Fiber supercontinuum sources (Invited) - OSA Publishing

Genty et al.

Vol. 24, No. 8 / August 2007 / J. Opt. Soc. Am. B

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Fiber supercontinuum sources (Invited) Goëry Genty,1 Stéphane Coen,2,* and John M. Dudley3 1

Helsinki University of Technology, Metrology Research Institute, P.O. Box 3000, FIN-02015 HUT, Finland 2 Department of Physics, University of Auckland, Private Bag 92019, Auckland, New Zealand 3 Département d’Optique, Institut FEMTO-ST, Université de Franche-Comté, Besançon, France *Corresponding author: [email protected] Received November 1, 2006; revised January 19, 2007; accepted January 22, 2007; posted January 30, 2007 (Doc. ID 76659); published July 19, 2007

We review supercontinuum generation in optical fibers for particular cases where the nonlinear spectral broadening is induced by pump radiation from fiber-format sources. Based on numerical simulations, our paper is intended to provide experimental design guidelines tailored ytterbium and erbium-based pumps around 1060 and 1550 nm, respectively. In particular, at 1060 nm, we consider conditions under which the generated spectra are phase and intensity stable, and we address the dependence of the supercontinuum coherence on the input pulse parameters and the fiber length. At 1550 nm, special attention is paid to the case of dispersion-flattened dispersion-decreasing fiber, where we revisit the underlying physics in detail and explicitly examine the use of such fiber for supercontinuum generation with pumps of peak power in the range 200– 1200 W and sub-10m fiber lengths. We show that supercontinuum generation under such conditions can be highly coherent and can be applied to nonlinear pulse compression. © 2007 Optical Society of America OCIS codes: 190.4370, 190.4380, 190.5650.

1. INTRODUCTION Supercontinuum (SC) generation describes the process where narrowband optical pulses undergo extreme spectral broadening in a nonlinear medium to yield a broadband spectrally continuous output. Almost immediately after its first observation in bulk media [1,2], it was apparent that its spatial coherence, high brightness, and broad bandwidth suggested numerous applications, and many significant results in diverse fields such as ultrafast spectroscopy and pulse compression were obtained. An extensive review of early research into SC generation is given in Part I of the monograph by Alfano [3]. SC generation in optical fibers was first observed in 1976 by Lin and Stolen for pumping in the normal groupvelocity dispersion (GVD) regime of standard silica fiber [4]. In this case the spectral broadening was attributed to a combination of Raman scattering, self- and cross-phase modulation, and four-wave mixing effects [5–7]. Subsequent research in standard fibers studied SC generation pumping around 1310 nm near the zero-dispersion wavelength (ZDW) [8–10], or around 1550 nm in the anomalous dispersion regime. In these cases it was found that the spectral broadening mechanisms were strongly influenced by soliton propagation dynamics, particularly the breakup of the injected pulse through the process known as soliton fission [11–13]. The generation of broadband spectra in the vicinity of the 1550 nm telecommunications window was of particular interest in the context of developing wavelength division multiplexed systems, and this motivated an intense research effort during the 1990’s [14–32]. These studies clarified the critical relationship between the input pulse wavelength and the dispersion profile of the fiber used. The importance of this relationship was highlighted even more dramatically with the advent of the photonic crystal 0740-3224/07/081771-15/$15.00

fiber (PCF) and the demonstration by Ranka et al. of SC generation spanning over an octave using a Ti:sapphire laser [33]. The elevated nonlinearity of PCF and the ability to readily engineer their GVD characteristics [34] has allowed SC generation in PCF to be observed over a much wider range of source parameters than has been possible with conventional fibers. The characteristics and applications of SC generation in PCF have been extensively described in a recent review [35]. The physics underlying SC generation is now generally well understood, and the emphasis is now shifting toward developing and refining applications and reducing the dependence on relatively complex and expensive pump sources. In this context, fiber-based sources clearly have an important role to play, because they provide lowfootprint alternatives to bulk laser sources in all the pumping configurations relevant to SC generation. However, with the majority of previous studies of SC generation focusing on pumping parameters associated with bulk mode-locked lasers such as Ti:sapphire, there is a need to review aspects of the SC generation physics and develop useful guidelines for the development of SC sources using parameters typical of fiber laser pumps. This is our main objective in this paper. More specifically, this paper is intended to complement our recent review of the general physics of SC generation [35] by addressing issues particular to fiber laser pumps as well as to provide an extended discussion of the conditions of SC stability. The paper is organized as follows. In Section 2, we present an overview of relevant previous studies of SC generation using fiber sources, considering all pumping configurations from the femtosecond to the continuouswave (CW) regime. It is our intention that this provides a comprehensive introduction to the previous work in the field. In Section 3, we present a brief review of the mod© 2007 Optical Society of America

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eling and physics of SC generation, and Section 4 then presents results of numerical simulations for femtosecond pulse pumping around 1060 nm (corresponding to Yb3+ gain media) and picosecond pulse pumping around 1550 nm (corresponding to Er3+ gain media). The aim here is to provide a clear discussion of the relevant broadening mechanisms for commonly available sources and to give guidelines for the generation of stable SC spectra. For the case of femtosecond pulse pumping around 1060 nm, we discuss the dependence of the SC coherence on the input pulse parameters and the fiber length and introduce a useful graphical representation in terms of input soliton number and a scaled propagation distance that allows regimes of coherent SC generation to be clearly identified for a particular fiber type. For SC generation around 1550 nm, we revisit and extend previous studies of SC generation in dispersion-decreasing fiber (DDF) and illustrate the benefits of such a pumping scheme. In contrast to previous studies of this scheme at power levels typical of telecommunications sources, we analyze the expected characteristics of the SC with peak power levels in the range 200– 1200 W, and we show that SC generation spanning ⬃800 nm around 1550 nm is expected when only sub-10 m fiber lengths are used. We also explicitly consider the coherence properties of SC generated under such conditions and show that suitably designed DDF can provide a route to the generation of coherent SC generation of picosecond pulses, even at kilowatt peak power levels. Section 5 presents a discussion of how these results apply to the appropriate choice of SC to particular applications and briefly concludes.

