5.4 Plots of approximate dependences b (x, ξ), i.e. based on Eqs. 5.34 and 5.39, .... nipulation of cells; and a light source for OCT and multi-photon microscopy. .... the much higher carrier frequency of light (1014 to 1015 Hz) in a lightwave system. ...... [2.18] Haus, H. A., âA Theory of forced mode lockingâ, IEEE J. of Quantum ...
Shaping ultra-short optical pulses by semiconductor structures in the mode-locking regime and technique of measuring their parameters by
Pedro Moreno Zarate A Dissertation Submitted to the program in Optics, “Optics Department” in partial fulfillment of the requirements for degree of
Doctor of Science in Optics at the National Institute for Astrophysics, Optics and Electronics October-2012 Tonantzintla, Puebla Supervised by: Dra. Svetlana Mansurova, Dr. Alexandre S. Shcherbakov, Titular Research Scientists, Optics Department, INAOE. c
INAOE 2012 The author hereby grants to INAOE permission to reproduces and to distribute copies of this thesis document in whole or in part
To God for giving me the privilege to meet you. To Yasmin Azucena Rodriguez Brito To be the best person whose has been crossing in my life. To Klein my little fat cat. To my mother and brothers Blanca Zarate Romero, Rene Moreno Zarate, Jose Moreno Zarate, Raul Ocelotl and Angela Ocelotl Zarate
Thanks i
ACKNOWLEDGEMENTS First of all, I want to thank to the National Council of Science and Technology (CONACyT), for supporting me financially with the scholarship number 160159, and the CONACyT projects N o. 61237 and N o. 84922. I would like to thanks all co-authors of “Articles” and “ Proceedings” that were detached from this thesis. From Spain; the Doctors Joaqun Campos Acosta, Alicia Pons Aglio. From Russian Federation; the Doctors Alexey Yu. Kosarsky, Sergey A. Nemov, Yurij V. Il’in, Il’ya S. Tarasov. For their direct contributions in my professional development, as well as, giving me the opportunity to participate in an international project. In particular, I would like to express my gratitude to the Doctors; Svetlana Mansurova and Alexander S. Shcherbakov, whose have given me the opportunity to develop this thesis work under their direction. Thanks for their dedication, effort, patience, support, comments, knowledge, corrections, teachings, for the trust you have given me and the chance to learn of you. “I am flaying” I would like to thanks my examiners. From the National Institute for Astrophysics, Optics and Electronics (INAOE); to the Doctors, Gabriel Martinez Niconoff, Julio Cesar Ramirez San Juan and Ruben Ramos Garcia. From the Benemeritus Autonomous University of Puebla (BUAP) to the Doctor Georgina Beltran Perez. From the Center of Investigation in Engineering and Applied Sciences (CIICAp) to the Doctor Darwin Mayorga Cruz, whose didn’t just make remarks to improve the thesis but also because of their support and advice. I also want to thank my friends and colleagues from which I have learned many things. Specially, those who were supporting me direct or indirectly in the process of this project; Raul Ochoa Valiente, Daniel Sanchez Lucero, Gustavo Martinez, Alejandro Ramos Cabrera, Ivan Hernandez Romano, Ana Luz Munos Zurita, Elizabeth Cruz Sosa, Zarina Pacheco Juarez, Soledad Luna Martinez, Carmen Garcia Espinosa de los Monteros, Luis Olivos Perez, Jaliel Lopez Popoca, Jose Francisco Ramirez Lucero, Cornelio Silva Lopez and my exstudents in NovaUniversitas; thanks for supporting me in the hard times and advising me in many others. Thank you for your friendship. Finally, I thanks to all people who helped me and whose names I might omitted. ii
A BSTRACT We discuss an approach for characterizing the train-average parameters of low-power picosecond optical pulses with the frequency chirp, arranged in high-repetition-frequency trains, in both time and frequency domains. This approach had been previously applied to rather important case of pulse generation when a single-mode semiconductor heterolaser operates in a multi-pulse regime of the active mode-locking with an external single-mode fiber cavity. This approach involves the joint Wigner time-frequency distributions, which can be created for those pulses due to exploitation of a novel interferometric technique. Also the modified scanning Michelson interferometer has been chosen for obtaining the field-strength auto-correlation functions, with the joint time frequency distribution approach for the pulses reconstruction. In fact, we have presented the key features of a new experimental technique for accurate and reliable measurements of the train-average temporal width and the frequency chirp of picosecond optical pulses in high-repetition rate trains. The InGaAsP/InP-heterolasers, operating at 1320 nm wavelength range, have been used within the experiments. In fact, the trains of optical dissipative solitary pulses, which appear under a double balance between mutually compensating actions of dispersion and nonlinearity as well as gain and optical losses, are under characterization. However, in the contrast with the previous studies, now we touch an opportunity of describing two chirped optical pulses together. The main reason of involving just a pair of pulses is caused by the simplest opportunity for simulating the properties of just a sequence of pulses rather then an isolated pulse. However, this step leads to a set of specific difficulty inherent generally in applying joint time-frequency distributions to groups of signals and consisting in manifestation of various false signals or artefacts. This is why the joint Chio-Williams time-frequency distribution and the technique of smoothing are under preliminary consideration here. Together with this, we present an advanced approach to describing low-power trains of bright picosecond optical dissipative solitary pulses with an internal frequency modulation, where the chosen schematic arrangement, process of the active mode-locking is caused by a hybrid nonlinear cavity consisting of this heterolaser and an external rather long single-mode optical fiber exhibiting square-law dispersion, cubic Kerr nonlinearity, iii
and small linear optical losses. The approach makes possible taking the modulating signals providing non-conventional composite regimes of a multi-pulse active mode-locking. Then the appearing nonlinear Ginzburg-Landau operator to the parameters of dissipative solitary pulses is described via exploiting an approximate variational procedure involving the technique of trial functions. Additionally, we propose estimating the average parameters of ultra short optical pulses with asymmetric envelopes, which are arranged in high-repetition trains. In so doing, one can utilize the advantages of applying the time triple auto-correlations to this problem. The potential advantages of similar, broadly speaking, more complicated auto-correlation function in comparison with the second order one consist in the capability of recovering signals almost unambiguously and in the possibility of revealing asymmetry of trainaverage pulse envelope as well as the phase. We estimate the principal parameters of such interferometer where the triple auto-correlation functions are measured using the effect of the third harmonic generation in two different optical configurations. One of them gives us the third harmonic directly within impinging a triplet of light beams, leaving that scanning interferometer, at nonlinear crystal. The other scheme implies a two cascade arrangement when, first, two beams generate the second harmonic and then, this second harmonic is mixed with the third beam of an initial frequency. The conversion efficiencies are analyzed and compared in various materials. Together with this, we examine the principal aspects of implementing the respective experimental arrangements. Another important issue, which is considered here, is a new application of the photoelectro-motive force detectors. We select this type of photo-detectors due to their high pass transfer function that gives us higher vibration stability and the ability to adapt to the compensated wave-fronts. Also, potential advantages and disadvantages for possible application of these photo-detectors to measuring the triple auto-correlations are studied. In particular, the experimental characterizations are presented for two different types of materials, namely, for gallium arsenide (GaAs) semiconductor and for poly-fluoren 6co-triphenyldiamine (P F 6 − T P D) photoconductor polymer, which both exhibit the photo-electro-motive force effect.
iv
Contents 1
2
Background and introduction
2
1.1
Ultra short optical pulse background . . . . . . . . . . . . . . . . . .
2
1.1.1
Introduction to the shaping ultra short optical pulse . . . . . . .
3
1.1.2
Introduction to Active Mode-locking technique . . . . . . . . .
3
1.1.3
Measurement techniques of ultrashort optical pulse . . . . . . .
8
1.1.4
Introduction of the Wigner distribution Function . . . . . . . . .
10
1.2
Problems to be resolved in this thesis are: . . . . . . . . . . . . . . . .
11
1.3
Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
1.4
Brief description of the thesis contents . . . . . . . . . . . . . . . . .
11
FUNDAMENTAL CONCEPTS 2.1
Mode Locking in semiconductor laser
. . . . . . . . . . . . . . . . .
17
Mode-locked semiconductor laser . . . . . . . . . . . . . . . .
18
2.2
Optical pulses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.3
Originating the Joint Wigner time-frequency distribution. . . . . . . . .
28
2.3.1
The joint time-frequency distribution . . . . . . . . . . . . . .
31
2.3.2
Local auto-correlation approach
. . . . . . . . . . . . . . . .
34
2.3.3
The Wigner time-frequency distribution for the Gaussian pulse. .
35
2.3.4
The Wigner time-frequency distribution for the Gaussian pulse
2.1.1
2.4 3
17
with a high-frequency filling. . . . . . . . . . . . . . . . . . .
36
The triple auto-correlation function for a one-dimensional signal . . . . .
41
Initial stage of the active mode-locking in semiconductor heterolasers
50
3.1
50
The initial stage of the active mode-locking description . . . . . . . . . v
CONTENTS 3.1.1 3.2
General theoretical consideration . . . . . . . . . . . . . . . .
Detection system of average time parameters for continuous trains of ultra-short optical pulses in NIR. . . . . . . . . . . . . . . . . . . . .
54
3.3
Algorithm of operation
. . . . . . . . . . . . . . . . . . . . . . . .
56
3.4
Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . .
59
3.5
Shaping a single optical pulses using a acousto-optics tunable filters modulation in transversal configuration 3.5.1
3.6 4
51
. . . . . . . . . . . . . . . . . .
60
Experimental results . . . . . . . . . . . . . . . . . . . . . .
63
Conclusion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
Characterization of the time-frequency parameter inherent in the radiation of semiconductor heterolasers using interferometric technique for a Gaussian pulses 4.1
4.2 4.3 5
68
The Joint Wigner distribution for a pair of Gaussian pulses
. . . . . . .
68
4.1.1
The joint Choi – Williams time-frequency distribution technique .
71
4.1.2
Procedure of smoothing the Wigner time-frequency distribution. .
75
Experimental studies . . . . . . . . . . . . . . . . . . . . . . . . . .
75
4.2.1
. . . . . . . . . . . . . . . . .
79
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
81
Characterizing Optical Pulses
Conclusion
Analysis of originating ultra-short optical dissipative solitary pulses in the actively mode-locked semiconductor heterolasers with an external fiber cavity.
85
5.1
Process of shaping regular trains of optical solitary pulses . . . . . . . .
5.2
A multi-pulse regime of the active mode-locking in semiconductor heterostructure with an external cavity . . . . . . . . . . . . . . . . . . .
5.3
85 87
Evolution equation governing the dynamics of shaping ultra-short optical dissipative solitary pulses trains in semiconductor heterolaser with an external fiber cavity . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1
91
Quasi-Linear Evolution equation governing the mode-locking process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
94
vi
CONTENTS 5.3.2
Application of an approximate variational procedure to estimating the parameters of optical dissipative solitary pulses in an external nonlinear fiber cavity with SQUARE-law dispersion and linear losses.
5.4
. . . . . . . . . . . . . . . . . . . . . . . . .
Measuring the train-average parameters of picosecond optical pulses in high-repetition-rate trains
5.5
6
. . . . . . . . . . . . . . . . . . . . . . . 102
Experimental studies . . . . . . . . . . . . . . . . . . . . . . . . . . 104 5.5.1
5.6
96
Characterization of optical dissipative solitary pulses. . . . . . . 106
Conclusion
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
Application of the correlation function in the characterization of the ultrashort optical pulses 6.1
113
The triple auto-correlation function for the chirped Gaussian pulse. 6.1.1
. . . 113
The algorithm of recovering the temporal signal from its triple auto-correlation function. . . . . . . . . . . . . . . . . . . . . 115
6.2
Principal aspect of the three-beam auto-correlator . . . . . . . . . . . . 117 6.2.1
Analysis of three-beam auto-correlator via direct THG . . . . . . 118
6.2.2
Theoretical aspect of the direct third harmonic generation efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
6.2.3 6.3
Analysis of three-beam auto-correlator via cascade THG . . . . . 125
Applying Non-Steady-State Photo-Emf Technique to Detection of Higher Order Auto Correlation Functions of Ultrashort Optical Pulses . . . . . . 132 6.3.1
Theoretical principles of the Photo-EMF effect . . . . . . . . . 132
6.3.2
Main aspect of the autocorrelation function in a Photo-EMF detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.3.3
Theoretical analisis of photo-EMF technique using three beam interferometer for detection of higher order correlation function. . 138
6.3.4 6.4 7
Conclusion
Statements 7.1
Characterization of Photo-EMF detector
. . . . . . . . . . . . 146
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 155
General conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 155
vii
CONTENTS 7.2
Statements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
viii
List of Tables 6.1
Estimating the conversion efficiency η for the direct THG
6.2
Estimating the conversion efficiency η for the cascade THG
6.3
Characteristic of the adaptive photo-detector based on the photo-EMF effect . . 150
ix
. . . . . . . . . . 124 . . . . . . . . . 132
List of Figures 1.1
Schematic diagram of a Pulse Shaper.[1.1]. . . . . . . . . . . . . . . . . .
3
1.2
Laser mode structure.[1.1]. . . . . . . . . . . . . . . . . . . . . . . . .
5
1.3
Ultrashort pulse mode locking shape process.[1.1].
. . . . . . . . . . . . .
6
1.4
Examples of short and ultrashort pulse light in the time domain with their corresponds physic parameter. [1.1]. . . . . . . . . . . . . . . . . . . . . . .
9
2.1
Ideal mode locking: the output of a laser with 10 completely locked modes [2.4].
18
2.2
Active mode locking of a diode laser using a current modulation: (a) the current waveforms, (b) carrier density, and (c) optical output against time [2.4].
2.3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
The Wigner time-frequency distribution for the Gaussian pulse with T = 1 and the varying parameter b: (a) b = 0 and (b) b = 2.
2.6
20
Configuration of a diode laser used in the traveling-wave numerical approach to active mode locking Ra is the residual reflectivity of the AR coating [2.4]. . . .
2.5
19
Schematic of a ring cavity used in self-consistent profile approach lo mode locking. [2.4].
2.4
. . . .
. . . . . . . . . . . . .
36
The Gaussian pulse with T = 1: the power density profile (a) and the spectral density profiles (b) with the varying parameter of the frequency chirp: blue line for b = 0, violet line for b = 2, olive line for b = 4, and green for b = 6.
2.7
. .
The plots of I (t) and J (t) with : (a) b = 0, T = 1, Ω = 10; (b) b = 4, T = 1, Ω = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8
38
Spectral intensity of Gaussian pulses: b = 0, T = 1, Ω = 10 blue line; b = 4, T = 1, Ω = 10 violet line . . . . . . . . . . . . . . . . . . . .
2.9
37
39
Field-strength auto-correlation functions for the Gaussian pulses with: (a) b = 0, T = 1, Ω = 40 ; (b) b = 4, T = 1,Ω = 40.
x
. . . . . . . . . . . . . . . . . .
41
LIST OF FIGURES 2.10 The envelope (a), triple auto-correlation (b), and bispectrum (c) of a Gaussian pulse 44 3.1
Exhibiting the diffusive instability. (a) The boundary curve G(ω) ; an area lying above G(ω) corresponds to growing the initial perturbation, while an area placing below is associated with attenuating the initial perturbations; (b) The evolution of an initial perturbation, i.e. the dynamics of pulse growing and stabilizing. 54
3.2
Scheme of the auto-manual opto-electronic detecting system of average time parameters inherent in continuous high repetition rate trains of ultra-short optical pulses in near-infrared range . . . . . . . . . . . . . . . . . . . . . . .
55
3.3
Principle scheme for shaping both auto- and cross-correlation functions. . . . .
56
3.4
Shaping the response function (solid line) conditioned by the incoming ultrashort Gaussian optical pulse (dashed line) and the exponential transfer function (dotted line); the scales of curves are changed to illustrate better.
3.5
. . . . . . .
The digitized oscilloscope trace for the auto-correlation function for a spikemode free oscillation with an average spike width of about 0.7 ps. . . . . . .
3.6
58 59
. The digitized oscilloscope traces related to a regular pulse train: (a) the autocorrelation function for an average pulse width of about 6.3 ps; (b) the output signal from a high-speed photodetector.
3.7
. . . . . . . . . . . . . . . . . .
The Bragg vector diagram and corresponding physical configuration for the diffraction of light from retreating sound wave [3.9]. . . . . . . . . . . . . . .
3.8
60 61
Collinear coupling between a z − polarized (extraordinary) and a y polarized
. . . . . . . . .
62
Wave-vector diagram for acousto-optic interaction in anisotropic [3.9]. . . . . .
63
3.10 The acousto-optics experimental set-up in a longitudinal configuration. . . . . .
64
(ordinary) optical beam by a shear Szy acoustic wave [3.9].
3.9
3.11 (a) and (b) show the experimental result in CaMoO4 crystal which is used like a Band-pass filter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
64
3.12 (a) show the electronic RF input signal, in the acousto optical crystal approximately in 1.5µs, (b) show the photodetector out put approximately in 15µs. . . .
4.1
65
The reduced joint Wigner time-frequency distribution for a pair of identical Gaussian pulses with T = 1, t0 = 6, and the varying parameter b of the frequency chirp: (a) b = 0 and (b) b = 3. . . . . . . . . . . . . . . . . .
70
xi
LIST OF FIGURES 4.2
Plots of partial 3D-distributions in time and frequency domains, respectively. A pair of the Gaussian pulses is taken with T = 1 and t0 = 6: the power density profile (a) and the spectral density profiles (b) with the varying parameter of the frequency chirp b. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3
71
Plots of 2D cross-sections for partial distributions in frequency domain. A pair of the Gaussian pulses with T = 1 and t0 = 6; two spectral density profiles with the fixed frequency chirp parameter: (a) b = 0 and (b) b = 3. . . . . . .
4.4
The illustrating representations for the Choi-Williams function KChW of kernel with s = 10: a 3D-plot of KChW (a) and two central 2D cross-sections (b). .
4.5
71 73
Two examples of the joint Choi-Williams time-frequency distributions for a pair of identical Gaussian pulses with T = 1, t0 = 6, and the varying parameter b of the frequency chirp: (a) b = 0 and (b) b = 3. . . . . . . . . . . . . . .
4.6
74
Various plots of the needed smoothing function L (t − t1 , ω − ω1 ): (a) α = 10, β = 1, γ = 0; (b) α = 100, β = 1, γ = 0; (c) α = 10, β = 1, γ = +3; and (d) α = 10, β = 1, γ = −3. . . . . . . . . . . . . . . . . . .
76
. . . . . . . . . . . . .
77
4.7
Schematic arrangement of the experimental set-up.
4.8
Radiation spectra inherent in semiconductor heterolaser operating at the wavelength λ = 1320 nm: (a) without an external modulation; (b) with an external sinusoidal modulation, i.e. in the active mode-locking regime.
4.9
. . . . . . . .
78
The digitized oscilloscope traces related to a regular pulse train: (a) the trainaverage auto-correlation function; the pulse width of this interferogram, measured on a level of 1/e for the intensity contour, was estimated by 4.4 ps; (b) the output signal from a high-speed photodetector; a train of the same ultra-short optical pulses with the repetition frequency f ≈ 718 M Hz was detected with the time resolution of about 300 ps. . . . . . . . . . . . . . . . . . . . .
79
4.10 A pair of the Wigner time-frequency distribution for the Gaussian pulses obtained from the performed estimation with T = 2.73 ps and the b = 0.84·10−4 as well as from the experiment with T = 2.2 ps and the b = 1.46 · 10−4 . . .
5.1
A set of N periodic sequences of the modulating pump currents; the constant
. . . . . . . . . . . . . . . . . . . . . .
89
Arrangement of the combined evolution equation 5.7. . . . . . . . . . . . .
91
current background is omitted.
5.2
81
xii
LIST OF FIGURES 5.3
Plots of exact (solid lines) and approximate (dashed lines)τ (x, ξ) for ξ = 0.3 (a) and ξ = − 0.3 (b). . . . . . . . . . . . . . . . . . . . . . . . . . . 101
5.4
Plots of approximate dependences b (x, ξ), i.e. based on Eqs. 5.34 and 5.39, for ξ = 0.3 (a) and ξ = − 0.3 (b).
5.5
. . . . . . . . . . . . . . . . . . . . . 102
Plots of approximate dependences |a (x, ξ)|, i.e. based on Eqs. 5.34 and 5.39, for ξ = 0.3 (a) and ξ = − 0.3 (b) . . . . . . . . . . . . . . . . . . . . 102
5.6
Schematic arrangement of the experimental set-up.
. . . . . . . . . . . . . 105
5.7
The digitized oscilloscope traces related to regular trains of optical dissipative solitary pulses with (a) N = 16 and (b) N = 32 detected by a high-speed photodetector with the time resolution of about 300 ps. . . . . . . . . . . . 106
5.8
The digitized oscilloscope traces related to regular trains of optical dissipative solitary pulses with N = 8 (here, 2 periods are shown) detected by a highspeed photodetector with the time resolution of about 300 ps (a) and (b) the auto-correlation function registered by the modified auto-correlator with a lowspeed photodetector. . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.9
A pair of the joint Wigner time-frequency distributions for the dissipative solitary optical pulses, having train-average Gaussian shapes of envelopes obtained from the numerical performed estimation with T = 7.3 ps and b = 5.06 · 10−4 as well as from the experiments with T = 8.5 ps and b ≈ 8.4 · 10−4 . . . . . 109
6.1
Triple auto-correlation function for the chirped Gaussian pulse with a = 0.5 and b = 1.0: (a) real-valued part, (b) imaginary-valued part. . . . . . . . . . 114
6.2
Bispectrum for the chirped Gaussian pulse with a = 0.5 and b = 1.0: (a) real-valued part, (b) imaginary-valued part. . . . . . . . . . . . . . . . . . 115
6.3
An example of the triple auto-correlation function inherent in a high-repetition train of asymmetric optical pulses. . . . . . . . . . . . . . . . . . . . . . 117
6.4
(a) Direct optical schemes for a third harmonic generation. (b) A three-beam scanning interferometer for registering the triple auto-correlation function of high-repetition-rate trains including ultra-short optical pulses direct THG.
6.5
. . . 118
Gaussian pulses moving with different linear velocities V1 and V2 relative to an immovable one (a);illustration to appearing the effective length of interaction Lef f for a triplet of the interacting optical pulses. . . . . . . . . . . . . . . 120
xiii
LIST OF FIGURES 6.6
(a) Cascade optical schemes for a third harmonic generation. (b) A three-beam scanning interferometer for registering the triple auto-correlation function of high-repetition-rate trains including ultra-short optical pulses via cascade THG .
6.7
125
Gaussian pulses moving with different linear velocities V1 and V2 relative to an immovable one; illustration to appearing the effective length of interaction Lef f for a double of the interacting optical pulses in cascade process. . . . . . . . . 126
6.8
Theoretical dependencies of the non-steady-state photo-EMF of (a) modulation frequency, (b) amplitude of phases modulation and spatial frequency . . . . . . 134
6.9
Typical experimental arrangement of the two Gaussian pulse correlator based on non-steady-state photo-EMF effect. . . . . . . . . . . . . . . . . . . . . 136
6.10 Example of the pulse correlation function obtained with the Photo-EMF detector, with b = 1 and T = 1. . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.11 Experimental arrangement to measurement of the high order correlation function with Photo-EMF detector using a transmission configuration . . . . . . . . . 140
6.12 . Figure (a) it si the correlation function calculated from Eq. 6.66 (c), and fig. (b) its bispectrum from Eq. 6.67; T=1 . . . . . . . . . . . . . . . . . . . 145
6.13 Experimental arrangement to the Photo-EMF characterization (a) transmission and (b)reflection configuration
. . . . . . . . . . . . . . . . . . . . . . 147
6.14 Experimental dependencies of the non-steady-state photo-EMF of (a) modulation frequency, (b) amplitude of phases modulation and (c) spatial frequency in GaAs, I0 = 40mW/cm2 , m = 0.99, RL = 10kOhms . . . . . . . . 148
6.15 Experimental dependencies of the non-steady-state photo-EMF of (a) modulation frequency, (b) amplitude of phases modulation and (c) spatial frequency in P F 6 − T P D, I0 = 40 mW/cm2 , m = 0.99, RL = 10 M Ohms and ∆ = 500mrad . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
1
Chapter 1
Background and introduction 1.1
Ultra short optical pulse background
In optics, an ultrashort pulse of light is an electromagnetic pulse whose time duration is on the order of the pico and sub-pico second [1.1]-[1.2]. Since the advent of the laser nearly 40 years ago, there has been a sustained interest in the quest to generate ultrashort laser pulses in the picosecond (10−12 s) and sub-pico-second (10−15 s) range. Reliable generation of pulses below 100 fs in duration occurred for the first time in 1981 with the invention of the colliding pulse modelocked (CPM) ring dye laser [1.2]. Subsequent nonlinear pulse compression of pulses from the CPM laser led to a series of even shorter pulses, [1.3]-[1.7] culminating in pulses as short as 6 fs, a record which stood for over a decade. Six femtoseconds in the visible corresponds to only three optical cycles, and therefore such pulse durations are approaching the fundamental single optical cycle limit. Further rapid progress occurred following the demonstration of femtosecond pulse generation from solidstate laser media in the 1990 time frame[1.8]. sub-pico-second solid-state lasers bring a number of important advantages compared to their liquid dye laser counterparts, including substantially improved output power and stability and new physical mechanisms for pulse generation advantageous for production of extremely short pulses. sub-pico-second solidstate laser technology has now advanced to the point that pulses below 6 fs can be generated directly from the laser [1.9]-[1.15]. Equally important, the use of solid-state gain media has also led to simple, turn-key sub-picosecond lasers, and many researchers are now setting their sights on practical and low cost ultrafast laser systems suitable for real-world applications [1.16]-[1.23]. Ultra-short optical pulses are used in a variety of fundamental research fields including nondestructive, noninvasive measurement for substance properties based on nonlinear optical responses; terahertz waves generation which is recently drawing attention as a drug identification technique and as a substitute for X ray inspection; as well as ultra-precision, thermal damage-free machining of glass, metal and semiconductor with the nanometer and micrometer-scale precision; biomedical applications such as cleavage and manipulation of cells; and a light source for OCT and multi-photon microscopy. In high-intensity physics, studying phenomena such as; multi-photon ionization, high harmonic generation, optics communication, or the generation of even shorter pulses with attosecond durations. Depending on the required pulse duration, pulse energy, and pulse repetition rate, different
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1.1. ULTRA SHORT OPTICAL PULSE BACKGROUND methods of; pulse shaping; pulse compression and pulse characterization are used; in generation; shaping and characterization of the optical pulse parameters.
1.1.1
Introduction to the shaping ultra short optical pulse
. The pulse shaping [1.1] is a technique that modifies the temporal profile of an ultrashort pulse from a laser. Pulse shaping can be used to shorten-elongate the duration of optical pulse, or to generate more complex pulses 1.1.
Figure 1.1: Schematic diagram of a Pulse Shaper.[1.1]. Generation of sequences of ultrashort optical pulses is key in realizing ultra high speed optical networks, Optical Code Division Multiple Access (OCDMA) systems, chemical and biological reaction triggering and monitoring etc. Based on the requirement, pulse shapers may be designed to stretch, compress or produce a train of pulses from a single input pulse. The ability to produce trains of pulses with sub-pico-second or picosecond separation implies transmission of optical information at very high speeds [1.1].
1.1.2
Introduction to Active Mode-locking technique
And other important technique to shape pulse is named; Mode-locking is a technique in optics by which a laser can be made to produce pulses of light of extremely short duration, on the order of picoseconds (10−12 s) or sub pico seconds (10−15 s). The basis of the technique is to induce a fixed phase relationship between the modes of the laser’s resonant cavity. The laser is then said to be phase-locked or mode-locked. Interference between these modes causes the laser light to be produced as a train of pulses. Depending on the properties of the laser, these pulses may be of extremely brief duration, as short as a few femtoseconds. Although laser light is perhaps the purest form of light, it is not of a single, pure frequency or wavelength. All lasers produce light over some natural bandwidth or range of frequencies.
3
1.1. ULTRA SHORT OPTICAL PULSE BACKGROUND A laser’s bandwidth of operation is determined primarily by the gain medium that the laser is constructed from, and the range of frequencies that a laser may operate over is known as the gain bandwidth. The second factor that determines a laser’s emission frequencies is the optical cavity or resonant cavity of the laser. In the simplest case, this consists of two plane (flat) mirrors facing each other, surrounding the gain medium of the laser. Since light is a wave, when bouncing between the mirrors of the cavity the light will constructively and destructively interfere with itself, leading to the formation of standing waves or modes between the mirrors. In a simple laser, each of these modes will oscillate independently, with no fixed relationship between each other, in essence like a set of independent lasers all emitting light at slightly different frequencies. The individual phase of the light waves in each mode is not fixed, and may vary randomly due to such things as thermal changes in materials of the laser. In lasers with only a few oscillating modes, interference between the modes can cause beating effects in the laser output, leading to random fluctuations in intensity; in lasers with many thousands of modes, these interference effects tend to average to a near-constant output intensity, and the laser operation is known as a c.w. or continuous wave. If instead of oscillating independently, each mode operates with a fixed phase between it and the other modes, the laser output behaves quite differently. Instead of a random or constant output intensity, the modes of the laser will periodically all constructively interfere with one another, producing an intense burst or pulse of light. Such a laser is said to be mode-locked or phaselocked. These pulses occur separated in time by τ = 2L/c, where τ is the time taken for the light to make exactly one round trip of the laser cavity. This time corresponds to a frequency exactly equal to the mode spacing of the laser, ∆τ = 1/ν 1.2. These standing waves form a discrete set of frequencies, known as the longitudinal modes of the cavity. These modes are the only frequencies of light which are self-regenerating and allowed to oscillate by the resonant cavity; all other frequencies of light are suppressed by destructive interference (fig. 1.3). For a simple plane-mirror cavity, the allowed modes are those for which the separation distance of the mirrors L is an exact multiple of half the wavelength of the light λ, such that L = qλ/2, where q is an integer known as the mode order. Methods for producing mode-locking in a laser may be classified as either active or passive. • Active methods typically involve using an external signal to induce a modulation of the intra-cavity light. • Passive methods do not use an external signal, but rely on placing some element into the
4
1.1. ULTRA SHORT OPTICAL PULSE BACKGROUND
Figure 1.2: Laser mode structure.[1.1]. laser cavity which causes self-modulation of the light. The most common active mode-locking technique places a standing wave acousto-optic modulator into the laser cavity. When driven with an electrical signal, this produces a sinusoidal amplitude modulation of the light in the cavity. Considering this in the frequency domain, if a mode has optical frequency ν, and is amplitude-modulated at a frequency f , the resulting signal has sidebands at optical frequencies ν − f and ν + f . If the modulator is driven at the same frequency as the cavity-mode spacing ∆ν, then these sidebands correspond to the two cavity modes adjacent to the original mode. Since the sidebands are driven in-phase, the central mode and the adjacent modes will be phase-locked together. Further operation of the modulator on the si-
5
1.1. ULTRA SHORT OPTICAL PULSE BACKGROUND debands produces phase-locking of the ν − 2f and ν + 2f modes, and so on until all modes in the gain bandwidth are locked. As said above, typical lasers are multi-mode and not seeded by a root mode. So multiple modes need to work out which phase to use. In a passive cavity with this locking applied there is no way to dump the entropy given by the original independent phases. This locking is better described as a coupling, leading to a complicated behavior and not clean pulses. The coupling is only dissipative because of the dissipative nature of the amplitude modulation. Otherwise the phase modulation would not work.
Figure 1.3: Ultrashort pulse mode locking shape process.[1.1]. This process can also be considered in the time domain. The amplitude modulator acts as a weak shutter to the light bouncing between the mirrors of the cavity, attenuating the light when it is “closed”, and letting it through when it is "open". If the modulation rate f is synchronised to the cavity round-trip time τ , then a single pulse of light will bounce back and forth in the cavity. The actual strength of the modulation does not have to be large; a modulator that attenuates 1% of the light when “closed” will mode-lock a laser, since the same part of the light is repeatedly attenuated as it traverses the cavity. Related to this amplitude modulation (AM) active mode-locking is frequency modulation (FM) mode-locking, which uses a modulator device based on the electro-optic effect. This device, when placed in a laser cavity and driven with an electrical signal, induces a small, sinusoidally varying frequency shift in the light passing through it. If the frequency of modulation is matched to the round-trip time of the cavity, then some light in the cavity sees repeated up-shifts in frequency, and some repeated down-shifts. After many repetitions, the up-shifted and down-shifted light is
6
1.1. ULTRA SHORT OPTICAL PULSE BACKGROUND swept out of the gain bandwidth of the laser. The only light which is unaffected is that which passes through the modulator when the induced frequency shift is zero, which forms a narrow pulse of light. The third method of active mode-locking is synchronous mode-locking, or synchronous pumping. In this, the pump source (energy source) for the laser is itself modulated, effectively turning the laser on and off to produce pulses. Typically, the pump source is itself another mode-locked laser. This technique requires accurately matching the cavity lengths of the pump laser and the driven laser.
Active mode-locking in a fiber optics cavity Active mode-locking is normally achieved by modulating the loss (or gain) of the laser cavity at a repetition rate equivalent to the cavity frequency, or a harmonic thereof. In practice, the modulator can be acousto-optic or electro-optic modulator, Mach-Zehnder integrated-optic modulators, or a semiconductor electroabsorption modulator (EAM). The principle of active mode-locking with a sinusoidal modulation. In this situation, optical pulses will form in such a way as to minimize the loss from the modulator. The peak of the pulse would automatically adjust in phase to be at the point of minimum loss from the modulator. Because of the slow variation of sinusoidal modulation, it is not very straightforward for generating ultrashort optical pulses (< 1ps) using this method. For stable operation the cavity length must precisely match the period of the modulation signal or some integer multiple of it. The most powerful technique to solve this is regenerative mode locking i.e. a part of the output signal of the mode-locked laser is detected; the beatnote at the round-trip frequency is filtered out from the detector, and sent to an amplifier, which drives the loss modulator in the laser cavity. This procedure enforces synchronism if the cavity length undergoes fluctuations due to acoustic vibrations or thermal expansion. By using this method, highly stable mode-locked lasers have been achieved. The major advantage of active mode-locking is that it allows synchronized operation of the mode-locked laser to an external radio frequency (RF) source. This is very useful for optical fiber communication where synchronization is normally required between optical signal and electronic control signal. Also active mode-locked fiber can provide much higher repetition rate than passive mode-locking. Currently, fiber lasers and semiconductor diode lasers are the two most important types of lasers where active mode-locking are applied.
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1.1. ULTRA SHORT OPTICAL PULSE BACKGROUND Semiconductor laser To date semiconductor lasers have been considered the best light sources for high-bit-rate optical fiber communication lines and ultrafast optical data-processing systems. Compared to solid-state, gas, and dye lasers, the semiconductor laser offers a considerably smaller size, higher efficiency, lower cost, and the unique ability to be modulated up to gigahertz rates by simply changing the driving current through the device. One of the main advantages of lightwave systems over electronic ones is their ultrahigh speed and transmission capacity. The fundamental reason for this is the much higher carrier frequency of light (1014 to 1015 Hz) in a lightwave system. To realize the potentially very high capacity of lightwave systems, the light source used should be able to generate ultrashort pulses at ultrahigh repetition rates and capable of being modulated and encoded. These requirements are completely satisfied by using semiconductor laser. The performance of optical transmission lines and optical data-processing systems in terms of their speed relies a great deal on the performance of optical pulses that are generated by diode lasers [1.1]. The Mode-locked semiconductor lasers of various kinds can generate light pulses with durations in the picosecond or femtosecond region, pulse energies from picojoules to several microjoules, and repetition rates between a few megahertz and many gigahertz.
