. We start with the usual case in which the sideband at m = −1 order provided the dominant force of repulsion. The optical spectrum of the hybrid-state supramolecule, as shown in Fig.7.19(a), exhibits a sideband at m = −1 order that dominates the others, with a peak power difference ΔP = +5.3 dB compared to the m = +1 order sideband. As we have previously denoted, the soliton located later in time is the first soliton (see Fig.7.19(b)). The force of repulsion within the doublesoliton unit was merely induced by perturbations of the dispersive wave shed from the first soliton, while the m = +1 order dispersive wave shed from the second soliton is much weaker. For a single-soliton unit to be trapped in one time-slot, that soliton would be located in the same position as that of the first soliton in the double-soliton unit. Fig.7.19(c) shows the structure of this hybrid soliton supramolecule including single- and double-soliton. In those single-soliton units (marked by white arrows), the solitons always locate at the upper row which corresponds to the later (the first soliton) position in Fig.7.19(b), being in accordance with our predictions.

160

Structural flexibility: Tailoring of long-range interactions

Spectral Power (dBm)

(a)

(b) 0

ΔP = +5.3 dB

7.3

1st soliton 2nd soliton

-20 Repulsion

-40 -60 1540 1560 1580 Wavelength (nm)

DW (m= −1)

t2

DW (m= +1)

t1

t

Relative time (ns)

(c) 0.4

0.2

0.0 0

20

40

60

80 100 Time-slot number

120

140

Fig. 7.19: The direction of repulsive force case 1. (a) The soliton spectrum with the dominant sideband at m = −1 order (+5.3 dB higher than the m = +1 order sideband). (b) The dominant dispersive waves (in blue) shed by the soliton located later in time (set as the first soliton) perturbs the soliton located earlier in time (the second soliton), leading to an effective force of repulsion. The minor dispersive wave (in yellow), on the contrary, can hardly perturb the first soliton in turn. (c) The hybrid soliton supramolecule with single- and double-soliton units. The positions of those single-soliton units are marked by the white arrows (the upper row of pulses).

In the second case, the two sidebands at m = ±1 orders were tuned to have comparable intensities of (with a peak power difference ΔP = +0.34 dB), as shown in Fig.7.20(a). As a result, two comparable perturbations from the dispersive waves were exerted by the two solitons upon each other (Fig.7.20(b)), which pushed both solitons away from the balanced position for a single-soliton unit. Thus, in this case, the single-soliton unit would then be located somewhere between the double-soliton unit in terms of relative positions in their own time-slots. This can also be readily revealed from the experimental observations as shown in Fig.7.20(c), in which the time-slots with single-soliton units are marked by the white arrows. In the third case, the lower-frequency sideband at m = +1 order dominated the force of repulsion, with ΔP = −2.1 dB as shown in the spectrum in Fig.7.21(a). In contrast to the first case shown in Fig.7.19, here we denote the soliton located earlier

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Ch. 7

SUPRAMOLECULAR ASSEMBLIES OF OPTICAL SOLITONS in time as the first soliton, because this particular soliton would now exert dominant the repulsive force. The dispersive wave shed from the first soliton travelled slower and then perturbed the second soliton located later in time (Fig.7.21(b)). As a consequence, in the hybrid-state supramolecule, the single-soliton unit would always be located in the lower row (see Fig.7.21(c)) which corresponds to the first (earlier) soliton position of the double-soliton unit. These experimental results with varied configurations did not alter the underlying physical picture of the soliton supramolecules: two long-range soliton interactions worked together to enable a large number of optical soliton to aggregate into supramolecular structures. Moreover, the results shown in Figs.7.19–7.21 exhibit the universality of the soliton supramolecules which can be formed within a broad range of system parameters.