2. LITERATURE REVIEW The objective in this section is to provide a review of previous work using fiber-based sources to induce SC generation. Although we refer where necessary to various details of the SC generation mechanisms, these are not treated in detail here, but are discussed in Section 3. The initial work in this field was pioneered by Morioka et al., who successfully demonstrated SC generation using an Er3+-based mode-locked ring laser operating at a 6.3 GHz repetition rate [20]. In these experiments, 1.7 W peak power pulses of 3.3 ps duration were injected into a 3 km length of dispersion-shifted fiber with a small anomalous dispersion at the pump wavelength. A spectrally flat 200 nm bandwidth SC from 1440 to 1640 nm (at the −20 dB level) was observed, and stable spectral slicing over four wavelength channels was demonstrated. Subsequent experiments using the same SC source quantified the stability of the generated SC by characterizing the time-domain properties of the spectrally sliced pulses obtained [18,22]. In another experiment using 3 km of fiber, an extended SC spanning 1150– 1770 nm was demonstrated by using a lower repetition rate of 42.5 MHz with a source generating 0.7 ps pulses with 1.2 kW of peak power [23]. In a pioneering systems experiment, a similar setup using a 10 GHz fiber-based source was used to successfully demonstrate 1 Tbit/ s transmission over a 40 km fiber link [36]. In these early studies the detailed dispersion profiles of the particular fibers used to generate the broadband SC

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were not reported. However, later research showed that this was, in fact, a critical factor in determining key SC characteristics such as the overall bandwidth and the spectral flatness. An important study by Lou et al. suggested that fibers with longitudinally varying dispersion could be advantageous in generating flatter and broader SC spectra compared to fibers with uniform longitudinal dispersion [37]. This was then examined in detail in Ref. [38] for the particular SC fiber that had been used in the many previous experiments reported in Refs. [20–22]. A major result of this work was the statement of clear fiber design criteria for the generation of a uniform SC spectra: (i) a dispersion profile presenting decreasing anomalous GVD with propagation and (ii) a dispersion-flattened convex wavelength variation of the GVD exhibiting two zerodispersion wavelengths at the fiber input and centered near the pump [39]. Detailed investigations of the dispersion-decreasing design were also reported by Okuno et al. [26] and by Mori et al. [32] In a systems context, stability properties were important considerations, particularly since SC fluctuations had previously been observed in a number of experiments using bulk sources [12,13]. Insight into the physical mechanism of the instabilities was provided by Nakazawa et al., who showed that noise on the input pulses could perturb higher-order soliton propagation dynamics through modulation instability (MI) [27]. Kubota et al. gave a direct numerical proof of this mechanism by showing that filtering input pulse noise in the vicinity of the MI gain maxima led to a strong improvement in the SC stability and spectral coherence [29]. Nakazawa et al. [27] also showed that the use of DDF could lead to improved stability by avoiding higher-order soliton breakup altogether and, in addition, suggested pumping the fiber in the normal GVD regime as an alternative means of avoiding MI. This latter approach was later successfully demonstrated experimentally [28,40]. Similarly, Nowak et al. proposed a scheme based on a very short fiber length in which the pump pulses propagate over only a fraction of a soliton period in such a way that the spectral broadening process is interrupted when the soliton has maximum bandwidth but before fission can occur [41]. These aspects of SC stability are discussed further in Section 3.C. The success of these various results occurred in large part because of the development of highly nonlinear fiber (HNLF) having tailored dispersion characteristics around the operating wavelengths of ultrafast sources at 1550 nm [42]. In parallel with these developments, the availability of PCF was leading to a dramatic revolution in the experimental study of SC generation in the near-infrared, because it was possible to engineer the fiber zero dispersion wavelength to coincide with the operating wavelength of high-power femtosecond Ti:sapphire lasers [33]. The widespread success of this work in opening up new applications in fields such as optical frequency metrology has been well documented [35], but it is appropriate here to review a selection of the most significant results in PCF that have been obtained by using fiber-based sources. The first experiments combining PCF with fiber-based ultrafast sources were obtained at 1060 nm using Yb3+-doped fiber laser pumps and exploited spectral broadening mechanisms (soliton fission dynamics) similar

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to that reported by Ranka et al. using a Ti:sapphire laser [33]. In a 2002 experiment, a stretched-pulse Yb3+-doped fiber laser operating around 1060 nm was used to pump a highly nonlinear small-core PCF with a ZDW around 800 nm [43]. Results obtained included soliton propagation (using 2.4 ps input pulses and PCF lengths in the range 0.5– 2.6 m) and SC generation over 400– 1700 nm (using 350 fs input pulses and a 7 m fiber length). In another experiment, a similarly configured Yb3+-doped fiber oscillator was used to pump a length of Yb3+-doped PCF to generate tunable Raman solitons over the range 1060– 1330 nm [44]. Other experiments using PCF and 100– 200 fs pulses from Yb3+ fiber lasers have reported tunable Raman soliton generation [45] and applications to optical coherence tomography [46]. An area of particular recent interest has been highbrightness SC generation using picosecond Yb3+ fiber sources with watt-level average powers and 10– 100 MHz repetition rates. For example, using 3 – 4 ps pulses at 1060 nm with peak powers in the 10– 50 kW range and using tens of meters of PCF with ZDWs near the pump wavelength, over 1000 nm spectral broadening and SC power spectral densities of approximately milliwatts per nanometer have been reported [47,48]. In this regime of picosecond pumping, four-wave mixing processes about the fiber ZDW play a dominant role in the observed spectral broadening, and optimizing the phase matching of this process through a decreasing ZDW with propagation has proved to be a successful means of extending the SC spectral broadening toward shorter wavelengths [49,50]. Finally, we note some other experiments of interest that use Yb3+-doped technology (albeit based on a solid-state Nd:glass oscillator as primary source), reporting highpower SC generation spanning hundreds of nanometers by using two different highly nonlinear PCFs with ZDWs of 975 and 1055 nm and injected pulses of 120 fs and 8.8 ps [51]. In parallel with these impressive results using Yb3+ doped fiber sources and PCF, there has been extensive continued interest in generating broadband SC by using Er3+-fiber-based SC sources around 1550 nm. Interestingly, because the development of PCF and advanced dispersion-tailored HNLF have occurred in parallel, significant results around 1550 nm have been obtained by using both classes of fiber. For example, a 2001 experiment reported SC generation using a femtosecond fiber laser generating 110 fs pulses at 1550 nm, injected into 1 – 200 m lengths of dispersion-shifted HNLF having small normal dispersion at the pump wavelength [52,53]. Spectra from 1100 to 2100 nm were reported at the longest fiber lengths, and these results were also notable for presenting the first temporal–spectral characterization of SC generation using the cross-correlation frequency-resolved optical gating (XFROG) technique. A study by Nicholson et al. (using ⬃200 fs pulses from a passively mode-locked femtosecond fiber laser at 1550 nm) reported improved SC generation of over an octave by using a series of four 1.5 m segments of HNLF to obtain a discrete map of decreasing anomalous GVD [54]. A number of other experiments using femtosecond pulses from Er3+-fiber sources to generate SC in HNLF were also reported [55–58]. With suitable optimi-