1.1.3
Measurement techniques of ultrashort optical pulse
In the case of the pulse characterization techniques, this evolved parallel to the advancement in ultrashort pulsed lasers. These techniques were initially only required to make estimates of the pulse width, but as the need for complete characterization increased, even phase information became necessary. Several techniques are available to measure ultrashort optical pulses and their parameters (see fig. 1.4) [1.24], [1.25]: • The pulse repetition rate is usually measured with a fast photodiode and an electronic spectrum analyzer. • The pulse duration can be measured with various methods, e.g. with an autocorrelator or a streak camera. Optical sampling techniques can be used when a shorter reference pulse is available. • The pulse energy may be measured directly or (for pulse trains) calculated from the average power and repetition rate.
8
1.1. ULTRA SHORT OPTICAL PULSE BACKGROUND
Figure 1.4: Examples of short and ultrashort pulse light in the time domain with their corresponds physic parameter. [1.1]. • The peak power may be directly measured with a photodiode or calculated from pulse energy, pulse duration and pulse shape. • The optical center frequency and spectral shape can be obtained with an optical spectrum analyzer. • The envelope offset frequency is of special interest in optical metrology, and may be measured with an f − 2f interferometer. • The chirp can be measured e.g. with frequency-resolved optical gating. • The timing jitter of a pulse train can be measured with various methods. • The coherence (e.g. of subsequent pulses) can be characterized e.g. with an interferometer. There are methods of complete pulse characterization [1.26]-[1.31] • Intensity autocorrelation: gives the pulse width when a particular pulse shape is assumed. • Spectral interferometry (SI): a linear technique that can be used when a pre-characterized reference pulse is available. Gives the intensity and phase. The algorithm that extracts the intensity and phase from the SI signal is direct. • Spectral phase interferometry for direct electric-field reconstruction (SPIDER) [1.31]: a nonlinear self-referencing technique based on spectral shearing interferometry. The method is similar to SI, except that the reference pulse is a spectrally shifted replica of itself, allowing one to obtain the spectral intensity and phase of the probe pulse via a direct FFT filtering routine similar to SI, but which requires integration of the phase extracted from the interferogram to obtain the probe pulse phase.
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1.1. ULTRA SHORT OPTICAL PULSE BACKGROUND • Frequency-resolved optical gating (FROG)[1.26]: a nonlinear technique that yields the intensity and phase of a pulse. It’s just a spectrally resolved autocorrelation. The algorithm that extracts the intensity and phase from a FROG trace is iterative. • Grating-eliminated no-nonsense observation of ultrafast incident laser light e-fields (GRENOUILLE), a simplified version of FROG. • Methods of characterizing and controlling the ultrashort optical pulses: MIIPS Multiphoton Intrapulse Interference Phase Scan, a method to characterize and manipulate the ultrashort pulse. The dawn of sub pico-second pulses brought to electronic pulse characterization techniques. This is mostly because electronic devices such as the semiconductor laser can only achieve rise times in the order of 10 pico-seconds (ps). For this reason, optical pulse characterization techniques were developed which exploit the fast nonlinear optical response of materials. All of these optical techniques utilize the correlation of the pulse with either itself (autocorrelation) or a different pulse (cross-correlation) within a non-linear optical (NLO) medium. The type of interaction within the NLO medium, together with the detection of this interaction, sets the premise for the different optical pulse characterization techniques. These techniques include but are not limited to the following: interference autocorrelation (IAC) [1.32], spectral phase interferometry and direct electric field reconstruction (SPIDER) [1.33], frequency resolved optical gating (FROG) [1.34] and modified spectrum auto interferometric correlation (MOSAIC) [1.35].
1.1.4
Introduction of the Wigner distribution Function
Finally, in this work, we have been studied the application of the Wigner distribution for the characterization of the ultra-short optical pulse. The Wigner distribution function (WDF) was first proposed in physics to account for quantum corrections to classical statistical mechanics in 1932 by Eugene Wigner, cf. Wigner quasi-probability distribution. Given the shared algebraic structure between position-momentum and time-frequency pairs, it may also usefully serve in signal processing, as a transform in time-frequency analysis. Compared to a short-time Fourier transform, such as the Gabor transform, the Wigner distribution function can furnish higher clarity in some cases [1.36]-[1.37].
The Mathematical definition There are several different definitions for the Wigner distribution function. The definition given here is specific to time-frequency analysis. The Wigner distribution function Wt (t, ω) is
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1.2. PROBLEMS TO BE RESOLVED IN THIS THESIS ARE: ∞
τ τ ∗ A t+ Wx (t, f ) = A t− exp (−i2πτ ω) dτ (1.1) 2 2 −∞ √ Where i = −1 is the imaginary unit. The WDF is essentially the Fourier transform of the Z
input signal’s autocorrelation function - the Fourier spectrum of the product between the signal and its delayed, time reversed copy, as a function of the delay.
1.2
Problems to be resolved in this thesis are:
We make an attempt to develop a novel approach to describing the initial stage of the active mode-locking in semiconductor laser structures based on analyzing the properties of dispersion relations in terms of stability for small initial perturbations. Nonlinear process of shaping optical pulses is interpreted as manifesting instability of diffusion type. Also, the schematic arrangement, the process of the active mode-locking is caused by a hybrid nonlinear cavity consisting of this heterolaser and an external rather long single-mode optical fiber exhibiting square-law dispersion, cubic Kerr nonlinearity, and small linear optical losses has been analyzing. The characterization of low-power bright picosecond optical pulses with an internal frequency modulation simultaneously in both time and frequency domains in the case of the Gaussian shape. This approach exploits the Wigner time-frequency distribution, which can be found for these bright pulses by using a novel interferometric technique. With the first analysis of the tow Gaussian pulses and their respectively Wigner time-frequency distribution approach. Finally the practical feasibility of measuring the train-average parameters of picosecond optical pulses being arranged in high-frequency repetition trains is investigated. For this purpose we consider exploiting the triple auto-correlations, whose Fourier transformations give us the bispectrum of a pulse train.
1.3
Objective
The objective of this work was to develop a new techniques and apparatus to shaping and characterization of ultrashort pulses emitted in infrared light. Pulse characterization will be shown through to the triple correlation implementation. The pulses are reconstructed using the joint Wigner time-frequency distributions using a dissipative solitary pulse. Together with this, we make a novel approach to describing the initial stage of the active mode-locking technique.
1.4
Brief description of the thesis contents
In this section we present a summary of the thesis contents.
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1.4. BRIEF DESCRIPTION OF THE THESIS CONTENTS • In the chapter 2 is presented the fundamental concepts to support this thesis. Here it is showed the principles theoretical aspect of; Semiconductor mode-Locking, Wigner distribution and it’s application in the optical pulses measurement and the fundamentals of triple correlation. • In the chapter 3 is shown an attempt to develop a novel approach to describing the initial stage of the active mode-locking in semiconductor laser structures based on analyzing the properties of dispersion relations in terms of stability for small initial perturbations. The detection system of average time parameters for continuous trains of ultra-short optical pulses was developed for a near infrared laser. Together with this a new Algorithm of operation is describing. • In this chapter 4 is presented a specific approach to characterizing the train-average parameters of low-power picosecond optical pulses with the frequency chirp. Together with this we develop this approach involves the joint Wigner time-frequency distributions. The experimental studies was develop for the InGaAsP/InP-heterolasers generating at the wavelength 1320 nm. • In the chapter 5; it is described the conditions of shaping regular trains of optical dissipative solitary pulses, in the actively mode-locked semiconductor laser heterostructure with an external long-haul single-mode silicon fiber. Within such a model, a contribution of the nonlinear Ginzburg-Landau operator to shaping the parameters of optical. Finally, the results of the illustrating proof-of-principle experiments are briefly presented and discussed. • In this chapter 6, it is presented the practical feasibility of measuring the train-average parameters of pico-second optical. The triple auto-correlation can be shaped by a three-beam scanning interferometer with the following one- or two-cascade triple harmonic generation. Also we present the principals details of the experimental arrangement, together whit the principal details of the Photo-EMF photo-detector.
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Bibliography [1.1] wikipedia, “en.wikipedia.org” wikipedia, date: 01 − 12 − 11. [1.2] Weiner, A. M.,“Femtosecond pulse shaping using spatial light modulators” Review Of Scientific Instruments‘ Volume 71, Number 5 [1.3] Fork, R. L., Greene, B. I. and Shank, C. V., “Generation of optical pulses shorter than 0.1 psec by colliding pulse mode locking” Appl. Phys. Lett. Vol. 38, pp 671-672, (1981). [1.4] Shank, C. V., Fork, R. L., Yen, R., Stolen, R. H. and Tomlinson, W. J., “Compression of femtosecond optical pulses” Appl. Phys. Lett. 40, pp 761-763, (1982). [1.5] Fujimoto, J. G., Weiner, A. M. and Ippen, E. P., “Generation and Measurement of Optical Pulses as Short as 16 fs” Appl. Phys. Lett. 44, pp 832-834, (1984). [1.6] Halbout J.-M. and Grischkowsky, D., “12-fs ultrashort optical pulse compression at a high repetition rate” Appl. Phys. Lett. 45, pp 1281, (1984). [1.7] Knox, W. H., Fork, R. L., Downer, M. C., Stolen, R. H., Shank, C. V., and Valdmanis, J. A., “Optical pulse compression to 8 fs at a 5-kHz repetition rate” Appl. Phys. Lett. 46, pp 1120, (1985). [1.8] Fork, R. L., Brito Cruz, C. H., Becker, P. C. and Shank, C. V., Compression of optical pulses to six femtoseconds by using cubic phase compensation Opt. Lett. 12, pp. 483-485, (1987). [1.9] Spence, D. E., Kean, P. N. and Sibbett, W., “60-fsec pulse generation from a self-modelocked Ti:sapphire laser” Opt. Lett. 16, pp 42-44, (1991). [1.10] Asaki, M. T., Huang, C.-P., Garvey, D., Zhou, J., Kapteyn, H. C. and Murnane, M. M., “Generation of 11-fs pulses from a self-mode-locked Ti:sapphire laser.” Opt. Lett. 18, pp. 977-979, (1993).
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BIBLIOGRAPHY [1.11] Zhou, J., Taft, G., Huang, C.-P., Murnane, M. M. and Kapteyn, H. C., “Pulse evolution in a broad-bandwidth Ti:sapphire laser” Opt. Lett. 19, pp 1149-1151, (1994). [1.12] Stingl, A., Lenzner, M., Spielmann, Ch., Krausz, F. and Szipocs, R., “Sub-10-fs mirrordispersion-controlled Ti:sapphire laser” Opt. Lett. 20, pp 602, (1995). [1.13] Jung, I. D., Kärtner, F. X., Matuschek, N., Sutter, D. H., Morier-Genoud, F., Zhang, G., Keller, U., Scheuer, V., Tilsch, M., Tschudi, T., “ Self-starting 6.5-fs pulses from a Ti:sapphire laser” Opt. Lett. 22, pp. 1009-1011, (1998). [1.14] Xu, L., Tempea, G., Spielmann, Ch., Krausz, F., Stingl, A., Ferencz, K. and Takano, S.,“Continuous-wave mode-locked Ti:sapphire laser focusable to 5 × 1013 W/cm2 ” Opt. Lett. 23, pp 789-791, (1998). [1.15] Morgner, U., Kärtner, F. X., Cho, S., H., Chen, Y., Haus., H. A., Fujimoto, J., G., Ippen, E., P., Scheuer, V., Angelow, G., Tschudi, T., “Sub-two-cycle pulses from a Kerr-lens modelocked Ti:sapphire laser”, Opt. Lett. 24, pp 411, (1999). [1.16] Sutter, D., H., Steinmeyer, G., Gallmann, L., Matuschek, N., Morier-Genoud, F., Keller, U., Scheuer, V., Angelow, G., Tschudi, T., “Semiconductor saturable-absorber mirror assisted Kerr-lens mode-locked Ti:sapphire laser producing pulses in the two-cycle regime” Opt. Lett. 24, pp .631-633, (1999). [1.17] Nakazawa, M., ed, IEICE Trans. E81 − C, pp. 93 (1998). [1.18] Keller, U. ed., Appl. Phys. B: Lasers Opt. 65, pp 113, (1997). [1.19] Barty, C. P. J., White, W., Sibbett, W. and Trebino, R., “Introduction To The Issue On Ultrafast Optics” eds., IEEE J. Sel. Top. Quantum Electron. 4, pp. 157-158, (1998). [1.20] Kaiser, W., “Ultrashort Laser Pulses and Applications” Springer, Berlin, (1988) [1.21] Diels, J. C. and Rudolph, W. “Ultrahort Laser Pulse Phenomena” Academic, San Diego, (1996). [1.22] Elsaesser, T., Fujimoto, J. G., Wiersma, D. A. and Zinth, W., “Ultrafast Phenomena XI”, Springer, Germany, (1998). [1.23] Rulliere, C., “Femtosecond Laser Pulses” Springer, Berlin, (1998).
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BIBLIOGRAPHY [1.24] Yan, C. and Diels, J. C. M., “Amplitude and phase recording of ultrashort pulses”, J. Opt. Soc. Am. B 8 (6), pp. 1259-1263, (1991). [1.25] Naganuma, K., Mogi, K. and Yamada, H., “General method for ultrashort light pulse chirp measurement”, IEEE J. of Quant. Elec. 25, pp. 1225 - 1233 (1989). [1.26] Kane, D. and Trebino, R., “Characterization of arbitrary femtosecond pulses using frequency-resolved optical gating”, IEEE J. Quantum Electron. 29 (2), pp. 571 - 579, (1993). [1.27] Chu, K. C., Heritage, J. P., Grant, R. S., Liu, K. X., Dienes, A., White, W. E. and Sullivan A., “Direct measurement of the spectral phase of femtosecond pulses”, Opt. Lett. 20 (8), pp. 904-906 (1995). [1.28] Walmsley I. A. and Wong V., “Characterization of the electric field of ultrashort optical pulses”, J. Opt. Soc. Am. B 13 (11), pp. 2453-2463 (1996) [1.29] Jung, I. D., Kärtner, F. X., Henkmann, J., Zhang, G., Keller, U.,“High-dynamic-range characterization of ultrashort pulses”, Appl. Phys. B 65, pp. 307-310, (1997) [1.30] Trebino, R., DeLong, K. W., Fittinghoff, D. N., Sweetser, J. N., Krumbugel, M. A., Richman, B. A. and Kane, D. J. “Measuring ultrashort laser pulses in the time-frequency domain using frequency-resolved optical gating”, Rev. Sci. Instrum. 68, pp. 3277-3295, (1997) [1.31] Iaconis, C. and Walmsley, I. A., “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses”, Opt. Lett. 23 (10), pp. 792-794, (1998) [1.32] Meshulach, D., Barad, Y. and Silberberg, Y., “Measurement of ultrashort optical pulses by third-harmonic generation”, Journal Of The Optical Society Of America B-Optical Physics, vol. 14(8), pp. 2122-2125, August (1997). [1.33] Iaconis, C. and Walmsley, I. A., “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses”, Optics Letters, vol. 23(10), pp. 792-794, May (1998). [1.34] Baltuska, A., Pshenichnikov, M. S. and Wiersma, D. A., “Amplitude and phase characterization of 4.5 − f s pulses by frequency-resolved optical gating”. Optics Letters, vol. 23(18), pp. 1474-1476, September (1998). [1.35] Hirayama, T. and Sheik-Bahae, M., “Real-time chirp diagnostic for ultrashort laser pulses”, Optics Letters, vol. 27(10), pp. 860-862, May (2002).
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BIBLIOGRAPHY [1.36] Boashash, B., “Note on the Use of the Wigner Distribution for Time Frequency Signal Analysis” IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. 36, No. 9, pp. 1518-1521, sep (1998) [1.37] Boashash, B., editor, “Time-Frequency Signal Analysis and Processing - A Comprehensive Reference” Elsevier Science, Oxford, (2003)
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Chapter 2
FUNDAMENTAL CONCEPTS 2.1
Mode Locking in semiconductor laser
Generally speaking, the generation of ultrashort laser pulses is based upon the confinement of the energy in a laser cavity into a small spatial region or the concentration of the optical power normally divided between the whole set of fluctuations into just one. The laser emission often consists of a set of resonator modes ωj , j = 1, 2, 3..... separated by δω [2.1]. The number of modes that oscillate is limited by the spectral bandwidth ∆ωg over which the laser gain exceeds the cavity loss. The output consists of a sum of frequency components that correspond to the oscillating modes, and the electric field is given as (without taking into account the spatial distribution) [2.1], [2.2] E(t) =
X
Am exp [i (ω0 + mδω) t + ϕj ] ,
(2.1)
m
where Am and ϕj represent the amplitude and phase of the mth mode. In general, relative phases between the modes are randomly fluctuating. If nothing fixes the phases φj the laser output will vary randomly in time, the average power being approximately equal to the simultaneous one. On the other hand, if the modes are forced to maintain a fixed phase and amplitude relationship, for example, δϕ = ϕj + ϕj−1 ,
(2.2)
Am exp (iϕj ) = A0 exp [i (ϕ0 + mδϕj )] ,
(2.3)
then the output of the laser will be a periodic function of time: E (t) = A0 exp (iω0 t)
sin [(k + 1) δωt1 /2] sin (δωt1 /2)
,
(2.4)
where k is the number of locked modes and t1 = t + δϕ/δω. Figure 2.1 illustrates the envelope of E 2 (t) for k = 10. The operating conditions given by eqs. 2.2 to 2.4 result in the generation of a train of regularly spaced optical pulses. This dynamic regime is called mode locking. The pulses have width ∆τ , which is proportional to 1/∆ωg . The pulse train has a
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2.1. MODE LOCKING IN SEMICONDUCTOR LASER temporal periodicity of T = 2Lµ /C. The ratio of the period T to the pulsewidth ∆τ is equal to the number of locked modes k. It is also possible to produce mode locking with several pulses, inside the cavity, that are spaced by a multiple of c/2Lµ .
Figure 2.1: Ideal mode locking: the output of a laser with 10 completely locked modes [2.4]. The basic advantage of mode locking is that it generates much shorter pulses than are produced by gain or Q-switching. Mode locking can be realized using a number of techniques, including active, passive, and hybrid methods.
2.1.1
Mode-locked semiconductor laser
Active mode locking is achieved by modulating the loss or gain of a semiconductor laser at a frequency that is equal to the inter modal spacing δν . In this case each spectral mode is driven by the modulation side bands of its neighbors [2.3]. As a result, phases of modes are locked by the external modulation. Because of very small dimensions of diode lasers (the cavity length is typically 250 to 400 m) δν is very high (more than 100 GHz). Consequently, it is quite difficult to modulate the gain at such a frequency. A straightforward decision of the problem is to increase the cavity length. If the cavity length is increased by placing the laser diode into an external resonator or by using an extended-upto-several-millimeters monolithic cavity, the intermodal spacing is decreased and active mode locking can easily be realized. Up to date, most actively mode-locked lasers have external-cavity configurations, with δν in the range of 0.3 to 20 GHz. Active mode locking is schematically illustrated in Figure 2.2. The modulation of the gain in the laser at the intermodal spacing frequency results from either modulating the driving current or using an external modulator. Sine wave oscillators are normally used for the current modulation. Alternatively, ultrashort electrical pulses from a comb generator
18
2.1. MODE LOCKING IN SEMICONDUCTOR LASER
Figure 2.2: Active mode locking of a diode laser using a current modulation: (a) the current waveforms, (b) carrier density, and (c) optical output against time [2.4]. can be used. The modulation is often superimposed on a dc bias I0 . The current modulation causes the modulation of the carrier density around the threshold value nth . Both the amplitude I1 of the current modulation and I0 are chosen to create a very short time window of net gain in the laser. This implies that the carrier density n (t) exceeds nth for a very short period of time during each modulation period. Due to the finite carrier lifetime there exists a phase difference between the electron response and the applied current modulation. The carrier density is a periodical function of time, but it consists of a set of harmonics of the modulation frequency 2π/T . The most strongly exited Fourier component of the carrier density is the fundamental at 2π/T , the harmonics of order j being reduced by approximately a factor l/j [2.5]. Mode locking is ultimate provided the period of modulation T is exactly equal to the round-trip time of the laser cavity or, in other words, the condition 1
, (2.5) δν is satisfied. The repetition rate of mode-locked pulses is obviously equal to the driving current T =
frequency. Detuning the modulation frequency from the resonance condition given by 2.5 leads to the degradation of mode-locked pulses; that is, the pulse width increases while the peak power
19
2.1. MODE LOCKING IN SEMICONDUCTOR LASER decreases with the increase of the frequency offset.
Theoretical Treatment A number of theories have been developed for actively mode-locked diode lasers [2.6]-[2.15]. Some of them are based upon the analytical treatment of laser equations [2.7]-[2.14], while other theories are numerical models of mode-Locked diode lasers when the laser dynamics are described by computer solutions of the rate equations. The analytic theories [2.7]-[2.9], which have been developed by Siegman and Kuizenga [2.16], [2.17] and Haus [2.18], basically follow the selfconsistent profile approach to forced mode locking of a laser. This approach is based upon the consideration of a ring laser configuration with a laser medium, a modulator of gain/loss, and a dispersive medium, with a traveling-wave excitation proceeding in one direction around the ring as shown in Figure 2.3
Figure 2.3: Schematic of a ring cavity used in self-consistent profile approach lo mode locking. [2.4]. It is assumed that an optical pulse has formed so that on the j − th pass around the cavity the linearly polarized electric field component of the pulse E may be written as
E(t) = νj (t) exp iω0 t + c.c., where νj (t) is the pulse envelope and ω0 is the carrier frequency of the optical pulse. The optical pulse is treated as a plane field without taking into account the transverse field pattern. The other assumptions of the approach developed by Haus [2.7], [2.18], [2.19]. are: • changes of the pulse envelope per pass in the cavity due to gain, loss, and modulation are small.
20
2.1. MODE LOCKING IN SEMICONDUCTOR LASER • the gain or loss are sinusoidal, that is, deviations of gain/loss waveform from sinusoidal due to the electron-photon interaction in the laser are neglected • the modulation frequency is tuned to resonance, that is, it is equal to the frequency separation of the cavity modes. According to this approach, modifying a pulse by different elements of the cavity is represented by the exponential operator as exp (gL) exp (α l) exp (iϕ) ,
(2.6)
operating on νj (t). In 2.6 we denote the gain and loss of the laser as g and α, respectively. If the change per pass is small, we can expand the operator as exp (gL) exp (−α l) exp (iϕ) ' 1 + gL − αl − iϕ,
(2.7)
The pulse νj (t), operated upon by the operator 2.7, is delayed by the travel time TR , so (1 + GL − αL − Iϕ) νj (t) produces νj+1 (t) on the next pass around the cavity. The steady state solution implies that νj+1 (t) must be equal to delayed νj (t) within a possible time shift δTr . If the time shift is small, we can write νj (t − δTR ) ' νj (t) − δTR
dνj (t)
. dt Using 2.7 and 2.8, one obtains the Master Equation of mode locking [2.5] and [2.19] 1 + GL − α − iϕ + δTR
d dt
(2.8)
νj (t) .
(2.9)
It is assumed that the loss is modulated as 2 2 α = α0 [1 + 2M (1 − cos (ωm t))] w α0 1 − M ωM t ,
(2.10)
where α0 is a constant loss, M is the modulation coefficient, and ωm is the modulation frequency. Next consider a Lorentzian profile of the gain g (ω) = g0 / 1 + i
(ω − ω0 ) ∆ωg
,
(2.11)
where the carrier frequency is equal to the central frequency of the gain profile. If we expand g (ω) in powers of (ω − ω0 ) /∆ωg and retain terms up to the second-order one, we obtain the following expression for the operator
21
2.1. MODE LOCKING IN SEMICONDUCTOR LASER " gL = g0 L 1 + i
ω − ω0
−
∆ωg
ω − ω0
2 # .
∆ωg
(2.12)
The operation performed on the Fourier transformed envelope νj (ω − ω0 ) by the multiplier i (ω − ω0 ) corresponds to differentiation with respect to time in time domain. According to 2.12, one obtains [2.19] gLνj (t) =⇒ g0 L 1 − i
d
1
∆ωg dt
+
1
d2
!
∆ωg2 dt2
νj (t) .
(2.13)
The right-hand side of 2.13 represents the action of the gain medium 2.13 on the pulse envelope, including the linear unsaturated gain g0 L, a small time delay (the first derivative term), and broadening of the pulse due to finite spectral gain bandwidth (the second derivative-a diffusion operator). The phase term ϕ (ω) in 2.9 can be expanded in powers of d2 k
lD (ω − ω0 )2 . (2.14) dω 2 A linear term is omitted in 2.14 because it corresponds to a time delay that has 2.12 been ϕ (ω) = ϕ0 +
already taken into account in 2.13. The second derivative of the propagation constant k represents the group velocity dispersion in a dispersive medium of the length lD . This term accounts for the contribution of all the dispersive components in the cavity. The quality factor Q of the laser cavity and the peak gain g are connected with α0 and g0 as α0 l =
ω 0 TR 2Q
,
(2.15)
g.
(2.16)
and g0 L =
ω0 TR 2Q
The self-consistent profile approach implies that after several passes around the cavity (m ≫ 1) the pulse envelope νj (t) should not depend on m and we can use ν(t) instead of νj (t). Combining the eq. 2.13 with eq. 2.14 and using eq. 2.9, one obtains the mode-locking equation [2.19]. 1 d2 2Q 2 2 g−1−i ϕ0 + 2 − M ωM t ν (t) = 0, ω0 TR ωD dt2
(2.17)
where
22
2.1. MODE LOCKING IN SEMICONDUCTOR LASER 1 2 ωD
=
g ∆ωg2
+i
2Q d2 k ω0 TR dt2
lD .
(2.18)
The solutions of eq. 2.17 are Hermite Gaussians ωP = Hν (ωp t) exp −ωp t2 /2 ,
(2.19)
ωP = M 1/4 (ωD ωM )1/2 ,
(2.20)
with
and g−1−i
2Q ω0 TR
ϕ0 =
2 ωP 2 ωD
.
(2.21)
Equation 2.20 determines the pulsewidth. According to this model of active mode locking, the pulse envelope νj (t) is Gaussian and the pulsewidth is inversely proportional to the square root of the effective spectral bandwidth ωD and the fourth root of the modulation depth M . The dispersion in the laser d2 k/dω 2 causes a wavelength chirp of the pulse. If we consider the mode locking of a diode laser in an external resonator, we should use the following approximation of the gain instead of the Lorentzian gain basic profile [2.7]. g0 , g (ω) = (ω − ω0 ) 2ωLµ 1+i 1 + Hg 1 − cos ∆ωg c
(2.22)
where 4Hg is the full normalized peak-to-peak variation of the periodic modulation of the gain by the diode of length L with refractive index . In this case we have the mode locking “in clusters model” and the solution of the master equation 2.9 is more complicated and can be found elsewhere [2.7, 2.20]. However, it is still possible to get the solution analytically. For example, the pulsewidth (FWHM) of pulses generated in an actively mode-locked uncoated diode in an external cavity is given by 1/2
∆τ = 2 (ln2)
µL ωM c
1/2
1
A ((1/r) + r)
2g0 M A2 + 2a/r + 1
1/4 ,
(2.23)
where A = (µ − 1/µ + 1) with r is the effective reflectivity of the external mirror. In the presence of noise the whole “ spectrum ” of eigenfunctions ( similar to solutions 2.19 with
23
2.1. MODE LOCKING IN SEMICONDUCTOR LASER different ν) may be excited. This corresponds to a pulse envelope with a noisy substructure that has the round-trip time as its period. The simplest traveling-wave model is the extension of Haus’s theory of the unidirectional ring laser cavity presented above using the following equations that describe the interaction of an optical pulse envelope S (z, t) with the gain medium [2.10]
a)
∂S ∂t ∂n
+ νg
∂S
= Γg (n) S − νg αi S + Γβ
∂z j (t) n b) = + + g (n) Sg . ∂t ed τs
n τs
, (2.24)
The dispersion of the laser cavity and bandwidth-limiting element (dispersion medium) can be treated separately as a frequency filter. The assumption of a small change in different elements of the cavity is not necessary any more, and gain saturation by the optical pulse is included in the model. To minimize the required computer memory and computation time, the following series expansion of the pulse can be used [2.10] S (η) =
X
cj η j−1 exp −η 2 /θ 2 ,
(2.25)
j
with η = t − z/νg . This model predicts the generation of nearly transform limited high power (≈ 1 − W peak) pulses with a FWHM in the range 3 to 10 ps at ≈ 1 − GHz repetition rates in mode-locked external cavity lasers. Due to the gain and loss saturation the pulse envelope becomes asymmetric, with the leading edge shorter than the trailing one. Strictly speaking, we should include both forward S + (z, t) and backward S − (z, t) traveling waves in the model. The traveling-wave rate equations in this case can be written as [2.12]-[2.22] ∂S + ∂t ∂S − ∂t ∂n
+ νg − νg
∂S + ∂z ∂S − ∂z
= Γg0 (n − nt ) , S + − υg αi S + + β = Γg0 (n − nt ) S − − υg αi S −
n j (t) = −g0 (n − nt ) S + + S − − + . ∂t ed Ts
n
τs n +β , τs (2.26)
Boundary conditions are necessary to solve the partial differential equations. For active mode locking in a diode laser with an AR coating in a linear external cavity (Fig. 2.4), the boundary conditions should include the reflections on all mirrors, which make up the composite cavity [2.22], [2.23].
24
2.1. MODE LOCKING IN SEMICONDUCTOR LASER
S + (0, t) = R1 S − (0, t) ,
(2.27)
S − (L, t) = Ra S + (L, t) + (1 − Ra ) R2 C 2 S + (L, t − τext ) ,
(2.28)
Figure 2.4: Configuration of a diode laser used in the traveling-wave numerical approach to active mode locking Ra is the residual reflectivity of the AR coating [2.4]. where R1 and R2 are the power reflectivities of the left and right mirrors of the composite cavity, Ra is the residual reflectivity of the AR-coated diode facet, C is the coupling coefficient to the external cavity, and τext is the external cavity round-trip time. The finite reflectivity of the AR-coated facet, that is Ra 6= 0, is very important because it results in multiple pulse formation, which is experimentally observed in mode-locked diode lasers with sub-picosecond pulse output. The nonlinear gain effects can be included in the model by using either this gain coefficient separate differential equation for the gain is next [2.22] G0 − G dn = −K S + + S − + + g0 , (2.29) dt τr dt where K is a gain reduction coefficient, τr is the nonlinear gain recovery time, and G0 = dG
g0 (n − nt ) is the gain in the absence of the nonlinear effects. A value of 0.4 ps is often used for the time constant τr , and values for K in the range of 107
to 1011 are used [2.22]. The numerical traveling-wave time-domain approach is particularly
effective in treating the mode locking of AR-coated diode lasers in external cavities with subpicosecond pulse output at repetition rates in excess of 10 GHz. It predicts the pulse shape of actively mode-locked pulses in the form of [2.15]-[2.23]
25
2.2. OPTICAL PULSES.
S (z, η) = where η = t − t0 (z) −
S0 (z) cosh2 [η/τ (z)]
,
(2.30)
z
. υg The multiple pulse formation caused by the imperfect AR coating of the diode facet and dynamic detuning can also be explained within the framework of the model.
2.2
Optical pulses.
Optical pulses are flashes of light, which are often generated with lasers (laser pulses) and delivered in the form of laser beams. Due to the enormously high optical frequencies, optical pulses can be extremely short (ultrashort), when their optical bandwidth spans a significant fraction of the mean frequency. Therefore, amplified ultrashort-pulses are very important for high-intensity physics, studying phenomena such as multi-photon ionization, high harmonic generation, or the generation of even shorter pulses with attosecond durations. Depending on the required pulse duration, pulse energy, and pulse repetition rate, different methods of pulse generation, pulse compression and pulse characterization are used, overall covering extremely wide parameter regimes. Pulse propagation in media has many interesting aspects. The peak of a pulse in a transparent medium propagates with the group velocity, not the phase velocity. Dispersion can cause temporal broadening (or sometimes compression) of pulses. For high peak intensities, optical nonlinearities can strongly affect the pulse propagation; often they lead to pulse broadening, but strong nonlinear compression is also possible [2.24]. There are various methods for measuring the pulse duration achieved or for pulse characterization in other respects. Particularly for measuring the duration of ultrashort pulses, purely optical techniques are very important, since electronics are too slow for such purposes. However, such a method is not efficient, since most of the light will be lost at the modulator, and also the pulse duration is limited by the speed (bandwidth) of the modulator. Pulses with much higher energies and much shorter durations can be generated in pulsed lasers [2.25]. In optics, an ultrashort pulse of light is an electromagnetic pulse whose time duration is on the order of the femtosecond (10−15 second). Such pulses have a broadband optical spectrum, and can be created by mode-locked oscillators. They are commonly referred to as ultrafast events [2.26]. They are characterized by a high peak intensity(or more correctly, irradiance) that usually leads to nonlinear interactions in various materials, including air. These processes are studied in the field of nonlinear optics [2.27]. The word “ultrashort” refers to the femtosecond (fs) to picosecond (ps) range, although such pulses no longer hold the record for the shortest pulses
26
2.2. OPTICAL PULSES. artificially generated. Indeed, pulse durations on the attosecond time scale have been reported. The real electric field corresponding to an ultrashort pulse is oscillating at an angular frequency ω0 corresponding to the central wavelength of the pulse. To facilitate calculations, a complex field E (t) is defined. Formally, it is defined as the analytic signal corresponding to the real field [2.28]. The central angular frequency ω0 is usually explicitly written in the complex field, which may be separated as an intensity function I (t) and a phase function ψ (t): E (t) =
p I (t) exp (iωo t) exp (iψ(t)) .
(2.31)
The expression of the complex electric field in the frequency domain is obtained from the Fourier transform of E (t): E (ω) = F (E (t)) .
(2.32)
Because of the presence of the exp (iωo t)term, E (ω) is centered around ω0 , and it is a common practice to refer to E (ω − ωo ) by writing just E (ω), just as in the time domain, an intensity and a phase function can be defined in the frequency domain: E (ω) =
p S (w) exp [iϕ (w)] .
(2.33)
The quantity S (ω)is the spectral density (or simply, the spectrum) of the pulse, and ϕ(ω)is the spectral phase. Example of spectral phase functions include the case where ϕ (ω) is a constant, in which case the pulse is called a bandwidth-limited pulse, or where ϕ (ω) is a quadratic function, in which case the pulse is called a chirped pulse because of the presence of an instantaneous frequency sweep. Such a chirp may be acquired as a pulse propagates through materials (like glass) and is due to their dispersion. It results in a temporal broadening of the pulse [2.29], [2.30]. The intensity functions I (t) and S (ω) determine the time duration and spectral bandwidth of the pulse. As stated by the uncertainty principle, their product (sometimes called the time-bandwidth product) has a lower bound. This minimum value depends on the definition used for the duration and on the shape of the pulse. For a given spectrum, the minimum time-bandwidth product, and therefore the shortest pulse, is obtained by a transform-limited pulse, i.e., for a constant spectral phase ϕ (ω). High values of the time-bandwidth product, on the other hand, indicate a more complex pulse [2.31].
27
2.3. ORIGINATING THE JOINT WIGNER TIME-FREQUENCY DISTRIBUTION.
2.3
Originating the Joint Wigner time-frequency distribution.