Spectral Power (dBm)

(a)

0

(b) Balanced position for single-soliton

ΔP = +0.34 dB

-20 -40

DW Repulsion (m= +1)

-60 1540 1560 1580 Wavelength (nm)

DW (m= −1)

Repulsion

t t-1

t+1

t0

Relative time (ns)

(c) 0.4

0.2

0.0 0

20

40

60

80 100 Time-slot number

120

140

160

Fig. 7.20: The direction of the repulsive force case 2. (a) The soliton spectrum with comparable sidebands at m = −1 and m = +1 orders, with a difference of only +0.34 dB. (b) Both sidebands (dispersive waves) will contribute the force of repulsion between the two solitons, and push both solitons away from the balanced position for the single-soliton slot. (c) The hybrid soliton supramolecule with single- and double-soliton units. The positions of those single-soliton units are marked by the white arrows.

162

Structural flexibility: Tailoring of long-range interactions

Spectral Power (dBm)

(a)

0

(b)

7.3

2nd soliton

ΔP = −2.1 dB 1st soliton

-20

DW (m= +1)

DW (m= −1)

-40 -60 1540 1560 1580 Wavelength (nm)

t1

Repulsion

t

t2

Relative time (ns)

(c) 0.4

0.2

0.0 0

20

40

60

80 100 Time-slot number

120

140

Fig. 7.21: The direction of the repulsive force case 3. (a) The soliton spectrum with the sideband at m = −1 order to be higher than the m = +1 order, with a difference of ΔP = −2.1 dB. (b) The dominant dispersive waves (in yellow) shed by the soliton located earlier in time (set as the first soliton) perturbs the soliton located earlier in time (second-soliton), leading to an effective force of repulsion, being opposite to the case 1 in Fig.7.19. (c) The hybrid soliton supramolecule with single- and double-soliton units. The positions of those single-soliton units are marked by the white arrows (the lower row of pulses).

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CHAPTER

8 SUMMARY AND OUTLOOK

Although the topic of using optoacoustic interactions for HML has been discussed some twenty years ago [39], it was not practically achieved until the advent of the solid-core PCF which can confine both optical and acoustic modes tightly inside its μm-sized core. In this thesis, we have demonstrated for the first time that, by using the enhanced optoacoustic interactions in the solid-core PCF, passively modelocked fibre lasers can stably operate at high harmonic-orders. This optoacoustic mode-locking mechanism has greatly expanded the parameter-space of laser pulses to regions that have been traditionally regarded to be difficult to achieve in passive fibre lasers: GHz-repetition-rate and meanwhile, depending on the cavity configurations, wideband tunability and broad pulse bandwidth. Moreover, such fibre lasers were found to be unique platforms for generating long-range bound-states of optical solitons, leading to experimental realizations beyond the homogeneous HHML state. Based on the theory of the SRLS developed in 2009 [95] and the preliminary experimental work done in 2013 [60], we have established a theoretical model for the optoacoustic mode-locking and meanwhile achieved remarkable optimizations and variations for GHz-rate HHML fibre lasers in the experiments. In the theory part, the conventional SRLS process that was usually studied in the frequency domain has been transformed into the time domain in order to interpret the optoacoustic mode-locking. We have developed the temporal trapping potential theory that interpreted the self-stablization mechanism of the optoacoustic mode-locking scheme. The presence of the GHz-acoustic resonance in the PCF-core has effectively divided the cavity into hundreds of time-slots and largely decoupled the pulse repetition rate