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zation of experimental parameters, such SC generated in HNLF by using femtosecond pulses were shown to present excellent stability and coherence characteristics [59], and the application to generating phase-stabilized fiber-laser-based frequency combs has been an area of intense subsequent study [60–63]. Low-noise SC generation from a femtosecond fiber laser source and HNLF was also applied to optical coherence tomography, yielding 5.5 ␮m resolution [64]. Using PCF for SC generation, Yusoff et al. generated a 36 channel 10 GHz spectrally sliced source by injecting amplified 2 ps pulses from a soliton mode-locked laser into 20 m of silica PCF [65]. The key to maintaining coherence in this experiment was the PCF design that presented normal dispersion of −30 ps/ nm/ km at 1550 nm. Yamamoto et al. [66] reported a similar approach using normal dispersion PCF (−0.23 ps/ nm/ km at 1550 nm), but the addition of Ge doping in the PCF core also resulted in a dispersion-flattened profile. The injection of 2.2 ps pulses from a 40 GHz fiber laser system resulted in the generation of a 40 nm broad SC spectrum [66]. In another experiment, a PCF based on extruded SF6 glass was used in conjunction with an Er3+-based oscillatoramplifier system generating 60 fs pulses to allow octavespanning SC generation at input pulse energies of only 200 pJ [67]. The experimental results above have been based on ultrafast femtosecond or picosecond fiber-based sources. However, the development of high-power CW fiber sources has also been applied to SC generation, and impressive results have been reported for both PCFs and HNLFs. In one experiment, Avodkhin et al. used a 100 m long PCF pumped at 1065 nm by a seeded Yb3+-doped fiber amplifier in a master oscillator power fiber amplifier configuration [68]. With a CW pump power as low as 8.7 W, a 3.8 W single-mode SC spanning the 1065– 1375 nm wavelength range was generated with a power density as high as 12 mW/ nm. The redshifted continuum resulted mainly from cascaded Raman scattering, but later experiments performed closer to the ZDW using cascaded Raman fiber lasers and kilometer lengths of HLNF identified the importance of four-wave mixing processes [55,69–71]. The case of normal GVD CW pumping was also investigated in HNLF [55,72]. Of particular significance when using a CW pump is the poor coherence properties of the generated SC [73–75], but, of course, this is not a disadvantage for some applications, and impressive results in optical coherence tomography have been obtained [76]. Finally, we consider recent developments in which the concepts of single-pass SC generation have been extended to develop a new generation of fiber laser source in which SC dynamics are central to the intracavity field evolution. Although earlier results considering this were associated with output instabilites [77], recent results have demonstrated the potential for stable broadband SC laser operation. In one experiment using a Yb3+ CW fiber laser as the primary pump, a series of fiber Bragg gratings were used to construct a nested linear cavity configuration involving both a cascaded Raman laser pump and a 1 km length of single-mode SC fiber [78]. Around 1550 nm, ring cavity configurations exploiting intracavity gain from Er3+ and

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Raman amplification have both been reported, and continuous SC spectra spanning ⬃200 nm about the pump wavelength have been observed [79,80]. Although these cavity SC generation experiments do not yet appear to be optimized, the results obtained nonetheless demonstrate the potential for additional control and manipulation of the SC characteristics.

3. SUPERCONTINUUM GENERATION PHYSICS A. Numerical Modeling Much of the understanding of the physics of SC generation has occurred because of the development of realistic numerical modeling of ultrashort pulse propagation in HNLF and PCF. Before discussing the physical mechanisms in more detail, we first describe the numerical model that has been used in many previous studies and that has been used to obtain the results in Section 4 [35,81–83]. This will also serve to define a number of useful parameters. Our numerical model uses a generalized propagation equation suitable for studying broadband pulse evolution in optical fibers [84]:

⳵A ⳵z

+

␣ 2

A−

兺

kⱖ2

冋冉冕

+⬁

⫻ A共z,t兲

−⬁

ik+1 k!

␤k

⳵ kA ⳵ tk

冉

= i␥ 1 + i␶shock

⳵ ⳵t

冊

冊册

R共t⬘兲兩A共z,t − t⬘兲兩2dt⬘ + i⌫R 共z,t兲 .

共1兲

Here, A = A共z , t兲 is the electric field envelope (dimensions of 冑w here), the ␤k’s are the fiber dispersion coefficients at center frequency ␻0, and ␥ = n2␻0 / 共cAeff兲 is the nonlinear coefficient, with n2 ⯝ 3.0⫻ 10−20 m2 / W the nonlinear refractive index and Aeff the fiber effective area evaluated at ␻0. The time derivative term describes effects such as self-steepening and optical shock formation, characterized by a time scale ␶shock = 1 / ␻0. In the context of fiber propagation, the frequency dependence of the modal effective area Aeff can introduce significant dispersion in the nonlinear response, and this can be included to a very good approximation through a first-order correction to ␶shock as discussed in Ref. [85]. The response function R共t兲 = 共1 − fR兲␦共t兲 + fRhR共t兲 includes both instantaneous electronic and delayed Raman contributions, with fR = 0.18 representing the contribution of the Raman response to the instantaneous nonlinear polarization. In our simulations, we use the experimental Raman response of fused silica. We assume a chirp-free input pulse of the form A共z = 0 , t兲 = 冑P0 sech共t / T0兲, where P0 is the peak power and T0 is related to the pulse intensity full width at half-maximum (FWHM) ⌬␶ through T0 = ⌬␶ / 1.7628. Characteristic length scales of importance in interpreting the simulation results are the nonlinear length LNL = 共␥P0兲−1 and the dispersive length LD = T02 / 兩␤2兩. Performing simulations in the presence of noise allows us to investigate how fluctuations on the input pulse (or noise introduced during propagation) influence the intensity and phase stability character-

istics of the output SC. Noise sources in the model include quantum-limited noise on the input pulse (through a phenomenological one photon per mode background) and a spontaneous Raman noise term ⌫ R, which has frequency-domain correlations * 共⍀⬘ , z⬘兲典 = 共2fR ប ␻0 / ␥兲兩 Im 关hR共⍀兲兴兩关nth共兩⍀ 兩兲具⌫R共⍀ , z兲 ⌫R + U共−⍀兲兴␦共z − z⬘兲␦共⍀ − ⍀⬘兲. Here ⍀ = ␻ − ␻0, the thermal Bose distribution nth共⍀兲 = 关exp共ប⍀ / kBT兲 − 1兴−1, and U is the Heaviside step function. In our simulations, we consider noise on the input pulse only at the quantum level; so our results are to be interpreted as suggesting bestcase scenarios. However, the criteria developed assuming quantum-limited input are nonetheless very usefully applied to the design of more realistic sources where both technical noise and amplifier noise are present. The particular approach used to study the noise sensitivity of SC generation is based on performing multiple simulations in the presence of different random noise seeds to generate an ensemble of output SC fields. Quantitative insight is obtained by examining the wavelength dependence of the modulus of the complex degree of firstorder coherence, defined at each wavelength in the SC by 共1兲兩g12 共␭,t1 − t2兲兩 =

冏冑

˜ * 共␭,t 兲A ˜ 共␭,t 兲典具A 1 2 2 1

˜ 共␭,t 兲兩2典具兩A ˜ 共␭,t 兲兩2典具兩A 1 1 2 2

冏

.