When the spectrum of signal varying in time is the subject of interest, it is rather worthwhile to refer to applying some joint function of the time and frequency, which would be able to describe the intensity distribution of this signal simultaneously in time domain as well as in frequency one. Such a distribution gives us opportunities for determining a relative part of energy at a given frequency in the required temporal interval or for finding the frequency distribution at a given instant of time. The method of deriving the time-frequency distribution can be based on usage of the corresponding characteristic function. Let us assume that some time-frequency distribution W (t, ω) exists and presents a function of two variables t and ω. The characteristic function M (θ, τ ) inherent in this distribution can be written as mathematical expectation of the value exp (iθt + iτ ω), i.e. as Z∞ Z∞ M (θ, τ ) = h exp (iθt + iτ ω) i =
W (t, ω) exp (iθt + iτ ω) dt dω. (2.34) −∞ −∞
In its turn, the time-frequency distribution W (t, ω) can be found from the characteristic function M (θ, τ ) as
W (t, ω) =
1
Z∞ Z∞ M (θ, τ ) exp (−iθt − iτ ω) dθ dτ.
4π 2
(2.35)
−∞ −∞
Due to the characteristic function is some averaged value, one can use quantum mechanics method of the associated operators with ordinary variables. If we have the function g1 (t) depending only on the time t, the average value for this function can be calculated by two ways, namely, exploiting the complex amplitude A (t) of a signal or its complex spectrum S (ω) as Z∞ hg1 (t)i =
2
Z∞
g1 (t) |A (t)| dt = −∞
d S (ω) dω, S (ω) g1 i dω ∗
(2.36)
−∞
because the time can be represented by the operator i d/dω in the frequency domain. Then, for the function g2 (ω) depending only on the frequency ω, the average value can be estimated by
28
2.3. ORIGINATING THE JOINT WIGNER TIME-FREQUENCY DISTRIBUTION.
Z∞
Z∞
2
hg2 (ω)i =
d A (t) g2 −i A (t) dt, dt ∗
g2 (t) |S (ω)| dω = −∞
(2.37)
−∞
because the frequency is represented by the operator −i d/dt in the time domain as well. Consequently, one can combine the time and frequency with the non-commutative operators = d d , < → ω in the frequency and 100 GHz) by mode locking”, IEEE J. of Quantum Electronics, vol. 26, pp. 250-261, (1990). [2.15] Schell, M., Weber, A. G., Bottcher, E. H., Scholl E. and Bimberg, D., “Theory of subpicosecond pulse generation by active mode-locking of a semiconductor laser amplifier in an external cavity: limits for the pulsewidth ”, IEEE J. of Quantum Electronics, vol. 27, pp. 402-409 (1991). [2.16] Siegman, A. E. and Kuizenga, D. J., “Simple Analytic Expressions for AM and FM ModeLocked Pulses in Homogeneous Lasers”, Applied Physics Letters, vol. 14, pp. 181-183 (1969). [2.17] Kuizenga, D. J. and Siegman, A. E., “FM and AM mode locking of the homogeneous laser Part I: Theory” IEEE J of Quantum Electronics, vol. 6, pp. 694-708, (1970). [2.18] Haus, H. A., “A Theory of forced mode locking”, IEEE J. of Quantum Electronics, vol. 11, pp. 323-330 (1975). [2.19] Haus, H. A., “Modelocking of semiconductor laser diodes”, Japanese J. of Applied Physics, vol. 20, pp. 1007-1020 (1981). [2.20] Haus, H. A., and Ho, P.-T., “Effect of noise on active mode locking of a diode laser”, IEEE J. of Quantum Electronics, vol. 15, pp. 1258-1265 (1979). [2.21] New, G. H. C. and Catherall, J. M., “Problems in the self-consistent profile approach to the theory of laser mode locking”, Optics Communications, vol. 50, pp. 111-116 (1984). [2.22] Morton, P. A., Helkey, R. J. and Bowers, J. E., “Dynamic detuning in actively mode-locked semiconductor lasers” IEEE J. of Quantum Electronics, vol. 25, pp. 2621- 2633 (1989).
46
BIBLIOGRAPHY [2.23] Schell, M., Weber, A. G., Scholl, E. and Bimberg, D., “Fundamental limits of sub PS generation by active mode locking of semiconductor lasers: The spectral gain width and the facet reflectivities”, IEEE J. of Quantum Electronics, vol. 27, pp. 1661-1668 (1991). [2.24] Krausz, F., Fermann, M. E., Brabec, T., Curley, P.F., Hofer, M., Ober, M. H., Spielmann, C., Wintner, E. and Schmidt, A.J., ”Femtosecond solid-state lasers” IEEE J. Quantum Electron, vol. 28, Issue 10, pp. 2097-2112, (1992). [2.25] Spielmann, C., Curley, P.F., Brabec, T. and Krausz, F., “Ultra broad band femtosecond lasers” IEEE J. Quantum Electron, vol. 30, Issue 4, pp. 1100-1114, (1994). [2.26] Arahira, S. Matsui, Y. and Ogawa, Y. , “Mode-locking at very high repetition rates more than terahertz in passively mode-locked distributed-Bragg-reflector laser diodes” IEEE J. Quantum Electron., vol. 32, Issue 7, pp. 1211-1224 (1996). [2.27] Morgner U., Kï¿ 12 rtner, F. X., Cho, S. H., Chen, Y., Haus, H. A., Fujimoto, J. G., Ippen, E. P., Scheuer, V., Angelow, G. and Tschudi, T., “ Sub-two cycle pulses from a Kerr-lens mode-locked Ti:sapphire laser”, Opt. Lett. vol. 24 (6), pp. 411-413, (1999); doi:10.1364/OL.24.000411. [2.28] Sutter, D. H., Steinmeyer, G., Gallmann, L., Matuschek, N., Morier-Genoud, F., Keller,U., Scheuer, V., Angelow,G. and Tschudi, T., “Semiconductor saturable-absorber mirror-assisted Kerr lens modelocked Ti:sapphire laser producing pulses in the two-cycle regime”, Opt. Lett. vol. 24 (9), pp. 631-633, (1999). doi:10.1364/OL.24.000631. [2.29] Hï¿ 12 nninger, C., Paschotta, R., Zhang, G., Biswal, S., Morier-Genoud, F., Moser, M., Giesen, A., Mourou, G. A., Nees, J. A., Seeber, W., Johannsen, I., Graf, M., Keller, U., Braun, Arthur R., “Ultrafast ytterbium-doped bulk lasers and laser amplifiers”, Appl. Phys. B, vol. 69 (1), pp. 3-17, (1999). doi:10.1007/s003400050762. [2.30] Paschotta, R., Hoenninger, C., der Au, J. A., Spuehler, G. J., Sutter, D. H., Matuschek, N., Loesel, F. H., Morier-Genoud, F., Keller, U., Moser, M., Hoevel, R., Scheuer, V., Angelow, G., Tschudi, T. T., Dymott, M. J, Kopf, D.,Meyer, J., Weingarten, K. J., Kmetec, J. D., Alexander, J. I. and Truong, G. D., “Progress on all-solid-state passively mode-locked ps and fs lasers”, Proc. SPIE vol. 3616, pp. 2, (1999). doi:10.1117/12.351820. [2.31] Sorokin, E.,Sorokina, I.T. and Wintner, E., “Diode-pumped ultra-short-pulse solid-state lasers”, Appl. Phys. B, vol. 72, pp. 3-14, (2001). doi:10.1007/s003400000464.
47
BIBLIOGRAPHY [2.32] Munoz-Zurita, A. L., “Physics and technique of detecting pulsed and continuous light radiation”, National Institute for Astrophysics Optics and Electronics, Ph.D. Thesis, (2009). [2.33] Sorokin, E., Sorokina, I.T. and Wintner, E., “Diode-pumped ultra-short-pulse solid-state lasers”, Appl. Phys. B vol. 72, pp. 3-14, (2001). doi:10.1007/s003400000464. [2.34] Ippen, E. P. and Schenk, C. V., “Picosecond techniques and applications”. in Ultrashort Light Pulses, Ed. by S.Shapiro Springer, Heidelberg, (1977). [2.35] Cohen, L., “Time-frequency distributions-a review”, Proc.IEEE, vol. 77, no. 7, pp. 941981, (1989). [2.36] Dragoman, D. and Dragoman,M., “Quantum coherent versus classical coherent light”, Optical and Quantum Electronics, vol. 33, no. 3, pp. 239-252, (2001). [2.37] Prudnikov, A. P., Brychkov, Y. A. and Marichev, O. I., “Integrals and series”, Gordon and Breach Science Publishers, Amsterdam, vol. 1, Elementary Functions, (1998). [2.38] Akhmanov, S. A., Vysloukh, V. A. and Chirkin,A. S., “Optics of femtosecond laser pulses”, Moscow, Nauka, Translated into English (New York: AIP), (1992). [2.39] Korn, G. A. and Korn, T. M., “Mathematical handbook”, McGraw-Hill Comp., New-York, chapter 21, (1968). [2.40] Kakarala, R., “Triple correlation on groups”, Ph.D. Thesis University of California Irvine, (1992). [2.41] Meshulach, D., Barad, Y. and Silberberg, Y., “Measurement of ultrashort optical pulses by third-harmonic generation”, Journal Of The Optical Society Of America B-Optical Physics, vol. 14(8), pp. 2122-2125, August (1997). [2.42] Iaconis, C. and Walmsley, I. A., “Spectral phase interferometry for direct electric-field reconstruction of ultrashort optical pulses”, Optics Letters, vol. 23(10), pp. 792-794, May (1998). [2.43] Baltuska, A., Pshenichnikov, M. S. and Wiersma, D. A., “Amplitude and phase characterization of 4.5 − f s pulses by frequency-resolved optical gating”. Optics Letters, vol. 23(18), pp. 1474-1476, September (1998).
48
BIBLIOGRAPHY [2.44] Hirayama, T. and Sheik-Bahae, M., “Real-time chirp diagnostic for ultrashort laser pulses”, Optics Letters, vol. 27(10), pp. 860-862, May (2002).
49
Chapter 3
Initial stage of the active mode-locking in semiconductor heterolasers In this chapter, we make an attempt to develop a novel approach to describing the initial stage of the active mode-locking in semiconductor laser structures based on analyzing the properties of dispersion relations in terms of stability for small initial perturbations. The detection system of average time parameters for continuous trains of ultra-short optical pulses was developed for a near infrared laser. Together with this a new Algorithm of operation is describing
3.1
The initial stage of the active mode-locking description
A novel approach to describing the initial stage of the active mode-locking in semiconductor heterolasers is proposed. It is based on analyzing properties of dispersion relations in terms of stability for small initial perturbations. Due to the light field in active medium is governed by the nonlinear Ginzburg-Landau diffusive equation, the dispersion relation between the wave number k and the frequency ω involves the light amplitude and has the simple root, including additional controlling parameter µ. When Imk (ω, µ) 6= 0, one can observe the diffusive instability accompanied by the growth ( Im k < 0) or attenuation ( Im k > 0) of light field; with Im k = 0, the neutral stability appears. Because Im k exceeds zero with one set of µ or it is under zero for the other set of µ, one can consider the dependence µ = G(ω) that gives a boundary curve dividing the areas of stability and instability in the (ω; k)-space. If each optical pulse, incoming into the active medium, is coinciding with maximum gain, one can find µ = − α0 + α1 − α2 , where α0 , α1 , and α2 are the factors of attenuation, gain, and absorption. Then, µ has meaning of the pure gain at a center of optical pulse and gives a parabolic boundary curve in the (ω; k)-space. The point (µC , ωC ) of a minimum on that boundary curve is a critical point. Exceeding the critical value µC = 0, the parameter µ determines the band of unstable frequencies near a critical frequency ωC = ω0 . Originating unstable frequencies, extracting energy from the medium, governs growing the amplitude of initial perturbation. Therefore, a condition of smallness for the amplitude no longer obeys, so the nonlinearity comes into force and competes with dispersion effects, restricting the amplitude and in shaping the optical pulse with rather stable envelope. The frequency
50
3.1. THE INITIAL STAGE OF THE ACTIVE MODE-LOCKING DESCRIPTION ωC is preferable and plays a role of the resonant frequency, because just it is the first, which turns to be unstable. Thus, nonlinear process of shaping optical pulses is interpreted as manifesting instability of diffusion type. Experimental confirmations of appearing the diffusive instability within the active mode-locking process in InGaAsP/InP structures are presented. There are two thresholds in semiconductor heterolasers operating in the active mode-locking regime, namely, one can recognize the threshold of spike-mode oscillation and the threshold of pulse shaping.
3.1.1
General theoretical consideration
Together with the spectral description [3.1] for a process of the active mode-locking in semiconductor laser structures, recently rather adequate models, based on considering the pulse evolution during sequential passing through the domains with optical gain and absorption [2.2] - [3.4], have been proposed. This paper is devoted to describing the initial stage of the above-mentioned processes via studying the properties of dispersion relations in terms of stability for small initial perturbations. For a slowly varying wave packet A (z, t) = A0 (z, t) exp [iθ (z, t)] ,
(3.1)
one can introduce the local angular frequency ω (z, t) = −∂θ/∂t and the local wave number k (z, t) = ∂θ/∂z [3.5]. Because the field A (z, t) in a medium is governed by some evolution equation, the functional dependence between k and ω can be found; and this functional dependence represents the dispersion relation, which can be written generally as P (ω; k) = 0. If a medium exhibits nonlinearity, the corresponding dispersion relation includes the dependence on the amplitude of wave packet P (ω; k; A0 ) = 0.
(3.2)
The physical system under consideration, i.e. semiconductor laser structure, includes the optical gain, so that eq. 3.2 represents the complex-valued relation between k and ω, which has the simple complex-valued root k (ω, µ) = kR (ω, µ) + ikI (ω, µ) ,
(3.3)
where µ is an additional controlling parameter [3.6]. With exploiting eq. 3.3, eq. 3.1 for a wave packet takes the form A (z, t) = A0 (z, t) exp [i (kR z − ωt)] exp (−kI z) + c.c..
(3.4)
51
3.1. THE INITIAL STAGE OF THE ACTIVE MODE-LOCKING DESCRIPTION Evolution equation, which governs the field A (z, t) described by eq. 3.4, has the character of diffusion equation. When kI (ω, µ) 6= 0, one can observe the diffusive instability being accompanied by exponentional grow (with kI (ω, µ) < 0) or attenuation (with kI (ω, µ) > 0 ) of field amplitude. In the case of kI (ω, µ) = 0, the neutral stability can be observed. Because kI (ω, µ) can exceed zero with one set of values of the parameter µ as well as it can be under zero for the other set of µ, one can formally consider the frequency dependence µ = G(ω) determined by eq. 3.5, which gives a boundary curve dividing areas of stability and instability in the (ω; k)space. To analyze the behavior of systems under consideration from the viewpoint of their stability relative to small initial perturbations let us exploit an approximation of one-directional (along, for example, positive direction of the z-axis) traveling for the light wave. This process is described by the following inhomogeneous evolution equation ∂A
+
1 ∂A
= f (A, z, t) , (3.5) ∂z c ∂t where c is the group velocity, while f (A, z, t) is some function reflecting possible contributions from the gain, dispersion, and non-linearity. Now, one can provide the analysis of dispersion relations for the solutions to eq. 3.5 in the form of A (z, t) = b exp {i [k − k(ω0 )] z − [ω − ω0 ] t} + c.c.,
(3.6)
where b and ω0 is small amplitude and the current frequency of a wave packet. To illustrate the proposed approach let us consider an example of the Ginzburg-Landau diffusive evolution equation, which appears usually during the analysis of various mode-locking problems [3.7]. In similar particular cases, the right side of eq. 3.5 takes the form
∂ 2A
2
+ γ |A| A , (3.7) ∂t2 where α, β, and γ are the factors of the gain (losses), dispersion, and nonlinearity, respectif (A, z, t) = −αA − i
β
vely. Substitution of eq. 3.6 into eq. 3.5 with the right side of eq. 3.7 leads to the complex-valued dispersion relation ω − ω0
+ β (ω − ω0 )2 − γ |b|2 + iα, (3.8) c which represents by it self an explicit function k of ω being similar to eq. 3.3. One can k = k(ω0 ) +
separate in eq. 3.8 the controlling parameter µ = α that determines the boundary curve, namely, the abscises axis in the (ω; k)-space. It is seen from eq. 3.8 that the system becomes to be diffusively unstable with α 6= 0.
52
3.1. THE INITIAL STAGE OF THE ACTIVE MODE-LOCKING DESCRIPTION Now, let us turn our attention to the process of active mode-locking in semiconductor laser at the approximation that each ingoing the optical pulse into an active medium is coinciding with a maximum of gain. In this case [2.2], the right hand side of eq. 3.5 takes the form f (A, z, t) =
1 ∂2 A, (3.9) −α0 + α1 [1 − M (1 − cos ωm t)] − α2 1 − 2 2 ωS ∂t
where M and ωm are the depth and frequency of modulation; α0 , α1 , and α2 are the factors of the attenuation, gain, and saturable absorption, respectively; ωS is the amplification (absorption) bandwidth. In so doing, smallness of the amplitude inherent in the initial perturbation is taken into account through exploiting the weak mode-locking approximation within deriving eq. 3.9. Similar to the above-considered example, the dispersion relation, corresponding eq. 3.5 with the right hand side in the form of eq. 3.9, represents an explicit complex-valued function ∆ω k = k(ω0 ) + − i {−α0 + α1 [1 − M (1 − cos ωm t)] − c ∆ω 2 , −α2 1 − ωS2
(3.10)
where ∆ω = ω − ω0 . It follows straightly from eq. 3.10 that within the regime of active mode-locking the combined system “ field-medium ” allows appearing the diffusive instability with the controlling parameter µ = − α0 + α1 − α2 .
(3.11)
Thus, one can separate the controlling parameter µ, which has a meaning of the pure gain at a center of the incoming optical pulse and determine a parabolic-like boundary curve in the (ω; k)space, see Fig. 3.1 (a). The point of a minimum (µC , ωC ) on that boundary curve is a critical point. Exceeding the critical value µC = 0 by the controlling parameter determines the band of unstable frequencies in the vicinity of a critical frequency ωC = ω0 . Originating the band of unstable frequency components, which are able to extract energy from active medium, governs growing the amplitude of initial perturbation. By this is meant that the condition of smallness for the amplitude no longer obeys, and the nonlinearity comes into force and competes with dispersion effects, resulting in the restriction of growing the amplitude and in shaping the optical pulse with rather stable envelope. The frequency ωC (or the mode ωC , if periodic boundary conditions are introduced) is preferable, because just this frequency component is the first, which turns to be unstable.
53
3.2. DETECTION SYSTEM OF AVERAGE TIME PARAMETERS FOR CONTINUOUS TRAINS OF ULTRA-SHORT OPTICAL PULSES IN NIR.
Figure 3.1: Exhibiting the diffusive instability. (a) The boundary curve G(ω) ; an area lying above G(ω) corresponds to growing the initial perturbation, while an area placing below is associated with attenuating the initial perturbations; (b) The evolution of an initial perturbation, i.e. the dynamics of pulse growing and stabilizing. Physically, just this frequency plays a role of the resonant frequency (or mode) in the system under consideration. Figure 3.1 (b) presents the results of computer simulation illustrating the development of initially low-power seeding optical fluctuation in semiconductor laser structure with the diffusive instability. The parameters inherent in active medium had been chosen in such a way that the magnitude of the controlling parameter exceeded a critical value. One can see a stage of growing the seeding fluctuation amplitude at the expense of the energy from active medium as well as a stage of stabilizing the optical pulse envelope due to a balance between contributions of the dispersion and the nonlinearity.
3.2
Detection system of average time parameters for continuous trains of ultra-short optical pulses in NIR.
To study continuous trains of ultra-short optical pulses the auto-correlation approach in parallel light beams was used. For this purpose an auto-manual opto-electronic detection system of trainaverage time parameters had been created. The scheme of this detection system consists of optical auto-correlator 1, electronic controller 2, and a pair of the checking units 3, see Fig. 3.2. The computer soft admits several regimes for the scheme operation: 1. calibration of optical auto-correlator, 2. measuring cycle, and
54
3.2. DETECTION SYSTEM OF AVERAGE TIME PARAMETERS FOR CONTINUOUS TRAINS OF ULTRA-SHORT OPTICAL PULSES IN NIR.
Figure 3.2: Scheme of the auto-manual opto-electronic detecting system of average time parameters inherent in continuous high repetition rate trains of ultra-short optical pulses in near-infrared range 3. data processing i.e. pulse width calculation, and data display. A high repetition rate ultra-short pulse trains arrive at the optical auto-correlator, i.e. at a two-beam scanning Michelson interferometer, which is formed by two total internal reflection prisms and a 50%-mirror. The selection of prisms as the reflecting components leads to an opportunities of both to control the time distribution of light radiation simultaneously and to analyze the auto-correlation functions of the second order inherent in optical pulse trains. Moreover, this prismatic optical circuit permits keeping out the backward scattering from basic reflecting planes of the optical auto-correlator. The rays trace difference in the scanning interferometer is determined by relative disposition of the stationary prism as well as the moving prism, which is fixed on a long-path speaker. During the measuring cycle, the moving prism changes its position step-by-step in relation to one another. The scheme utilizes an interferometric technique of detection: the reflected optical signals form an interferogram, which is registered by a slow-speed photodetector. The given photodetector output signal is amplified, then is converted into digital code, and finally goes into computer that controls the whole process. As a result, all the data related to cycle time-average auto-correlation function of pulse trains is stored in the computer memory and the train-average
55
3.3. ALGORITHM OF OPERATION pulse width is calculated. For visual displaying of auto-correlation function the detected electronic signal arrives at an external memory oscilloscope. The checking units receive another part of pulsed radiation after interferometer. By using a high-speed photodetector and a sampling oscilloscope, one can observe the character of light radiation. The temporal picture is characterized by the fact that optical pulse interval is true, but the pulse envelope is determined by the response time inherent in a high-speed photodetector, whose bandwidth was about 3.5 GHz, so that adequate pulse width measuring was possible only when auto-correlation responses were exploited. This auto-manual opto-electronic scheme is acceptable for ultra-short pulse width measurements within the range from one to a few tens picoseconds, and the record time for an individual interferogram is no more than 1 second.
3.3
Algorithm of operation
The correlation methods of signal processing are successfully in use for a long time. In the simplest case of the second order correlations similar technique is based on the following rather general algorithm see Fig. 3.3.
Figure 3.3: Principle scheme for shaping both auto- and cross-correlation functions. The initial signal S (t) is applied to the input port 1 of a multiplier. The input port 2 is activated by either the additional signal H (t) or the same initial signal S (t), but they both have some variable temporal shift τ due to passing through a delay line and take the forms of H (t + τ ) and S (t + τ ), respectively. The product of a pair of the input signals is integrated with respect of time by the integrator. Thus, the auto- and cross-correlation functions are given by Z∞ a) G (τ ) =
Z∞ dt S (t) S (t + τ ) ,
−∞
dt S (t) H (t + τ ) . (3.12)
, b) K (τ ) = −∞
These formulas represent the correlation functions of just the above-mentioned lowest possible order, i.e. the second one. Under some additional conditions, but definitely not always, the
56
3.3. ALGORITHM OF OPERATION availability, for example, of the function G (τ ) makes it possible to identify the time dependence of the input signal S (t). If H (t) is extremely short pulse, which can be approximated by the Dirac δ-function, and an area of this function H (t) is normalized to unity, one can find that K(t) = S (t). The simplest optical auto-correlator, which can be exploited for estimating time parameters of ultra-short optical pulses in high-repetition-frequency trains, is a two-beam scanning Michelson interferometer with a slow-speed photodetector, see Fig. 3.2 (part 1). In principle, it makes possible detecting the auto-correlation function for the light field strength and, after conversion, the Fourier spectral density of light radiation and find the train-average spectral width of radiation. Two fields E1 (t) = E (t) and E2 (t) = E (t + τ ) related to ultra-short optical signals reflected from the stationary and moving mirrors, respectively, are summarized by a slow-speed photodetector. The delay time τ from pulse to pulse is varied by scanning the moving mirror under electronic control. The output signal is proportional to the energy W on a slow-speed photodetector under condition that the time of integration is long enough. This energy can be estimated as Z∞ a) W ∝
dt (E1 + E2 )2 ≈ [GE (0) + GE (τ )] ,
−∞
Z∞ b) GE (τ ) =
dt E (t) E(t + τ ) = −∞
1
Z∞
2π
dω |E(ω)|2 e−iωτ ,
(3.13)
−∞
where GE (τ ) is the auto-correlation function of the field strength, while |E(ω)|2 is proportional to the Fourier spectral density of light radiation. However, during such a measurement (as well as with exploiting another Fourier spectrometers) the information about the phase of the field E(ω) becomes to be lost. That is why one cannot make correct (unambiguous) conclusion about the train-average pulse width. The ultra-short optical pulse width can be determined from the spectral width rather accurately only if it is known in advance that optical pulse is spectrally (transform) limited, i.e. does not include any internal frequency modulation. This takes place when the phase of field strength along the pulse width grows linearly, so that only the shape of optical pulse envelope determines the spectrum width. The half-width ∆ω of the spectrum of power density for spectrally limited pulses and the half-width τL for the dependence of power on time inherent in the spectrally (transform) limited pulses are connected by ∆ω · τL / (2π) = CB , where CB is the constant determined by the pulse shape. In general
57
3.3. ALGORITHM OF OPERATION case, when pulse is not transform (spectrally) limited, the left hand side of the last formula exceeds CB . To determine the train-average ultra-short optical pulse width, one needs two independent measures of the spectrum width and the internal frequency modulation. Within the direct photodetection, the time resolution is restricted by inertia of various components and an effect of storage associated with this inertia. The response function R (t), inherent in even rather high-speed photodetector , is not perfectly identical to the incoming optical signal S (t), because this response is conditioned by a transfer function B (t). As a result, one has to write Z∞ dt1 S (t1 ) B (t1 − t) ,
R (t) =
(3.14)
−∞
in linear systems. Moreover, B (t1 − t) = 0 with t1 > t due to the causality principle. One can see that the response function R (t) is coinciding with the signal S (t) again only if the transfer function B (t) is the Dirac δ-function. Usually, the normalized transfer functions of high-speed photodiodes can be mathematically approximated by functions of two kinds, namely, by the exponential function exp (−t/T ) or the hyperbolic-like function 1/ 1 + T −1 tm with the power, where the characteristic parameter T is determined by properties of each individual type of photodetector. Figure 3.4 illustrates principally appearing the response function conditioned by the incoming Gaussian optical pulse and the exponential transfer function.
Figure 3.4: Shaping the response function (solid line) conditioned by the incoming ultra-short Gaussian optical pulse (dashed line) and the exponential transfer function (dotted line); the scales of curves are changed to illustrate better.
58
3.4. EXPERIMENTAL RESULTS
3.4
Experimental results
Within using this detection system some measurements of ultra-short pulse continuous trains generated by single-mode semiconductor laser structures at the wavelength of 1320 nm and 1550 nm were done. Our experimental studies have demonstrated that within mode-locking single-mode InGaAsP-laser heterostructures at a threshold of self-excitation (practically, it was realized at a pump current of about 50 mA), only a spike-mode free oscillation regime had been observed with an individual spike width of about 0.7 − 0.9 ps. The investigation of these spikes has shown that each individual spike includes an irregular set of intensity fluctuations. Figure 3.5 represents an example of the digitized oscilloscope trace for the auto-correlation function related to a spike-mode free oscillation when an average spike width is close to 0.7 ps.
Figure 3.5: The digitized oscilloscope trace for the auto-correlation function for a spike-mode free oscillation with an average spike width of about 0.7 ps. Shaping a continuous sequence of stable regular ultra-short optical pulses with duration of about 2 − 10 ps can be achieved only after exceeding a threshold of self-excitation by 10 − 20 %. In so doing, one can observe increasing the energy of oscillation about 10 times, so that the peak-power of regular optical pulses approaches 0.2 − 1.0 W . Figure 3.6 demonstrates an example of the digitized oscilloscope trace for the auto-correlation function related to a regular pulse sequence when an average pulse width is about 6.3 ps.
59
3.5. SHAPING A SINGLE OPTICAL PULSES USING A ACOUSTO-OPTICS TUNABLE FILTERS MODULATION IN TRANSVERSAL CONFIGURATION
Figure 3.6: . The digitized oscilloscope traces related to a regular pulse train: (a) the autocorrelation function for an average pulse width of about 6.3 ps; (b) the output signal from a high-speed photodetector.
3.5
Shaping a single optical pulses using a acousto-optics tunable filters modulation in transversal configuration
Acousto-optics is a branch of physics that studies the interactions between sound waves and light waves, especially the diffraction of laser light by ultrasound or sound in general. An acousto-optic modulator (AOM ), also called a Bragg cell, uses the acousto-optic effect to diffract and shift the frequency of light using sound waves (usually at radio-frequency). The principle behind the operation of acousto-optic filters is based on the wavelength of the diffracted light being dependent on the acoustic frequency. By tuning the frequency of the acoustic wave, the desired wavelength of the optical wave can be diffracted acousto-optically. There are two types of the acousto-optic filters, the collinear and non-collinear filters. The type of filter depends on geometry of acoustooptic interaction. The polarization of the incident light can be either ordinary or extraordinary. The phasematching condition is defined by ki = ks + kd
(3.15)
Where ki is the vector for the field incident, ks the vector of the sound field, and kd is the vector of the diffracted field. The interaction in this case can be viewed in the following terms. A photo with energy hωi , and momentum hki , is incident on a sound wave of frequency ω s and
60
3.5. SHAPING A SINGLE OPTICAL PULSES USING A ACOUSTO-OPTICS TUNABLE FILTERS MODULATION IN TRANSVERSAL CONFIGURATION wave momentum hks . The incident photon is annihilated, giving rise instead to a new photon at ωd , kd and a phonon ωs , ks . The equation 3.15 is a statement of total momentum conservation whereas ωi = ωs + ωd is that of energy conservation. This situation is depicted in figure 3.7 (a) yields drawn, for simplicity, for the case of ni = nd . In this case, since ωs ωi ; ki ≈ kd = k.
Figure 3.7: The Bragg vector diagram and corresponding physical configuration for the diffraction of light from retreating sound wave [3.9]. The vector triangle which is showing in the Figure 3.7 (a) yields ks = 2ksinθ
(3.16)
that, if we use ks = 2λs , k = 2πn/k, can be written as λ 2λs k sin θ = n
(3.17)
This condition is identical to the first-order Bragg condition for the scattering of x rays in crystal. The Bragg scattering treated above is described by a vector diagram figure 3.7(a) involving an isosceles triangle. This is a consequence of the basic momentum conservation condition ki = kd ± kS
(3.18)
61
3.5. SHAPING A SINGLE OPTICAL PULSES USING A ACOUSTO-OPTICS TUNABLE FILTERS MODULATION IN TRANSVERSAL CONFIGURATION when ki ≈ kd = (ωd /c) n. More generally, ni 6= nd , and, consequently, ki 6= kd . The relative directions of the incident and diffracted beams are found from the vector triangle ki = kd ± kS ,. For a given optical frequency and specified directions ki , kd , the acoustic wave frequency and direction are thus determined. A simple example, which follows, may best illustrate this point. If we consider the configuration sketched in Figure 3.8 for coupling by an acoustic wave, two orthogonally polarized optical beams propagating along the x axis of CaM oO4 .
Figure 3.8: Collinear coupling between a z − polarized (extraordinary) and a y polarized (ordinary) optical beam by a shear Szy acoustic wave [3.9]. The crystal possesses a non vanishing p45 photo-elastic element that makes it possible, according to
1 0 0 ∆P (r, t) = − εo εi εd pidkl Ed (rd ) 4 [exp [i (ωd t − kd r)] + c.c.] Skl [exp [i (ωs t − ks r)] + c.c.](3.19) to couple collinearly an ordinary ray and an extraordinary ray as illustrated. The collinear phase matching condition for this case becomes, according to 3.15 ω c
(no − ne ) =
ωs vs
(3.20)
and is demonstrated in Figure 3.9. Here, we associate i with the z polarized extraordinary ray, whereas d is the y-polarized ordinary ray. For (he two waves to be coupled by the acoustic wave, it is thus necessary, according to
62
3.5. SHAPING A SINGLE OPTICAL PULSES USING A ACOUSTO-OPTICS TUNABLE FILTERS MODULATION IN TRANSVERSAL CONFIGURATION
Figure 3.9: Wave-vector diagram for acousto-optic interaction in anisotropic [3.9].
dE drj
=
i q 0 0 ω µε0 εi εd pidkl Skl Ed exp [i (kd + kS − ki ) r] , 4
(3.21)
that pzykl 6= 0. Since in CaM oO4 , pzyzx = p45 is finite, the coupling is accomplished by a shear Szx wave propagating along the x direction. The mathematical description of this configuration is given by E (ri ) = Ei (0) cos (ηri ) + iEi (0) sin (ηri )
(3.22)
E (rd ) = Ed (0) cos (ηrd ) + iEi (0) sin (ηrd )
(3.23)
with ri = rd = x. If we take the input field Eo as polarized along the z axis and for an acoustic frequency satisfying 3.20, the interaction is given by a)Ez (z) = cos (ηx)
πl η= √ λ 2
b)Ez (z) = i sin (ηx) s
n6 p245 ρvshear
Iacoustic
(3.24)
(3.25)
The coupling described above can be “turned off” by changing ωs , since this causes a violation of the momentum conservation condition (2)
3.5.1
Experimental results
We characterized the CaM oO4 crystal like wavelength Band-pass filter [3.9]-[3.14]. For this propose we uses the acousto-optics effect in a longitudinal configuration. The experimental arrangement 3.10
63
3.6. CONCLUSION
Figure 3.10: The acousto-optics experimental set-up in a longitudinal configuration. consist in a green laser with wavelength in 532 nm, two crossed polarizer, RF signal generator, pulses generator, oscilloscope and CaM oO4 crystal. The experimental result is showing next.
Figure 3.11: (a) and (b) show the experimental result in CaMoO4 crystal which is used like a Band-pass filter. The experimental results show that is possible to uses the material like a Band-pass filter width bandwidth in approximately 60kHz. Now we need to know the time to takes the pulse. Now we need to know the time it takes pulse to travel into the crystal. In this particular case the result 15µs.
3.6
Conclusion
Evidently, the proposed approach makes it possible to consider the initial stage of the active modelocking in semiconductor laser structures through analyzing the properties of dispersion relations
64
3.6. CONCLUSION
Figure 3.12: (a) show the electronic RF input signal, in the acousto optical crystal approximately in 1.5µs, (b) show the photodetector out put approximately in 15µs. in terms of stability for small initial perturbations. Within such an analysis, the nonlinear process of shaping optical pulses can be interpreted as manifesting instability of diffusion type. We have observed both the stage of spike-mode oscillation and the stage of pulse shaping. Results of the performed analysis are in coincidence with both the data of numerical simulations and the obtained experimental data. Finally the first approximation of the shaping a ultrashort optical pulse using a acousto-optics tunable filter has been implemented.