165

Ch. 8

SUMMARY AND OUTLOOK from the cavity length. The index modulation wave induced by the acoustic vibrations in the PCF, acting in concert with cavity GVD, trapped and accommodated the multiple pulses inside individual time-slots and meanwhile highly suppressed the pulse timing-jitter. Moreover, the non-interferometric nature of optoacoustic mode-locking has been revealed in the experiments. This nature has rendered the optoacoustic mode-locking to be wavelength-independent, while the multi-pulses trapped in different time-slots were found to be uncorrelated sub-laser pulses that can be manipulated individually. In the experiment part, GHz-rate HHML Er-fibre lasers based on the optoacoustic mode-locking and polarization APM have been developed which can generate clean sub-ps soliton train with low noise and stable performance. Wavelengthtunability has also been incorporated into the GHz-rate soliton laser using a F-P based tunable filter. A HHML laser based on Tm-doped gain fibre has been realized. Moreover, significant scaling of pulse bandwidth (duration) has been achieved beyond the previous soliton HHML lasers, with a novel cavity configuration named as the stretched-soliton laser. As a result, the HHML fibre lase can deliver sub-100-fs pulses at GHz-repetition-rate at only moderate pump power, which was previously regarded difficult to achieve in conventional fibre lasers. Due to the fact that the multi-pulses in the HHML laser are uncorrelated sublaser pulses, independent on/off state of an individual pulse trapped in each time-slot can establish a diversity of optomechanical bound-states of intra-cavity pulses, leading to the realization of an all-optical bit-storage in the laser cavity. We developed a technique that can selectively erase the intra-cavity pulses in the HHML fibre laser using an externally-launched addressing pulse sequence. The pulse sequence left in the optomechanical lattice of the cavity, which is the complimentary pattern of the addressing pulse sequence, can carry bit-information and be stored in the laser cavity for over 100 hours—a record-long preservation time of an arbitrary pulse sequence in a fibre laser cavity. In addition to such optomechanical bound-states, we have realized supramolecular assemblies of solitons in the laser cavity by trapping more than one soliton in each time-slot of the optomechanical lattice. In contrast to the conventional soliton pairs or crystals based on XPM, the multi-solitons bound within each time-slot has internal spacings being hundreds of times longer than their individual durations, due to the stable balance between two types of long-range interactions. Additionally, the hundreds of solitons, extending throughout the entire cavity, with different long-range bound-states filled in the time-slots, mutually form

166

an optical structure analogy to the chemical supramolecule with configurational flexibility, reversibility and dynamic stability. The HHML fibre laser has thus acquired unprecedented diversity in output formats, and the mode-locked fibre cavity has found to be a unique platform for interplay of long-range interactions between laser pulses. Despite the substantial developments and unexpected expansions of the optoacoustic mode-locking technique, several problems remain unsolved. Concerning the foundation of the optoacoustic interactions in the PCF, the damping mechanism of the GHz acoustic vibration remains unclear and requires more in-depth studies. Besides, the acoustic resonance is still limited to only a few-GHz range (less than 3 GHz). Novel structural designs for further scaling of the PCF core-resonance frequency are desired to achieve higher pulse repetition rate. Concerning the optoacoustic mode-locking theory, although the time-domain interpretation has partially unveiled the self-stabilization mechanism of the multi-pulses, the physical picture in the frequency domain is still incomplete. The random phase relation between different sets of frequency combs, each being coherently coupled by the acoustic phonons in the SRLS process, does not fully explain the time domain picture of phase-uncorrelated pulses. Such “incoherent” mode-locking demands a clearer physical picture in the frequency domain. At last, the supramolecular assembly of the optical solitons relies on stable binding of optical solitons due to the balance between the long-range forces exerted by the acoustic wave and the dispersive waves. However, the underlying mechanism for such attractor-like behaviours remains unclear, and we still only have some preliminary knowledge of the balanced interactions. The combination of this non-interferometric mode-locking and interferometric mode-locking (based on sub-cavity) may become an interesting topic for future researches. The possibility exists that, by coupling this optoacoustic mode-locking laser with a sub-cavity that matches the intra-cavity pulse spacing, the multi-pulses in the main-cavity could become in-phase due to the presence of sub-cavity, while the acoustic wave in the PCF may ensure the self-stabilization of the pulse spacings. In addition, the simultaneous presences of two acoustic resonances in the mode-locked laser, which can be introduced by two pieces of different solid-core PCF in the cavity (or two acoustic modes of the same PCF), may also generate interesting phenomena. It is likely to obtain the multi-pulses being modulated by both resonances, leading to periodical variations of the pulse spacings due to the inhomogeneous optoacoustic lattice.