共2兲

The angle brackets denote an ensemble average over independently generated pairs of SC spectra ˜ 共␭ , t兲 , A ˜ 共␭ , t兲兴 obtained from a large number of simula关A 1 2 tions, and t is the time measured at the scale of the temporal resolution of the spectrometer used to resolve these spectra. Also, since we are interested mainly in the wavelength dependence of the coherence, we can calculate the 共1兲 modulus 兩g12 兩 at t1 − t2 = 0, which corresponds to the fringe visibility at zero path difference in a Young’s two source experiment performed between independent SC spectra [86]. It is also useful to introduce a spectrally averaged co共1兲共1兲 ˜ 共␭兲兩2d␭ / 兰兩A ˜ 共␭兲兩2d␭ to herence gav = 具兩g12 兩典 = 兰兩g12 共␭ , 0兲兩兩A provide a useful measure of the global coherence across the SC spectrum. B. Propagation Dynamics and Broadening Mechanisms The different regimes of SC generation can be broadly distinguished by considering anomalous versus normal GVD pumping, as well as short (subpicosecond) versus long (picosecond, nanosecond, and CW) pump pulses. In the following, we briefly summarize the particular features of the physics of each regime, but the interested reader should refer to Ref. [35] for more detail. Additional discussion of the spectral broadening processes is also given below in Section 4 for the particular cases considered at 1060 and 1550 nm. To focus on the principal dynamical mechanisms, we restrict our discussion to scalar propagation. Considering first the anomalous GVD regime with short pump pulses, spectral broadening arises from soliton-related dynamics and occurs broadly in three phases. The power of the pump pulses is generally high enough for those pulses to be considered high-order solitons, of order N = 关LD / LNL兴1/2 ⬎ 1, so that they initially undergo an initial period of spectral broadening and tempo-

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ral compression. Subsequently, because of perturbations such as high-order dispersion and stimulated Raman scattering (SRS) [87,88], the dynamics departs from the recurrent behavior expected of ideal high-order solitons, and the pulse breaks up (“fissions”) into a series of N distinct fundamental soliton components. The initial propagation of these fundamental solitons is associated with the generation of dispersive wave spectral components through resonant transfer of energy across the ZDW [87,89]. The resonance is due to high-order dispersion and is intrinsically narrowband, which explains the abrupt short-wavelength edge typically seen on SC spectra in this regime. Continued propagation of the solitons also results in a continuous shift to longer wavelengths through the Raman soliton self-frequency shift [90]. Finally, the generated Raman soliton and dispersive waves can couple through cross-phase modulation to generate additional frequency components that increase the overall bandwidth [91]. For pulses of longer durations with high peak power such that the soliton order becomes very large 共N 10兲 the soliton fission process described above becomes progressively less important during initial propagation. This is because the characteristic length scale over which it occurs 共Lfiss = LD / N ⬀ ⌬␶兲 increases with the pump pulse duration [35,92]. Instead, MI (equivalently, the generation of four-wave mixing parametric sidebands) occurs on the same length scale regardless of the pulse duration and therefore begins to dominate the initial propagation phase. Under these conditions, the initial MI dynamics leads to the temporal breakup of the pump pulse into a large number of subpulses. Subsequently, further spectral broadening arises essentially in the same way as with short, low-order pump pulses; i.e., each subpulse can undergo further fission, self-frequency shift, and dispersive wave generation. In comparison with short pump pulses, however, pumping too far in the anomalous GVD regime results in reduced spectral broadening because the initial MI dynamics does not generate sufficient bandwidth to efficiently seed dispersive wave transfer into the normal GVD regime. These conclusions can also be extended to the CW regime, where the partial coherence of the pump also seeds the beam breakup, and collisions between ejected solitons can also play an important role in reducing the duration of the MI-generated pulses [74,93,94]. However, the fact that MI actually develops spontaneously from noise at frequencies that do not overlap with the broadened bandwidth of the propagating pulse has dramatic consequences for the SC coherence. This is discussed some more in Subsection 3.C. In the normal GVD regime, the spectral broadening dynamics for subpicosecond pulses arises from the interaction of self-phase modulation and the normal GVD of the fiber, with shorter pulses inducing greater nonlinear broadening [95]. Since this leads to significant temporal broadening and rapid decrease of peak power over the first few centimeters of propagation, the extent of nonlinear spectral broadening is necessarily limited. For pump wavelengths approaching the ZDW, however, the initial spectral broadening due to self-phase modulation transfers spectral content into the vicinity of the ZDW and


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across into the anomalous GVD regime, and soliton dynamics then play an increasingly important role as well. Parametric generation and stimulated Raman scattering can also contribute significantly to the transfer of energy into the anomalous GVD regime, depending on the particular pumping conditions and fiber characteristics. For longer (picosecond) pulses and CW radiation for which self-phase-modulation does not play an important role, significant initial spectral broadening can develop from four-wave mixing and Raman scattering. Raman effects dominate when pumping far into the normal GVD regime, because the parametric sidebands are too much detuned from the pump. Closer to the ZDW, four-wave mixing becomes progressively more important, since the parametric gain is higher than the Raman gain [95,96]. In the case where this broadening begins to overlap with the ZDW, soliton effects can again contribute to the overall dynamics [97]. C. Stability Considerations As discussed above in the Literature Review (Section 2), issues of SC stability were extensively considered by a number of groups in a telecommunications context. These studies have since been complemented by work carried out in the framework of optical frequency metrology, where there is a similar requirement for high-stability SC [98]. We consider first stability issues for short (subpicosecond) pulses in the anomalous dispersion regime. In practice, instability arises because of the sensitivity of the spectral broadening mechanisms to noise that is inevitably present on the input pulse [27], and in the anomalous dispersion regime this noise can be significantly amplified along the fiber through MI gain. The MI gain in this case competes with soliton fission, and two broad stability regimes can be distinguished [35]. When soliton fission dominates the initial dynamics, the input pulses become temporally compressed so quickly that the spectral extent of the emerging SC can overlap with the frequencies of maximum MI gain before significant amplification of the noise background has taken place. This results in a coherent seeding of the MI gain bandwidth, and the SC that is generated is stable. On the other hand, when the MI amplified noise background becomes a dominant feature of the propagation dynamics, the initial spectral broadening is seeded from noise, and the SC generated exhibits spectral and temporal instabilities from pulse to pulse [99]. Hence, when MI dominates the initial spectral broadening dynamics, the generated SC is incoherent. We consider certain aspects of SC stability for anomalous dispersion regime pumping in more detail below in Subsection 4.A. Of course, as MI does not occur in the normal GVD regime, SC spectra generated with subpicosecond pulses with normal GVD pumping are, in principle, always highly coherent. However, the drawback is that the SC spectral width (at the same peak power) is comparatively much smaller because of the rapid initial temporal spreading of the pump pulses [27]. In the case of pumping with longer pulses or in the CW regime, SC generation is initiated with a fast modulation of the pump envelope. With normal dispersion regime pumping this modulation develops as a result of stimu-

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lated Raman scattering, while it can arise from both Raman scattering and/or MI in the anomalous dispersion regime. In both cases, however, the modulation arises spontaneously from noise at frequencies that do not overlap with the pump bandwidth, and the generated SC is incoherent.