65
Bibliography [3.1] Svelto, O., “Principles of laser”. fourth edition, Springer, chapter 8 and 11, New-York (1998) [2.2] van der Ziel, J.P., “The mode-locking in semiconductor lasers. Semiconductors and Semimetals”, vol. 22, Lightwave Communication Technology, vol. Editor W.T. Tsang Academic Press, chapter 1, Orlando (1985) [3.3] Haus, H. A., “Theory of mode-locking with a fast saturable absorber”, J. Appl. Phys., vol. 46, no.7, pp. 3049-3058 (1975). [3.4] Haus, H. A., “Mode locking of semiconductor laser diodes”, Japan J. Appl. Phys., vol.20, no.6, pp. 1007-1020 (1981). [3.5] Whitham, G. B., “Linear and nonlinear waves”. Jhon Wiley and Sons, New-York, (1974). [3.6] Dodd, R. K., Eilbeck, J. C., Gibbon, J. D. and Morris H. C., “Solitons and nonlinear wave equations”. Academic Press, London (1982). [3.7] Haus, H. A., “Waves and fields in opto-electronics”. Prentice-Hall, Upper Saddle River, N.J. (1984) [3.8] Shcherbakov, A. S., Kosarsky, A. Y., Campos Acosta, J., Moreno Zarate, P., Mansurova, S., "Initial stage of the active mode-locking in semiconductor heterolasers", Proc. of SPIE vol. 7386, pp. 73862Z-1-73862Z-9, (2009); doi:10.1117/12.839296 [3.9] Goutzoulis, P. J., Pape, R. D., “Design and Fabrication of Acousto-optic Devices”, Taylor and Francis, New York,(1994). [3.10] Yariv, A., “Quantum Electronics Wiley, 3a Ed., New York, (1988)
66
BIBLIOGRAPHY [3.11] Harris, S. E., S. T. K. Nieh, and D. K. Winslow, “Electronically Tunable Acoustooptic Filter”, Appl. Phys. Letters 15,pp 325 (1969). [3.12] Harris, S. E. S. T. K. Nieh. and R. Feigelson. “CaM oO4 Electronically Tunable” Optical Filter: Appl. Phys. Letters 17, pp 223, (1970). [3.13] Pinnow. D. A., L. G. Van Uitert. A. W. Wamer, and W. A. Bonner, “CaM oO4 ; A Melt Grown Crystal with a High Figure of Merit for Acoustooptï¿ 12 c Device Applications”, App. Phys. Letters 15,pp 83, (1969). [3.14] Yariv. A. and P. Yeh, “Optical Waves in Crystals” Wiley, New York (1984).
67
Chapter 4
Characterization of the time-frequency parameter inherent in the radiation of semiconductor heterolasers using interferometric technique for a Gaussian pulses In this chapter we touch an opportunity of describing two chirped optical pulses together. The main reason of involving just a pair of pulses is caused by the simplest opportunity for simulating the properties of just a sequence of pulses rather then an isolated pulse. However, this step leads to a set of specific difficulty inherent generally in applying joint time-frequency distributions to groups of signals and consisting in manifestation of various false signals or artefacts. This is why the joint Chio-Williams time-frequency distribution and the technique of smoothing are under preliminary consideration here. Finally we present a specific approach to characterizing the train-average parameters of low-power picosecond optical pulses. Together with this we develop this approach involves the joint Wigner time-frequency distributions, which can be created for those pulses due to exploitation of a novel interferometric technique under discussion. The experimental studies was develop for the InGaAsP/InP-heterolasers generating at the wavelength 1320 nm. Then the opportunity of reconstructing the corresponding joint Wigner time-frequency distributions was successfully demonstrated in this section. Also the modified scanning Michelson interferometer has been chosen for obtaining the field-strength auto-correlation functions, with the joint time frequency distribution approach for the pulses reconstruction.
4.1
The Joint Wigner distribution for a pair of Gaussian pulses
The combined complex amplitude for a pair of the chirped solitary optical pulses with Gaussian shape of envelope (at the same carrier light frequency, which is omitted for simplicity sake) can
68
4.1. THE JOINT WIGNER DISTRIBUTION FOR A PAIR OF GAUSSIAN PULSES be written as " AG (t) = A1 exp
−
(1 + ib1 ) (t − t0 )2 2T12
#
" + A2 exp −
(1 + ib2 ) (t + t0 )2 2T22
# , (4.1)
where T1,2 are the Gaussian pulses half-widths measured at a level of 1/e for the intensity contour, t0 is the time shift from the central position, and b1,2 is the parameter of the internal frequency modulation, i.e. the frequency chirp. In the case of similar pulse pair, the joint Wigner time-frequency distribution, see eq. 2.53, is given by " # A21 T1 (t − t0 )2 b1 (t − t0 ) 2 WG (t, ω) = √ exp − − ωT1 + + π T12 T1 " √ ! # (t + t0 )2 b2 (t + t0 ) 2 A22 T2 A1 A2 T1 T2 2 − ωT2 + + × + √ exp − √ π T22 T2 π 2(b1 −i)(1−ib2 )t2 +2ω [(1−ib2 )(t+t0 )T12 −(1+ib1 )(t−t0 )T22 ]−2iω 2 T12 T22 exp (i+b2 )T12 −(b1 −i)T22 p × + (1 + ib1 ) T22 + (1 − ib2 ) T12 √ ! A 1 A 2 T1 T2 2 × + √ π 2(i+b1 )(1+ib2 )t2 +2ω [(1+ib2 )(t+t0 )T12 +(ib1 −1)(t−t0 )T22 ]+2iω 2 T12 T22 exp (b2 −i)T12 −(i+b1 )T22 p × . (4.2) (1 − ib1 ) T22 + (1 + ib2 ) T12 The Wigner distribution 4.2 for a pair of the Gaussian pulses is positive-valued. In practically important case of the pulse train including identical chirped Gaussian pulses with the same amplitudes, durations, and frequency chirps, i.e. with the following values: A1,2 = 1, T1,2 = T , b1,2 = b, and t0 6= 0, eq. 4.2 can be reduced at " # (t − t0 )2 b (t − t0 ) 2 T − ωT + + WGR (t, ω) = √ exp − π T2 T " # T (t + t0 )2 b (t + t0 ) 2 + √ exp − − ωT + + π T2 T T [1 + exp (4iωt0 )] t2 2 2 2 + exp − 2 1 + b − 2bωt + 2iωt0 − ω T . (4.3) √ π T Two examples illustrating the reduced joint Wigner time-frequency distribution WGR (t, ω), which is defined by eq. 4.3 with T = 1, t0 = 6, and b = 0 or 3 are presented in 4.1.
69
4.1. THE JOINT WIGNER DISTRIBUTION FOR A PAIR OF GAUSSIAN PULSES
Figure 4.1: The reduced joint Wigner time-frequency distribution for a pair of identical Gaussian pulses with T = 1, t0 = 6, and the varying parameter b of the frequency chirp: (a) b = 0 and (b) b = 3. Performing integrations of the reduced joint Wigner time-frequency distribution WGR (t, ω) with respect to the corresponding variables give the reduced partial (marginal) one-dimensional Wigner distributions for a pair of identical Gaussian pulses over the time or frequency domains separately
|SGR (ω)|
2
Z∞
T 2ω2 exp − × WGR (t, ω) dt = √ = WGF = 1 + b2 2 1 + b2 −∞ ( " # t0 1 + b2 + bT ω + × 3 + 4 cos (2t0 ω) + Erf √ T 1 + b2 " #) t0 1 + b2 + bT ω +Erf c , (4.4) √ T 1 + b2 2
T2
Z∞
|AGR (t)| = WGT =
WGR (t, ω) dω.
(4.5)
−∞
By mathematical technical reasons, the partial reduced one-dimensional distribution WGT in time domain cannot be currently explained analytically, so that only result of numerical integration is presented in Fig. 4.2 a. It is seen that this 3D-distribution exhibits the absence of any dependence on the frequency chirp parameter b. At the same time, the partial reduced distribution WGF in frequency domain includes the frequency chirp parameter b as an additional coordinate (and degree of freedom).
70
4.1. THE JOINT WIGNER DISTRIBUTION FOR A PAIR OF GAUSSIAN PULSES
Figure 4.2: Plots of partial 3D-distributions in time and frequency domains, respectively. A pair of the Gaussian pulses is taken with T = 1 and t0 = 6: the power density profile (a) and the spectral density profiles (b) with the varying parameter of the frequency chirp b. These distributions are formally determined by Eqs. 4.4, 4.5 and shown in Figs. 4.2 and 4.3 in the particular case of T = 1 and t0 = 6.
Figure 4.3: Plots of 2D cross-sections for partial distributions in frequency domain. A pair of the Gaussian pulses with T = 1 and t0 = 6; two spectral density profiles with the fixed frequency chirp parameter: (a) b = 0 and (b) b = 3.
4.1.1
The joint Choi – Williams time-frequency distribution technique
It is seen from, for instance, 4.1 that the Wigner distribution exhibits the presence of some energy distribution in the central area of these plots where the energy of signal should be intuitively
71
4.1. THE JOINT WIGNER DISTRIBUTION FOR A PAIR OF GAUSSIAN PULSES equal to zero. In connection with this fact, an opinion exists that these false (mistaken) signals or artefacts are conditioned by the mathematical circumstances, in particular, by presence of mutual cross-terms in the Wigner distribution due to its original bilinear nature. Evidently, this difficulty can be resolved within various mathematical approaches. The main idea inherent in one of similar approaches is the following: instead of eliminating the artefacts from Wigner distribution, one can try to find another joint time-frequency distribution, which will minimize these false signals. For this purpose one has to consider the generalized version of similar distributions
P (t, ω) =
Z∞ Z∞ Z∞
1 4π 2
τ τ A∗ u − × φ (θ, τ ) A u + 2 2
−∞ −∞ −∞
× exp (−iθt − iτ ω + iθu) du dτ dθ,
(4.6)
where φ(θ, τ ) is the kernel of this transformation. With φ(θ, τ ) = 1, one can integrate eq. 4.6 first with respect to θ and obtain the Dirac delta-function δ(u − t) under the remaining integrals. Then, performing the second integration with respect to u, one arrives just at the standard Wigner time-frequency distribution. Thus, it is clearly seen that the properties of time-frequency distribution in eq. 4.6 are mainly determined by the properties of its kernel. If the signal under consideration includes, for instance, two sub-signals, one can write A = A1 + A2 , and the substitution of this signal in eq. 4.6 gives P (t, ω) = [P11 (t, ω) + P22 (t, ω)] + [P12 (t, ω) + P21 (t, ω)] .
(4.7)
Here, the first square brackets include the sum of autonomous distributions, while the second square brackets collect a pair of mutual distributions. Each of them can be explained as
Pkm (t, ω) =
1
Z∞ Z∞ Z∞
4π 2
τ τ ∗ φ(θ, τ ) Ak u + Am u − × 2 2
−∞ −∞ −∞
× exp (−iθt − iτ ω + iθu) du dτ dθ,
(4.8)
with (k, m) = (1, 2). Analysis shows that the kernel can be chosen in such a way that the contributions of mutual terms will be minimized, whereas autonomous term will keep their main useful properties. This approach had been studied exploiting, in particular including the generalized uncertainty principle, and one acceptable opportunity is connected with applying the kernel φ(θ, τ ) = exp (− sθ 2 τ 2 ), where s a constant, which can be chosen arbitrarily. After
72
4.1. THE JOINT WIGNER DISTRIBUTION FOR A PAIR OF GAUSSIAN PULSES substituting eq. 4.8 into eq. 4.6, one can perform the integration with respect to θ and find the Choi-Williams time-frequency distribution
PCW (t, ω) =
KChW
1
Z∞ Z∞
KChW exp (−iτ ω) × 4π 3/2 −∞ −∞ τ τ ∗ A u− dudτ, ×A u + 2 2 1 (u − t)2 . = √ exp − 4sτ 2 sτ 2
(4.9)
(4.10)
The kernel function KChW in eq. 4.10 reaches its maximum at u = t, so that an arbitrary constant s can be utilized for the control over relative contribution of the variable τ . In particular, varying s one can manipulate by the level of cross-terms or, what is the same, mutual distributions in eq. 4.7. The kernel, defined by eq. 4.9, leads to real-valued time-frequency distribution. The 3D and 2D plots of the kernel function KChW with s = 10 are shown in Fig. 4.4.
Figure 4.4: The illustrating representations for the Choi-Williams function KChW of kernel with s = 10: a 3D-plot of KChW (a) and two central 2D cross-sections (b). The solid line in Fig. 4.4b is related to the KChW (u − t) cross-section in the limit τ → 0, √ while the dashed line in Fig. 4.4b is for the KChW ( sτ 2 ) cross-section in the limit (u − t) → 0. Now, one can take eq. 4.1 for a pair of the Gaussian pulses in practically important case of the pulse train including identical chirped Gaussian pulses with A1,2 = 1, T1,2 = T , b1,2 = b, t0 6= 0, and substitute it into eq. 4.9. The following calculations are rather complicated, however the first integration with respect to the variable u can be performed analytically and it gives
73
4.1. THE JOINT WIGNER DISTRIBUTION FOR A PAIR OF GAUSSIAN PULSES
PCW (t, ω) =
Z∞
1
√
2π −∞
" × exp − ( × exp
"
T 4sτ 2 + T 2 t2
4sτ 2
−
×
(4.11)
t20 − τ 2 /4 + t0 τ (1 + ib)
t0 τ (2 + ib)
T2
# − iτ ω ×
2 # tT 2 + 2ibsτ 2 (2t0 − τ )
+ 4T 2 τ 2 s (T 2 + 4sτ 2 ) " 2 2 # ibt0 τ tT − 2ibsτ 2 (2t0 + τ ) + exp + + T2 4T 2 τ 2 s (T 2 + 4sτ 2 ) " 2 2 # t0 τ tT − 2sτ 2 (2t0 + ibτ ) + + exp + T2 4T 2 τ 2 s (T 2 + 4sτ 2 ) " 2 2 #)! t0 τ (1 + 2ib) tT + 2sτ 2 (2t0 − ibτ ) + exp + dτ. T2 4T 2 τ 2 s (T 2 + 4sτ 2 ) T2
+
Formally speaking, this expression should be integrated with respect to τ for finding the final analytical version of the joint Choi-Williams time-frequency distribution. Currently, however, we are not able to perform similar integration analytically, so that only the results of illustrative numerical calculations are presented in Fig. 4.5.
Figure 4.5: Two examples of the joint Choi-Williams time-frequency distributions for a pair of identical Gaussian pulses with T = 1, t0 = 6, and the varying parameter b of the frequency chirp: (a) b = 0 and (b) b = 3. One can obviously see that artefacts are significantly sup-pressed in comparison with the corresponding Wigner distributions in 4.1.
74
4.2. EXPERIMENTAL STUDIES
4.1.2
Procedure of smoothing the Wigner time-frequency distribution.
It has been already seen that the Wigner distribution gives false signals due to the presence of mutual cross-terms. The other option, which provides sup-pressing artefacts is connected with so-called smoothing procedure. The most effective realization of this idea, i.e. the procedure of smoothing, is based usually on exploiting the operation of some additional integral transformation. This operation represents in fact a two-dimensional convolution given by Z Z∞ L (t − t1 , ω − ω1 ) W (t1 , ω1 ) dt1 dω1 ,
V (t, ω) =
(4.12)
−∞
where L(t, ω) is a two-dimensional soothing kernel of this convolution. It should be noted that, if the chosen function L(t, ω) does not depend on a signal, one has to donate by the partial distributions to obtain the positively-defined resulting distribution. One of the most frequently applied smoothing functions is
a) L (t − t1 , ω − ω1 ) = h i = exp −α (t − t1 )2 − β (ω − ω1 )2 − 2γ (t − t1 ) (ω − ω1 ) , b) Q = αβ − γ 2 ≤ 1.
(4.13)
A few plots of the smoothing function L are shown in Fig. 4.6. One can see in Figs. 4.6 (a) and 4.6 (b) that with γ = 0 the area of determining the carriers (or arguments) for the function L is positioned perfectly symmetrical relative to the initial axes (t−t1 ) and (ω −ω1 ), so that, for example, the growth of either α and β leads to compressing the smoothing function L along the corresponding axis. When γ 6= 0, see Figs. 4.6 (c) and 4.6 (d), the smoothing function exhibits the dropping behavior as far as the condition 4.13 is satisfied and turns to be remarkably rotated relative to the initial axes. Potentially the procedure of exploiting similar smoothing functions is able to eliminate false signals or artefacts from the resulting joint timefrequency distribution. Then, the question appears about comparison of efficiencies in applying the smoothing procedure and the distributions with modified kernels.
4.2
Experimental studies
Semiconductor lasers have a broad gain band (about ∆ν ≈ 1013 Hz), so that by this is meant that their operation in the regime of active mode-locking makes it possible to expect generating
75
4.2. EXPERIMENTAL STUDIES
Figure 4.6: Various plots of the needed smoothing function L (t − t1 , ω − ω1 ): (a) α = 10, β = 1, γ = 0; (b) α = 100, β = 1, γ = 0; (c) α = 10, β = 1, γ = +3; and (d) α = 10, β = 1, γ = −3. ultra-short optical pulses with a duration of about τ0 ≈ 1/∆ν lying in a picosecond time range. Generally, the active mode-locking process provides shaping stable trains of wave packets with rather good reproducibility from pulse to pulse. Recently, this regime has been practically realized utilizing a periodic modulation of gain inherent in the active medium through injecting the pump current with a frequency equal or multiple to the frequency spacing between longitudinal modes of the laser cavity. Within this discussion, the single-mode InGaAsP/InP semiconductor heterolasers are considered. They have been designed with one antireflection-coated facet and an external single-mode optical-fiber cavity. To obtain the shortest possible optical pulses the facets of semiconductor crystals facing the fiber cavity were coated via deposition of a SiO2 − f ilm, so that the reflection coefficient was typically less than 1%. An external cavity was made of a single-mode silica optical fiber with the refractive index n ≈ 1.5 and the length L ≈ 1 meter with an additional mirror at its far end, providing the optical feedback. The corresponding feedback factor was estimated by 15% due to about 40%-efficiency of exiting the light radiation
76
4.2. EXPERIMENTAL STUDIES in that optical fiber by semiconductor laser structure with the refractive index nS ≈ 3.3. The fiber cavity length L corresponded to the frequency spacing about f0 ≈ 100 M Hz between its longitudinal optical modes because of f = c/ (2n), where c is the light velocity. The scheme of our experiments is presented in Fig. 4.7.
Figure 4.7: Schematic arrangement of the experimental set-up. Periodic modulation of optical losses in a cavity was provided through modulating the pump current from an external source of the electronic sinusoidal RF-signal within the frequency range 400 − 800 M Hz. The electronic port of semiconductor heterolaser was matched with a 50 − Ohm output of that source via specially designed strip-line waveguiding circuit. The regular operation of semiconductor heterolasers was provided by thermo-stabilizing system at a temperature of 160 C with an accuracy of ± 0.20 C. The regime of operation was controlled by the diffractive optical spectrometer. Figures 4.8 illustrate profiles of light radiation spectra at the wavelength λ = 1320 nm without an external RF-modulation as well as with periodic RF-modulation applied at the semiconductor heterolaser, i.e. in the active mode-locking regime. The extended spectrum width within o
the active mode-locking regime was estimated by about ∆λ ≈ 100A. In the frequency domain, this estimation gives ∆ν = (∆λ)c/λ2 ≈ 1.72 T Hz that makes it possible to expect
77
4.2. EXPERIMENTAL STUDIES generating trains of ultra-short optical pulses with characteristic durations lying in the picosecond range.
Figure 4.8: Radiation spectra inherent in semiconductor heterolaser operating at the wavelength λ = 1320 nm: (a) without an external modulation; (b) with an external sinusoidal modulation, i.e. in the active mode-locking regime. A bit rugged profile inherent in the spectrum in fig. 4.8 (b) is affected evidently by the presence of the laser diode cavity by itself and connected with residual reflections from the coated diode facet, which is facing the fiber cavity. Measuring the time-frequency parameters were carried out exploiting the modified interferometric technique described in Ref.[4.1]. During the performed proof-of-principle experiments, oriented on shaping stable trains of rather powerful (about 1 W in a peak) picosecond optical pulses with predictable pulse parameters and with the repetition frequency multiple to the frequency spacing of longitudinal optical modes in fiber cavity, an opportunity of estimating the train-average pulse duration as well as the train-average frequency chirp. The active mode-locking regime on multiple repetition frequencies can be associated with the cases of circulating more than one optical pulse in a long-haul cavity. A number of the circulating optical pulses N can be estimated as N = 2nf L/c, and experimentally the cases with N = 1 − 8 had been successfully realized. It can be noted that the interferogram widths, measured on a level of 1/e for the intensity contour, were decreasing from 12.2 ps to 3.9 ps as the number N was growing from 1 to 8. The absolute frequency bandwidth, being available for the observation of mode-locking, was varying in the range 0.2 − 0.5 M Hz, so that the relative frequency locking band was a little bit less than 10−3 . Figure 4.9 (a) represents the digitized interferogram of the second order auto-correlation function for a high-repetition-rate train of optical pulses; the width of this interferogram was esti-
78
4.2. EXPERIMENTAL STUDIES mated by 4.4 ps, while Fig. 4.9 (b) shows the digitized oscilloscope trace for a train of ultra-short pulses with the repetition frequency f ≈ 7f0 = 718 M Hz, which was identified as the most stable during the experiments performed. The parameter b, related to the frequency chirp, was estimated with applying the above-mentioned technique by b ≈ 1.46 · 10−4 . This is a train of picosecond pulses detected with the time resolution of about 300 ps, which is associated with the transfer function of a high-speed photodetector exploited. The off-duty ratio for optical pulses depicted in Fig. 4.9 (b) is in correspondence to the ration between the repetition period 1/f and the above-mentioned time resolution of that high-speed photodetector.
Figure 4.9: The digitized oscilloscope traces related to a regular pulse train: (a) the train-average auto-correlation function; the pulse width of this interferogram, measured on a level of 1/e for the intensity contour, was estimated by 4.4 ps; (b) the output signal from a high-speed photodetector; a train of the same ultra-short optical pulses with the repetition frequency f ≈ 718 M Hz was detected with the time resolution of about 300 ps.
4.2.1
Characterizing Optical Pulses
In the active mode-locking regime, optical pulses are self-reproducing after each path through the cavity. Restoration of pulse parameters is conditioned by properties of the active medium, and because the cavity exhibits an optical dispersion, one of the necessary conditions for reproducibility of pulses is the presence of frequency chirp. Presently known mechanisms of interacting optical pulses with semiconductors allow us to simplify the theoretical model of shaping an ultra-short pulse with the complex field amplitude E(t) = A(t) exp (iω0 t) + c. c. in a heterostructure. The pulse, grown during the process of active mode-locking, has a Gaussian shape and can be described by eq. 5.40.
79
4.2. EXPERIMENTAL STUDIES (1 + ib) t2 , AG (T ) = A0 exp − 2T 2
(4.14)
The pulse width, measured on a level of 1/e for the intensity contour, is given by [4.1] T = (g m)−1/4 (ωm ωS )−1/2 ,
(4.15)
where g is the maximal gain at t = 0, m is the factor of external modulation of the losses in a cavity, ωm is the external modulation frequency, and ωS is the gain contour width. Finally, the frequency chirp can be expressed as [4.1] 2
a) b = 2T β ,
b) β =
LD ωm ωS2
√ m
4 [gω0 TC /(2Q)]3/2
d2 k dω 2
,
(4.16)
where β is the dimensional factor of frequency chirp, LD is the length of high-dispersion components (for example, the laser crystal), ω0 is the central frequency of emission, Q is the quality factor inherent in a cavity, TC is the transit time of a pulse through a cavity, and k is the wave number. In fact, Eqs. 4.15 and 4.16 can be practically used to estimate the parameters of the optical pulses generated. Using the values characteristic of the experiments: g = 3, m = 0.25, ωm = 2π · 718 · 106 rad/s, and ωS = 2π · 1013 rad/s, one can obtain T ≈ 2.73 ps from eq. 4.15, which can be considered as rather good agreement with the experimental data. The frequency chirp that arises within establishing the self-reproducing pulses can be estimated with eq. 4.16. For LD ≈ 0.5 mm, ω0 = 2.1 · 1015 rad/s (at λ = 1320nm), TC = 10−8 s, Q = 105 , and d2 k/dω 2 = 3.7 · 10−24 s2 /m, one can obtain β = 7.3 · 1018 s−2 from eq. 4.16 (b). Nevertheless, this dimensional magnitude of the estimated frequency chirp is relatively small, because one can find from eq. 4.16 (a) in dimensionless values that b ≈ 0.84 · 10−4 1. In practically reasonable assumption that the envelopes of optical pulses under consideration can be described rather adequately by Gaussian functions, these estimations make it possible to create the corresponding theoretical version of Wigner time-frequency distribution with the abovecalculated parameters T and b. Together with this, the experimental version of similar timefrequency distribution can be designed with experimentally obtained parameters T ≈ 2.2 ps and b ≈ 1.46 · 10−4 in the same approximation by Gaussian functions. The resulting plots of two Wigner distributions for the Gaussian-like optical pulses, obtained from estimations and from experiment, are shown in Fig. 4.10.
80
4.3. CONCLUSION
Figure 4.10: A pair of the Wigner time-frequency distribution for the Gaussian pulses obtained from the performed estimation with T = 2.73 ps and the b = 0.84 · 10−4 as well as from the experiment with T = 2.2 ps and the b = 1.46 · 10−4 .
4.3
Conclusion
A specific approach for the average characterization of high-repetition-rate trains including lowpower bright picosecond optical dissipative solitary pulses with an internal frequency modulation is developed in both time and frequency domains in more practical manner that before [4.2]-[4.4]. The presented approach is oriented to using the joint time-frequency distributions for a pair of solitary pulses, because similar distributions consist of both autonomous and mutual contributions peculiar not to the isolated pulses, but to the combined signals or/and pulse trains. In spite of practical mathematical difficulties that had been met in this case, the algorithm of general characterization has been nevertheless sequentially followed. Within this analysis, the significant part of attention has been paid to rather specific problem of appearing false signals or artefacts caused by mutual cross-terms inherent in the joint time-frequency distributions generally as well as in the Wigner distribution in particular. In a view of improving this obvious disadvantage peculiar to just Wigner time-frequency distribution, two possibilities have been considered to resolve or avoid the problem of appearing the artefacts. • One of them is based on involving reasonable modifications into the kernel of Wigner dis-
81
4.3. CONCLUSION tribution, so that the joint Choi-Williams time-frequency distribution has been included into our approach to the above-mentioned average characterization. • The other possibility is related to the procedure of smoothing via constructing an additional 2D-convolution with the needed smoothing function. Both these avenues have been preliminary touched, i.e. open slightly and briefly looked through. Practical application of this technique had been recently discussed and consists in characterizing the parameters of ultra-short optical dissipative solitary pulses in the actively mode-locked semiconductor laser with an external fiber cavity. Also the modified scanning Michelson interferometer has been chosen for obtaining the fieldstrength auto-correlation functions, with the joint time frequency distribution approach for the pulses reconstruction. In fact, we have presented the key features of a new experimental technique for accurate and reliable measurements of the train-average temporal width and the frequency chirp of picosecond optical pulses in high-repetition rate trains. This technique makes it possible to find the parameters needed for reconstructing the joint Wigner distributions inherent in optical pulses. The InGaAsP/InP-heterolasers, operating at 1320 nm wavelength range, have been used within the experiments. When the optical signal consists of contiguous pulses with the repetition frequency close to 1 GHz, conditioned by operating semiconductor lasers in the active mode-locking regime, typical requirements for measurements and operating with the Wigner distributions have been satisfied, so that the train-average pulse parameters have been successfully characterized.
82
Bibliography [4.1] van der Ziel, J. P. “ The mode-locking in semiconductor lasers”. Semiconductors and Semimetals, vol. 22 ( Lightwave Communication Technology), volume editor W.T. Tsang Academic Press, Orlando, chapter 1, (1985). [4.2] Shcherbakov, A. S. , Munoz Zurita A. L. , Kosarsky A.Yu.and Campos Acosta J. “Determining the time - frequency parameters of low-power bright picosecond optical pulses using interferometric technique”. OPTIK - Int. J.Electron. Opt. Elsevier, Germany; doi:10.1016/j.ijleo.2008.07.033. [4.3] Shcherbakov, A. S., Moreno Zarate, P., Campos Acosta, J., Il’n, Y. V., Tarasov, I. S., "Characterization of the time-frequency parameters inherent in the radiation of semiconductor heterolasers using interferometric technique", Proc. of SPIE vol. 7386, pp. 73862H-1-73862H-9 (2009); doi:10.1117/12.838469 [4.4] Shcherbakov, A. S., Moreno Zarate, P., Campos Acosta, J., Il’n, Y. V., Tarasov, I. S., “Applying the joint Wigner time-frequency distribution to characterization of ultrashort optical dissipative solitary pulses in the actively mode-locked semiconductor laser with an external single-mode fiber cavity” Proc. of SPIE vol. 7597, pp. 75971B-175971B-10 (2010); doi:10.1117/12.839445 [4.5] Shcherbakov, A. S., Kosarsky, A. Y., Moreno Zarate, P., Campos Acosta, J., Il’n, Y. V., Tarasov, I. S., “Characterization of the train-average time-frequency parameters inherent in the low-power picosecond optical pulses generated by the actively mode-locked semiconductor laser with an external single-mode fiber cavity”, Opt. Int. J. Light Electron. Opt. (Optik, Elsevier, Germany), pp. 1-7, (2009) DOI: 10.1016/j.ijleo.2009.11.020. [4.6] Shcherbakov, A. S., Kosarsky, A. Y., Campos Acosta, J., Moreno Zarate, P., Mansurova, S., Il’n, Y. V., Tarasov, I. S., “Characterizing the parameters of ultra-short optical
83
BIBLIOGRAPHY dissipative solitary pulses in the actively mode-locked semiconductor laser with an external fiber cavity”, Proc. of SPIE Tracking No. PN10-PN100-122, pp. 1-9 (2010)
84
Chapter 5
Analysis of originating ultra-short optical dissipative solitary pulses in the actively mode-locked semiconductor heterolasers with an external fiber cavity. We describe the conditions of shaping regular trains of optical dissipative solitary pulses, excited by multi-pulse sequences of periodic modulating signals, in the actively mode-locked semiconductor laser heterostructure with an external long-haul single-mode silicon fiber exhibiting square-law dispersion, cubic Kerr nonlinearity, and linear optical losses. The presented model for the analysis includes three principal contributions associated with the modulated gain, optical losses, as well as linear and nonlinear phase shifts. In fact, the trains of optical dissipative solitary pulses appear within simultaneous presenting and a balance of mutually compensating interactions between the second-order dispersion and cubic-law Kerr nonlinearity as well as between active medium gain and linear optical losses in the combined cavity. Within such a model, a contribution of the nonlinear Ginzburg-Landau operator to shaping the parameters of optical dissipative solitary pulses is described via exploiting an approximate variational procedure involving the technique of trial functions. Finally, the results of the illustrating proof-of-principle experiments are briefly presented and discussed in terms of optical dissipative solitary pulses
5.1
Process of shaping regular trains of optical solitary pulses
The active mode-locking regime in semiconductor heterolaser with an external cavity caused by a periodic signal. Within the spectral approach, one has to investigate an ensemble of the standing optical waves corresponding to the eigen-modes inherent in a combined laser cavity. The frequency spacing ∆νm between eigen-modes with the optical frequencies νm and νm+1 is given
85
5.1. PROCESS OF SHAPING REGULAR TRAINS OF OPTICAL SOLITARY PULSES by ∆νm = c (2
P
i
ni li )− 1 , where m is the number of the longitudinal optical eigen-mode
peculiar to a combined cavity, c is the light velocity, while the second multiplicand represents full optical length of a combined cavity formed by a sum of domains of the geometric length li with the refractive index ni . Nonlinear factors, such as the time-dependent amplification and nonlinear dispersion, induce appearing side components of polarization spaced from the initial mode by the frequency intervals multiple to ∆νm . The presence of similar polarization components at these frequencies provides the energy exchange and originating the fixed phase relations between modes. The mode-locking takes place within a limited spectral bandwidth ∆ν, measured usually at the chosen intensity level, so that the number M of the locked eigen-modes can be estimated as M = ∆ν/∆νm . In this case, the optical pulse width τP , measured at the same level as ∆ν and inherent in just a transform-limited pulse, can be estimated by τP ≈ α/∆ν, where the numerical factor α depends on that pulse shape. The corresponding theory describes evolving the pulse envelope as far as it propagates through a cavity. In a steady-state regime, one can observe the self reproduction of pulse envelope during each particular path along a cavity. Within each similar path taken alone, optical losses in a cavity are compensated by amplification, while effects, conditioned by dispersion and finiteness of the spectral bandwidth, are responsible for spectrum and phase structure of the resulting optical pulse. When the mode-locking process is realized inside a cavity and the circulating pulse arrives at the active medium, very intensive burst of the stimulated radiation and dumping the accumulated inversion. Thus, one need only provide a small portion of current amplification for a pulse to compensate current optical losses inside the cavity. Together with this, one can say that both the dynamics of photon density as well as the behavior of charge carriers in a laser have not too essential effect on the output pulse duration. Such a conclusion follows from the fact that within the mode-locking only very short time interval of pure amplification exists, which is conditioned by a balance between depleting the gain and acting the total losses in a cavity. Just the observation of picosecond optical pulses in experiments shows that the last processes develop about two orders faster than the relaxation processes caused by the dynamics of photon density and charge carriers.