167

Ch. 8

SUMMARY AND OUTLOOK The self-starting process of the HHML laser should also be investigated in the future for a deeper understanding of this optoacoustic mode-locking laser. Such investigations would potentially contribute to the popular topic of the self-organization phenomena. Meanwhile, as the opposite process, the collapse of the HHML state due to intended or random perturbations, as we have occasionally observed in the experiments, is also worth in-depth studies. It will be interesting to know what exactly causes the self-organization and the collapse of these highly-ordered structures. These studies would perhaps reveal a subtle metaphor of life and other self-assembled systems.

168

LIST OF FIGURES

1.1 1.2

Schematic of the fundamental and the harmonic mode-locked state . . SEM photo of a typical solid-core PCF. . . . . . . . . . . . . . . . . .

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

The absorption dynamics of fast and slow saturable absorbers. Schematic of the conventional APM action . . . . . . . . . . . Schematic of a NALM . . . . . . . . . . . . . . . . . . . . . . Schematic of the polarization APM principle . . . . . . . . . . Sinusoidal response of the Kerr-based saturable absorber . . . Schematic of the Kerr-lens mode-locking principle . . . . . . . Schematic of the Kelly-sidebands phase-matching condition . . Map of self-consistency of typical dissipative soliton lasers. . .

. . . . . . . .

. . . . . . . .

14 16 18 19 21 22 31 36

3.1 3.2 3.3 3.4 3.5

Principle of the backward SBS . . . . . . . . . . . . . . . . . . . . . Principle of the forward SBS . . . . . . . . . . . . . . . . . . . . . . Principle of the SRLS pumped with a dual-frequency CW light . . . SRLS pump with pulse train: on and off resonance driving . . . . . The balance of the red-shift and the blue-shift in an optoacoustic mode-locked laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . The re-timing principle of pulses in optoacoustic mode-locking . . . The temporal trapping potential . . . . . . . . . . . . . . . . . . . . The phase-relations between the cavity axial modes in different modelocking regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . .

42 44 45 51

3.6 3.7 3.8

4.1

. . . . . . . .

. . . . . . . .

3 6

. 54 . 55 . 58 . 68

The schematic procedure of PCF fabrication . . . . . . . . . . . . . . 70

169

List of Figures 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21 5.1 5.2 5.3 5.4 5.5 5.6

170

The solid-core PCF structure, with the optical and the acoustic mode of it . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The dependency of PCF-core resonance frequency upon the fibre structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The set-up for measuring the optoacoustic gain spectrum . . . . . . . The optoacoustic gain spectrum measured at R01 acoustic resonance . The optoacoustic gain spectrum of PCF with and without polymer coating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The optoacoustic gain spectrum of PCF in ambient and in vacuum . The PCF acoustic resonance tuning using temperature variation . . . The GVD of solid-core PCF with different core diameters . . . . . . . The splicing method between solid-core PCF and SMF-28 . . . . . . The schematic of the HHML soliton ring-fibre laser cavity . . . . . . The HHML soliton laser output: the pulse train . . . . . . . . . . . . The HHML soliton laser output: wavelength tuning . . . . . . . . . . The HHML soliton laser output: supermode suppression and repetitionrate drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The HHML soliton laser output: SSB noise spectra . . . . . . . . . . The heterodyne measurement of the HHML laser: the set-up and the principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The heterodyne measurement of the HHML laser: the RF spectrum and the modulated pulse train . . . . . . . . . . . . . . . . . . . . . . The interference pattern between the HHML laser and local oscillator: round-trips plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The nano-bore PCF structure, the optical and the acoustic modes . . Sketch of the Tm-doped fibre laser cavity with the nano-bore PCF . . The output of the HHML-TDFL . . . . . . . . . . . . . . . . . . . . . The self-consistency map of the stretched-soliton laser . . . . . . . . . Sketch of the experimental set-up of the stretched-soliton fibre laser. . The stretched-soliton laser results: the GHz-rate pulse train . . . . . The stretched-soliton laser results: the laser noise and the long-term stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The stretched-soliton laser results: the optical spectra and AC traces. The stretched-soliton fibre laser output under fundamental modelocked state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