4. DESIGNING SUPERCONTINUA WITH FIBER SOURCES A. Supercontinuum Generation around 1060 nm In this section, we use numerical simulations to discuss particular aspects of SC generation relevant to pumping with Yb3+-fiber-based sources around 1060 nm. We focus here on ultrafast fiber sources generating pulses in the 50– 250 fs range, spanning a range that is readily available experimentally. The particular PCF that we consider is based on the commercially available fiber SC-5.0-1040 from Crystal Fiber, possessing a hexagonal structure with hole diameter ␾ = 1.6 ␮m and pitch ⌳ = 3.2 ␮m. The GVD of this fiber can be readily calculated by using standard methods and is shown in Fig. 1. The Taylor series coefficients corresponding to the dispersion of this fiber are listed in Table 1, and we also note that our modeling used a nonlinearity coefficient ␥ = 14.1 W−1 km−1 and a shock time scale of ␶shock = 0.658 fs. The fiber ZDW is around 1040 nm, and the pump wavelength is 1060 nm. We consider anomalous dispersion regime pumping, because that has been the case studied in most experiments and generally leads to the broadest generated SC at lowest input power. Our main objectives here are to illustrate the dependence of the SC stability characteristics on the choice of input pulse parameters and to develop useful guidelines that can be used for experimental design when the generation of a stable SC is paramount. To this end, we first present simulation results that illustrate extreme cases of anomalous dispersion SC dynamics observed at low and high input soliton numbers. Specifically, Fig. 2 uses a false-color representation to show simulated spectral and temporal evolution along a 20 cm length of PCF for input pulses of (a) 50 fs FWHM and 10 kW peak power for and (b) 250 fs FWHM and 100 kW peak power. These are parameters that could correspond to Yb3+ ultrafast oscillator or oscillator–amplifier systems, respectively. For the 50 fs pulse, the corresponding soliton order is N = 5.6, the fission distance is Lfiss = 3.95 cm, and the char-

Fig. 1. GVD curve showing (a) D (ps/nm/km) and (b) ␤2 共ps2 / km兲 for the PCF used in simulations at 1060 nm.

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Table 1. Dispersion Coefficients for the GVD of the PCF Used in Simulations at 1060 nm ␤2 = −3.6517 ps2 / km ␤3 = 7.3401⫻ 10−2 ps3 / km ␤4 = −1.0603⫻ 10−4 ps4 / km ␤5 = 2.9671⫻ 10−7 ps5 / km ␤6 = −1.1768⫻ 10−9 ps6 / km ␤7 = 4.4803⫻ 10−12 ps7 / km ␤8 = −1.2034⫻ 10−14 ps8 / km ␤9 = 1.9675⫻ 10−17 ps9 / km ␤10 = −1.5088⫻ 10−20 ps10 / km

acteristic MI distance is LMI = 11.3 cm (see next page). The evolution in this case is dominated by soliton fission dynamics, with initial spectral broadening and temporal compression, followed by the orderly ejection of discrete solitons and associated dispersive wave generation. These features are all apparent in Fig. 2(a). In contrast, for the 250 fs pulse, the corresponding soliton order N = 88.2, the fission distance Lfiss = 6.25 cm, and the MI distance LMI = 1.13 cm. In this case, we observe significantly different propagation dynamics. Specifically, the longer initial pulse duration results in a slower rate of initial coherent spectral broadening and, after around 1 cm propagation, we note the growth from noise of MI sidebands outside the broadening pump bandwidth at ⬃820 and ⬃1500 nm. The subsequent evolution shows the breakup of both the spectral and temporal structures, but, although there are some similarities with the orderly soliton ejection features seen in Fig. 2(a), it is clear that the evolution contains spectral and temporal structure on much finer scales.

Fig. 2. (Color online) For pulses of (a) 50 fs and 10 kW peak power and (b) 250 fs and 100 kW peak power, the figure shows the SC spectral (left) and temporal (right) evolution over a propagation distance of 20 cm of PCF. Note the different temporal spans between (a) and (b).

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To examine the consequences of these dynamical differences on the stability of the SC at the fiber output, Fig. 3 shows results from an ensemble of 20 simulations carried out with different initial random noise seeds for each input pulse. For Fig. 3(a) the 50 fs pulse, and 3(b), the 250 fs pulse, each subfigure shows the spectra obtained from all the simulations in the ensemble (plotted in gray), the mean spectrum calculated from this ensemble (superimposed as the solid black curve), and the corresponding degree of coherence calculated by using Eq. (2). For the 50 fs, N = 5.6 input pulse, it is clear that the solitonfission-dominated dynamics leads to negligible shot-toshot variation in the spectral characteristics and unity coherence across the output spectrum. In contrast, there are significant shot-to-shot spectral fluctuations in the spectra for the 250 fs, N = 88.2 input pulse, and the output spectrum is essentially incoherent everywhere except for a small region around the pump, which is reflected in a near-zero average coherence of gav ⬇ 0.07. These simulation results confirm the qualitative discussion in Subsection 3.C concerning the dependence of SC stability on the input pulse characteristics. However, our simulations can be used to examine the SC stability characteristics more quantitatively by testing whether it is possible to develop general scaling laws relating the output SC coherence to the input pulse and fiber parameters. In particular, it is reasonable to consider that coherent SC generation in the anomalous dispersion regime requires that the characteristic soliton fission distance 共Lfiss ⬃ LD / N兲 be less than the distance over which MI has amplified input noise to a level where it significantly impacts on the evolution dynamics. By adapting criteria discussed in Ref. [100], we can estimate such a characteristic distance for MI as LMI ⬃ 16LNL. Specifically, the requirement Lfiss LMI then suggests that a condition such as N 16 may represent a general condition on the input pulse soliton number that ensures a high level of SC coherence over a wide range of experimental conditions. To test this hypothesis, we have carried out extensive additional simulations at 1060, 1070, and 1080 nm for a fixed fiber length of 25 cm, pulse durations in the range 50– 500 fs, and peak powers in the range 1 – 10 kW. This

Fig. 3. For pulses of (a) 50 fs and 10 kW peak power and (b) 250 fs and 100 kW peak power, the figure shows the output spectra (bottom curves, left axis) and corresponding degree of coherence (top curves, right axis) after propagation in 20 cm of PCF. For the spectral plots, the gray curves show the individual spectra from the ensemble, while the solid curve shows the calculated mean.


