Nonlinear Optical Cavity Dynamics E-Book
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- Herausgeber: Wiley-VCH
- Kategorie: Wissenschaft und neue Technologien
- Sprache: Englisch
By recirculating light in a nonlinear propagation medium, the nonlinear optical cavity allows for countless options of light transformation and manipulation. In passive media, optical bistability and frequency conversion are central figures. In active media, laser light can be generated with versatile underlying dynamics. Emphasizing on ultrafast dynamics, the vital arena for the information technology, the soliton is a common conceptual keyword, thriving into its modern developments with the closely related denominations of dissipative solitons and cavity solitons. Recent technological breakthroughs in optical cavities, from micro-resonators to ultra-long fiber cavities, have entitled the exploration of nonlinear optical dynamics over unprecedented spatial and temporal orders of magnitude. By gathering key contributions by renowned experts, this book aims at bridging the gap between recent research topics with a view to foster cross-fertilization between research areas and stimulating creative optical engineering design.
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Veröffentlichungsjahr: 2015
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Table of Contents
Cover
Related Titles
Title Page
Copyright
List of Contributors
Foreword
Chapter 1: Introduction
References
Chapter 2: Temporal Cavity Solitons in Kerr Media
2.1 Introduction
2.2 Mean-Field Equation of Coherently Driven Passive Kerr Resonators
2.3 Steady-State Solutions of the Mean-Field Equation
2.4 Existence and Characteristics of One-Dimensional Kerr Cavity Solitons
2.5 Original Experimental Observation of Temporal Kerr Cavity Solitons
2.6 Interactions of Temporal CSs
2.7 Breathing Temporal CSs
2.8 Emission of Dispersive Waves by Temporal CSs
2.9 Conclusion
References
Chapter 3: Dynamics and Interaction of Laser Cavity Solitonsin Broad-Area Semiconductor Lasers
3.1 Introduction
3.2 Devices and Setup
3.3 Basic Observations and Dispersive Optical Bistability
3.4 Modelling of LS and Theoretical Expectations in Homogenous System
3.5 Phase and Frequency Locking of Trapped Laser Cavity Solitons
3.6 Dynamics of Single Solitons
3.7 Summary and Outlook
Acknowledgments
References
Chapter 4: Localized States in Semiconductor Microcavities, from Transverse to Longitudinal Structures and Delayed Systems
4.1 Introduction
4.2 Lasing Localized States
4.3 Localized States in Nonlinear Element with Delayed Retroaction
4.4 Conclusion and Outlook
Acknowledgements
References
Chapter 5: Dynamics of Dissipative Solitons in Presence of Inhomogeneities and Drift
5.1 Introduction
5.2 General Theory: Swift–Hohenberg Equation with Inhomogeneities and Drift
5.3 Excitability Regimes
5.4 Fiber Cavities and Microresonators: The Lugiato–Lefever model
5.5 Periodically Pumped Ring Cavities
5.6 Effects of Drift in a Periodically Pumped Ring Cavity
5.7 Summary
Acknowledgments
References
Chapter 6: Dissipative Kerr Solitons in Optical Microresonators
6.1 Introduction to Optical Microresonator Kerr-Frequency Combs
6.2 Resonator Platforms
6.3 Physics of the Kerr-comb Formation Process
6.4 Dissipative Kerr Solitons in Optical Microresonators
6.5 Signatures of Dissipative Kerr Soliton Formation in Crystalline Resonators
6.6 Laser Tuning into the Dissipative Kerr Soliton States
6.7 Simulating Soliton Formation in Microresonators
6.8 Characterization of Temporal Dissipative Solitons in Crystalline Microresonators
6.9 Resonator Mode Structure and Soliton Formation
6.10 Using Dissipative Kerr solitons to Count the Cycles of Light
6.11 Temporal Solitons and Soliton-Induced Cherenkov Radiation in an Photonic Chip
6.12 Summary
References
Chapter 7: Dynamical Regimes in Kerr Optical Frequency Combs: Theory and Experiments
7.1 Introduction
7.2 The System
7.3 The Models
7.4 Dynamical States
7.5 Conclusion
7.6 Acknowledgements
References
Chapter 8: Nonlinear Effects in Microfibers and Microcoil Resonators
8.1 Introduction
8.2 Linear Optical Properties of Optical Microfibers
8.3 Linear Properties of Optical Microcoil Resonators
8.4 Bistability in Nonlinear Optical Microcoil Resonators
8.5 Harmonic Generation in Optical Microfibers and Microloop Resonators
8.6 Conclusions and Outlook
References
Chapter 9: Harmonic Laser Mode-Locking Based on Nonlinear Microresonators
9.1 Introduction
9.2 Modeling
9.3 Experiments
9.4 Conclusions
References
Chapter 10: Collective Dissipative Soliton Dynamics in Passively Mode-Locked Fiber Lasers
10.1 Introduction
10.2 Multistability and Hysteresis Phenomena
10.3 Soliton Crystals
10.4 Toward the Control of Harmonic Mode-Locking by Optical Injection
10.5 Complex Soliton Dynamics
10.6 Summary
Acknowledgments
References
Chapter 11: Exploding Solitons and Rogue Waves in Optical Cavities
11.1 Introduction
11.2 Passively Mode-Locked Laser Model
11.3 The Results of Numerical Simulations
11.4 Probability Density Function
11.5 Conclusions
11.6 Acknowledgments
References
Chapter 12: SRS-Driven Evolution of Dissipative Solitons in Fiber Lasers
12.1 Introduction
12.2 Generation of Highly Chirped Dissipative Solitons in Fiber Laser Cavity
12.3 Scaling of Dissipative Solitons in All-Fiber Configuration
12.4 SRS-Driven Evolution of Dissipative Solitons in Fiber Laser Cavity
12.5 Conclusions and Future Developments
References
Chapter 13: Synchronization in Vectorial Solid-State Lasers
13.1 Introduction
13.2 Self-Locking in Dual-Polarization Lasers
13.3 Dynamics of Solid-State Lasers Submitted to a Frequency-Shifted
13.4 Conclusion
Acknowledgments
References
Chapter 14: Vector Patterns and Dynamics in Fiber Laser Cavities
14.1 Introduction
14.2 Fiber Laser Models
14.3 Experiments of Vector Dynamics
14.4 Summary
Acknowledgments
References
Chapter 15: Cavity Polariton Solitons
15.1 Introduction
15.2 Mathematical Model
15.3 One-Dimensional Bright Cavity Polariton Solitons
15.4 Two-Dimensional Parametric Polariton Solitons
15.5 Two-Dimensional Moving Bright CPSs
15.6 Summary
Acknowledgments
References
Chapter 16: Data Methods and Computational Tools for Characterizing Complex Cavity Dynamics
16.1 Introduction
16.2 Data Methods
16.3 Adaptive, Equation-Free Control Architecture
16.4 Prototypical Example: Self-Tuning Mode-Locked Fiber Lasers
16.5 Broader Applications of Self-Tuning Complex Systems
16.6 Conclusions and Technological Outlook
Acknowledgments
References
Chapter 17: Conclusion and Outlook
References
Index
End User License Agreement
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Guide
Cover
Table of Contents
Foreword
Begin Reading
List of Illustrations
Chapter 2: Temporal Cavity Solitons in Kerr Media
Figure 2.1 Schematic of a CW-driven nonlinear, passive, optical fiber ring cavity.