86
5.2. A MULTI-PULSE REGIME OF THE ACTIVE MODE-LOCKING IN SEMICONDUCTOR HETEROSTRUCTURE WITH AN EXTERNAL CAVITY
5.2
A multi-pulse regime of the active mode-locking in semiconductor heterostructure with an external cavity
Let us consider the active mode-locking regime in semiconductor laser with an external cavity caused by a periodic signal. A set of the well-known mechanisms [5.1], [5.2] governing the interaction between optical pulses and semiconductors make it possible to elaborate a model of shaping an ultra-short pulse in the form of E(t) = A(t) exp(iω0 t) + c.c. in heterostructure; where A(t) and ω0 are the amplitude and carrier frequency of a pulse. The process of reshaping a pulse can ˆ (t). Usually, the semiconductor hetebe described within an action of some resulting operator M ˆ while action of the total rostructure includes an active domain with the gain function operator G, ˆ and the dispersion operator Φ ˆ are distributed along the combined cavity of the losses operator L ˆ (t) has to be described as total length l0 , so that the resulting operator M
l Z0 n o h i ˆ ˆ ˆ ˆ ˆ (t) = exp G(x, t) − L(x, t) − iΦ(x, t) dx = exp D(t) , M
(5.1)
0
If varying the optical pulse envelope is small enough for each pass through a cavity, one can write that h i ˆ ˆ + D(t) Aj+1 (t) ≈ 1 Aj (t),
(5.2)
where j is the number of a pass. Then, in a steady-state regime, the pulse becomes to be self-reproducible after each individual pass through a cavity, so that the relation ˆ D(t) A(t) = 0,
(5.3)
ˆ will be true. Consequently, the differential operator D(t) is characterizing the evolution of a pulse in a linear cavity. The homogeneity of eq. 5.3 allows a superposition of a few processes of the active mode-locking simultaneously in the same linear cavity. Such phenomena are possible within imposing a few external periodical modulations, and what’s more, the dynamics of shaping pulses, corresponding to the different external modulations, can differ from each other. Similar superposition of a few processes of the active mode-locking can be described by a sum of the PN ˆ = ˆ k . When any influence of one modulation onto partial differential operators as D D k=1
87
5.2. A MULTI-PULSE REGIME OF THE ACTIVE MODE-LOCKING IN SEMICONDUCTOR HETEROSTRUCTURE WITH AN EXTERNAL CAVITY another is practically (or physically) absent due to, for instance, their spacing in time, one can realize N independent on each other processes of the active mode-locking in parallel. Moreover, these independent processes of the active mode-locking can take place even in a nonlinear cavity, which is nevertheless able to keep their mutual independence on each other due to again their spacing in time. In this case, eq. 5.3 represents a linear sum of partial relations ˆ k (t) Ak (t) = 0, D
(5.4)
where Ak (t) is the envelope of a k − th partial optical pulse train, which is shaped under ˆ k (t) can action of a k − th external partial periodic modulation, while each partial operator D now exhibit some nonlinearity in behavior. A multi-pulse regime of the active mode-locking can be initiated, in particular, by a set of N shifted in phase independent on each other electronic excitations with the same repetition frequency ωm to keep periodicity of this process as a whole. The role of these excitations can be played by an amount of the partial injection currents representing an overall pump for the semiconductor laser heterosrtucture under operation. Of course, there are a lot of possibilities how to choose similar set of mutually independent periodic excitations; however, the most practical approach is based on utilizing just periodic functions. For instance, one can choose the even powers of trigonometric functions providing just positive-valued strings of well-separated pulse-like modulating signals. In so doing, one can write the following sum of just trigonometric function-like exciting signals I(t) = I0 +
XN k=1
Ik cos2n (ωm t − φk ),
(5.5)
where n 1 is the factor determining a separation of the neighboring electronic modulations, I0 is the amplitude of a constant pump current background, Ik is the current amplitude of a k − th external partial periodic modulation, φk = πk/N is the needed phase shift. An illustrating set of N periodic sequences of the modulating pump current is shown in fig. 5.1; the constant current background is omitted. Broadly speaking, the modulating excitation expressed by eq. 5.5 can be considered from the possible viewpoint of applications to high-bit-rate optical communication systems as the modulation by a multi-bit digital code, which is capable generating the encoded and periodically repeating string of N optical pulses. The property of this string is such that each Ik = 0 corresponds to the bit symbol “zero”, while each Ik = Ik0 6= 0 is associated with the bit symbol “unity”. It is very attractive terminologically to describe the processes peculiar to a multi-pulse active mode-locking in terms of binary encoded signals. Mutual independence of individual processes of the active
88
5.2. A MULTI-PULSE REGIME OF THE ACTIVE MODE-LOCKING IN SEMICONDUCTOR HETEROSTRUCTURE WITH AN EXTERNAL CAVITY
Figure 5.1: A set of N periodic sequences of the modulating pump currents; the constant current background is omitted. mode-locking requires that intervals between the stages of amplification related to different external modulations should be large enough for recovering the gain in semiconductor heterosrtucture. Assuming that external modulations are equidistant in their sequence, one can estimate that the repetition frequency ωm is restricted by the inequality ωm < 1/(τ1 N ). The above-listed power factor n determining a separation of the neighboring electronic modulations plays rather important role, so that it should be optimally chosen when the chosen modulating excitations are described by the even powers of trigonometric functions. Within n 1, one can expect that the gain can be provided for a short time interval in comparison with the pulse-repetition period, while the effect of modulation is close to zero for the rest of the time. An advantage of the chosen presentation for the joint modulating electronic signal in the form of eq. 5.5 is connected with the fact that the modulating signal can be approximated by a parabolic dependence near each maximum of amplification. This circumstance makes it possible to analyze our problem in the approximation of so-called weak active mode-locking [5.2]. Just a parabolic approximation for the modulating excitations allows us to estimate the duration of the amplification stage at half width level Tm by p −1 2/n. This is why, when the condition the value of ωm |φk − φk+1 | =
π
r > ω m Tm ≈
2
, (5.6) N n is satisfied, one can say that the neighboring periodic electronic signals are imposed independently on each other. In the other hand one can be exploiting such a form of a multi-pulse excitation, the corresponding analysis can be performed [5.3] to obtain the combined evolution equation, governing the dynamics of shaping the trains of ultra-short optical dissipative solitary pulses in semiconduc-
89
5.2. A MULTI-PULSE REGIME OF THE ACTIVE MODE-LOCKING IN SEMICONDUCTOR HETEROSTRUCTURE WITH AN EXTERNAL CAVITY tor laser heterostructures with an external long-haul single-mode fiber cavity, has the following differential form λ2 ∂ 2 ∂ πn2 2 Ak (x, t) + − + i D1 |Ak (x, t)| − LC + i ∂x 2πc ∂t2 Sλ h X 2 2 Lp δ(x − xp )+ + g(x) (1 − 2nmk ωm t ) − Lg − p LS λ2 ∂2 + − iΦ0 Ak (x, t) = 0 − iD 2 ωS2 2πc ∂t2
(5.7)
Here, λ and c are the wavelength and velocity of light; D1,2 are the dispersion factors for πn2 |Ak (x, t)|2 is caused the fiber span and semiconductor domain, respectively. The term Sλ by nonlinear contribution from the self-phase modulation of a pulse when it is passing through a fiber cavity with the cubic-law Kerr nonlinearity, where n2 is the Kerr nonlinear refractive index, S is the cross-section of a fiber core. This nonlinear term is directly proportional to the instant power of optical pulse in a k − th partial sequence. It should be noted once again the corresponding nonlinear contributions in each k − th partial sequence of pulses can be considered here as perfectly independent on one another, so that general representation of eq. 5.3 in terms of well-separated Eqs.5.4 is definitely true. Then, g(x) is the gain distribution at t = 0, mk is the modulation depth for a k − th partial component of the pump current. The distributed non-dispersive losses Lg and LC characterize the light scattering in the combined cavity and the dissipation inherent in just an external fiber cavity, respectively. To describe the spectral-selective losses LS one can exploit a parabolic approximation for the Lorentz-profile contour of losses and consider ωS as the gain contour width. The lumped optical losses at p discrete points of the P combined cavity are taken into account with the term p Lp δ(x − xp ) , where δ(x) is the Dirac delta function. The term Φ0 reflects the non-dispersive phase shift. Now, it can be noted additionally, see fig. 5.2, that this combined evolution equation includes the terms, isolated by the figure brackets in eq. 5.7, that represent the nonlinear operator of so-called Schroedinger-like type. In fact, this is the cubic Ginzburg-Landau operator, which describes the evolution of optical dissipative solitary pulses in a sufficiently long single-mode cavity with square-law dispersion, cubic-law Kerr nonlinearity, and linear optical losses. The corresponding nonlinear GinzburgLandau evolution equation will be analyzed in section 4.5. Then, the square brackets in eq. 5.7 isolate a quasi-linear operator, which describes in its turn a partial process of the active mode-locking in semiconductor heterolaser with just a linear cavity.
90
5.3. EVOLUTION EQUATION GOVERNING THE DYNAMICS OF SHAPING ULTRA-SHORT OPTICAL DISSIPATIVE SOLITARY PULSES TRAINS IN SEMICONDUCTOR HETEROLASER WITH AN EXTERNAL FIBER CAVITY
Figure 5.2: Arrangement of the combined evolution equation 5.7. Equating the last quasi-linear operator to zero leads to the reduced evolution equation, which is considered in the next section 4.3
5.3
Evolution equation governing the dynamics of shaping ultra-short optical dissipative solitary pulses trains in semiconductor heterolaser with an external fiber cavity
Evidently, the approximation of a weak active mode-locking [5.2], i.e. the assumption of rather weak exhaustion of the gain by the stimulated radiation, gives the most sequential description for the dynamics of this process, so that it can be exploited here. In the case under consideration, one has to carry out the analysis taking into account the raised level of peak power inherent in optical pulses in a cavity. Moreover, if an external cavity includes optically nonlinear material such as a semiconductor heterostructure and/or a long-haul single-mode fused silica fiber, manifesting the optical selfaction effects can be expected. Keeping this in mind, let us rewrite eq. 5.2 with eq. 5.4 in the differential form
a)
∂Ak (x, t) ∂x
ˆ k (x, t) Ak (x, t) =D
ˆ k (t) = b) D
Zl0
ˆ k (x, t) dx, D
(5.8)
0
ˆ k (x, t) should be found from the determination Now, the obvious form for the operator D ˆ k (x, t) is comwithin eq. 5.1. Starting from the gain operator, one can say that the operator G pletely caused by the electronic modulating signal. In the case of the above-chosen amplitude ˆ k (x, t) = g(x) {1 − modulation, see eq. 5.5, this operator has the almost standard form G
91
5.3. EVOLUTION EQUATION GOVERNING THE DYNAMICS OF SHAPING ULTRA-SHORT OPTICAL DISSIPATIVE SOLITARY PULSES TRAINS IN SEMICONDUCTOR HETEROLASER WITH AN EXTERNAL FIBER CAVITY 4nmk [1 − cos (ωm t)]}, where g(x) is the gain distribution at t = 0, mk is the modulation ˆ k (x, t) in a power series depth for a k − th partial component of the pump current. Expanding G of the parameter ω0 t under the condition t < τ0 , one can find a parabolic approximation ˆ k (x, t) = g(x) (1 − 2nmk ω 2 t2 ), G m
(5.9)
The factor 2n in eq. 5.9 reflects a significant steeping of the modulating excitation edges due to their chosen shape. Together with this, one can take 2n = 1 within a harmonic excitation, so that the modulating frequency is halved. ˆ k (x, t) are conditioned by non-dispersive as well as The main contributions to the operator L spectral-selective optical losses. The non-dispersive losses include both the losses on mirrors and the imperfection of coupling between a heterostructure and an external cavity can be described as P a set of the lumped optical losses at p discrete points of the combined cavity, i.e. p Lp δ(x − xp ) , where δ(x) is the Dirac delta function. The distributed non-dispersive losses L0 and LC characterize the light scattering in the combined cavity and the dissipation inherent in just an external long-haul fiber cavity, respectively. To describe the spectral-selective losses LS one can exploit a parabolic approximation for the Lorentz-profile contour of losses. In this approximation, the ratio (∆ω0 /ωS ) 1 can be taken as the parameter of smallness for the corresponding power series expansion. Summarizing all these contributions, one can write the losses operator as ˆ k (x, t) = Lg + LC + L
X p
Lp δ(x − xp ) −
LS ∂ 2 ωS2 ∂t2
,
(5.10)
ˆ k (x, t) represents the following sum of three contributions The phase operator Φ ˆ k (x, t) = Φ0 + Φ ˆ 1 (t) + Φ ˆ 2 (t) + Φ ˆ 3 (x, t), Φ
(5.11)
ˆ 1,2 (t) describe The term Φ0 reflects the non-dispersive phase shift, while the summands Φ the dispersion of an active medium, i.e. a semiconductor heterostructure, and the dispersion of a long-haul single-mode fiber, i.e. an external cavity, respectively. They can be explained in a quite similar way as ˆ 1,2 (t) = − D1,2 Φ
λ2
∂2
, (5.12) 2πc ∂t2 where D1,2 are the dispersion factors for the corresponding spans; λ and c are the wavelength and velocity of light. The last term in eq. 5.11 is caused by nonlinear contribution from the selfphase modulation of a pulse when it is passing through a fiber cavity with the cubic-law Kerr nonlinearity, so that one can write
92
5.3. EVOLUTION EQUATION GOVERNING THE DYNAMICS OF SHAPING ULTRA-SHORT OPTICAL DISSIPATIVE SOLITARY PULSES TRAINS IN SEMICONDUCTOR HETEROLASER WITH AN EXTERNAL FIBER CAVITY πn2
|Ak (x, t)|2 , (5.13) Sλ where n2 is the Kerr nonlinear refractive index, S is the cross-section of a fiber core. The ˆ 3 (x, t) is directly proportional to the instant power of optical pulse in a nonlinear operator Φ ˆ 3 (x, t) = − Φ
k − th partial sequence. It should be noted once again that nonlinear contributions caused by ˆ 3 (x, t) in each k − th partial sequence of pulses are considered here the nonlinear operators Φ as perfectly independent on one another, so that general representation in terms of well-separated Eqs.5.3 and 5.4 is true. Finally, one can say that the combined evolution equation, governing the dynamics of shaping the trains of ultra-short optical dissipative solitary pulses in semiconductor laser heterostructures with an external long-haul single-mode fiber cavity, has the following differential form
−
∂ ∂x
+ i D1
λ2 ∂ 2 2πc ∂t2
+ i
πn2 Sλ
2
|Ak (x, t)| − LC
Ak (x, t) +
h
X 2 2 g (x) 1 − 2nmk ωm t − Lg − Lp δ (x − xp ) + p ∂2 LS λ2 − iD + − iΦ0 Ak (x, t) = 0, 2 ωS2 2πc ∂t2
(5.14)
Now, it can be noted additionally, that this combined evolution equation includes the term, isolated by the figure brackets in eq. 5.14, that represents the nonlinear operator of so-called Schroedinger-like type. In fact, this is the cubic Ginzburg-Landau operator, which describes the evolution of optical dissipative solitary pulses in a long-haul single-mode cavity with square-law dispersion, cubic-law Kerr nonlinearity, and linear optical losses. Then, the square brackets in eq. 5.14 isolate some linear operator, which describes in its turn a partial process of the active mode-locking in semiconductor heterolaser with just a linear cavity. Equating the last linear operator to zero leads to the reduced evolution equation whose solutions can be explained in terms of the standard Hermit-Gauss functions. Fortunately, the eigen-function for this linear operator, corresponding to the simplest dispersion-free regime of generating the unchirped optical pulses (i.e. pulses free of any internal frequency modulation) via the active mode-locking process, is represented by an ordinary Gaussian function 4t2 ln 2 , a) ALk (t) = A0k exp − 2 τ0k √ b) τ0k = 2 ln 2 (2nmg)−1/4 (ωm ωS )−1/2 .
(5.15)
93
5.3. EVOLUTION EQUATION GOVERNING THE DYNAMICS OF SHAPING ULTRA-SHORT OPTICAL DISSIPATIVE SOLITARY PULSES TRAINS IN SEMICONDUCTOR HETEROLASER WITH AN EXTERNAL FIBER CAVITY Nevertheless, one has to keep in mind that the partial amplitude A0k of an individual optical pulse cannot be chosen in an arbitrary way in eq. 5.15 (a) due to the presence of the nonlinear Ginzburg-Landau operator in eq. 5.14.
5.3.1
Quasi-Linear Evolution equation governing the mode-locking process
The quasi-linear operator related to the square brackets in eq. 5.7 leads to the linear differential equation
a) a
d2 Ak dt2
+ p − rt2 Ak = 0,
c) p = g (x) − Lg −
X p
LS
b) a = a1 − ia2 =
Lp δ (x − xp ) − iΦ0 ,
ωS2
− iD2
λ2 2πc
, (5.16)
2 d) r = 2g (x) nmk ωm .
General solution to eq. 5.16(a) with the initial conditions Ak (t = 0) = Ak,0 and dAk /dt | (t = 0) = 0 has the form
Ak (x, t) = Ak,0 exp
−
t2
r r
2
a
· 1F1
r ! √ ar − p 1 2 r ; ; t , √ 4 ar 2 a
(5.17)
where ar 6= 0 and
1F1 (µ; ϑ ; z) = 1 +
µz ϑ
+
µ (µ + 1) ϑ (ϑ + 1)
·
z2 2!
+ ··· =
X∞ S=0
(µ)S (ϑ)S
zS s!
is the standard Kummer confluent hypergeometric function. The obtained solution, described by eq. 5.17, will describe a pulse with a Gaussian shape in two theoretically possible cases. The first case is connected with the formal option 1F1 (µ; ϑ ; z) = 1, which gives Gaussian shape of pulse envelope √ √ Ak (x, t) = Ak,0 exp −t2 r/(2 a)
(5.18) √ due to µ = 0. Therefore, one has to consider the complex-valued equality p = + ar, were, r is real-valued positive parameter, see eq. 5.16(d). The square root is taken here with the positive sign, which is conditioned by choosing just the positive sign for the square root within √ √ calculating a, even when the magnitude of a by itself is complex-valued. Squaring this
94
5.3. EVOLUTION EQUATION GOVERNING THE DYNAMICS OF SHAPING ULTRA-SHORT OPTICAL DISSIPATIVE SOLITARY PULSES TRAINS IN SEMICONDUCTOR HETEROLASER WITH AN EXTERNAL FIBER CAVITY equality and dividing its real and imaginary parts lead to the following pair of two real-valued algebraic equations
i2 h X 2 LS Lp δ(x − xp ) − Φ20 = 2g (x) nmk ωm a) g (x) − Lg − p ωS2 i h X λ2 2 D2 Lp δ (x − xp ) Φ0 (5.19) b) g (x) nmk ωm = g (x) − Lg − p 2πc √ √ The second case is related to the option 1F1 (µ; ϑ ; z) = exp t2 r/ a when µ = √ ϑ, which gives the other complex-valued relation p = − ar. To obtain Gaussian shape √ √ Ak (x, t) = Ak,0 exp [−t2 r/(2 a)] within this second case one has to consider a pos√ sibility of choosing the negative sign for the square root within calculating a. Hence, eq. 5.18 can be rewritten as
a) Ak (x, t) = Ak,0 exp +
t2
r r
2
a
· 1F1
r ! √ ar + p 1 r ; ; −t2 , √ 4 ar 2 a
√ b) p = − ar
(5.20)
Taking the square of eq. 5.20(b), one will arrive at Eqs.5.19 again. Thus, one can see that both the analyzed cases, connected in fact with formal possibility of two representations: √ √ √ √ a = a a = − a − a even if a is complex-valued, formulate the same set of requirements to the physical parameters resulting in Gaussian shape of pulse envelope. To check this analysis, one can now substitute the anzatz r t2 r Ak (x, t) = Ak,0 exp − 2 a
(5.21)
directly in eq. 5.16. Such a substitution shows that eq. 5.16 can be satisfied by this anzatz √ under conditions p = ± ar, which are exactly equivalent to the above-performed considerations. Thus, finally, one can exploit eq. 5.21 as the representation for the complex amplitude of ultrashort pulses with the Gaussian shape of envelope acceptable for the linear part of complex cavity under consideration. Usually, the complex amplitude of Gaussian pulse includes the expoh i 2 nential multiplicand in the standard form of exp −t2 (1 + ib) / 2τk,0 , where τk,0 is the pulse duration at the intensity level of 1/e and bk is the frequency chirp parameter. Keeping in mind this form, one has to perform the following
95
5.3. EVOLUTION EQUATION GOVERNING THE DYNAMICS OF SHAPING ULTRA-SHORT OPTICAL DISSIPATIVE SOLITARY PULSES TRAINS IN SEMICONDUCTOR HETEROLASER WITH AN EXTERNAL FIBER CAVITY √ 1 a1 + ia2 √ = p 2 a a1 + a22
=
1
(5.22) × 2 a21 + a22 " rq ! !# rq 2 2 2 2 × a1 + a2 + a1 + i a1 + a2 − a1 q
Now, one divide the right hand side of eq. 5.22 by the real-valued term in square brackets, substitute the result into the power of exponent in eq. 5.21, and write
−
r t2 r 2
a
−1 q 2 2 2 a1 + a2 t2 r × = − p 2 r a21 + a22 + a1 !1/2 p a21 + a22 − a1 × 1 + i p 2 a1 + a22 + a1
(5.23)
Comparing this formula with the above-noted standard form of the exponential multiplicand, one can identify that
2 a) τk,0
q 1/2 q 2 a21 + a22 a21 2 2 2 a1 + a2 − a1 1+ 2 , = r = r p a2 2 2 r a1 + a2 + a1 p
b) bk =
p
a21 + a22 − a1 a21 + a22 + a1
!1/2 =
a−1 2
q 2 2 a1 + a2 − a1
(5.24)
Thus, now both τk,0 and bk can be explained in terms of physical parameters using eq. 5.16. Nevertheless, one has to keep in mind that the partial amplitude A0k of an individual optical pulse cannot be chosen in an arbitrary way in eq. 5.17 due to the presence of the nonlinear GinzburgLandau operator in eq. 5.13.
5.3.2
Application of an approximate variational procedure to estimating the parameters of optical dissipative solitary pulses in an external nonlinear fiber cavity with SQUARE-law dispersion and linear losses.
The contribution of the nonlinear Ginzburg-Landau operator to shaping the parameters of optical dissipative solitary pulses in a long-haul single-mode cavity with square-law dispersion, cubic-law
96
5.3. EVOLUTION EQUATION GOVERNING THE DYNAMICS OF SHAPING ULTRA-SHORT OPTICAL DISSIPATIVE SOLITARY PULSES TRAINS IN SEMICONDUCTOR HETEROLASER WITH AN EXTERNAL FIBER CAVITY Kerr nonlinearity, and linear optical losses can be estimated via exploiting some mathematical variational procedure [5.4] [5.5]. The corresponding nonlinear Ginzburg-Landau evolution equation can be written as ∂Ak
k2 ∂ 2 Ak
−n ˜ 2 |Ak |2 Ak − iγAk , (5.25) ∂x 2 ∂t2 where the simplifying notations: k2 = D1 λ2 / (πc), n ˜ 2 = πn2 / (Sλ), and γ = LC , i
=
being in fact much more common in optical soliton physics, are inserted in eq.(5.25). Using the substitution Ak (x, t) = F (x, t) exp (−γx), one can rewrite eq. 5.25 as ∂F
k2 ∂ 2 F
−n ˜ 2 |F |2 F exp (−2γx) , (5.26) ∂x 2 ∂t2 This equation can be considered as a mathematical variational problem in terms of the Lagrani
=
gian L0 , namely, k2 ∂F 2 n ˜2 F −F − exp (−2γx) |F |4 , − L0 = 2 ∂x ∂x 2 ∂t 2 i
∂F ∗
∗ ∂F
(5.27)
For simplicity sake here, one can exploit an approximate variational procedure involving the technique of trial functions [5.4]-[5.5]. By this is meant that the variational principle can be written in the form ZZ δ
Z L0 dtdx = δ
L dx = 0,
(5.28)
in the assumption that the reduced Lagrangian L can be found through integrating the original Lagrangian L0 with respect to the time t after substituting the chosen trial function in eq.(5.27). An approximate character of this variational procedure lies in the assumption that the envelope of the chosen trial function does not vary, while its free parameters can evolve with the distance and, naturally, depend on the coordinate x. In particular, the evolution of initially hyperbolic secant-like pulse, can be considered with the following trial non-stationary function Fk (x, t) = ak (x) exp −
t2 2τk2
(x)
exp ibk (x)t2 ,
(5.29)
Here, the amplitude ak (x), width τk (x), and frequency chirp parameter bk (x) depend on the distance of pulse propagation. Taken alone, the fact of exploiting the trial function involves an approximation in this analysis. That is why it will be rather worthwhile to estimate the contribution of linear optical losses associated with the exponential term in the last summand of the
97
5.3. EVOLUTION EQUATION GOVERNING THE DYNAMICS OF SHAPING ULTRA-SHORT OPTICAL DISSIPATIVE SOLITARY PULSES TRAINS IN SEMICONDUCTOR HETEROLASER WITH AN EXTERNAL FIBER CAVITY reduced Lagrangian. Typically, the double-length of a fiber cavity is about 100 m. At the optical wavelength λ = 1320 nm, one can find γ = 0.28 dB/km ≈ 0.065 km
−1
in fused silica
single-mode fiber [5.14], so that 2γx ≈ 0.013. Consequently, the exponential term can be omitted in Eqs. 5.26 and 5.27 with an accuracy better than. Now, exploiting eq. 5.29, one can find the reduced Lagrangian
L0
√ √ √ π da∗k π k2 π 2 3 dbk ∗ dak = ak − ak + |ak | τk − |ak |2 − 2 dx dx 2 dx 4τk r √ n2 π 2 3 2 |ak |4 τk (5.30) −k2 π |ak | τk bk + 2 2 iτk
In so doing, one can now calculate the following set of equalities for variational derivatives
∂L ∂L − = 0, ∗ ∂ak δa∗k ∂x ∂a∗k ∂ ∂x δL δL c) = 0, d) =0 δτk δbk
a)
δL
=
∂
b)
δL δak
= 0,
(5.31)
These equalities lead to a triplet of the relatively simplified equations
a) τk (x) |ak (x)|2 = τC,k |aC,k |2 = C0 − const, b) c)
∂ 2 τk (x) ∂x2
=
k22 τk3
(x)
−
∂τk (x) ∂x
n2 k2 √ |ak (x)|2 τk (x) 2
= −2k2 bk (x) τk (x) , (5.32)
Let us convert the last summand in eq. 5.32(c) as: |ak (x)|2 τk−1 (x) = C0 |ak (x)|2 τk−2 (x) . Both the values τC,k and aC,k in eq. 5.32(a) have not been yet determined. To determine them one can suppose that they both present so-called soliton parameters inherent in the concrete singlemode silica optical fiber selected. With this in mind, one can take into account an approximate √ 2 relation k2 = n2 / 2 τC,k |aC,k |2 , peculiar to soliton-like pulses with just the Gaussian shapes of envelopes, which form a family of the idealized solitary optical pulses [5.6]. This is why eq. 5.32(c) can be rewritten as
98
5.3. EVOLUTION EQUATION GOVERNING THE DYNAMICS OF SHAPING ULTRA-SHORT OPTICAL DISSIPATIVE SOLITARY PULSES TRAINS IN SEMICONDUCTOR HETEROLASER WITH AN EXTERNAL FIBER CAVITY
∂ 2 τk (x) ∂x2
" = k22
1 τk−3 (x)
−
#
1
(5.33)
τk−2 (x) · τC,k
With the help of Eqs. 5.32(a) and 5.32(b), the solution to eq. 5.33 gives the evolutions of both the pulse amplitude modulus and the frequency chirp as
1/2
a) |ak (x)| = C0
−1/2
τk
(x) ,
b) bk (x) = −
1
∂τk (x)
2τk (x)
∂x
(5.34)
Now, one can integrate eq. 5.33, and the integration constant C1 characterizes the full energy in our conservative dynamic system. This constant can be chosen from the standard physical requirement “the particle starts from the rest” , i.e. from the boundary conditions τk (x = 0) = τ0 and (dτk /dx) | (x = 0) = 0. The second condition means physically that initially the pulse under consideration is unchirped. Thus,
a)
1 k22
∂τk ∂x
2 =
2 τk · τC,k
−
1 τk2
+ C1 ,
b) C1 =
1 τ02
−
2 τ0 · τC,k
.
(5.35)
Potentially, in the case of the initially chirped pulse, one has to take (dτk /dx) | (x = 0) = − 2τ0 b0 , where b0 = bk (x = 0), so that the integration constant C1 should be modified. Nevertheless, just the case of the unchirped optical pulses in fiber cavity is analyzed. At this stage, the value τC,k has to be discussed in more detail and determined. q Let√us put that initially τk (x = 0) = τ0 and ak (x = 0) = a0 + δ, where a0 τ0 = k2 2/n2 , i.e. a0 and τ0 belongs to a family of ideal Schroedinger solitons with the hyperbolic-secant shape of envelopes, while δ characterizes the initial amplitude perturbation, |δ| < 0.5. It is known, unfortunately, only for pulses with a hyperbolic-secant shape [5.15] that under similar boundary conditions one will observe an asymptotic soliton pulse with a∞ τ∞ = q √ k2 2/n2 and a∞ = a0 + 2δ at the infinity x → ∞. Then, from these relations one can calculate τ∞ = τ0 [1 − (2δ/a∞ )] = τ0 / (1 + ξ) with ξ = 2δ/a0 . Nevertheless, there are no other options and we have to use these relations for Gaussian pulses. Because τ∞ belongs asymptotically to ideal soliton parameters from the above-taken family, one can put τC,k = τ∞ . Thereafter one can normalize the variables in eq. 5.35 as τ = τk /τ0 and y = k2 x/τ02 . In terms of these new variables, eq. 5.35 can be rewritten as
99
5.3. EVOLUTION EQUATION GOVERNING THE DYNAMICS OF SHAPING ULTRA-SHORT OPTICAL DISSIPATIVE SOLITARY PULSES TRAINS IN SEMICONDUCTOR HETEROLASER WITH AN EXTERNAL FIBER CAVITY
a)
∂y Z
b)
∂τ
2 =
1 τ2
(τ − 1) [1 − (1 + 2ξ) τ ] ,
τ dτ p = ± (y + y0 ) , (τ − 1) [1 − (1 + 2ξ) τ ]
(5.36)
where y0 is the integration constant. The result of direct integration in eq. 5.36(b) is given by
τ dτ
Z p
n o−1 p = (1 + 2ξ)3/2 (τ − 1) [1 − (1 + 2ξ) τ ]
(τ − 1) [1 − (1 + 2ξ) τ ] n p (1 + 2ξ)1/2 (τ − 1) [(1 + 2ξ) τ − 1] + (1 + ξ) (τ − 1) [(1 + 2ξ) τ − 1]× p 2 [(1 + 2ξ) τ − ξ − 1] × ln 2 (τ − 1) [(1 + 2ξ) τ − 1] + = J0 . (5.37) √ 1 + 2ξ Using eq. 5.37, one can calculate lim J0 (t → 1) = ± i (1 + ξ) (1 + 2ξ)−3/2 , which can be associated with the integration constant y0 . Taken into account these results, one can write finally y = J (τ, ξ) = J0 − lim J0 (t → 1). In terms of the “mixed” variables (x, τ ), this relation takes the form
a) x = τ02 |k2 |−1 J (τ, ξ) , p i (1 + ξ) (τ − 1) [(1 + 2ξ) τ − 1] ± × b)J (τ, ξ) = − 1 + 2ξ (1 + 2ξ)3/2 n o 1 p × ln (τ − 1) [(1 + 2ξ) τ − 1] + (1 + 2ξ) τ − ξ − 1 . ξ
(5.38)
In a medium characterized by the dispersion of the second order, the dispersion length can be (2)
introduced via the ratio τ02 / |k2 | = LD . This is why one can obviously see that the right hand side of eq. 5.38 has the dimension of a length. Moreover, some numerical estimation can be done now. At the optical wavelength λ = 1320 nm, one can find k2 ≈ − 2 ps2 /km in fused silica single-mode fiber [5.8], so that, for example, with an optical pulse of τ0 = 2 ps one can estimate (2)
that LD = τ02 / |k2 | ≈ 2 km. Equation 5.38(a) includes non-obvious form of the dependence τ (x, ξ), which can be already exploited practically. Nevertheless, for characterizing the spatial dependencies inherent in the pulse amplitude and the frequency chirp one has to convert eq. 5.38 to more or less accurate but just obvious form of the approximate dependence τ (x, ξ) if even only for individually chosen value of the ratio ξon a limited distance of propagation x. In so doing, let
100
5.3. EVOLUTION EQUATION GOVERNING THE DYNAMICS OF SHAPING ULTRA-SHORT OPTICAL DISSIPATIVE SOLITARY PULSES TRAINS IN SEMICONDUCTOR HETEROLASER WITH AN EXTERNAL FIBER CAVITY us take, for example, ξ = 0.3 and perform rather approximate illustrative conversion of eq. 5.38 as a) τ ≈ 0.811 +
1 5.3
· sin dτ
x + 1.04 0.66
=
0.66
, so that
· cos
x + 1.04
, dx 5.3 0.66 5 x − 4.33 b) ξ = − 0.3; τ ≈ 2.345 + sin , so that 6.7 2.8 dτ 5 x − 4.33 = · cos , dx 18.76 2.8
(5.39)
The corresponding plots for both exact and approximate dependencies τ (x, ξ) are presented in Fig. 5.3.
Figure 5.3: Plots of exact (solid lines) and approximate (dashed lines)τ (x, ξ) for ξ = 0.3 (a) and ξ = − 0.3 (b). Equations 5.39 make it possible to exploit Eqs. 5.34 for finding |ak (x)| and bk (x), see Figs. 5.4 and 5.5. It should be noted that approximate dashed plots in Figs. 5.3 as well as the plots in Figs. 5.4 and 5.5, based on Eqs. 5.39, are able to serve only as illustrations of more or less realistic behavior of these parameters governed by nonlinear operator. Thus, rather accurate balance between the data from Eqs. 5.14 and the values from Eqs. 5.33 – 5.34 and/or the exactly calculated dependences, being perfectly analogous to Figs. 5.3 - 5.5 with the needed refinings, should be found to describe the steady-state regime of pulse generation [5.14] in the system under consideration.
101
5.4. MEASURING THE TRAIN-AVERAGE PARAMETERS OF PICOSECOND OPTICAL PULSES IN HIGH-REPETITION-RATE TRAINS
Figure 5.4: Plots of approximate dependences b (x, ξ), i.e. based on Eqs. 5.34 and 5.39, for ξ = 0.3 (a) and ξ = − 0.3 (b).
Figure 5.5: Plots of approximate dependences |a (x, ξ)|, i.e. based on Eqs. 5.34 and 5.39, for ξ = 0.3 (a) and ξ = − 0.3 (b)
5.4
Measuring the train-average parameters of picosecond optical pulses in high-repetition-rate trains
In many cases, for example, with the investigations of evolving the optical solitons in active and passive waveguide structures, a simple method is frequently required for measuring current timefrequency parameters of low-power pico- and sub-picosecond optical pulses traveling in highrepetition-rate trains. Most widely used is a method based on the formation of a train-average auto-correlation function of the field strength, which is coupled through the Fourier transform with the spectral power density. From the recorded power spectral density, one can determine an average width of the radiation spectrum. However, in this case, information on the average field phase is lost and it is impossible to determine the time variation of the field amplitude A (t). Exact determination of the train-average pulse duration from the width of the radiation spectrum is only possible when the shape of pulse envelope is known a priori and, in addition, the pulse spectrum is limited [5.9]. An approximate estimation of the pulse duration is also correct, if the frequency chirp is sufficiently small [5.10]. Here, we demonstrate an opportunity of providing ex-
102
5.4. MEASURING THE TRAIN-AVERAGE PARAMETERS OF PICOSECOND OPTICAL PULSES IN HIGH-REPETITION-RATE TRAINS perimental conditions, under which the train-average auto-correlation function of the field strength can serve as a source of exact and reliable information on the average values of both duration and frequency chirp of a low-power optical pulses traveling in high-repetition-rate trains. As usually, let us proceed from the assumption that all pulses in a train are identical pulses with a Gaussian envelope, see eq. 5.40 (1 + ib) t2 , (5.40) AG (T ) = A0 exp − 2T 2 √ with the amplitude A0 = P , where P is the incoming pulse peak power. As it was listed above for a Gaussian envelope, the relationships between the train-average pulse parameters T and b and the width τ0 of the corresponding auto-correlation function, measured on a level of 1/e for the intensity contour, are given by p τ = τ0 = 2T / 1 + b2 .
(5.41)
Usually, the real-time auto-correlation function of the field strength averaged over a train of optical pulses is obtained with a scanning Michelson interferometer [5.10, 5.11], which allows measuring the value of τ0 . However, information on the width τ0 of the field strength autocorrelation function is insufficient to determine the time-frequency parameters of pulse train. That is why one can propose performing two additional measurements of the auto-correlation function width with the help of a scanning Michelson interferometer. During the second and third measurements, supplementary optical components, changing the parameters T and b in a predetermined way but not influencing the envelope of the investigated pulses, should be placed in front of the beam-splitting mirror of the interferometer. The autocorrelation function widths τm (m = 1, 2) obtained from the repeated measurements are coupled with the new values of the pulse duration Tm and the frequency chirp bm by eq. 5.41. One can assume that Tm = αm T0 and bm = b0 + βm , where T0 and b0 are unknown values of the parameters T and b, while the quantities αm and βm are determined by supplementary optical components. Using the above-noted relations, one can write two different algebraic quadratic equations for a quantity of b0 . The corresponding solutions are given by a pair of the following formulas
b0 =
qm α2m
−1
−1
βm
q 2 2 2 4 ± qm αm βm + 2 − qm αm + 1 ,
(5.42)
103
5.5. EXPERIMENTAL STUDIES 2 and τ where qm = τ02 /τm m is the width of the field strength auto-correlation function
obtained without supplementary optical components. For (m = 1, 2), eq. 5.42 gives four values of b0 , of which two coincide with each other and correspond to just the true value of the trainaverage frequency chirp of the pulses. The proposed measurement method allows one to determine not only the value, but the sign of the frequency chirp as well, which is often impossible even with the help of substantially more complicated methods, such as, for example, the method described in Ref.[5.12]. Once the pulse frequency chirp b0 is determined, one can use formula (5.42) to calculate the pulse duration T by using τ0 and b = b0 . For the supplementary electronically controlled optical component, one can propose exploiting a specific device based on an InGaAsP single-mode traveling-wave semiconductor heterolaser, which is quite similar to a saturable-absorber laser with clarified facets [5.13].