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allows us to span an input soliton order range up to N ⬃ 60. For each set of input pulse parameters, the average SC coherence is calculated and plotted against the input pulse soliton order, and Fig. 4 is a scatter plot of the results obtained. Although from this graph it is not possible to identify one unique scaling law in terms of N that universally applies to any arbitrary combination of initial conditions, it appears that, within the range of parameters we have considered, input pulses where N ⬍ 16 possess high coherence, and input pulses where N ⬎ 40 possess low coherence. This choice of threshold values for the coherent and incoherent regions of SC obtained with Yb3+ sources must be interpreted conservatively. To illustrate this explicitly, the simulation results at 1060, 1070, and 1080 nm are plotted separately in Fig. 4, where it can be seen that a different soliton number condition could be specified for each wavelength if desired. In particular, it is clear that, with pump wavelengths closer to the ZDW, higher values of N may be used while still maintaining the coherence. Additionally, we must stress that the precise condition on N that is obtained from simulations is likely to vary quantitatively depending on factors such as the fiber type (particularly the nonlinearity and higher-order dispersion) and the fiber length. For example, when considering SC generation under conditions typical of Ti:sapphire laser pumping around 800 nm in Ref. [35] the particular soliton number condition obtained is slightly different 共N ⬍ 10兲, arising from the different fiber and source parameters considered. So, when considering any particular range of PCF and source parameters likely to be encountered in any specific experiment, simulations should always be carried out to examine the parameter dependence of the SC coherence in detail. Nonetheless, given the largely different parameter ranges examined in Ref. [35] and in our current work, and the closeness of the conditions obtained, it is clear that examining the coherence properties of SC generation in terms of a soliton number condition provides useful physical insight and can be very usefully applied to experimental design. To further extend this analysis, we have specifically considered the sensitive dependence of SC coherence on the fiber length within our parameter range of Yb+3-pumped fiber [82]. To this end, we carried out addi-

Fig. 4. Average coherence of the SC versus soliton order N calculated for pump wavelengths of 1060 (circles), 1070 (crosses), and 1080 nm (diamonds) and with pulse durations in the range 50– 500 fs and peak powers 1 – 10 kW. Propagation in 25 cm of PCF is considered.

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tional simulations to study the dependence of the SC average coherence on both the input pulse parameters and the propagation distance, and the results are shown in Fig. 5. In these simulations, we determined the maximum input soliton order N that could be injected into a fiber of a given length L and still yield a coherent SC with average coherence gav ⬎ 0.95. At two different pulse durations (50 and 200 fs) and two different wavelengths (1060 and 1080 nm) as shown in the figure, the solid curve plots the maximum value of input soliton order N thus determined against propagation distance. Note that here it is convenient to normalize the propagation distance relative to the fission distance Lfiss. For a given propagation distance, values of N less than the maximum value plotted yield coherent spectra, so we can consider the regime of coherence as being to the left of the plotted curves, as indicated by the arrows. These results illustrate two general trends. First, for fixed pump wavelength and soliton order N, shorter pulses can propagate over longer distances while still maintaining high coherence. Second, for fixed pulse duration and soliton order N, a larger detuning from the ZDW into the anomalous dispersion region yields a decrease in the propagation distance over which coherence is maintained. As well as providing useful physical insight, Fig. 5 can also be applied to allow the optimization of specific experimental conditions. For example, considering pulses at 1060 nm of peak power P0 = 5 kW and duration (FWHM) 200 fs 共T0 = 113.5 fs兲, we have N = 15.8 and Lfiss = 22.3 cm, and, based on Fig. 5(a), we might expect coherence to be maintained under such conditions over several characteristic fission lengths. In this regard, however, we stress that these results assume only quantum-limited input noise, and technical and/or amplified spontaneous emission noise may induce coherence degradation over shorter distances. In addition, specific numerical simulations are indispensable in ensuring that high coherence is obtained while generating a bandwidth sufficient for the particular application in question. Nonetheless, the results in Fig. 5 can be very usefully applied in the initial design of SC experiments where coherence is desired. The representation in the figure can be

Fig. 5. For input wavelengths of (a) 1060 and (b) 1080 nm and pulse durations as indicated, the graph shows the maximum input soliton number N to use for a given normalized propagation distance L / Lfiss to obtain an average coherence gav ⬎ 0.95. The regime of coherence is to the left of the plotted curves as indicated by the arrows.

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used either to estimate the required fiber length given an input soliton order for a given pulse duration or to place constraints on the input soliton order if the fiber length is fixed. Moreover, in cases where a range of both source parameters and PCF fiber types are available, the use of plots such as those in Fig. 5 can allow an efficient convergence on a source–fiber combination that generates a SC with the desired characteristics, thus minimizing costly and time-consuming experimental trial and error. B. Supercontinuum Generation around 1550 nm In the simulations above at 1060 nm, we considered the common experimental configuration where the PCF used possesses only one ZDW and the pump wavelength lies slightly in the anomalous dispersion regime. However, it is also possible to fabricate PCF possessing two ZDWs, and theoretical and experimental studies using femtosecond pump pulses injected in such PCF have reported a number of novel effects. Specifically, the presence of a second ZDW has been shown to lead to both redshifted and blueshifted dispersive waves and simultaneous energy transfer across both ZDWs [101–104], a modified spectral recoil phenomenon that can stabilize frequency-shifting processes [105,106], and the generation of coupled soliton pairs [107]. These studies in PCF are also notable for shedding new light on previous experiments using dispersion-flattened fibers around 1550 nm [108,109] and, specifically, in providing an improved interpretation of the experiments studying SC generation in dispersion-flattened dispersion-decreasing fiber (DF-DFF) that were briefly discussed in Section 2. Moreover, although SC generation around 1550 nm is certainly possible in fibers with only one ZDW [110], we focus here on SC generation in DFDDF because, as we shall see, it presents a particular advantage in generating both spectrally uniform and highly coherent SC spectra. Aspects of SC generation in DF-DDF at peak power levels in the range 1 – 2 W have been considered in previous studies [32], but our aim here is to revisit the spectral broadening mechanisms in such fibers in detail, particularly in light of the additional insight provided by the many recent studies in PCF. We begin by considering lowpower SC generation with parameters typical of those used in previous studies [32], but we significantly extend this work to (i) provide a more detailed discussion of the SC dynamics, (ii) show explicitly that this form of SC generation is associated with high-coherence properties, (iii) examine the use of such DF-DDF for coherent SC generation at peak power levels in the much higher peak power range 200– 1200 W, and (iv) examine applications in pulse compression. We first consider input pulses at 1550 nm with peak power P0 = 1 W and duration (FWHM) ⌬␶ = 5 ps propagating in a 1 km length of DF-DDF. The input pulse parameters correspond to an average power of 50 mW at a repetition rate of 10 GHz, typical of amplified pulses from high-repetition-rate harmonically mode-locked fiber lasers. The dispersion parameters used in Eq. (1) are calculated from a longitudinally varying dispersion model [32], given by D共z兲 = D0共1 − z / L0兲 + 共1 / 2兲D2共␭ − ␭0兲2 with D0 = 6 ps nm−1 km−1, D2 = −2 ⫻ 10−4 ps nm−3 km−1, and L0