72 72 73 74 75 76 77 78 80 83 85 87 88 89 90 90 91 92 93 95 99 100 101 102 103 104

List of Figures 5.7

The simulation results of intra-cavity pulse evolution in the stretchedsoliton fibre laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Experimental validations of the stretched-soliton simulation results 5.9 Characterization of the wavelength-dependent attenuator . . . . . . 5.10 The calculated phase-shift of the stretched-soliton propagation in one cavity round-trip . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. 106 . 107 . 109 . 111

6.1 6.2

Optomechanical bound-states in a mode-locked fibre laser. . . . . . . 115 The principle of selective erasure of intra-cavity pulses using external addressing pulses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 6.3 The mechanism of intra-cavity pulses erasure using external addressing pulse. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.4 The experimental set-up for the all-optical bit-storage . . . . . . . . . 119 6.5 A experimental demonstration of using an addressing pulse sequence to generate a complementary pulse sequence in the HHML laser cavity.121 6.6 The RF spectrum and SSB noise spectra of the intra-cavity pulse sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 6.7 The measured dynamics of intra-cavity erasing process. . . . . . . . . 123 6.8 Demonstrations of different bit-patterns stored over 2 hours. . . . . . 125 6.9 Intra-cavity pulse pattern with 64 evenly-spaced pulses erased, which extends to the entire cavity, stored for over 2 hours. . . . . . . . . . . 126 6.10 An 100-hour bit-storage demonstration. . . . . . . . . . . . . . . . . . 127 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9

The principle of a supramolecular assembly of optical solitons. . . . . 132 One double-soliton unit within one acoustic cycle. . . . . . . . . . . . 134 The long-range soliton interaction due to dispersive wave perturbations.138 Schematic illustrations of the internal spacing tuning in a doublesoliton unit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 Experimental set-up for supramolecular assembly of optical solitons. . 145 Illustrations of the all-double-soliton supramolecule in three difference ways. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 The ADS supramolecule: pulse trace under persistence mode, FFT power spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 The ADS supramolecule: the AC-trace and the spectral details. . . . 149 The ATS supramolecule: the single round-trip structure, the pulse trace under persistence-mode, and the FFT power spectrum . . . . . 150

171

List of Figures 7.10 The ATS supramolecule: the AC-trace and the spectral details. . . 7.11 A soliton supramolecule under the hybrid-state with null-soliton up to triple-soliton states. . . . . . . . . . . . . . . . . . . . . . . . . . 7.12 The long-term preservation of a soliton supramolecule consisting of null-, single-, double-, and triple-soliton states. . . . . . . . . . . . . 7.13 The long-term preservation of a soliton supramolecule consisting of double-, triple-, and quadruple-soliton states. . . . . . . . . . . . . . 7.14 The long-term preservation of a soliton supramolecule consisting of triple-, and quadruple-soliton states. . . . . . . . . . . . . . . . . . 7.15 The continuous tuning of the internal-spacing of double-soliton unit by tailoring of dispersive waves. . . . . . . . . . . . . . . . . . . . . 7.16 The continuous tuning of internal-spacing of triple-soliton by tailoring of dispersive waves. . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.17 The continuous tuning of the internal spacing of double-soliton by tailoring of acoustic waves. . . . . . . . . . . . . . . . . . . . . . . . 7.18 The spectrum of the ADS state soliton supramolecule during the repetition-rate tuning. . . . . . . . . . . . . . . . . . . . . . . . . . 7.19 The direction of the repulsive force case 1: with dominant m = −1 sideband. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.20 The direction of repulsive force case 2: with comparable m = ±1 orders of sidebands. . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.21 The direction of repulsive force case 3: with dominant m = +1 order sideband. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