5.5
Experimental studies
It is well-known that semiconductor heterolasers have a broad gain band (about ωS ≈ 1013 Hz), so that by this is meant that their operation in the regime of the active mode-locking makes it possible to expect generating ultra-short optical pulses with a duration τ0 lying in a picosecond time range. Generally, the active mode-locking process provides shaping stable trains of wave packets with rather good reproducibility from pulse to pulse. The main peculiarity of these studies is connected with an opportunity to characterize the generated multi-pulse trains of ultra-short optical dissipative solitary pulses utilizing a set of well-separated periodic modulations being shifted in time domain and perfectly independent on each other [5.2] I(t) = I0 +
XN k=1
Ik cos2n (ωm t − φk ) ,
(5.43)
where n 1 is the factor determining a separation of the neighboring electronic modulations, I0 is the amplitude of a constant pump current background, Ik is the current amplitude of a k − th external partial periodic modulation, φk = πk/N is the needed phase shift. In so doing, one has to provide periodic modulation of gain inherent in a semiconductor active medium through injecting a set of the partial pump currents, each with a frequency equal or multiple to the frequency spacing between longitudinal modes of the laser cavity. Within this discussion, a single-mode InGaAsP/InP semiconductor heterolaser is considered. It had been designed with one antireflection-coated facet and an external single-mode optical-fiber cavity. To obtain the shortest possible optical pulses the facets of semiconductor crystals facing the fiber cavity were coated via
104
5.5. EXPERIMENTAL STUDIES deposition of a SiO2 -film, so that the reflection coefficient was typically less than 1 %. An external long-haul cavity was made of a single-mode silica optical fiber with the refractive index nF ≈ 1.5 with an additional mirror at its far end, providing the optical feedback. The length L of fiber cavity was chosen proceeding from the expected number N of pulse trains and the repetition frequency f of optical pulses in a cavity, because of L = cN/ (2nF f ). When N = 32 and f = 80 M Hz, one can find L ≈ 40 m. The bandwidth δf of mode-locking dictates the needed accuracy δL of keeping the fiber cavity length L and determines the limits of this length variations as δL = cN/ 2nF f 2 δf . With δf ≈ 100 KHz, one yields δL = 5 cm. The factor n, determining a separation of the neighboring electronic cosine-like modulations in eq. 5.43, can be estimated from Eqs. 2.63 and 2.64 of [5.8] as n > 2N 2 /π 2 , so that with the above-taken number N = 32 one can find that n > 200. The corresponding feedback factor was estimated by 15 % due to about 40 % -efficiency of exiting the light radiation in that optical fiber by semiconductor laser structure with the refractive index nS ≈ 3.3. The electronic port of semiconductor heterolaser was matched with a 50-Ohm output of that source via specially designed strip-line wave-guiding circuit. The regular operation of semiconductor heterolasers was provided by thermo-stabilizing system at a temperature of 160 C with an accuracy of ± 0.20 C. The functional scheme of the experiments is presented in Fig. 5.6.
Figure 5.6: Schematic arrangement of the experimental set-up. The realized active mode-locking regimes can be associated with the cases of circulating a set of optical dissipative solitary pulses in a long-haul fiber cavity. A number N of the circulating optical solitary pulses was determined by the generator of digital modulating signals, which pro-
105
5.5. EXPERIMENTAL STUDIES vided issuing regular homogeneous sequences of almost rectangular narrow (about 300 ns each) electronic video-pulses. Experimentally the regimes with N = 1 − 32 had been successfully realized. Estimating the time-frequency parameters were carried out via exploiting the modified interferometric technique described in Ref. [5.6]. The performed proof-of-principle experiments were oriented on shaping multi-pulse trains of rather powerful (about 1 W in a peak) picosecond optical dissipative solitary pulses with stable pulse parameters and the partial repetition frequency of about 100 M Hz. The absolute frequency bandwidth, being available for the observing a multi-pulse active mode-locking, was varying in the range 0.1 − 0.3 M Hz depending on the number N of optical dissipative solitary pulses simultaneously circulating through fiber cavity, so that the relative frequency locking band was a little bit more than 10−3 .
5.5.1
Characterization of optical dissipative solitary pulses.
Figure 5.7 shows the digitized oscilloscope trace for two trains of ultra-short optical dissipative solitary pulses with N = 16 and N = 32, which were identified as the most stable during the experiments performed. These trains of picosecond pulses had been detected with the time resolution of about 300 ps, which is associated with the transfer function of the high-speed photodetector exploited. The off-duty ratio for optical pulses depicted in Fig. 5.7 is in correspondence to the ratio between each partial repetition period 1/f and the above-mentioned time resolution of that high-speed photodetector. The train-average duration of an individual optical pulse was estimated by 8.5 ps using the modified auto-correlator with a low-speed photodetector; more detailed characterizing these pulses is presented in Ref.[5.7].
Figure 5.7: The digitized oscilloscope traces related to regular trains of optical dissipative solitary pulses with (a) N = 16 and (b) N = 32 detected by a high-speed photodetector with the time resolution of about 300 ps.
106
5.5. EXPERIMENTAL STUDIES Figure 5.8a shows the digitized oscilloscope traces for ultra-short optical dissipative solitary pulses with N = 8, which were identified as the most available for measurements during the experiments performed. Similar trains of picosecond solitary pulses had been detected with the time resolution of about 300 ps, which is associated with the transfer function of the high-speed photodetector exploited. This plot includes two neighboring fractions of the oscilloscope traces, which are rather similar to each other resulting in a demonstration of relative stability inherent in the process of generating these pulse trains. The off-duty ratio for optical pulses depicted in Fig. 5.8a is in correspondence to the ratio between each partial repetition period 1/f and the abovementioned time resolution of that high-speed photodetector. Figure 5.8b represents the digitized train-average interferogram of the second order auto-correlation function for a high-repetition-rate train of optical dissipative solitary pulses with N = 8. The width of this interferogram, measured on a level of 1/e for the intensity contour, was estimated by 8.5 ps. The dimensionless parameter b, related to the frequency chirp, was estimated with applying the above-mentioned technique by b ≈ 8.4 · 10−4
Figure 5.8: The digitized oscilloscope traces related to regular trains of optical dissipative solitary pulses with N = 8 (here, 2 periods are shown) detected by a high-speed photodetector with the time resolution of about 300 ps (a) and (b) the auto-correlation function registered by the modified auto-correlator with a low-speed photodetector. In a multi-pulse regime of the active mode-locking, optical dissipative solitary pulses, belonging to every independent train are self-reproducing after each path through the lengthy cavity. Restoration of dissipative pulse parameters is conditioned by properties of the active medium and is realized due to a double balance between optical square-law dispersion and cubic Kerr nonlinearity as well as between gain and linear losses. One of the necessary conditions for reproducibility of solitary pulses is the presence of frequency chirp. Presently known mechanisms of interacting optical pulses with single-mode semiconductor heterostructures allow us to
107
5.5. EXPERIMENTAL STUDIES simplify the theoretical model of shaping an ultra-short pulse with the complex field amplitude E (t) = A (t) exp (iω0 t) + c. c. in a heterostructure. The pulse, grown during a multipulse process of the active mode-locking, has an envelope close to the Gaussian shape and can be described by eq. 2.69. The pulse width, measured on a level of 1/e for the intensity contour, is given by [5.8] T = (2ng m)−1/4 (ωm ωS )−1/2 ,
(5.44)
where g is the maximal gain at t = 0, m is the factor of an external amplitude modulation of the losses in a lengthy cavity, ωm is the external modulation frequency, and ωS is the gain contour width. Finally, the frequency chirp can be expressed as [5.8]. 2
a) b = 2T β
b) β =
ωm ωS3
√ m
4 [gω0 TC / (2Q)]3/2
X j
Lj
d2 kj dω 2
,
(5.45)
where β is the dimensional factor of frequency chirp, Lj is the length of a j − th highdispersion component (one can take, for example, j = 1 for a semiconductor laser heterostructure and j = 2 for a single-mode lengthy fiber cavity), (d2 kj /dω 2 ) is the corresponding quantity of the second order group-velocity optical dispersion, ω0 is the central frequency of emission, Q is the quality factor inherent in a cavity, TC ≈ 2LnF /c is the transit time of an individual pulse through a silica fiber cavity with L ≈ 40 m and nF ≈ 1.5; k is the wave number. In fact, Eqs. 5.44 and 5.45 can be practically used to estimate the parameters of the optical dissipative pulses generated. Using the values characteristic of the experiments: g = 3, m = 0.25, ωS = 2π · 1013
rad/s, f = 80 M Hz, and N = 32, so that 2n = 500 and ωm = 2π · f /N =
2π · 2.5 · 106 rad/s, one can obtain T ≈ 7.3 ps from eq. 5.44, which can be considered as rather good agreement with the experimental data. The frequency chirp that arises within establishing the self-reproducing dissipative solitary pulses can be estimated with eq. 5.45. For ω0 = 2πc/λ ≈ 1.43 · 1015 rad/s (at λ = 1320 nm), TC = 4 · 10−7 s, Q ≈ 105 , one can take L1 ≈ 1.0 mm, d2 k1 /dω 2 = 3.7 · 10−24 s2 /m and L2 = 2L ≈ 80 m, (d2 k2 /dω 2 ) = 3 · 10−27 s2 /m (i.e. 3 ps2 /km at λ = 1320 nm)[5.7, 5.8], one can obtain β = 4.75 · 1018 s−2 from eq. 5.45b). By the way, it should be noted that the disper sion contribution L2 d2 k2 /dω 2 = 2.4 · 10−25 s2 from a single-mode lengthy fiber cavity far exceeds the corresponding dispersion contribution L1 (d2 k1 /dω 2 ) = 3.7 · 10−27 s2 from a semiconductor heterostructure. Nevertheless, this dimensional magnitude of the estimated frequency chirp is relatively small, because one can find from eq.(5.45a) in dimensionless values that
108
5.6. CONCLUSION b ≈ 5.06 · 10−4 1. In practically reasonable assumption that the envelopes of optical pulses under consideration can be described rather adequately by Gaussian functions, these estimations make it possible to create the corresponding theoretical version of Wigner time-frequency distribution with the above-calculated parameters T and b. Together with this, the experimental version of similar time-frequency distribution can be designed with experimentally obtained parameters T ≈ 8.5 ps and b ≈ 8.4·10−4 in the same approximation by Gaussian functions. The resulting plots of two Wigner distributions for the Gaussian-like optical pulses, obtained from estimations and from experiment, are shown in Fig. 5.9.
Figure 5.9: A pair of the joint Wigner time-frequency distributions for the dissipative solitary optical pulses, having train-average Gaussian shapes of envelopes obtained from the numerical performed estimation with T = 7.3 ps and b = 5.06 · 10−4 as well as from the experiments with T = 8.5 ps and b ≈ 8.4 · 10−4
5.6
Conclusion
A detailed contribution to the physical conception of the dissipative solitons is developed for describing the process of originating regular trains of picosecond optical pulses with Gaussian shape of envelope, excited by multi-pulse sequences of periodically repeating modulating signals, in the actively mode-locked semiconductor laser heterostructure with an external cavity. In the case under consideration, the cavity is represented by a rather long single-mode silica fiber exhibiting square-law dispersion, cubic-law Kerr nonlinearity, and linear optical losses. Our analytical model takes into account principal contributions associated with the externally modulated gain, the linear
109
5.6. CONCLUSION optical losses as well as with linear and nonlinear phase effects. As usually for just dissipative solitons, regular trains of optical solitary pulses appear due to a pair of balances. The first of them is caused by mutually compensating influences of the pulse broadening second-order anomalous dispersion and the pulse compressing (focusing in time) cubic-law Kerr nonlinearity. While, both the pulse amplifying heterostructure gain and the pulse attenuating linear optical losses in the combined cavity govern the second balance. Here, two contributions have been recognized within these processes. One contribution is associated with a quasi-linear operator describing just a process of the active mode-locking in heterolaser with quasi- linear external cavity. The realized eigen-functions of this quasi-linear operator can be explained in terms of Gauss functions under certain conditions. Then, the cubically nonlinear Ginzburg-Landau operator governs the second contribution. The action of this nonlinear operator has been described using an approximate variational procedure based on the technique of trial functions. This contribution is responsible for appearing dissipative solitary pulses evolving in adiabatic solitons-like regime through the combined cavity. When cavity is not too lengthy in the scale of exhibiting both linear and nonlinear phenomena, the initially introduced non-stationary trial function leads to rather weak variations of parameters inherent in each individual pulse, so that one can expect compensating these variations by the semiconductor laser heterostructure and shaping rather stable trains of ultra-short optical dissipative solitary pulses. The presented just illustrating results taken from proof-of-principle experiments are looking like a confirmation for such a conclusion and demonstrating the physical possibility of appearing optical dissipative solitary pulses within a multi-pulse regime of the active mode-locking in the actively mode-locked semiconductor laser with an external long-haul single-mode fiber cavity. This technique makes it possible to find the parameters needed for reconstructing the trainaverage joint Wigner distributions inherent in these ultra-short optical pulses. The InGaAsP/InPheterolaser, operating at a 1320 nm optical wavelength range, has been used within the illustrating experiments. When the optical signal consists of contiguous pulses with a high repetition frequency, conditioned by operating semiconductor heterolaser in a multi-pulse regime of the active mode-locking, typical requirements for measurements and operating with the joint Wigner timefrequency distributions have been satisfied, so that the train-average pulse parameters have been successfully characterized.
110
Bibliography [5.1] Agrawal, G. P. and Dutta, N. K., “Semiconductor lasers”, 2-nd Ed., Van Nostrand Reinhold, New-York, (1993). [5.2] van der Ziel, J. P., “The mode-locking in semiconductor lasers. // Semiconductors and Semimetals”, (Lightwave Communication Technology), Volume Editor W. T. Tsang, Academic Press vol. 22, Orlando, chapter 1, (1985). [5.3] Anderson, D., “ High transmission rate communication systems using lossy optical solitons,” Optics Communications, vol. 48, no. 2,pp. 107-112, (1993). [5.4] Anderson, D., “ Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev A, vol.27, no.6, pp. 3135-3145 (1983) [5.5] Anderson, D., “ High transmission rate communication systems using lossy optical solitons,” Optics Communication, vol.48, no.2, pp. 107-112 (1983). [5.6] Shcherbakov, A. S., Munoz Zurita, A. L., Kosarsky, A. Yu. and Campos Acosta, J., “ Determining the time-frequency parameters of low-power bright picosecond optical pulses using interferometric technique,” OPTIK-Int. J. Electron. Opt. Elsevier, (Germany),(2008);doi:10.1016/j.ijleo.2008.07.033. [5.7] Shcherbakov, A. S., Moreno Zarate, P., Campos Acosta, J., Il’in, Yu. V. and Tarasov, I. S., “‘ Applying the joint Wigner time-frequency distribution to characterization of ultra-short optical dissipative solitary pulses in the actively mode-locked semiconductor laser with an external single-mode fiber cavity,” Proc. of SPIE vol. 7597,pp. 75971B-175971B-10 (2010); doi:10.1117/12.839445 [5.8] Shcherbakov, A. S., Moreno Zarate, P., “Dynamics of shaping ultrashort optical dissipative solitary pulses in the actively mode-locked semiconductor laser with an exter-
111
BIBLIOGRAPHY nal long-haul single-mode fiber cavity” Proc. SPIE, vol. 7600, pp. 76001H-1-76001H-10 (2010); doi:10.1117/12.839446 [5.9] Ippen, E. P. and Schenk, C .V.. “Picosecond techniques and applications”. in Ultrashort Light Pulses, Ed. S.Shapiro, Springer, Heidelberg, (1977). [5.10] Shcherbakov, A. S., “Synchronization of a radio-interferometer by the high-repetition-rate picosecond solitons”. Tech. Phys. Lett., vol.19, pp. 615-616 (1993). [5.11] Herrmann, J. and Wilhelmi, B., “Laser fur ultrakurze lichtimpulse”. Akademi-Verlag, Berlin, (1984). [5.12] Nagamuna, K., Mogi, K. and Yamada, H. “General method for ultrashort light pulse chirp measurement. IEEE J. Quantum Electron., vol.25, pp. 1225-1233 (1989). [5.13] Shcherbakov, A. S. , Munoz Zurita, A. L. , Kosarsky, A. Yu. and Campos Acosta J. “Determining the time - frequency parameters of low-power bright picosecond optical pulses using interferometric technique”. OPTIK - Int. J.Electron. Opt. Elsevier, Germany; doi:10.1016/j.ijleo.2008.07.033. [5.14] Agrawal, G. P.,“Nonlinear Fiber Optics”, 3-rd Ed., Academic Press, San Diego, (2001). [5.15] Satsuma, Ju. and Yajima, N,. “Initial value problem of one-dimensional self-modulation of nonlinear waves in disper-sive media”. Suppl. of the Progress of Theor. Phys., no.55,pp. 284-306 (1974). [5.16] Shcherbakov, A. S., Moreno Zarate, P., Campos Acosta, J., Il’n, Y. V., Tarasov, I. S., “Applying the joint Wigner time-frequency distribution to characterization of ultrashort optical dissipative solitary pulses in the actively mode-locked semiconductor laser with an external single-mode fiber cavity” Proc. of SPIE vol. 7597,pp. 75971B-175971B-10 (2010); doi:10.1117/12.839445 [5.17] Shcherbakov, A. S., Kosarsky, A. Y., Moreno Zarate, P., Campos Acosta, J., Il’n, Y. V., Tarasov, I. S., “Characterization of the train-average time-frequency parameters inherent in the low-power picosecond optical pulses generated by the actively mode-locked semiconductor laser with an external single-mode fiber cavity”, Opt. Int. J. Light Electron. Opt. (Optik, Elsevier, Germany), pp. 1-6 (2009). doi: 10.1016/j.ijleo.2009.11.020.
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Chapter 6
Application of the correlation function in the characterization of the ultrashort optical pulses In this chapter, we present the practical feasibility of measuring the train-average parameters of pico-second optical pulses being arranged in high-frequency repetition trains is investigated. For this purpose we consider exploiting the triple auto-correlations, whose Fourier transformations give us the bispectrum of a pulse train. The main merit of similar approach consists in the capability of recovering signals and revealing asymmetry of pulse envelopes. The triple auto-correlation can be shaped by a three-beam scanning interferometer with the following one- or two-cascade triple harmonic generation. The efficiencies of these processes are studding in different materials. Also we present the principals details of the experimental arrangement, together whit the principal details of the Photo-EMF photo-detector. In particular, the experimental characterizations are presented for two different types of materials, namely, for the Gallium Arsenide (GaAs) semiconductor and for the poly-fluoren 6-co-triphenyldiamine (P F 6 − T P D) photo-conductor polymer, which both exhibit the photo-electro-motive force effect.
6.1
The triple auto-correlation function for the chirped Gaussian pulse.
Now, one can consider an example of the triple auto-correlation and bispectrum for the chirped Gaussian pulse. In this case, let us explain the intensity profile as F G (t) = exp (−a + ib) t2 ,
(6.1)
where the parameters a and b characterize the pulse width and the frequency chirp, respectively. Substituting eq.(6.1) into eq.(2.78), one can find that
s a)
F3G (t1 , t2 )
=
2 2 2 exp − (a + ib) t + t − t t . 1 2 1 2 3 (a2 + b2 ) 3 π (a + ib)
113
(6.2)
6.1. THE TRIPLE AUTO-CORRELATION FUNCTION FOR THE CHIRPED GAUSSIAN PULSE. Then, one can use eq.(2.79) to obtain the corresponding bispectrum
b)
F3G (f1 , f2 )
=
π (a + ib)
3/2
× a2 + b2 2π 2 (a + ib) × exp − a2 + b2
f12
+
f22
+ f1 f2
,
(6.3)
Dividing real and imaginary parts of eq.(6.2), one can write for the triple correlation, see fig. 6.1
Re F3G (t1 , t2 ) =
Im F3G (t1 , t2 ) =
2a 2 2 t + t2 − t1 t2 × exp − √ 3 1 3 a2 + b2 2b 2 b 1 2 × cos − t1 + t2 − t1 t2 + arctan , 3 2 a r
π
π
r
2a
t21
t22
exp − + − t1 t2 × √ 3 a2 + b2 2b 2 1 b 2 × sin − t1 + t2 − t1 t2 + arctan , 3 2 a 3
(6.4)
(6.5)
Figure 6.1: Triple auto-correlation function for the chirped Gaussian pulse with a = 0.5 and b = 1.0: (a) real-valued part, (b) imaginary-valued part. Then, dividing real and imaginary parts of eq.(6.3), one can write for the corresponding bispectrum as show in Fig .6.2
114
6.1. THE TRIPLE AUTO-CORRELATION FUNCTION FOR THE CHIRPED GAUSSIAN PULSE.
2π 2 a(f12 + f22 + f1 f2 ) × exp − (a2 + b2 )3/4 a2 + b2 2π 2 b(f12 + f22 + f1 f2 ) 3 b × cos − + arctan , (6.6) a2 + b2 2 a Re F3G (f1 , f2 ) =
Im F3G (f1 , f2 ) =
π 3/2
π 3/2
2π 2 a(f12 + f22 + f1 f2 )
exp − (a2 + b2 )3/4 a2 + b2 2π 2 b(f12 + f22 + f1 f2 ) 3 b × sin − + arctan . 2 2 a +b 2 a
× (6.7)
Figure 6.2: Bispectrum for the chirped Gaussian pulse with a = 0.5 and b = 1.0: (a) realvalued part, (b) imaginary-valued part.
6.1.1
The algorithm of recovering the temporal signal from its triple auto-correlation function.
If the temporal signal F (t) is, for example, real-valued as well as is of a finite extent, it can be retrieved from its triple auto-correlation function F3 (t1 , t2 ) almost uniquely apart from a shift. For a real signal F (t) of finite extent, its spectrum F (f ) can be analytically continued by extending the frequency f to the complex variable z = z 0 + iz 00 . The analytic continuation F (z) is determined by its complex zeros zn and can be written as a Hadamard product F (z) = exp (α + βz)
Y n
(z − zn ) exp
z zn
.
(6.8)
115
6.1. THE TRIPLE AUTO-CORRELATION FUNCTION FOR THE CHIRPED GAUSSIAN PULSE. where α and β are some constants. This fundamental equation from the theory of complex functions cannot be applied directly to any arbitrary function of two variables. However, in the case of triple correlations, it is known how the two-dimensional function F3 (z1 , z2 ) is related to the one-dimensional function F (z), because one can exploit eq.(2.79), so that one can write F3 (z1 , z2 ) = F (z1 )F (z2 )F (−z1 − z2 ).
(6.9)
Consequently, one can insert eq.(6.8) into eq.(6.9) and obtain
F3 (z1 , z2 ) = exp (3α)
Y
(z1 − zn ) (z2 − zn ) (−z1 − z2 − zn ) .
(6.10)
n
Using eq.(6.10), one can derive the particular complex zeros of F (z) from the complex zero subspaces of F3 (z1 , z2 ) = 0 . Ones the zeros zn are known, one can compute F (z), hence F (f ), and then F (t). The detailed consideration of this proof shows that for the general case of the complex-valued function F (t), its spectrum F (f ) can be reconstructed up to the exponential factor exp (α + βf ) , where the factor β is an arbitrary, broadly speaking complex-valued, constant, while α = {0, 2πi − (1/3), 2πi − (2/3)}. One can consider a retrieval algorithm. For the sake of simplicity let us assume reality of the signal F (t) and hence Hermitian symmetry of its spectrum F (f ), see eq. 2.81 (a). We begin by assuming that f2 = 0 in a bispectrum F3 (f1 , f2 ) , consequently F3 (f1 , 0) = F (f1 )F (0)F (−f1 ) = |F (f1 )|2 F (0),
(6.11)
and therefore the Fourier amplitude |F (f )| is available directly on the f1 -axis of F3 (f1 , f2 ). Then, one can retrieve the Fourier phase ϕ(f ), defined by F (f ) = |F (f )| exp [ϕ(f )]. To that end we concentrate on a straight line, being parallel to the f1 -axis, but above it by one sampling step δf
F3 (f1 , δf ) = F (f1 )F (δf )F (−f1 − δf ) = F (δf ) × × |F (f1 )F (−f1 − δf )| exp (iϕ(f1 ) − iϕ(f1 + δf )) . (6.12) Defining ϕ3 (f1 , f2 ) as the phase of the bispectrum F3 (f1 , f2 ) , one can derive from eq.(6.12) the following phase equation ϕ3 (f1 , f2 ) = ϕ(δf ) + [ϕ(f1 ) − ϕ(f1 + δf )] .
(6.13)
116
6.2. PRINCIPAL ASPECT OF THE THREE-BEAM AUTO-CORRELATOR Finally, one can extract ϕ(f1 ) itself, apart from an additive constant, and a term liner in f1 , that reflects the lack of knowledge about t0 in F (t − t0 ). Consequently now is possible to use the triple auto correlation function of the three beam intensities that it is defined by Z∞ I3 (t1 , t2 ) =
I(t)I(t + t1 )I(t + t2 )dt,
(6.14)
−∞
for reconstructing the pulse using the algorithm described in this section. Here we show an example of the triple auto-correlation inherent in a high-repetition train of asymmetric optical pulses, see fig. 6.3.
Figure 6.3: An example of the triple auto-correlation function inherent in a high-repetition train of asymmetric optical pulses.
6.2
Principal aspect of the three-beam auto-correlator
A few tenths years ago it became possible to generate optical pulses whose widths lie in the picoand sub-pico-second ranges. Now the optical pulse is a topic research of considerable interest in different knowledge fields. Nevertheless, up to now there are no detectors fast enough to measure such ultra-short pulses directly. That is why a lot of the elaborated methods of measuring are based on the analysis of several auto-correlation functions. Unfortunately, the auto-correlation functions of the second order are symmetric in behavior, so that they cannot give any information about asymmetry of optical pulses under investigation. The basic parameters inherent in a three-beam auto-correlator [6.1], providing the observation of the triple auto-correlations, are analyzed below. Here, we consider two possible configurations for arranging a three-beam auto-correlator, which exploit two slightly different mechanisms of the
117
6.2. PRINCIPAL ASPECT OF THE THREE-BEAM AUTO-CORRELATOR third harmonic generation (THG) for characterizing ultra-short optical pulses. Both these configurations include three arms shaping mutually delayed partial trains of pulses whose intensities are I (t), I (t + t1 ) and I (t + t2 ), where the time intervals t1 and t2 describe the corresponding delays. These delays relative to the partial train are produced by the lengths d1 , . . . , d4 of the autocorrelator’s arms as well as by the moving mirrors 1 and 2, see Figs. 6.4 (b) and 6.6 (b) , due to scanning one and the same initial optical pulse with different velocities V1 and V2 , so that V1 V2 . Mixing the obtained triplet of pulse trains in the specially selected non-linear medium results in the THG. Generally, a few options exist for realizing the THG in non-linear media. Two of them, related to direct and cascade generations are presented here.
6.2.1
Analysis of three-beam auto-correlator via direct THG
The first case that is consider here, it relate to the direct THG as is possible to see 6.4 (a). The triple auto-correlator can be exploited to record the raw data of the experiments with a sequence of ultra-short optical pulses is shown in fig. 6.4(b).
Figure 6.4: (a) Direct optical schemes for a third harmonic generation. (b) A three-beam scanning interferometer for registering the triple auto-correlation function of high-repetition-rate trains including ultra-short optical pulses direct THG. Within the mechanism for the direct THG, three partial fragments of the initial pulse should arrive at the facet of non-linear medium at the same time when both the moving mirrors are in the medium positions. In a point of fact, the optical path differences are ideally equal to zero. In fact, by this is mind that one can one has to consider three auto-correlator arms. For the figure 6.4(b), the first arm of the length L1 is related to the moving mirror 1 and includes double path
118
6.2. PRINCIPAL ASPECT OF THE THREE-BEAM AUTO-CORRELATOR of the distance d1 , double path of the first beam splitter of the thickness dB , and single path of the distance d5 , i.e. L1 = 2d1 + 2dB + d5 . The second arm of the length L2 is related to the moving mirror 2. This arm consists of double paths of the distances d2 and d3 , four paths through beam splitters 1 and 2, and single path of the distance d5 , i.e. L2 = 2d2 + 2d3 + 4dB + d5 Finally, the third arm of the length L3 is related to the fixed mirror. It involves double paths of the distances d3 and d4 , four paths through beam splitters 1 and 2, and single path of the distance d5 , i.e. L3 = 2d3 + 2d4 + 4dB + d5 . To estimate potential contributions from the beam splitters let us take, for example, the splitter BS016 (Thorlabs) made of BK7 glass. It has a well-determined geometric thickness of 20 mm, while its refractive index n is slightly dispersive. This dispersion can be illustrated the following data: λ = 480 nm, n = 1.5228, λ = 589 nm, n = 1.5167, λ = 656 nm, n = 1.5143, λ = 1060 nm, n = 1.5067, and λ = 1526 nm, n = 1.5009 [6.2]. Selecting, for instance, a near infra-red line, one can estimate n ≈ 1.5, so that the beam splitter optical length can be estimated as dB ≈ 30 mm at the chosen light wavelength. The thin o
chrome film, which is realizing the splitting, has about 400 A in thickness and a refractive index of 2.5, so that these values lead to very small optical length of approximately 1 nm, which can be omitted in practice. Additionally, it should be noted that the estimations related to the contributions from beam splitters will be true even in the case when light beams are oriented not exactly at the centers of cubes. The above-listed condition, related to the medium positions of the moving mirrors, requires the equality L1 = L2 = L3 . Consequently, one can consider a triplet of the following partial equalities. In a case of figure 6.4(a), the first is L1 = L2 that leads to the condition d1 = d2 + d3 + 2dB . The second is L1 = L3 , which gives d1 = d3 + d4 + 2dB . The third is L2 = L3 , so that one can find d2 = d4 . One can estimate both time and spatial scales of appearing triple correlation response. Initially unchirped Gaussian pulse with the parameters a = 4/τ02 and b = 0 in Eq6.3 is characterized by the intensity contour I(t) = exp (−4t2 /τ02 ), where τ0 is the full time duration at a level of 1/e ≈ 0.368. One can easily find the corresponding spatial pulse width d0 = cτ0 /n, where c is the light velocity and n is the refractive index of non-linear medium. In the case of the minimal pulse duration τ0 = 1.0 ps, which can be practically analyzed by this triple auto-correlator, and n = 2, one can estimate that d0 = 150 µm at the same level of 1/e. Due to obvious mechanism of shaping the triple correlation shown in fig. 6.5, where V1 and V2 are linear parts of velocities caused by scanning the corresponding moving mirrors 1 and 2 relative to an immovable mirror.
119
6.2. PRINCIPAL ASPECT OF THE THREE-BEAM AUTO-CORRELATOR
Figure 6.5: Gaussian pulses moving with different linear velocities V1 and V2 relative to an immovable one (a);illustration to appearing the effective length of interaction Lef f for a triplet of the interacting optical pulses. One can see that the values of τ0 and d0 determine both time and spatial scales of originating the triple correlations in this scheme. The obtained spatial scale makes it possible to estimate the needed accuracy of adjusting optical components within the chosen scheme of auto-correlator. For this purpose one can say that, for example, just 10 samples are needed to detect the triplecorrelation function really adequate. With this in mind, one arrives at the accuracy (resulted in a nonlinear medium) of about 100 fs and 15 µm in time and spatial domains, respectively. Together with this one can introduce the effective length of interaction Lef f between these pulses, which can be used for estimating efficiency of the THG. If we will use Lef f = d0 , connected with the full time duration at a level of 1/e, one can calculate that Z a) S =
∞
I (t) dt =
τ0
−∞ Z τ0 /2
b) SA =
I ( t ) dt = −τ0 /2 Z ∞
c) Si = 2
I (t) dt = τ0 /2
d) NA = SA /S ≈ 0.84
√
π
2 τ0 τ0
≈ 0.886 τ0 ,
√ π 2 √
Erf ( 1 ) ≈ 0.747 τ0 ,
π
[1 − Erf ( 1 )] ≈ 0.139 τ0 , 2 e) Ni = Si /S ≈ 0.16.
(6.15)
These estimations show that with Lef f = d0 only about NA ≈ 84% of photons will be involved in the interaction, while about Ni ≈ 16% of them will be lost. This is why one need consider another case, which can be conditioned by a level of 1/e2 ≈ 0.135. One has to consider √ √ τ0 2 as the full time duration and d0 2 the corresponding spatial pulse width. In so doing, one can arrive to
120
6.2. PRINCIPAL ASPECT OF THE THREE-BEAM AUTO-CORRELATOR
∞
Z
I (t) dt =
a) S = −∞
Z
τ/
√
b) SA = −τ0 / Z ∞
c) Si = 2 τ/
2
I ( t ) dt = 2
√
π
2
I ( t ) dt =
d) NA = SA /S ≈ 0.9545, Thus, with Lef f = d0
√
2
√
√
τ0
τ0
≈ 0.886 τ0 , √ π
2 √ τ0 π 2
Erf (
√
2 ) ≈ 0.846 τ0 ,
[ 1 − Erf (
√
2 ) ] ≈ 0.040 τ0 ,
e) Ni = Si /S ≈ 0.0455.
(6.16)
2 one can have about NA ≈ 95.5 of them will be lost. Finally,
the maximal effective length of interaction Lef f max is connected with the maximal pulse width τ0 max , which can be lie practically in the range of 20 . . . 30 ps.
Operation of the experimental arrangement via direct THG. In fact the triple auto-correlator is based in double Michelson interferometer as was shown in the figure 6.4 (b). In general the triplet arms of the triple auto-correlator provide mutually time delayed pulse trains. The last is obtained in the interferometer considering, the beam from the sources is dividend in two beams by the beam splitter. One of them arrive to the Moving Mirror 1 where the intensity I (t + t1 ) is delayed in time and reflected. For our analysis, the intensity I (t + t1 ) travel the optical path L1 before to arrive to the nonlinear medium. The other beam again is dividing in two parts. The first part of the beam I (t + t1 ) arrive to the moving mirror 2 where it is delay and reflected, this beam travel to the optical path L2 before to arrive to the nonlinear medium. The last beam I(t) come in to the fixed mirror traveling the optical path L3 before to arrive to the nonlinear medium. The time delayed is obtained by two moving mirror driving by two speaker, which scanning the central pulses. Mixing these three pulses (the intensities) trains in a nonlinear medium will result in harmonic generation. The experimental arrangement was selected for its feasibility to measurement the three beam autocorrelation function. In practices is possible to build this experimental set up, first considering the wave length used, to select the appropriate optical components. Here we submit the principal optical components of the experimental arrangement; 1. For the mirrors we select a BB2 − E02 dielectric mirrors from thorlabs 2. For dividing the beam we proposed a BS016 50 : 50 non-polarizing beam splitter cube 3. The intensities delays is obtained using two speakers with mirror from Pioneer.