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= 900 m. Figure 6 plots the dispersion parameter D at selected distances to illustrate the longitudinal evolution of the convex dispersion from anomalous to normal values in the vicinity of the pump. We also note that for values of z ⬍ L0, the profile presents two ZDWs whose separation decreases with distance. At distances exceeding L0, the dispersion is normal at all wavelengths. We assume that the pump wavelength corresponds to ␭0, and we use values of ␥ = 5 W−1 km−1 and ␶shock = 0.822 fs. Figure 7 shows the spectral and temporal evolution observed in this case. A comparison with Fig. 2 immediately reveals that the dynamics of the SC formation in a DFDDF differs substantially from that observed in a fiber with single ZDW. The initial pump pulse here corresponds to a second-order soliton 共N = 2.3兲, and the adiabatically decreasing dispersion induces an initial phase of symmetrical spectral broadening and associated temporal compression. In this regard, we note that although such adiabatic pulse compression in DDF is usually analyzed for injected fundamental solitons [111,112], higher-order soliton compression in DDF also leads to similar dynamics [113]. It is instructive to examine the evolution of the key pulse parameters along the length of the DF-DDF as shown in Fig. 8(a), as this allows us to explicitly identify different phases of the SC generation process. Specifically, we see how the smooth increase in peak power and decrease in pulse duration are associated with the initial evolution of the propagating pulse toward a fundamental soliton. It is this adiabatic evolution in the soliton parameters that precludes the breakup of the pulse due to soliton fission. With further evolution toward the propagation distance of L0, the dispersion at the pump wavelength tends to zero, and the soliton order rapidly increases in parallel with significant increase in pulse peak power and significant temporal compression. For propagation beyond L0 the bandwidth stabilizes somewhat, while the temporal intensity broadens and develops strong modulation. Although we discuss these dynamics in more detail in the next paragraph, it is apparent even at this point that the absence of soliton fission would suggest that this propagation regime would be highly coherent. Indeed, this is the case and is shown explicitly in Fig. 8(b) based on an ensemble of 20 simulations performed with differ-

Fig. 6. Dispersion profile for the DF-DDF as described in the text, at selected fiber lengths as shown. The zero line is shown to highlight the variation in the separation between the two ZDWs with distance.


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Fig. 7. (Color online) For one simulation, evolution of spectral (left) and temporal intensity (right) for a 5 ps duration, 1 W peak power pulse at 1550 nm propagating in a DF-DDF as described in the text.

ent random noise seeds. We note negligible pulse-to-pulse intensity noise and near-perfect coherence across the SC bandwidth. The slight asymmetry observed in the output spectra arises from stimulated Raman scattering. We now analyze the physics of the spectral broadening mechanisms in this regime in more detail. The initial phase of propagation is associated with adiabatic compression and spectral broadening, but (after a propagation distance of around ⬃850 m) the spectral bandwidth increases sufficiently to induce dispersive wave generation symmetrically with respect to the pump wavelength [107]. In the case of a fiber with a single ZDW, dispersive wave generation is associated with a modification of the pump spectrum because of spectral recoil [105], but the simultaneous generation of two dispersive waves in the DFDDF fiber considered here leads to opposite spectral recoils, which balance. This prevents any shift in the spectrum of the pump and contributes to yield a continuous spectrum from the edges toward the center. Two other factors influence the flatness of the spectrum. First, although the phase-matching condition for dispersive wave generation is intrinsically narrowband, the longitudinally varying separation between the pump and the ZDWs results in a continuous and broadband range of phase-matched wavelengths. Second, after propagating beyond L0 when the dispersion is normal ev-

Fig. 8. Further details of the dynamical evolution shown in Fig. 7. (a) Evolution from one simulation of peak power P0共z兲, pulse duration (FWHM) ⌬␶共z兲 and (for propagation distance z ⬍ L0) the associated soliton order. (b) Results from 20 simulations showing the output spectra (bottom curves, left axis) and corresponding degree of coherence (top curve, right axis). For the spectral plots, the gray curves show the individual spectra from the ensemble while the solid line shows the calculated mean.

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erywhere, the residual pump components temporally broaden and overlap with the frequency-shifted dispersive waves, facilitating interaction through cross-phase modulation [91,114]. The combined result of these dynamics is a broad and flat SC almost free of fine structure. The residual spectral structure at the pump wavelength is a typical feature of higher-order soliton compression and corresponds to the broad time-domain pedestal (see also Fig. 9 below.) But it is very clear that the generation of coherent and flat SC spectra using DF-DFF is in sharp contrast to SC generation using single-ZDW PCF (such as shown in Fig. 3), where high coherence is often accompanied by a highly structured spectrum. Aspects of these dynamics can be conveniently visualized in the projected-axis spectrogram [115] plotted in Fig. 9. The specific dispersion profile of the fiber manifests itself in the fact that the dispersive waves are spread in time-frequency space in an S-like structure. A careful inspection of Fig. 9 also reveals the physical origin of the strong temporal modulation in the pulse wings as arising from the beating of the unconverted and dispersed pump and the dispersive wave components. The spectrogram clearly reveals how the unconverted pump is sufficiently temporally broadened to overlap with all the dispersive wave components. Of additional interest is the fact that the S-like characteristic of the spectrogram leads to a temporal chirp of this intensity modulation, and this is explicitly illustrated in the expanded portion of the intensity profile that is shown. Although propagation in DF-DDF was initially studied in the context of low-power SC generation for telecommunications applications, suitable scaling of the fiber dispersion can lead to the similar generation of broadband flat SC with input pulses over a much wider parameter range

Fig. 9. (Color online) Projected axis spectrogram of the output pulse of Fig. 7 illustrating how the flattened spectrum is associated with a complex modulated temporal structure. The gate function used in the spectrogram calculation had a duration (FWHM) of ⌬␶ = 1 ps, the same as the initial pump pulse.

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with peak powers over 2 orders of magnitude greater than those typically used in a telecommunications context. These results are summarized in Fig. 10. Here we plot results over 20 simulations for picosecond pulses around 1550 nm with significantly higher peak power than considered above. In Fig. 10(a) we consider 200 W peak power pulses of duration (FWHM) 1 ps propagating in a 7 m length of DF-DDF with D0 = 10 ps nm−1 km−1, D2 = −2 ⫻ 10−4 ps nm−3 km−1, L0 = 6 m, and ␥ = 5 W−1 km−1. This yields similar spectral characteristics as above, again with excellent coherence. The input soliton order here is N = 5.0. In Fig. 10(b) we show results for 1200 W peak power pulses of duration (FWHM) 1 ps propagating in a 1.4 m length of DF-DDF with D0 = 30 ps nm−1 km−1, D2 = −2 ⫻ 10−4 ps nm−3 km−1, L0 = 1.2 m, and ␥ = 5 W−1 km−1. Although we do not see the same degree of spectral flatness at this very high power level, we can still observe broadband spectral generation. The input soliton order here is N = 7.1. Also shown in these figures are the corresponding soliton evolution along the fiber to illustrate the similar smooth evolution dynamics to the low-power case in Fig. 8. A significant potential application of the flattened SC spectra generated in DF-DDF is in the field of pulse compression. The initial nonlinear evolution phase in DFDDF has previously been studied in detail to directly yield a temporally compressed pulse [113], but we have found that the extension of the initial propagation into the SC generation regime and the use of broadband spectral phase compensation would be expected to yield vastly improved compressed pulse quality. To illustrate this, we reconsider the propagation of the 200 W, 1 ps pulses in the DF-DDF described above and examine two possible configurations for compression of these pulses. The results are shown in Fig. 11. In the first case, we consider adiabatic soliton compression propagat-

Fig. 10. For input pulses at higher peak powers as shown, top, results from 20 simulations showing the output spectra (bottom curves, left axis) and corresponding degree of coherence (top curve, right axis). For the spectral plots, the gray curves show the individual spectra from the ensemble while the solid line shows the calculated mean. Bottom, evolution of the soliton number with propagation for distances less than L0.