172

. 151 . 152 . 153 . 153 . 154 . 156 . 157 . 158 . 159 . 161 . 162 . 163

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192

LIST OF PUBLICATIONS

Journal papers • W. He, M. Pang, J. Huang, C. R. Menyuk, and P. St.J. Russell, “Supramolecular assembly of optical solitons via long-range interactions,” arXiv:1710.01034 (2017). • W. He, M. Pang, C. R. Menyuk, and P. St.J. Russell,“Sub-100-fs 1.87 GHz mode-locked fiber laser using stretched-soliton effects,” Optica 3, 1366-1372 (2016). • W. He, M. Pang, and P. St.J. Russell, “Wideband-tunable soliton fiber laser mode-locked at 1.88 GHz by optoacoustic interactions in solid-core PCF,” Opt. Express 23, 24945-24954 (2015). • W. He, M. Leich, S. Grimm, J. Kobelke, Y. Zhu, H. Bartelt, and M. Jäger, “Very large mode area ytterbium fiber amplifier with aluminum-doped pump cladding made by powder sinter technology,” Laser Phys. Lett. 12, 015103 (2015) • M. Pang, W. He, and P. St.J. Russell, “Gigahertz-repetition-rate Tm-doped fiber laser passively mode-locked by optoacoustic effects in nanobore photonic crystal fiber,” Opt. Lett. 41, 4601-4604 (2016). • M. Pang, W. He, X. Jiang, and P. St.J. Russell, “All-optical bit storage in a fibre laser by optomechanically bound states of solitons,” Nat. Photon. 10, 454-458 (2016).

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Bibliography List of Publications • M. Pang, X. Jiang, W. He, G. K. L. Wong, G. Onishchukov, N. Y. Joly, G. Ahmed, C. R. Menyuk, and P. St.J. Russell, “Stable subpicosecond soliton fiber laser passively mode-locked by gigahertz acoustic resonance in photonic crystal fiber core,” Optica 2, 339-342 (2015).

Conference papers • W. He, M. Pang, and P. St.J. Russell, “Multi-soliton bound states in fibre laser harmonically mode-locked at GHz-rates by optoacoustic effects in PCF,” CLEO/Europe-EQEC, Munich, Germany, paper CJ_13_5 (2017) • W. He, M. Pang, and P. St.J. Russell, “Sub-100 fs pulses from Er-fiber laser passively mode-locked at 1.872 GHz by acoustic resonance in solid-core PCF,” CLEO, San Jose, USA, paper SM1I.2 (2016) • W. He, M. Pang, and P. St.J. Russell, “Wideband-tunable, soliton fibre laser mode-locked at 1.88 GHz by optoacoustic interactions in solid-core PCF,” CLEO/Europe-EQEC, Munich, Germany, paper CJ_12_4 (2015) • M. Pang, W. He, and P. St.J. Russell, “Programmable generation and storage of soliton sequences in fibre laser cavity locked to gigahertz core resonance in PCF,” ECOC, Valencia, Spain, pp. 1-3. (2015) • M. Leich, W. He, S. Grimm, J. Kobelke, Y. Zhu, B. Müller, J. Bierlich, H. Bartelt, and M. Jäger, “High peak power amplification in large-core allsolid Yb fibers with an index-elevated pump clad and a low numerical aperture core,” SPIE LASE, San Francisco, United States, paper 93440T (2015)

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ACKNOWLEDGEMENTS

I would like to thank the following people who have contributed to the accomplishment of this work. First and foremost I would like to thank my supervisor Prof. Dr. Philip St.J. Russell. It has been a great honour to become a PhD student in his group and it has also been a great luck that to follow up the topic of optoacoustic mode-locking based on solid-core PCF which he established many year ago. I have always been interested in the fibre optics and fibre laser topics, and I am very excited to have the opportunity to realized this beautiful idea. I have been benefited so much during conversations with from the weekly meetings, receiving suggestions and guidances from him throughout my entire research stages. Not only have I learned a lot about the research procedures about this specific topic, but also have I learned some very important aspects about scientific research in a more general sense from him. I have progressed a lot in the thinking about the physics instead of merely looking at the phenomena, which is something that he emphasized for many times during the last four years. This is something I believe that I will continuously benefit from in my following years of scientific career. Moreover, I have been benefiting a lot from the cooperative atmosphere in this group, and he has managed to run this group very efficiently, making everyone to learn and benefit from the research and communication activities. In general, when look back to that specific moment some 4 years ago, I can say now that I have made a right decision to apply and then join his research group at that time. Next I wish to offer my great gratitude to Dr. Meng Pang, who has been the secondary supervisor for my PhD study and has guided me on a daily basis throughout the research projects one after another during the last four years. He