121
6.2. PRINCIPAL ASPECT OF THE THREE-BEAM AUTO-CORRELATOR
6.2.2
Theoretical aspect of the direct third harmonic generation efficiency
As it has been already told, rather effective optical generation of the third-harmonic frequency may be achieved by the direct application of a third-order nonlinear optical process as well as by cascading second-order ones. For the direct third-harmonic generation the process ω1 +ω1 +ω1 = ω3 has the phase-matching condition , where angular frequencies ωi and the wave numbers are related to the i−th optical harmonic; here, (i = 1, 2, 3) and ni are the corresponding refractive indices. For the cascade third-harmonic generation both the second-harmonic generation ω1 + ω1 = ω2 and the frequency mixing ω1 +ω2 = ω3 should be phase matched by ∆ kSHG = k2 −2 k1 and ∆ kF M = k3 − k2 − k1 . By the way, one can calculate that ∆ kT HG − ∆ kF M = ∆ kSHG . These processes can be described by the following set of equations [6.3].
∂ E2
= −i
ω2
(2)
χ E1a E1b exp ( i ∆ kSHG z0 ), 2 n2 c SHG ω3 (2) = −i χF M E2 E1c exp ( i ∆ kSF z0 ), b) ∂ z0 2 n3 c ω3 ∂ E3 , T HG (3) = −i χ c) E1a E1b E1c exp ( i ∆ kT HG z0 ). ∂ z0 2 n3 c T HG a)
∂ z0 ∂E3 F M
(6.17)
Here, E1a, b, c are the electrical field strengths of the ordinary and extraordinary field components according to the interaction processes ( as is shown in the ref [6.3] Table 1). E2 are the electric field second harmonic; E3,F M and E3,T HG are the electric fields amplitudes of the third (2)
(2)
(3)
harmonics generated through cascade and direct processes respectively; χSHG , χSF and χT HG are the corresponding components inherent in susceptibility tensors of the second and third orders; is the spatial coordinated along nonlinear media. They all are valid in the slowly varying amplitude approximation as well as in the approximation of a given fields in the right hand sides implying that the initial pulse depletion can be neglected. In case of direct third harmonic generation, one need to fund the solution of eq. 6.17(c), that has the form
(3)
E3,T HG (z0 ) = −i
ω3 χT HG z 2nc
E13 exp (i∆kT HG z0 /2)
sin(∆kT HG z0 /2) ∆kT HG z0 /2
.
(6.18)
Now, the phase matching condition is ∆kT HG → 0. The total third-harmonic signal is the sum E3 = E3,SF + E3,T HG over the both processes simultaneously. It may be written as
122
6.2. PRINCIPAL ASPECT OF THE THREE-BEAM AUTO-CORRELATOR (3)
E3 (z0 ) = −i
ω3 χef f z0 2n3 c
E13 exp (i∆kz0 /2)
sin(∆kz0 /2) ∆k z0 /2
,
(6.19)
(3)
whereχef f = χ(3) casc + χT HG is the sum of the direct and cascaded third order nonlinear (2)
(2)
susceptibility. χ(3) casc is proportional to the product χSHG χSF . The exact expression of χ(3) casc depends on the particular phase matching mechanism. For (3)
phase-matched direct third harmonic generation ∆kT HG → 0, one arrives at χef f = χT HG . The third harmonic intensity generated in a non-linear media of length L is obtained by use of the relation I3 = (ni ε0 c/2) |Ei |2 . The result is I3 (L) =
2 2 3 sin (∆kL) χ I . n3 n31 c4 ε20 ef f 1 (∆kL)2 ω32 L2
(6.20)
For Gaussian pulse of width r0 and duration t0 the intensity is x2 + y 2 4t2 Ii (x, y, t) = Ii exp − . exp − 2 r0 t0
(6.21)
The energy conversion efficiency of third-harmonic light generation is given by ∞ R
η=
W3 (L) W1 (0)
=
−∞ ∞ R −∞
dx dx
∞ R −∞ ∞ R
dy dy
−∞
∞ R −∞ ∞ R
dtI3 (x, y, L, t) (6.22) dtI1 (x, y, 0, t).
−∞
For Gaussian input pulses the energy conversion efficiency is:
ηT H G =
(3) 2 2 2 L2 2 ω χ 1 3 ef f ef f I1 sin (∆kL) 33/2
n3 n31 c4 ε20
(∆kL)2
.
(6.23)
The last equation is concert to conversion efficiency of the third harmonic generation, and it is possible to use in the direct or cascade THG. Next we will consider our particular experimental situation for estimation of the efficiency of third harmonic generation and assume the phase matching conditions of ∆k → 0, which reduces eq.(6.23). The effective length of interaction Lef f in our case is effective distance along which the overlapping of interacting pulses occurs, rather than the nonlinear media length. This characteristic length was estimated above as approximately150 µm for 1 ps pulse. Equation (6.23) shows that in order to obtain high third harmonic generation efficiency the input beam intensity need to be increased, which can be achieved by tight input beam focusing. For divergent ∆ θ pump pulses, however, phase matching
123
6.2. PRINCIPAL ASPECT OF THE THREE-BEAM AUTO-CORRELATOR is achieved only for the central component of the pulse, which results on the reduced conversion efficiency. On the other side, the spectral width ∆ ν of the pump pulses also reduces the energy conversion efficiency, since phase-matching is achieved only for the central laser frequency. All these factors must be taken into account when a particular triple correlation scheme is considered.
Estimation of the conversion efficiency by the direct THG. Direct phase-matched third harmonic generation has been done in a number of materials, including crystals [6.4], liquid crystals [6.5] and polymer based organic thin films [6.6]. Let us consider the case with exploiting a single-mode semiconductor heterolaser, which is capable of generating about 10mW of light power peculiar to the fundamental harmonic in a continuous-wave regime. If similar heterolaser is operating, for instance, in the active mode-locking regime, one can expect appearing a train of ultra-short optical pulses with the time duration of about 1 ps with the repetition rate of up to 10GHz. The peak power of each individual pulse in this case can be estimated by approximately 1W . When such pulsed radiation is focused into a point of about in diameter, i.e. in an area of almost 80µm2 , one can find that the corresponding power density (or, what is the same, the intensity) will reach I1 ≈ 1.27M W/cm2 . Leaving aside the wavelengths of each concrete semiconductor heterolaser generation, one can take a few examples of the direct third harmonic generation using the needed fundamental harmonic wavelength λ1 in visible and infrared ranges [6.7], see Table 6.1.
Table 6.1: Estimating the conversion efficiency η for the direct THG (3)
Nonlinear medium
λ1 nm
χef f m2 /V 2
n1
n3
η
Buckministerfullerence
1064
280 × 10−20
2
2
4.24 × 10−6
p-toluene sulfonate
651.5
1.26 × 10−17
3
3
4.52 × 10−5
Cadmium sulfide selenide
1064
7.44 × 10−22
1.5
1.5
9.46 × 10−13
Gallium phosphide
577
2.94 × 10−18
3.47
3.47
1.75 × 10−6
The choice of the first group is motivated by relatively high third order susceptibility usually attributed to organic materials, while the second group, including semiconductors, is a characterized by their outstanding properties in practically important wavelength region of fiber communication systems (λ1 = 1.3 − 1.7µm). It is seen the best efficiency of the direct THG is obtained in one of organic crystals.
124
6.2. PRINCIPAL ASPECT OF THE THREE-BEAM AUTO-CORRELATOR
6.2.3
Analysis of three-beam auto-correlator via cascade THG
The second case that is considered here is related to the THG via cascade process as is shown in the fig. 6.6 (a). The triple auto-correlator in cascade processes can be exploited to record the raw data of the experiments with a sequence of ultra-short optical pulses is shown in fig. 6.6(b).
Figure 6.6: (a) Cascade optical schemes for a third harmonic generation. (b) A three-beam scanning interferometer for registering the triple auto-correlation function of high-repetition-rate trains including ultra-short optical pulses via cascade THG . Inside of the mechanism for the cascade THG, two partial fragments of the initial pulse should arrive at the facet of the first nonlinear medium at the same time when the moving mirror 2 is in the medium position. In this point the SH is generated. After this the beam from this crystal I (t, t2 ) is mixing in the second crystal with the delayed fundamental harmonic I (t + t1 ), which the both beam arrive to the same time to the second sample I (t, t2 ) when the moving mirror 1 is in the medium position, and the THG is generated I (t1 , t2 ). Once again, the principal consideration is the optical paths differences ideally is equal to zero. In fact, by this is mind that one can one has to consider three auto-correlator arms. For the figure 6.6(b), the first arm of the length L1 is equivalent to the case of the direct THG, but include the single path d8 , i.e. L1 = 2d1 + 2dB + d5 + d8 . For the second arm related to the length L2 again is equivalent to that the direct THG, but it is including the paths dSHG ,df , d6 and d7 , and three paths through beam splitters 1 and 2 , i.e. L2 = 2d2 + 2d3 + 3dB + df + d5 + dSHG + d6 + d7 . Finally, the third arm of the length L3 also is equal to the previous case, but also it is including the paths dSHG ,df ,d6 and d7 and three paths through beam splitters 1 and 2, i.e. L3 = 2d3 + 2d4 + 3dB + d5 + dSHG + d6 + df + d7 . Another important contribution one
125
6.2. PRINCIPAL ASPECT OF THE THREE-BEAM AUTO-CORRELATOR need to take in to account in the cascade processes in different crystal(fig. 6.6(a)), is the dispersion obtained for the interaction of the nonlinear medium SHG. This interaction depend directly of the size crystal and its respectively refractive index. Now let us to consider the case showed in the fig. 6.6(b), where the first condition is L1 = L2 that leads to the result d1 = d2 + d3 +
1 2
(dB + dSHG + d6 + d7 + df − d8 ). The second
is L1 = L3 , which gives d1 = d3 + d4 +
1 2
(dB + dSHG + d6 + d7 + df − d8 ). The
third consideration is L2 = L3 , also one can find d2 = d4 . At present moment, we calculate both time and spatial scales of appearing triple correlation response in the particular case of the THG cascade processes. Again one consider unchirped Gaussian pulse with the same parameters that the last time was consider with the intensity contour I(t) = exp (−4t2 /τ02 ). The corresponding spatial pulse width in d0 = 150 µm at the same level of 1/e. Ascribable to the mechanism of shaping the triple correlation via cascade processes show in fig. 6.7, where V1 and V2 are linear parts of velocities caused by scanning the corresponding moving mirrors 1 and 2 relative to an immovable mirror, but in this particular cases, the interaction is consider two times by the cascade process.
Figure 6.7: Gaussian pulses moving with different linear velocities V1 and V2 relative to an immovable one; illustration to appearing the effective length of interaction Lef f for a double of the interacting optical pulses in cascade process. As the last time, the values of τ0 and d0 determine both time and spatial scales of originating
126
6.2. PRINCIPAL ASPECT OF THE THREE-BEAM AUTO-CORRELATOR the triple correlations in this scheme. The obtained spatial scale makes it possible to estimate the needed accuracy of adjusting optical components within the chosen scheme of auto-correlator. Making the same assumption the last case, one can say that, one arrives at the accuracy of about 100 f s and 15 µm in time and spatial domains, respectively. Together with this one can introduce the effective length of interaction Lef f between these pulses, which can be used for estimating efficiency in this particular case. Now let us make the same assumption that the last time Lef f = d0 , related to the full time duration at a level of 1/e of the first and second nonlinear medium, one can estimate that. ∞
Z
Z
∞
I (t) dt ×
a) S = −∞
τ0 /
Z b) SA =
√
−∞ 2
√ −τ0 / 2
Z I ( t ) dt ×
τ0 /2
Z
I ( t ) d t ≈ 0.785 τ02 , √ τ0 / 2
√ −τ0 / 2
Z
τ0 /2
I (t) dt ×
c) Si = 4 −τ0 /2
−τ0 /2
d) NA = SA /S ≈ 0.710
I ( t ) dt ≈ 0.558 τ02 ,
I ( t ) d t ≈ 0.019 τ02 ,
e) Ni = Si /S ≈ 0.019.
(6.24)
These estimations show that with Lef f = d0 only about NA ≈ 71% of photons will be involved in the interaction, while about Ni ≈ 1.93% of them will be lost. Considering the other √ case which can be conditioned by a level of 1/e2 ≈ 0.130. One has to consider τ0 2 as the √ full time duration and d0 2 the corresponding spatial pulse width. one arrive at ∞
Z
Z
∞
I (t) dt ×
a) S = −∞
Z
τ/
√
b) SA = −τ0 / Z ∞
c) Si = 4 τ/
−∞ 2
√
√
2
2
I ( t ) dt ≈ 0.715 τ02 , Z ∞ −3 2 I ( t ) dt × τ0 , √ I ( t ) dt ≈ 1.6 × 10
d) NA = SA /S ≈ 0.911, Thus, with Lef f = d0
I ( t ) d t =≈ 0.784 τ02 ,
τ/
2
e) Ni = Si /S ≈ 2.23 × 10−3 .
(6.25)
√ 2 one can have about NA ≈ 91.1% of them will be lost. Finally,
the maximal effective length of interaction Lef f max is connected that the last time with the maximal pulse width τ0 max .
127
6.2. PRINCIPAL ASPECT OF THE THREE-BEAM AUTO-CORRELATOR Operation of the experimental arrangement via cascade THG. In fact the triple auto-correlator is based again in a double Michelson interferometer as was shown in the figure 6.6 (b). In general the triplet arms of the triple auto-correlator provide mutually time delayed pulse trains. The last is obtained in the interferometer considering, the wave from the sources is dividend in two wave by the beam splitter. One of them arrive to the Moving Mirror 1 where the intensity I (t + t1 ) is delayed in time and reflected. For our analysis, the intensity I (t + t1 ) travel the optical path L1 before to arrive to the nonlinear medium, where the THG is generated. The other beam again is dividing in two parts. The first part of the beam arrive to the moving mirror 2 where it is delay and reflected, I (t + t2 ) this beam travel to the optical path L2 before to arrive to the nonlinear medium where is summing with the wave from the fixed mirror I (t). The last beam I(t) come in to the fixed mirror traveling the optical path L3 before to arrive to the nonlinear crystal where is summing with the wave I (t + t2 ) and the second harmonic is generated I (t , t2 ). After this, the third harmonic is generated I (t1 , t2 ) by the second crystal where is summing the wave I (t + t1 ) with the second harmonic I (t , t2 ). Once more the time delayed is obtained by two moving mirror driving by two speakers. We deem to be possible to build this experimental arrangement using the same meaning the previous case. First one need to consider the wave length used, to select the appropriate optical components. The optical components that is consider are the same that the direct THG case.
Principals theoretical aspect of the efficiency of THG via cascade process In the cases of the THG via cascade processes, let us start to the solution to eq.(6.17(a)) E2 (z0 ) = −
ω2
(2)
χSHG E1a E1b
exp ( i ∆ kSHG z0 ) − 1
. (6.26) 2 n2 c ∆ kSHG Insertion of (6.26) into (6.17b) gives the amplitude of the third harmonic within a cascade process
1 ω2 ω3 (2) (2) E3,F M (z0 ) = χSHG χSF E1a E1b E1c × 2n2 n3 c ∆kSHG exp [i ( ∆kSHG + ∆kF M ) z0 ] − 1 exp ( i∆kF M z0 ) − 1 × − .(6.27) ∆kSHG + ∆kSF ∆kF M Next, we consider the different phase matching conditions for cascading process. For ∆kF M → 0 in phase-matched frequency mixing eq. 6.27 reduces to:
128
6.2. PRINCIPAL ASPECT OF THE THREE-BEAM AUTO-CORRELATOR
E3,F M
= −i
ω2 ω3 2n2 n3 c
(2)
(2)
χSHG χF M E1a E1b E1c
z0
∆kSHG sin(∆kF M z0 /2) , × exp (i∆kF M z0 /2) ∆kSF z0 /2
× (6.28)
with sin(∆kF M z0 /2)/(∆kF M z0 /2) → 1. For ∆kSHG → 0 in phase-matched second harmonic generation eq. 6.27 reduces to:
E3,F M
= −i
ω2 ω3
(2)
2n2 n3 c
(2)
χSHG χF M E1a E1b E1c
z0
∆kF M sin(∆kF M z0 /2) × exp (i∆kF M z0 /2) , ∆kF M z0 /2
× (6.29)
with sin(∆kSF z0 /2)/(∆kSF z0 /2) 1. A comparison of 6.28 and 6.29 shows that the third-harmonic generation via phase-matched second-harmonic generation is negligibly small compared to third-harmonic generation via phasematched frequency mixing. In case of the simultaneous phase matching, one has to assume that the condition of matching leads to ∆kSHG → 0, ∆kSF → 0 and, consequently, to ∆kSHG +∆kSF = ∆kT HG → 0. Than the eq. 6.27 could be simplifies to:
E3,F M
= −i
ω2 ω3 2n2 n3 c
(2)
(2)
χSHG χSF E1a E1b E1c
z0
∆kSF sin(∆kT HG z0 /2) , exp(i∆kT HG z0 /2) ∆kT HG z0 /2
(6.30)
with sin(∆kT HG z0 /2)/(∆kT HG z0 /2) → 1. E3,F M of eq. 6.28 (∆kF M → 0) and E3,F M of eq. eq: 6.30 ∆kT HG → 0 are of the same magnitude. As we mention in the previous cases the total third-harmonic signal is the sum E3 = E3,SF + E3,T HG over the both processes simultaneously. It may be written as (3)
E3 (z0 ) = −i
ω3 χef f z0 2n3 c
E13 exp (i∆kz0 /2)
sin(∆kz0 /2) ∆k z0 /2
,
(6.31)
(3)
whereχef f = χ(3) casc + χT HG is the sum of the direct and cascaded third order nonlinear (2)
(2)
(3) susceptibility. χ(3) casc is proportional to the product χSHG χSF . The exact expression of χcasc
depends on the particular phase matching mechanism.
129
6.2. PRINCIPAL ASPECT OF THE THREE-BEAM AUTO-CORRELATOR For phase-matched frequency-mixing interaction ∆kSF → 0 it is (2)
χ(3) casc =
(2)
ω2 χSHG χF M 2n2 c∆kSHG
,
(6.32)
where ∆k = ∆kSF . For phase-matched second harmonic interaction ∆kSHG → 0: (2)
χ(3) casc
=
(2)
ω2 χSHG χSF 2n2 c∆kF M
,
(6.33)
in this cases ∆k = ∆kF M . For mixed cascaded and direct third harmonic generation ∆kSHG + ∆kSF = ∆kT HG → 0, one can write (2)
(3)
χef f = χT HG +
(2)
ω2 χSHG χF M n2 c∆kF M
.
(6.34)
in this cases ∆k = ∆kT HG Then the energy conversion efficiency of third-harmonic light generation is this particular cases of the Gaussian pulses is given by ηT H G =
1 ω32 L2ef f I12 (3) 2 sin2 (∆kL) . χ 33/2 n3 n31 c4 ε20 cas (∆kL)2
(6.35)
where χ(3) casc is defined specifically for phase-matched frequency-mixing interaction ∆kSF → 0, and second harmonic interaction ∆kSHG → 0: (2)
χ(3) casc =
(2)
ω2 χSHG χF M
(2)
,
χ(3) casc =
(2)
ω2 χSHG χSF
, 2n2 c∆kSHG 2n2 c∆kF M In general cases for the cascade processes one need to consider the phases matched and need more specific analysis as was shown in the last set equation 6.32 to 6.33. Now it possible to predict that the higher efficiency is obtained in THG cascade process, the last conclusion it is viable, since, it is possible to select the set of crystal to uses in this processes, than the SHG (based on the THG cascade process) is more efficiency than the direct THG. In other (2)
(2)
(3)
words if we select correctly the material the product of χSHG χSF, F M > χT HG
130
6.2. PRINCIPAL ASPECT OF THE THREE-BEAM AUTO-CORRELATOR Estimation of the conversion efficiency by the third harmonic generation via cascade process A simultaneous phase matching for both the second harmonic generation and the sum frequency mixing is not possible in a single bulk material. The light generation at the third-harmonic frequency by phase-matched second-harmonic generation and phase-matched frequency mixing is possible by (a) the successive application of two crystals which are differently oriented and (b) using the single nonlinear media, where both the phase matching condition is fulfilled simultaneously or both nonlinear processes. The application of two separately phase-matched crystals is experimentally more complex than the application of a single crystal. Third harmonic generation in single quadratic media is one of the most extensively investigated cascading schemes. The first proposals for single crystal THG on the base of second order nonlinearities can be found in the Ref.[6.8]. The first attempts [6.9], [6.10] to fulfill simultaneously two phase-matching conditions were not successful. Nowadays the situation is totally different due to the methods for designing nonlinear media with periodical and quasi-periodical spatial modulation of the quadratic nonlinearity [6.11], [6.12]. Several methods that allow simultaneous phase-matching of two or more second order processes have been suggested [6.12]-[6.14]. The estimation of the energy conversion efficiency for the cascaded process is more com(3)
plex due to the fact that the value of χef f depends pretty much on the particular experimental conditions and can hardly be found in the literature. Here, we use the data on cascaded third harmonic generation in periodically polled LiN bO3 [6.15]. For the reported experimental condition (λ1 = 1064nm, crystal length 2.5cm) the expected efficiency was η ≈ 7.5 × 10−6 [ I1 ( M W/cm2 ) ] 2 . Taking into account the difference in the interaction length and the intensity of fundamental radiation in that experiment, the expected conversion efficiency is η ≈ 2.4 × 10−9 . Probably, higher efficiency can be obtained in an experimental configuration with two separately phase matched processes where two nonlinear crystals are used, but this experimental configuration require additional analysis. For give us an idea to the values of efficiency in cascade processes, let us to present some estimation in general cases, which an ideal phases matching is considering. Analyzing this configuration is made with the same experimental condition as the last case. Now it possible to see, if we compare the cascades versus direct processes of the THG [6.16]. It is possible to get best efficiencies in the cascade THG considering not only the photon will be converted but also comparing with the direct results in some materials (compared in the result obtained directly in the tables 6.1 and 6.2). Another advantage is the viability to mixing the
131
6.3. APPLYING NON-STEADY-STATE PHOTO-EMF TECHNIQUE TO DETECTION OF HIGHER ORDER AUTO CORRELATION FUNCTIONS OF ULTRASHORT OPTICAL PULSES Table 6.2: Estimating the conversion efficiency η for the cascade THG (2) Nonlinear medium Symmetry class λ1 nm d1, 2 pm/V , n1 n2
η
DAST
m
1064
d11 = 600
1.59
1.76
332.73 × 10−3
NPAN
mm2
1064
d12 = 200
nz = 1.82
nz = 1.99
1.87 × 10−3
Potassium malate
m
630
d11 = 6.62
1.56
1.60
5.23 × 10−8
DMNP
mm2
884
d33 = 29
nx = 1.51
nx = 1.56
5.96 × 10−6
materials, second, the best efficiencies is obtained in SHG based of the cascade process.
6.3
Applying Non-Steady-State Photo-Emf Technique to Detection of Higher Order Auto Correlation Functions of Ultrashort Optical Pulses
Here we examine a new application of the adaptive detectors based on non-steady-state photoelectro-motive force effect for the detection of higher order correlation functions, aiming the estimation of the parameters of ultra short optical pulses arranged in high-repetition trains. For this purpose three beam interferometer scheme with two signal beams modulated at different frequencies is proposed. Theoretical analysis of non-steady-state photo-EMF current [6.17] generated by light distribution formed by superposition of three waves is performed and the possibility to detect simultaneously second and higher order correlation function is demonstrated [6.18]. Potential advantages and disadvantages of such detection scheme for measuring the higher order auto-correlations functions are discussed.
6.3.1
Theoretical principles of the Photo-EMF effect
Semiconductor based non-steady state photo-induced EMF sensors can generate a time-varying output current that is proportional to the phase modulation in one of the beams. When two laser beams interfere on the surface of a photo-conductive crystal they form a periodic intensity pattern. Photo generated carriers are produced that diffuse away from the regions of intense illumination [6.17]. The carriers are trapped and form a stationary charge pattern and a corresponding spacecharge field. In the absence of any change in the interference pattern no net current flows in the circuit. However, if one of the laser beams is modulated by electro-optic (EO) phase modulator, the fringes (as well as the light-induced charges) move relative to the stationary space-charge field
132
6.3. APPLYING NON-STEADY-STATE PHOTO-EMF TECHNIQUE TO DETECTION OF HIGHER ORDER AUTO CORRELATION FUNCTIONS OF ULTRASHORT OPTICAL PULSES grating. If the intensity pattern moves faster than the material response time, an induced EMF is produced across the material. Using non steady photo-EMF phase sensitive photodetectors as an integrated optical interferometer for the detection of optical signal modulated in phase is quite a breakthrough idea being under development for more than decade. The inherent adaptive (spatial and temporal) properties of such detectors make them promising e.g. for nondestructive testing of vibrations from rough surfaces in industrial environment. Due to its robustness, simplicity, and high sensitivity the photo-EMF based configuration was proposed for numerous practical applications such as laser vibrometry [6.19], laser induced ultrasonic detection [6.20], profilometry of rough surfaces [6.21], etc. [6.22]-[6.25]. The expression of the current density of the detectors based on the non-steady state photoelectromotive-force (photo-EMF) induced by the dynamic interference pattern is shown next[6.17]. j1 = √
m2 (τ ) ∆ 2
σ0 ED
iΨτdi 1 + iΨτdi
×
1 1 + K 2 L2D
,
(6.36)
where m (τ ) is the contrast of the interferences pattern, ED is the diffusion field, LD = Dτc is the diffusion length with diffusion coefficient D and carrier lifetime τc , and σ0 is
the average photoconductivity of the sample, ∆ is an amplitude of the modulation frequency defined by the Bessel product J0 (∆) J1 (∆), Ψ is the periodical modulation frequency, finally −1 ∝ Ψ0 is the dielectric relaxation time that it is proportional to the cut-off frequency [6.17]. τdi
Figure 6.8 show principal the theoretical dependencies which describes the behaviour of PhotoEMF signal. The figure 6.8 (a) show the frequency transfer function of the Photo-EMF signal. One can see that, the transfer function is similar to the characteristic of high-pass filter which explain the ability to compensate for slow environment phases drift, based on a special “high-pass” frequency transfer function of the photo-EMF effect. We can see from here that relatively low modulation frequencies Ψ < Ψ0 are effectively suppressed in the output electrical signal. Note that the effect observed is not equivalent to simple passing of the output signal through a high-pass linear filter with the same cutoff frequency Ψ0 . In the cases of the sensitivity, it necessary to mention the low responsivity (or low energetic efficiency) does not always mean low sensitivity of the detecting system. Let us estimate the sensitivity of the Photo-EMF photo-detector, that is, the minimal detected amplitude (M DA) of phase modulation. Whereas we know the normalized (i.e., approximated to ∆ = 1 ∼ rad as show in the figure 6.8 (b) output current amplitude J 1 , this implies the evaluation of the dominating noise
133
6.3. APPLYING NON-STEADY-STATE PHOTO-EMF TECHNIQUE TO DETECTION OF HIGHER ORDER AUTO CORRELATION FUNCTIONS OF ULTRASHORT OPTICAL PULSES
Figure 6.8: Theoretical dependencies of the non-steady-state photo-EMF of (a) modulation frequency, (b) amplitude of phases modulation and spatial frequency current for some frequency band width and calculation of their ratio. The dependence of the signal that on modulation amplitude is shown in the fig. 6.8 (b) is the product of the Bessel function J0 (∆) J1 (∆). For a small signal amplitude of modulation ∆ 1rad the Photo-EMF signal has a lienal dependence. Then the first maximum appear when the modulation amplitude is ∆ = 1rad, after the maximum signal this decrease lineally up to the negative maximum. In the figure 6.8(c) one can observe the Photo-EMF signal as a function of spatial frequency k One can observe that the signal increase up to maximum, and after this maximum signal, the responses of the detector decreases lineally. The expression of the responsivity of the Photo-EMF detector is shown next J1 ∼ =
J0
K 2 L2D
8πN 1 + K 2 L2D
,
(6.37)
where
134
6.3. APPLYING NON-STEADY-STATE PHOTO-EMF TECHNIQUE TO DETECTION OF HIGHER ORDER AUTO CORRELATION FUNCTIONS OF ULTRASHORT OPTICAL PULSES
J0 =
eP0 Φ
, (6.38) ~ω is the primary current (multiplied by e the total number of mobile carriers generated per second) N is the number of interference fringes in the inter-electrode spacing. P0 is the total energy absorbed, Φ is the charge quantum efficiency generation. Then the only factors that determine the responsivity they are the primary current and J0 and the N = L/Λ number of interference periods Λ in the inter electrode spacing L, and is the fringe spacing. For the same number of absorbed photons the primary current is bigger in the detector based on the semiconductor, because they efficiency quantum is near to 1. Here, The detector sensitivity is determined by the minimal amplitude detecting M AD = J noise /J 1 , that mean the minimal amplitude of phase modulation is described by the following equation producing the signal to noises rating equal to 1 √ M AD ∼ = 16 2
r ∆f
P0
, (6.39) ~ω as one can maintain, the M AD signal only depends of the number photons absorbed by the detector. If we compare the sensitivity of the Photo-EMF detector with a typical photodiode, √ the Photo-EMF detector is 16 2 times lower than the photodiode. The Photo-EMF detector in generally less sensitive compared to conventional photo-detector. Here we have considered the functioning of photo-EMF detectors basically in 1D approximation (high abortion), without taking into account real, 2D design of the devices. One potential application of photo-EMF detector can be associated with the measurements of interference pattern visibility m [6.22]-[6.25], which makes possible evaluation of mutual coherence of two waves ΓR,S . The basic idea of this application lays in the fact that for all other experimental parameters (wavelength, amplitude and frequency of periodical modulation, period of interference patterns, light intensity, etc.) being fixed the amplitude of the photo-EMF signal is proportional to the square of interference pattern visibility ∝ m2 . Hence, it must be proportional to the second power of a degree of mutual temporal coherence between two interfering beams, because in general [6.26], √
where IR,S
IR IS
ΓR,S (τ ) (6.40) I0 are the intensities of reference and signal beams respectively, I0 = IR + IS is m (gR,S ) =
the average light intensity and is the delay time Ding et al. [6.22], demonstrated experimentally that measuring photo-EMF output signal amplitude as a function of a delay time τ it is possible to evaluate the second power of autocorrelation
135
6.3. APPLYING NON-STEADY-STATE PHOTO-EMF TECHNIQUE TO DETECTION OF HIGHER ORDER AUTO CORRELATION FUNCTIONS OF ULTRASHORT OPTICAL PULSES functions of the electric field envelope of the laser pulses
6.3.2
Main aspect of the autocorrelation function in a Photo-EMF detectors
The theory and application of the two Gaussian pulse correlator which is using a Photo-EMF detector had been explored by Ding et al [6.22]. We develop the theory of the two beam correlator using a Photo-EMF detector. Here we present only a brief description of the working principle of photo-EMF based correlator. Basic experimental set up (Fig. 6.9) for such correlator consists of two beam interferometer with temporal delay among them and additional periodical phase modulation in one of the arms
Figure 6.9: Typical experimental arrangement of the two Gaussian pulse correlator based on non-steady-state photo-EMF effect. Then one can consider, two ultrashort laser beams (reference and signal) enter a detector with an incidence angle θ = θS = θR with a time delay τ on the reference beam. And so, one can write the total electric as
136
6.3. APPLYING NON-STEADY-STATE PHOTO-EMF TECHNIQUE TO DETECTION OF HIGHER ORDER AUTO CORRELATION FUNCTIONS OF ULTRASHORT OPTICAL PULSES
E (t, τ ) = ES (t) + ER (t + τ ) = = ER fR (t + τ ) exp [i (kR z0 + ψ (t))] + +ES fS (t) exp [i (kS z0 + ψ (t) + ψ (t + τ ))] ,
(6.41)
with field shapes fS (t) and fR (t + τ ). For a Gaussian pulse, one has 2 fR (t) = exp −bt/TR
fS (t) = exp −b(t + τ )/TS2 ,
(6.42)
where or is the full width at half-maximum (FWHM) of the signal or the reference pulse, respectively, is the Gaussian pulse frequency chirp. The intensity distribution is written
I (t, τ ) = IR A + IS B +
p
IR IS gSR (τ ) exp {i [Kz0 − ψ (t + τ )]} + c.c.,
(6.43)
where K = 2π/Λ = 2k sin (θ) is the amplitude of the grating vector K = kR − kS traveling in a plane. Then one can define 2 a) A = fR (t)
b) B = fS2 (t) ,
and c) gR,S (τ ) = fR (t) fS∗ (t + τ ) .
(6.44)
Finally it is considering that the time deled is provided by a periodic modulation frequencies, the phase temporal dependencies are rewritten as ψ (t + t1 ) = ∆ sin (Ψt). One can get the equation of the current density for the first harmonic to trough the sample. J Ψ (τ ) =
m2 (gR,S ) ∆ 2
σ0 ED
iΨτdi 1 + iΨτdi 1 + K2 L2D
,
(6.45)
where the contrast of the interference pater is written as √ I1 I2
gR,S (τ ) = mS,R gS,R (τ ) . (6.46) I0 Taking in our consideration the electronic detector is too slow to the electric field with optical m (gR,S ) =
frequency, then the intensity seen by the detector is the time average it is given by
137
6.3. APPLYING NON-STEADY-STATE PHOTO-EMF TECHNIQUE TO DETECTION OF HIGHER ORDER AUTO CORRELATION FUNCTIONS OF ULTRASHORT OPTICAL PULSES
Z∞ I (τ ) =
I (t, τ ) dt = I0 1 +
∞
1 2
m0 gS,R exp (iωL τ ) exp (iKz0 ) + c.c. , (6.47)
where I0 is the total intensity, I0 = IR ΓR + IS ΓS , with average reference IR and signal IS intensity, then one can see that ΓR and are constants which is calculated for a Gaussian pulses Z
∞
a) ΓR =
r
2
fR (t) dt = −∞
π 2b
∞
Z TR ,
b)ΓS =
2
r
fS (t) dt = −∞
π 2b
TS ,
(6.48)
One can see the time varying modulation index of the interferences m (τ ) = m0 gS,R with √ the nominal modulation m0 = 2 IS IR /I0 , depends directly to the electric-field correlation function ΓR,S (τ ) of the two beams which is given by Γ (τ ) =
m20
Z
∞
2
|gR,S | dt = −∞
τ2 exp −b 2 . 2b T
T m2R,S
2π
(6.49)
Such a non-steady-state interference pattern gives rise to photo-EMF current density. Since the photo-EMF current density in the first harmonic is proportional to the square of the correlation function ΓS,R (τ ) (as was showing in 6.45) JΨ (t) ∝ Γ2R,S (τ ) .
(6.50)
By measuring the photo-EMF current as a function of the time delay, of optical pulses, one can deduce information about the signal pulses, such as pulse width, given a known reference pulse. This method gives the same information as the interferometric electric-field correlation does but without the need for data processing. In the case of transform limited pulses the photo-EMF current trace as a function of the delay is identical to the conventional background free intensity correlation, which gives the pulse width directly as it is showing in the fig 6.10.
6.3.3
Theoretical analisis of photo-EMF technique using three beam interferometer for detection of higher order correlation function.