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5. DISCUSSION AND CONCLUSIONS

Fig. 11. (a) Spectral and (b) temporal profiles illustrating pulse compression. The dashed curves show the spectra and nonlinearly compressed pulse after a propagation distance of 5 m in the soliton compression regime. The shading in the temporal plot is used to illustrate the broad pedestal. The solid curves show the spectra and compressed pulse after ideal spectral phase compensation after a propagation distance of 7 m in the SC generation regime with significant dispersive wave generation.

ing to the point where the temporal pulse duration is nonlinearly compressed to its minimum value. This corresponds to a fiber length of 5 m in this case and yields a pulse of duration (FWHM) 20 fs and of peak power 2.95 kW. However, this is accompanied by a very broad low-amplitude pedestal spanning over ⬃2 ps. These results are shown as the dashed curves (and the shaded region) in Figs. 11(a) and 11(b). In the second case, propagation is allowed to continue over the full fiber length of 7 m so that significant additional bandwidth is generated through dispersive wave dynamics. Although the output temporal pulse in this case is complex and could not be exploited directly, the high coherence across the spectrum would, however, be expected to lead to very high-quality compressed pulses after appropriate spectral phase compensation [116,117]. To illustrate this, Fig. 11(a) shows the spectrum after 7 m, and Fig. 11(b) shows the corresponding compressed pulse after numerical phase compensation. The compressed pulse here has a duration of 12 fs and peak power of 10 kW and a pedestal structure that is temporally localized within only ⬃150 fs of the central pulse. The improvement compared with only adiabatic soliton compression is apparent. We note that the dispersion parameters used in the simulations presented above are within the realistic range of what might be expected with recent advances in dispersion-flattened and PCF technology around 1550 nm [34,42,103,118]. Comparable work has also explored the development of PCF with dual ZDWs centered around 1 ␮m [119,120], so that exploring tapered PCF design in this wavelength range could provide further possibilities for coherent SC generation for longer pulses with Yb3+-based sources. When considering tapered PCF, however, the ideal dynamics described above can be modified by the variation in effective area along the propagation direction, but this can be included numerically in a straightforward manner [121]. Finally, we note that, when using dispersion-flattened fibers based on HNLF where splicing techniques are straightforward, the use of a discrete comblike dispersion map might be expected to provide a convenient and inexpensive experimental means of approximating the continuous DDF case [122,123].

A. Supercontinuum Design Criteria Although SC generation can certainly involve complex physics, it is nonetheless relatively straightforward to provide clear guidelines in terms of source and fiber parameters that will allow the generation of a SC having the desired properties in terms of bandwidth, spectral brightness, and coherence. Probably the most important first consideration relates to the desired coherence and stability, as this directly affects the type of pump source required. If one aims for a coherent SC with the maximum possible bandwidth (e.g., spanning over an octave for metrology applications), good results are obtained by using a sub-50 fs pulses of ⬃nanojoule energy with a pump wavelength slightly in the anomalous GVD regime of a highly nonlinear PCF possessing only one ZDW [33,35]. Assuming that one wishes to operate the laser source with maximum peak power, the fiber length should be cut to around the characteristic fission length where the initial phase of higherorder soliton evolution has already led to a broadband SC. If the fiber length is not matched to the source peak power, then either the coherence or the generated bandwidth may be nonoptimal. Specifically, with a fiber length that is too short, the SC spectral width does not attain its maximum possible value, whereas a fiber length that is too long may result in a larger bandwidth but at the expense of some coherence degradation. It is often unavoidable experimentally that one begins with a length slightly longer than Lfiss and successively cuts back to achieve optimal SC characteristics. An alternative approach to generate coherent SC is to use femtosecond pumping deep in the normal dispersion region, but, although this typically does lead to highly coherent SC, the bandwidth is relatively small. For some applications, achieving spectral flatness or strong spectral content in particular wavelength ranges (e.g., toward the blue or red edges of the SC) can be desirable. In this case, more careful engineering of the fiber dispersion characteristics is required. We have seen how the use of a longitudinally varying dispersion profile (DFDDF) can yield highly coherent flat SC in the picosecond regime, and this is a very promising solution where such fiber is available. However, even when there is no available DDF technology, the use of PCFs with two ZDWs provides the possibility to tailor spectral symmetry about the pump to some degree. In the femtosecond regime, this occurs as a result of processes such as soliton dispersive wave cross-phase modulation or simultaneous blueshifted and redshifted dispersive wave generation and would be expected to yield coherent spectra in the case of sub-50 fs pumping [35,124] although the spectral flatness may not be as good as with the use of DF-DDF. For applications where coherence properties are unimportant, the utility of the SC source resides in its high brightness and broad bandwidth. In this case, any conveniently available high-power source in the femtosecond to the CW regime can in principle be combined with an appropriate PCF, with the SC bandwidth generated being critically dependent on the proximity of the pump wavelength to the fiber ZDW. Optimal bandwidth would be ex-

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pected for pumping close to the ZDW with small anomalous GVD, and it is also possible to engineer dispersion to yield improved spectral flatness [96]. B. Conclusion Our intention here has been to provide a brief survey of SC generation as relevant to commonly encountered fiberbased pump sources around 1060 and 1550 nm. It has not been possible to be completely exhaustive, but a number of important points have been covered, and specific scenarios for SC generation around both of these pump wavelength regimes have been examined. Although much work studying SC generation focuses on the fiber nonlinearity characteristics, it is likely that future work in this field will establish more systematic guidelines for dispersion design and optimization. With fiber tapering technologies and the use of comblike dispersion profiling both becoming commonplace, we anticipate that the study of longitudinal dispersion management strategies will be increasingly important in the development of fiber SC sources.

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10. 11. 12.

13.

14.

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ACKNOWLEDGMENTS G. Genty acknowledges the Academy of Finland and the Emil Aaltonen foundation. S. Coen thanks the New Zealand Foundation for Research, Science & Technology and The Marsden Fund of The Royal Society of New Zealand. J. M. Dudley acknowledges financial support from the French National Ministry of Education, Research and Technology, the Centre National de la Recherche Scientifique (CNRS) and the Institut Universitaire de France. We also note that where the names of commercial products are given in this paper, it is for purposes of completeness and does not imply formal endorsement by the authors.

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