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Bibliography Acknowledgements has been a great supervisor and colleague at the same time for me. I have learned so much about almost every single aspect of this research topic and it has been a great experience to work with him. During the initial period of my research he has patiently guided me and introduced to me the various details and elements about this topic. He has been very accurate in the sense of novelty in physics, and he always knows what is the right thing to do during the research. We usually spent hours and hours together in the office and labs, either talking about the exciting physics behind our newly discovered phenomena, or arguing over some difficult problems during the research. He is also a very hard-working and clever researcher, who has greatly inspired me through the difficult or depressing time. I can never over-claim his contribution to this research topic and his assistance during my PhD study, and I will always look forward someday to becoming an energetic, diligent and wise scientific researcher as well as a caring and beneficial teacher as he is now. I would like to thank Dr. Johannes Köhler for this continuous help during my experiments. When I first started my experiments, He offered lots of helps when I tried to find some components or tried to learn how to use a specific device. He is an amazingly hard-working and meanwhile helpful colleague, from whom I have learned a lot about experimental details as well as research procedures. I cannot remember for how many times that he managed to help me through some difficult problems with experiments. Besides I would like to thank him a lot to help me translate the abstract of this thesis into the German "Zusammenfassung". I would also like to offer my special thanks to both him and his father who have helped me twice in my house-moving. I will always be grateful to their kind and timely help, which has made a comfortable start for my life here in Erlangen. I would like to thank Dr. Xin Jiang and other members in the fibre fabrication team, including Dr. Michael Frosz, Fehim Babic, Goron Ahmed, etc. for making such beautiful and extremely satisfactory PCF. I cannot imagine how my research would ever be able to start without these specially-fabricated PCF. He has also guided me a lot with the basic procedures in the fibre drawing. He has been a nice and helpful colleague, a greatly experienced scientists and meanwhile a good friend. I would like to thank Prof. Curtis R. Menyuk for his participations and contributions in our research topics. He has offered good suggestions and careful revisions to our manuscripts for publications. He has been an impressive scientist who has very thorough understanding on both optics theory and the relevant mathematics. He has made a great help to both Meng and me in our research projects.

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Bibliography Acknowledgements I would like to offer my thanks to Prof. Nicolas Joly, who has also participated in this research topics and offered many helpful suggestions. Besides, I am also very gratitude to be given the opportunity to teach the tutorial class for his Nonlinear Optics course, from which I gained a lot of experience in the communications with students. I would also like to thank other colleagues in our group for their kind help and accompany during my research work: Jiapeng Huang, Dr. Shangran Xie, Dr. Xiaoming Xi, Riccardo Pennetta, Richard Zeltner, Ramin Beravat, Zheqi Wang, Dr. Gordon Wong, Felix Köttig, Mehmet Can Günendi, Manoj Mridha, Pooria Hosseini, Rafal Sopalla, Marco Cassataro, Jonas Hammer, Sona Davtyan, Nitin Edavalath, etc. I have been in a very friendly and helpful atmosphere during my research work, and I will also miss a lot the time that I spent together with them on those many group meetings, international conferences and summer schools. It has been a great experience to be a group member among these amazingly nice and helpful people. At last but definitely not the least, I would like to offer my sincere gratitude to my parents who have unconditionally supported me throughout my entire period of study abroad during the last six years. I have always had a very warm feeling when having them to talk with during the weekends, receiving encouragement and comfort from them. Being simple but powerful, these weekly remote communications with them have been a great source of motivations for me to keep on with my study and my pursuit of future objectives.

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