Here we develop the theory of the photo-EMF applied in the detection higher order correlation function for Gaussian pulses. For this, we propose the new configuration in to the application of the photo-EMF effect, which is based in the three beams interferometer 6.11 In general, the three beam Gaussian pulse correlator is based in a modified Michelson interferometer. The triplet arms of this three beam interferometer provide the scanning of the pulse
138
6.3. APPLYING NON-STEADY-STATE PHOTO-EMF TECHNIQUE TO DETECTION OF HIGHER ORDER AUTO CORRELATION FUNCTIONS OF ULTRASHORT OPTICAL PULSES
Foto-EMF Correlation function 1.5
GHΤL
1.0
0.5
0.0 -2
-1
0
1
2
Time HsL
Figure 6.10: Example of the pulse correlation function obtained with the Photo-EMF detector, with b = 1 and T = 1. trains. The beam from the source is divided in two beams by the beam splitter. One of them it is arriving to the Moving Mirror 1 where it is delayed in time I (t + τ1 ) and reflected. The other beam again is divided in two parts. The second part of the beam arrive to the moving mirror 2 where it is delaied I (t + τ2 ) and reflected. The last beam with the intensity I (t) comes in to the fixed mirror. Periodical phase modulation can be introduced in first two beams e.g. by attaching the mirror in the delay line to the piezoelectric transducer. Finally, three pulse trains are mixed on photo-EMF detector, whose output current is detected by lock-in amplifier As in the previous section, one can consider a Gaussian pulses beam as with electric field defined by,
a) E1 (t) = E1 f1 (t) exp [i (k1 z0 + ψ (t))] , b) E2 (t + t1 ) = E2 f2 (t + t1 ) exp [i (k2 z0 + ψ (t) + ψ (t + t1 ))] , (6.51) c) E3 (t + t2 ) = E3 f3 (t + t2 ) exp [i (k3 z0 + ψ (t) + ψ (t + t2 ))] , where the functions f1, 2, 3 is given by a) f1 (t) = exp −
t2 T2
"
,
b) f2 (t + t1 ) = exp −
(t + t1 )2 T2
# ,
and " c) f3 (t + t2 ) = exp −
(t + t2 )2 T2
# ,
(6.52)
139
6.3. APPLYING NON-STEADY-STATE PHOTO-EMF TECHNIQUE TO DETECTION OF HIGHER ORDER AUTO CORRELATION FUNCTIONS OF ULTRASHORT OPTICAL PULSES
Figure 6.11: Experimental arrangement to measurement of the high order correlation function with Photo-EMF detector using a transmission configuration where t1,2 are the respectively pulses time delay provided by two moving vibrating mirror. Then the interference pattern which describe the interferences between three beams is given by
IT (t) = I1 |f1 (t)|2 + I2 |f2 (t + t1 )|2 + I3 |f3 (t + t2 )|2 + p + I1 I2 f1 (t) f2∗ (t + t1 ) exp {i [K1 z0 − ψ (t + t1 )]} + p + I1 I3 f1 (t) f3∗ (t + t3 ) exp {i [K2 z0 − ψ (t + t2 )]} + p + I2 I3 f2 (t + t1 ) f3∗ (t + t2 ) ×
(6.53)
× exp (i {K3 z0 − [ψ (t + t1 ) − ψ (t + t2 )]}) + c.c., where K1 = k2 − k1 , K2 = k3 − k1 and K3 = k2 − k3 . It is easy to fix the grating vectors so that K1 = K0 , K2 = −K0 and K3 = 2K0 . Taken the previous consideration, it is possible to rewrite the eq. 6.53 as
140
6.3. APPLYING NON-STEADY-STATE PHOTO-EMF TECHNIQUE TO DETECTION OF HIGHER ORDER AUTO CORRELATION FUNCTIONS OF ULTRASHORT OPTICAL PULSES
IT (t) = AI1 + BI2 + CI3 + (6.54) p + I1 I2 g1,2 exp {i [K0 z0 − ψ (t + t1 )]} + p + I1 I3 g1,3 exp {−i [K0 z0 + ψ (t + t2 )]} + p + I2 I3 g2,3 exp (i {2K0 z0 − [ψ (t + t1 ) − ψ (t + t2 )]}) + +c.c., where g1,2 = f1 (t) f2∗ (t + t1 ), g1,3 = f1 (t) f3∗ (t + t2 ) and g2,3 = f2 (t + t1 ) f3∗ (t + t2 ). For simplicity we also define A; B; C; as A = |f1 (t)|2 , B = |f2 (t + t1 )|2 and C = |f3 (t + t2 )|2 . We consider the time delay which is produced by two moving modulated mirrors; in form of periodical modulation; a)ψ (t + t1 ) = ∆ sin (Ψ1 t)
b)ψ (t + t2 ) = ∆ sin (Ψ2 t) .
(6.55)
Considering, the amplitude of modulation frequency is approximately ∆ 1. One can analyze separately the complex amplitudes from the equation 6.54;
exp (iK0 z0 ) exp [−i∆ sin (Ψ1 t)] + c.c. ' exp (iΨ1 t) − exp (−iΨ1 t) + C.C, ' exp (iK0 z0 ) 1 − ∆ 2
(6.56)
exp (−iK0 z0 ) exp [−i∆ sin (Ψ2 t)] + c.c. ' ' exp (−iK0 z0 ) 1 − ∆
exp (iΨ2 t) − exp (−iΨ2 t) 2
+ c.c.,
(6.57)
and finally
exp (i2K0 z0 ) exp {−i∆ [sin (Ψ1 t) − sin (Ψ2 t)]} + C.C ' exp (iΨ1 t) − exp (−iΨ1 t) ' exp (i2K0 z0 ) 1 − ∆ − 2 exp (iΨ2 t) − exp (−iΨ2 t) − + c.c.. 2
(6.58)
141
6.3. APPLYING NON-STEADY-STATE PHOTO-EMF TECHNIQUE TO DETECTION OF HIGHER ORDER AUTO CORRELATION FUNCTIONS OF ULTRASHORT OPTICAL PULSES Substituting the previous set of equations (eq. 6.56 to 6.58), in to the eq. 6.54 the interference pattern in terms of the complex amplitudes is redefined as
IT (t) = I0 + I11,0,0 m (t1, 2 ) exp (iK0 z0 ) + +I11,1,0 m (g1, 2 ) exp (iK0 z0 ) exp (iΨ1 t) + +I11,−1,0 m (g1, 2 ) exp (iK0 z0 ) exp (−iΨ1 t) + +I21,0,0 m∗ (g1, 3 ) exp (iK0 z0 ) + +I21,0,1 m∗ (g1, 3 ) exp (iK0 z0 ) exp (iΨ2 t) + +I21,0,−1 m∗ (g1, 3 ) exp (iK0 z0 ) exp (−iΨ2 t) + +I32,0,0 m (g2, 3 ) exp (i2K0 z0 ) + +I32,1,0 m (g2, 3 ) exp (i2K0 z0 ) exp (iΨ1 t) + +I32,−1,0 m (g2, 3 ) exp (i2K0 z0 ) exp (−iΨ1 t) + +I32,0,1 m (g2, 3 ) exp (i2K0 z0 ) exp (iΨ2 t) + +I32,0,−1 m (g2, 3 ) exp (i2K0 z0 ) exp (−iΨ2 t) + c.c.,
(6.59)
where I0 = AI1 + BI2 + CI3 . In this notation, the I 1,#,# is related to the corresponding spatial harmonic. And I #,1,1 indicates the corresponding harmonic of the temporal periodic modulation, and a) m (g1, 2 ) =
√ I1 I2 I0 √
b) m (g1, 3 ) =
c) m (g2, 3 ) =
I1 I3 I0
g1,2 = m01,2 g1,2 ,
g1,3 = m01,3 g1,3 ,
(6.60)
√ I2 I3
g2,3 = m02,3 g2,3 , I0 Analyzing the expression 6.59, one can deduce that have five gratings at the same time, one of them is considering as the statical grating. The other 4 gratings are vibrating at the frequencies Ψ1 ; Ψ2 . Then, one have to consider the space-charge field ESC (t) and the photo-carrier concentration n (t) induced by the interaction of the interferences pattern in to the detector, which has in amplitude the same distribution of the interferences pattern. One can estimate the density through the sample that is equal to the average current density over the space between the electrodes, which is determined by
142
6.3. APPLYING NON-STEADY-STATE PHOTO-EMF TECHNIQUE TO DETECTION OF HIGHER ORDER AUTO CORRELATION FUNCTIONS OF ULTRASHORT OPTICAL PULSES
j (t) = eµ
1
Z
Λ
n (t) ESC (t) dz0 , Λ 0 where e is the electronic charge, µ is the charge mobility and Λ is the period grids.
(6.61)
Substituting the expression for n (x; t) and ESC (x; t), one can get the expression of the current density to give us the sample
j (t) =
eµ 0 j + j 1,0 exp (iΨ1 t) + j −1,0 exp (−iΨ1 t) + Λ +j 2,0 exp (2iΨ1 t) + j −2,0 exp (−2iΨ1 t) + j 0,1 exp (iΨ2 t) + +j 0,−1 exp (−iΨ2 t) + j 0,2 exp (2iΨ2 t) + j 0,−2 exp (−2iΨ2 t) + +j 1,1 exp [it (Ψ1 + Ψ2 )] + j −1,−1 exp [−it (Ψ1 + Ψ2 )] + +j 1,−1 exp [it (Ψ1 − Ψ2 )] + j −1,1 exp [−it (Ψ1 − Ψ2 )] ,
(6.62)
where j 0 , j 1,0 , j 2,0 , j 0,1 , j 0,2 are the fundamental, first and second harmonics of the current densities respectively defined as; Here we are interested on the current densities concerning of the firsts harmonics: J Ψ (t) = j 1,0 + j 0,1 + j 1,1 + j 1,−1 .
(6.63)
The expressions of the spaces-charge ESC field and the photo-carrier concentration n can be taken from ref. [6.17]
a) n−1 = mσ b) E −1
1 − iΨτdi
1 − iΨτdi (1 + K 2 L2D ) 1 , = −im∆ED 1 − iΨτdi (1 + K 2 L2D )
(6.64)
where the diffusion field is ED = KkB T /e, and kB is the Boltzmann constant. Substituting the corresponding expression in to the respectively current density by the first harmonic (eq. 6.63) provided by the eq. 6.62. One can obtain the equation of the photo-EMF current density produced by the interferences of the three beams has four components corresponding to the frequencies Ψ1 , Ψ2 , Ψ1 ± Ψ2 Ψ1 a) Jp−emf (t) =
m2 (g1,2 ) ∆ 2
σ0 ED
iΨ1 τdi 1 + iΨ1 τdi (1 + K2 L2D )
143
6.3. APPLYING NON-STEADY-STATE PHOTO-EMF TECHNIQUE TO DETECTION OF HIGHER ORDER AUTO CORRELATION FUNCTIONS OF ULTRASHORT OPTICAL PULSES Ψ2 b) Jp−emf (t) = Ψ1 −Ψ2 c) Jp−emf (t) =
m2 (g1,3 ) ∆ 2
m (g1,2 ) m (g2,3 ) ∆ 2
Ψ1 +Ψ2 d) Jp−emf (t) =
σ0 ED
σ0 ED
m (g1,2 ) m (g2,3 ) ∆ 2
iΨ2 τdi 1 + iΨ2 τdi (1 + K2 L2D ) i (Ψ1 − Ψ2 ) τdi
1 + i (Ψ1 − Ψ2 ) τdi (1 + K2 L2D )
σ0 ED
(6.65)
i (Ψ1 + Ψ2 ) τdi 1 + i (Ψ1 + Ψ2 ) τdi (1 + K2 L2D )
.
Taken the assumption: the response of our detector is too slow to the electric field with optical frequency. Then the intensity seen by the detector in time average is obtained by. Z∞ IT (τ ) =
IT (t) dt, −∞
then one can supposes the modulation frequencies Ψ1 and Ψ2 are fixed. Since the only parameter which is changed it is the time responses of the Photo-EMF detector when the pulses are coherence between them, and the interferences pattern is appearing. One can introduce the correlation function to see the detector by the action of the pulses and the moving mirrors. As it was showing in the previous section, the current density described in the equation 6.65 is strongly depended of the visibility of the interferences pattern: m2 (g1,2 ), m2 (g1,3 ) and m (g1,2 ) m (g2,3 ). And so, one can obtain the respectively correlation function which see the detector that is defined by: Z∞
2
m (g1,2 ) =
a) Γ (t, t1 ) = −∞
=
2
|g1,2 | dt =
m2012
−∞
Z∞
f1 (t) f ∗ (t + t1 ) 2 dt 2
−∞
t21 exp − 2 , 2b T
T m2012 Z∞
b) Γ (t, t2 ) =
2π
m2 (g1,3 ) = m2013
−∞
=
Z∞
m2012
Z∞ −∞
T m2013
2π
2b
exp −
t22 T2
2
|g1,3 | dt = m2013
Z∞
f1 (t) f ∗ (t + t2 ) 2 dt 2
−∞
,
(6.66)
And finally, we obtain the signal provide by the high order correlation function
144
6.3. APPLYING NON-STEADY-STATE PHOTO-EMF TECHNIQUE TO DETECTION OF HIGHER ORDER AUTO CORRELATION FUNCTIONS OF ULTRASHORT OPTICAL PULSES
Z∞ c) ΓDG (t1 , t2 ) =
m (g1,2 ) m (g1,3 ) −∞
Z∞ = m012 m013
f1 (t) f2 (t + t1 ) f1 (t) f3 (t + t2 ) dt −∞
= m012 m013
T 2
r
π d
exp −
3 4T 2
t21
+
t22
−
2 3
t1 t2
.
As it’s showing in the first two terms (Eq. 6.65 and 6.66 (a) and (b)) carry the same information as in the case of two-pulse correlator discussed in the previous section and gives the possibility to measure the square of the second order electric field correlation function. As to the last two terms, they are proportional to the higher order correlation function, namely the 4-th order correlation function [6.18], [6.23]. Then one can calculated the corresponding bispectrum which is given by
r ΓDG (f1 , f2 ) = m01,2 m01,3 T
π 2
exp −
3π 2 T 2 2
f12
+
f22
−
2 3
f1 f2
(6.67)
Figure 6.12: . Figure (a) it si the correlation function calculated from Eq. 6.66 (c), and fig. (b) its bispectrum from Eq. 6.67; T=1 Thus, the proposed three beam interferometer allows to measure both second and higher order correlation function in a single experimental arrangement using non-steady-state photo-EMF phase sensitive detector. The main advantage of this technique (compared with the conventional
145
6.3. APPLYING NON-STEADY-STATE PHOTO-EMF TECHNIQUE TO DETECTION OF HIGHER ORDER AUTO CORRELATION FUNCTIONS OF ULTRASHORT OPTICAL PULSES measurements of pattern contrast) is the simplicity of the configuration and the potentially high speed of operation. Indeed, the output signal from the lock-in amplifier gives the value of the mutual coherence directly without any signal processing. Characteristic time which is needed to measure the value of autocorrelation function for a particular optical path is basically limited by the characteristic time of space charge grating formation which can be as small as 10−7 s in GaAs detector in the visible and infrared region of spectra. Other important advantage of this technique consists on the possibility to work with complicated wavefronts, even with speckle-like pattern instead of perfect Gaussian interfering wavefronts. Finally, unlike standard interferometer photo-EMF base correlator can operate in a presence of environmental vibration and noises. Among the disadvantage, one can mention that the detected correlation function is of the even order, which make impossible the extraction of information about the phase of the pulses. Another limitation of photo-EMF based detector is that the distance between electrodes should be small enough; in particular the interelectrode spacing should fit the maximal number of fringes which can be produced for given experimental condition, this last one, in general, is defined by the ratio λ/∆λ of the source
6.3.4
Characterization of Photo-EMF detector
The experimental arrangement to the Photo-EMF characterization is based in a Mach-Zender interferometer, it is shown in the fig. 6.13. The experimental set up using in the Photo-EMF characterization samples is dividing in three parts • First, the optical system is a Mach-Zender interferometer. where the vertically polarized beam, from the He-Ne laser (power 40mW/cm2 ,) is divided by a beam splitter (50 : 50). One of them is denoted reference beam and the other signal beam that is modulated by the electro optical modulator. After a series of mirrors, the tow beams are interfering in to the photo-detector where the interferences pattern is generated. The gallium arsenide is characterizing using the transmission experimental arrangement, as was shown in the fig. 6.8 (a). For the experimental set up that is used to characterize the poly-fluoren 6co-triphenyldiamine (P F 6 − T P D) photoconductor polymer the tow beams incident in reflection configuration. • Second, the electronic phases modulation system is via electro optics modulator with this respectively driver, which applied a sinusoidal modulated to the beam signal.
146
6.3. APPLYING NON-STEADY-STATE PHOTO-EMF TECHNIQUE TO DETECTION OF HIGHER ORDER AUTO CORRELATION FUNCTIONS OF ULTRASHORT OPTICAL PULSES
Figure 6.13: Experimental arrangement to the Photo-EMF characterization (a) transmission and (b)reflection configuration • Third, the electronic detection system is via Lock-in amplifier.
Samples description In the cases of the GaAs, we assumed that the sample is unlimited along the z and y axes. This approximation is obviously valid if the characteristic sizes of the sample along these axes are large enough to compare with the fringe spacing i.e., if Lz,y K −1 . In all experiments performed, the condition Ly K −1 is practically always fulfilled. The typical light penetration depth in GaAs in the region of fundamental absorption is about dp = α−1 ≈ 1 − 0.3µm. In fact, real GaAs photo-detectors have some physical electrical contacts and some limited physical thickness. It seems that the contacts with the photoconductive sample and the front surface of the sample can significantly influence the parameters of the photo-EMF detectors. However, very strong optical absorption does not main really shallow penetration of photoconductivity into the bulk of the photoconductor. This advantage of the Photo-EMF detectors is that it needs only a few micrometers of material or the penetration depth; thus dispersion that is due to finite crystal size is
147
6.3. APPLYING NON-STEADY-STATE PHOTO-EMF TECHNIQUE TO DETECTION OF HIGHER ORDER AUTO CORRELATION FUNCTIONS OF ULTRASHORT OPTICAL PULSES not a crucial issue in our technique, whereas the problem of dispersion limits the thickness of the SHG and THG crystal in standard nonlinear characterization techniques [6.27]. The polymer sample has an absorption spectrum in 10cm−1 of 830nm [6.28], the uniform thickness is 7µm. the potential polymer oxidation of P F 6 − T P D is 0.65 eV . The predominant carriers in the material is holes. In particular photo-refractive gratings recorded on this material exhibit high diffraction efficiency and times response around 30 ms to the wavelength of 830 nm near-infrared.
Principal experimental dependencies The figure 6.14 show the principal experimental dependencies of the photo-EMF signal in GaAs.
Figure 6.14: Experimental dependencies of the non-steady-state photo-EMF of (a) modulation frequency, (b) amplitude of phases modulation and (c) spatial frequency in GaAs, I0 = 40mW/cm2 , m = 0.99, RL = 10kOhms
148
6.3. APPLYING NON-STEADY-STATE PHOTO-EMF TECHNIQUE TO DETECTION OF HIGHER ORDER AUTO CORRELATION FUNCTIONS OF ULTRASHORT OPTICAL PULSES The experimentally the cut-off frequency Ψ0 is determined in the fig. 6.14 (a). Ψ0 is approximately in 1Khz using a Raleigh criterion ( 80% of the maximum signal). In this cases the signal is growing lineal with the modulation frequency Ψ, after a cut-off frequency Ψ0 the signal is saturated. In the next measurement Ψ = 10kHz.. The fig. 6.14 (b) show the dependencies of the amplitude of modulation. In this cases the signal increases lineally with the amplitude of modulation ∆. Comparing whit the theoretical dependence, fig. 6.8 (b), one can conclude that the range of measurement was doing in ∆ 1. The last dependence (fig. 6.14 (c)) give us the information to determine the optimal numerical aperture. In this case the signal growth in typical Photo-EFM behavior, lineally in small spatial frequencies, and decay as 1/K after the maximum. Experimentally with this information one can determine the angle θ in to our experimental arrangement. The figure 6.15 show the principal experimental dependencies of the photo-EMF signal in P F 6 − T P D.
Figure 6.15: Experimental dependencies of the non-steady-state photo-EMF of (a) modulation frequency, (b) amplitude of phases modulation and (c) spatial frequency in P F 6 − T P D, I0 = 40 mW/cm2 , m = 0.99, RL = 10 M Ohms and ∆ = 500mrad As was show in the fig 6.15 (a), one have two cut-off frequency Ψ0 one of them around to 300 Hz. the second is in 1KHz., in this cases when the light intensity is decreases the cut-off frequency is constant. Previous research in photoconductive polymers [6.29] indicate the presence of the second cut-off frequency independent of the intensity decay is due to the concentration of free carriers. In the particular case of low spatial frequency KLD 1 the characteristic time of the process is the carriers lifetime in the conduction band. The numerical aperture is fixed in K = 3.4 × 10−5 cm−1
149
6.4. CONCLUSION As shown in Figure 6.15 (b), in a low modulation amplitudes region ∆ 1rad, one can see a growth linear signal. For big values of ∆, approximately ∆ ≈ 1 rad the signal is saturated. The principal characteristic of the Photo-EMF detector is shown next in the table 6.3:
Table 6.3: Characteristic of the adaptive photo-detector based on the photo-EMF effect j1
=
m2 ∆ iΨτdi × σ0 ED 1+iΨτ di 2
1 1+K 2 L2 D
The ability to compensate for slow environmental phase drifts This is an interferometric technique Invariant to the mechanical vibration. The ability to adaptive to the wave front. Experimental band width in GaAs in 1 − 10 M Hz.
j 1 ∝ 1 + K 2 L2 D
−1
m (τ ) = m0 gS,R
Experimental numerical aperture in GaAs in KOP T = 2500cm1 The advantage of this technique is that it produces the correlation function directly without the need for data processing [6.22]
J1 ∼ =
J0 8πN
2
K L2 D 1+K 2 L2 D
Then the only factors that determine the responsivity are the primary current J0 and the N = L/Λ number of interference fringes in the inter electrode and spacing L, and Λ is the fringe spacing
√ M AD ∼ = 16 2
r
P0 ∆f ~ω
The Photo-EFM detector in generally is less sensitive compared to conventional √ photo-detector 16 2 time approximately This detector work to use nonlinear process type Kerr.
6.4
Conclusion
We propose the methodology to estimate the average parameters of ultra short optical pulses with asymmetric envelopes, which are arranged in high-repetition trains. We using higher order correlation function, first of all the method based on third harmonic feneration and cascade nonlinear processes were analyzed. The conversion efficiencies are analyzed and compared in various materials, obtaining the best efficiencies in the cascade processes. We estimate the principal parameters of such interferometer where the triple auto-correlation functions are measured using the effect of the third harmonic generation in two different optical configurations. Together with this, We propose the methodology to measure higher order correlation functions of short optical pulses using non-steady-state photo-EMF detector. For this purpose three beam interferometer scheme with two signal beams modulated at different frequency is proposed. Theoretical analysis of non-steady-state photo-EMF current generated by light distribution formed by
150
6.4. CONCLUSION superposition of three waves is performed and the possibility to detect simultaneously second and higher order correlation function is demonstrated. The theoretical analysis has been performed as first step to implementing this technique, the experimental characterizations of the Gallium Arsenide (GaAs ) semiconductor and for the polyfluoren 6-co-triphenyldiamine (P F 6 − T P D ) photoconductor polymer was performed.
151
Bibliography [6.1] Kakarala, R., “Triple correlation on groups,” Ph.D. Thesis University of California, Irvine (1992). [6.2] Escoproducs, “web page; http://www.escoproducts.com/html/bk-7_optical_glass.html.”, BK7 Indices of Refraction, USA, 17-Febraury-2010. [6.3] Qiu, P., and Penzkofer, A., “Picosecond Third-Harmonic Light Generation in -BaB204” Appl. Phys. B vol. 45,pp. 225-236, (1988). [6.4] Stegeman, G. I., Hagan, D. J. and Torner, L., “χ(2) cascading phenomena and their applications to all-optical signal processing, mode-locking, pulse compression and solitons” Opt. Quantum Electron. vol. 28,pp. 1691-1740 (1996). [6.5] Tomov, I. V., VanWonterghem, B. and Rentzepis, P.M., “Third-harmonic generation in barium borate”, Appl. Opt. vol. 31, pp. 4172 -4174, (1992) [6.6] Yelin, D., Silberberg, Y., Barad, Y. and Patel, J. S. “Phase-matched thirdharmonic generation in a nematic liquid crystal cell”, Phys. Rev. Lett., vol. 82, pp. 3046-3049, Apr. (1999). [6.7] Sutherland, R. L., McLean, D. G. and Kirkpatrick S., “Handbook of Nonlinear Optics”, 2-nd Ed Revised and Expanded, Marcel Dekker, New York, chapter 8 (2003) [6.8] Kintaka, K., Fujimura, M., Suhara, T. and Nishihara, H., “Third harmonic generation of Nd:YAG laser light in periodically poled LiNbO3 waveguide” Electron. Lett., vol. 33, Issue 17,pp. 1459-1461, (1997). [6.9] Akhmanov, S. A. and Khokhlov, R. V., “Problemy nelineinoi optiki (Problems of Nonlinear Optics)”, (Izd. VINITI, Moscow), 1964; Engl. Transl. (Gordon & Breach, New York), pp. 189, (1972)
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BIBLIOGRAPHY [6.10] Sukhorukov, A. P. and Tomov I. V., “On simultaneous synchornous generaton of the second and the third harmonics in crystals in quadratic nonlinearity”, Izv.Vyssh., Uchebn. Zaved. Radiofiz. vol. 13, pp. 266-270, (1970). [6.11] Orlov, R. U., Sukhorukov, A. P. and Tomov Annu, I. V., Sofia Univ. Phys., vol. 65, pp. 283, (1972). [6.12] Fejer, M. M., Magel, G. A., Jundt, D. H. and Byer, R. L., “Quasi-phase-matched second harmonic generation: tuning and tolerances” IEEE J. Quantum Electron., vol. 28 Issue 11, pp 2631-2654, (1992) [6.13] Zhu, Y.Y., and Ming, N.B., “ Dielectric superlattices for nonlinear optical effects” Opt. Quantum Electron, vol. 31 N 12, pp. 1093-1128, (1999). [6.14] Fradkin-Kashi, K., Arie, A., Urenski, P. and Rosenman, G., “Multiple Nonlinear Optical Interactions with Arbitrary Wave Vector Differences”, Phys. Rev. Lett. vol. 88, Issue 2, pp. 023903-1 023903-4, (2001) [6.15] Das, S.K., Mukhopadhyay, S., Sinha, N., Saha, A., Datta, P.K., Saltiel, S.M. and Andreani, L.C., “Direct third harmonic generation due to quadratic cascaded processes in periodically poled crystals,” Optics Communications, vol. 262, pp. 108-113, (2006); DOI: 10.1103/PhysRevLett.88.023903. [6.16] Shcherbakov, A. S., Moreno Zarate, P., Campos Acosta, J., Mansurova, S., Munoz Zurita, A. L., Nemov, S. A., “Practical aspects of applying triple correlations to the characterization of high-frequency repetition trains of picosecond optical pulses” Proc. SPIE, vol. 7582, pp. 75821G-1 - 75821G-11, (2010); doi:10.1117/12.839447 [6.17] Stepanov, S., “Handbook of advanced electronic and photonic materials and devices”. Two volume set, H.S. Nalwa, (2001). [6.18] Moreno Zarate, P., Shcherbakov, A. S., J., Mansurova, S., “Applying Non-SteadyState Photo-Emf Technique to Detection of Higher Order Auto Correlation Functions of Ultrashort Optical Pulses” Journal of Applied Research and Technology (JART) vol. 10, No. 2, pp 143-151, (2012) [6.19] Rodriguez P., Trivedi S.,Wang CC., Stepanov S., Elliott G., Meyers J., Lee J. and Khurgin J. “Pulsed-laser vibrometer using photoelectromotive-force sensors”. Applied Physics Letters vol. 83, pp 1893-1895, (2003).
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BIBLIOGRAPHY [6.20] Castillo Mixcoatl, J., Rodriguez Montero, P., Stepanov, S., Mansurova, S. “Increasing the sensitivity of laser ultrasonic detection in GaAs photo-EMF configuration under external DC bias” Applied Physics Letters vol. 80, page 3697 - 3699 (2002). [6.21] Stepanov, S., Plata, M., “Measuring correlation between speckle patterns reflected from rough surfaces at different wavelengths by adaptive photo-EMF detector” Proc. SPIE vol 4447, pp. 140 (2001); doi:10.1117/12.446728 [6.22] Ding, Y., Lahiri, I. and Nolte, D. D., “Electric-field correlation of femtosecond pulses by use of a photoelectromotive-force detector”, J. Opt. Soc. Am. B, vol. 15, No. 7, pp. 20132017, (1998). [6.23] Shcherbakov, A. S., Mansurova, S., Moreno Zarate, P., Campos Acosta, J. and Nemov, S. A., "Performing the triple auto-correlation of picosecond optical pulse train with a photo electromotive forces detector", Proc. SPIE 7917, 79170S (2011); doi:10.1117/12.873977. [6.24] Moreno Zarate, P., Rodriguez-Montero, P., Koeber, S., Meerholz, K. Mansurova, S., “Adaptive detection of Doppler frequency shift using ac non-steady-state photo-EMF effect”, CLEO/QELS, OSA paper JThA 50 ,pp. 1-2, Technical Digest (CD) (OSA, 2008). [6.25] Moreno Zarate, P., Mansurova, S., Rodriguez-Montero, P., Espinosa, M., “Velocimetro basado en detectores con Efecto foto-FEM no Estacionario” LI Congress of The Mexican Physics Society (SMF) and the XXI Annual Meeting of The Mexican Association of Optics (XXI-RAO), pp. 1-9, Technical Digest (CD) (AMO, 2008). [6.26] Born, M. and Wolf, E., "Principles of Optics; Electromagnetic Theory of Propagation, Interference and Diffraction of Light ", Cambridge University Press; 7th edition (1999) [6.27] Bittner, R., Bruchle, C. and Meerholz, K. “Influence of the glass-transition temperature and the chromophore content on the grating buildup dynamics of poly (n-vinylcarbazole)-based photorefractive polymers”. App Optics. vol. 37. pp. 2843-2851. (1998). [6.28] Koeber, S. “Degree Thesis”. Koeln, Germany, (2006) [6.29] Sheet Resistances, “ITO-Layer Thicknesses, Substrate Materials and Substrate Thicknesses, web page; http://www.pgo-online.com/intl/katalog/ito.html”, przisions glas and opik, Germany, 17/june/(2006).
154
Chapter 7
Statements 7.1
General conclusions
In this thesis, have been develop the specific approach for the average characterization of highrepetition-rate trains, including low-power bright picosecond optical dissipative solitary pulses with an internal frequency modulation is developed, in both time and frequency domains in practically important case of operating the semiconductor laser with an external singlemode fiber cavity in near-infrared range in the active mode-locking regime has been presented. The presented approach is oriented to using the joint time-frequency distributions for a pair of solitary pulses, because similar distributions consist of both autonomous and mutual contributions peculiar not to the isolated pulses, but to the combined signals or/and pulse trains. In spite of practical mathematical difficulties that had been met in this case, the algorithm of general characterization has been nevertheless sequentially followed. Within this analysis, the significant part of attention has been paid to rather specific problem of appearing false signals or artefacts caused by mutual cross-terms inherent in the joint time-frequency distributions generally as well as in the Wigner distribution in particular. In a view of improving this obvious disadvantage peculiar to just Wigner time-frequency distribution, two possibilities have been considered to resolve or avoid the problem of appearing the artefacts. One of them is based on involving reasonable modifications into the kernel of Wigner distribution, so that the joint Choi-Williams time-frequency distribution has been included into our approach to the above-mentioned average characterization. The other possibility is related to the procedure of smoothing via constructing an additional 2D-convolution with the needed smoothing function. Both these avenues have been preliminary touched, i.e. open slightly and briefly looked through. Practical application of this technique had been recently discussed and consists in characterizing the parameters of ultra-short optical dissipative solitary pulses in the actively mode-locked semiconductor laser with an external fiber cavity. Together with this, the cavity is represented by a rather long single-mode silica fiber exhibiting square-law dispersion, cubic-law Kerr nonlinearity, and linear optical losses. Our analytical model takes into account principal contributions associated with the externally modulated gain, the linear optical losses as well as with linear and nonlinear phase effects. As usually for just dissipative solitons, regular trains of optical solitary pulses appear due to a pair of balances. The first of
155
7.1. GENERAL CONCLUSIONS them is caused by mutually compensating influences of the pulse broadening second-order anomalous dispersion and the pulse compressing (focusing in time) cubic-law Kerr nonlinearity. While, both the pulse amplifying heterostructure gain and the pulse attenuating linear optical losses in the combined cavity govern the second balance. Here, two contributions have been recognized within these processes. One contribution is associated with a quasi-linear operator describing just a process of the active mode-locking in heterolaser with quasi- linear external cavity. The realized eigen-functions of this quasi-linear operator can be explained in terms of Gauss functions under certain conditions. Then, the cubically nonlinear Ginzburg-Landau operator governs the second contribution. The action of this nonlinear operator has been described using an approximate variational procedure based on the technique of trial functions. This contribution is responsible for appearing dissipative solitary pulses evolving in adiabatic solitons-like regime through the combined cavity. When cavity is not too lengthy in the scale of exhibiting both linear and nonlinear phenomena, the initially introduced non-stationary trial function leads to rather weak variations of parameters inherent in each individual pulse, so that one can expect compensating these variations by the semiconductor laser heterostructure and shaping rather stable trains of ultra-short optical dissipative solitary pulses. In the experimental part, the modified scanning Michelson interferometer has been chosen for obtaining the field-strength auto-correlation functions. In fact, we have presented the key features of a new experimental technique for accurate and reliable measurements of the trainaverage temporal width and the frequency chirp of picosecond optical pulses in high-repetitionrate trains. The InGaAsP/InP-heterolaser, operating at 1320 nm wavelength range, has been used within the experiments. Together with this, we propose the methodology to estimate the average parameters of ultra short optical pulses with asymmetric envelopes, which are arranged in high-repetition trains. We using higher order correlation function, first of all the method based on third harmonic feneration and cascade nonlinear processes were analyzed. We estimate the principal parameters of such interferometer where the triple auto-correlation functions are measured using the effect of the third harmonic generation in two different optical configurations. Conjointly, we examine the principal aspects of implementing the respective experimental arrangements. The theoretical analysis of Photo-EMF effect in three beam interferences configuration has been performed. Finally, we discussed the viability of application of the photo-electro-motive force detectors as higher correlator.
156
7.2. STATEMENTS
7.2
Statements
• Physically the proposed interferometric technique is capable to measurement accurate the train-average pulse width as well as the value and sign of the frequency chirp of picosecond optical pulses with Gaussian shapes in high- repetition-rate trains. This it makes possible creating the corresponding joint Wigner time- frequency distributions not only for a single pulse but also for a pair of them. • Also, the novel approach to describing the initial stage of the active mode-locking in semiconductor laser structures based on analyzing the properties of dispersion relations in terms of stability for small initial perturbations has been studied. Jointly, the schematic arrangement which the process of the active mode-locking is caused by a hybrid nonlinear cavity consisting of this heterolaser and an external rather long single-mode optical fiber exhibiting square-law dispersion, cubic Kerr nonlinearity, and small linear optical losses has been analyzing. • Finally, the principal key of measuring the train-average parameters of picosecond optical pulses being arranged in highfrequency repetition trains has been proposed. In this order the triple auto-correlations, whose Fourier transformations give us the bispectrum of a pulse train was investigated. In this order, the triple auto-correlation can be shaped by a three-beam scanning interferometer with the following one- or two-cascade triple harmonic generation. The efficiencies of these processes depend on the number of cascades and differ in various materials are analyzed.
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