Figure 2.2 Optical bistability of the homogeneous state. (a) Intracavity power versus driving power for various values of the detuning . (b) Corresponding response but versus , for fixed values of (normalized to for clarity). The circle corresponds to the peak of the resonance at . Dotted parts are unstable.
Figure 2.3 Spontaneous emergence of a periodic pattern through MI from a CW background power level with and (anomalous dispersion).
Figure 2.4 Illustration of how the CS solution results from the coexistence of the CW and patterned solution in different parts of the cavity. Both solutions are calculated for the same parameters, and (anomalous dispersion).
Figure 2.5 Existence and stability charts of CS (red) solutions superimposed on the CW response (black). Dotted parts are unstable. (a) Peak intracavity power as a function of driving power for a fixed detuning . (b) Same as (a) but versus , for a fixed driving power, . The middle insets show temporal intensity profiles of stable (top) and unstable (bottom) CSs that exist for the same parameters (, ) and indicated in the main plots with the cross and the solid circle, respectively.
Figure 2.6 Numerical simulation illustrating individual addressing of CSs. Starting from an empty cavity, the field first converges to the steady-state CW solution. At , a pulse at is superimposed on the background, exciting a single CS. At , two further CSs are similarly excited at and . At , the first CS is erased using a perturbation with a phase shift relative to that used for excitation.
Figure 2.7 Experimental setup used for the first observation of temporal CSs by Leo
et al.
EDFA, Erbium-doped fiber amplifier; BPF, bandpass filter; PD, photodiode (adapted from [32]).
Figure 2.8 Original 2010 observation of an isolated temporal CS in the form of an oscilloscope recording (linear scale) of the light intensity at the cavity output. The delay between subsequent pulses matches the 1.85- cavity roundtrip time and confirmed the presence inside the cavity of a single persistent pulse circulating repeatedly, only sustained by the CW driving beam (adapted from [32]).
Figure 2.9 (a) Experimental (blue) intensity autocorrelation trace of the intracavity pulse observed by Leo
et al.
It is compatible with a 4-ps-long temporal CS, as revealed by its comparison with a numerically simulated trace (red) that corresponds to the temporal intensity profile of the CS shown in the inset. (b) Experimental (blue) and simulated (red) optical spectra. The simulated spectrum corresponds to the CS shown in the inset in (a) (adapted from [32]).
Figure 2.10 The acronym of ULB (Université Libre de Bruxelles) encoded with temporal CSs as a 15-bit data stream. Each letter is represented with five bits by its ordinal position in the alphabet (U = 21, L = 12, B = 2) (adapted from [32]).
Figure 2.11 Color plots made up of successive oscilloscope measurements at the cavity output showing the temporal evolution of two temporal CSs as they interact. (a) For an initial separation of 100 ps (38 soliton widths), the interaction is repulsive until a stable separation of 420 ps is attained. (b) An attractive interaction is observed when the initial separation is increased to 1500 ps (577 soliton widths), eventually leading to the same final separation of 420 ps. (c) For an initial separation of 1800 ps, the interaction is very weakly repulsive (adapted from [34]).
Figure 2.12 (a) Theoretical impulse response of the refractive index acoustic perturbation calculated for values pertinent to the experiment. Subsequent spikes (numbered 1-5) are separated by ns and arise from consecutive reflections of the acoustic wave from the fiber cladding-coating boundary, as schematically illustrated in the inset. (b) Close-up of the first peak of the acoustic response. Ranges of repulsion and attraction of a trailing temporal CS are highlighted. The maximum corresponds to a stable separation (adapted from [34]).
Figure 2.13 (a) Close-up of the first echo of the acoustic impulse response (see spike labeled 2 in Figure 2.12a). (b) Experimental color plot of the trailing cavity soliton in each of the regions highlighted in (a), as in Figure 2.11. The plot consists of the juxtaposition of four different independent measurements (separated by vertical lines) obtained for different initial separations between the two cavity solitons. Red curves are numerical simulations (adapted from [34]).
Figure 2.14 Oscilloscope traces (linear scaling) of temporal CSs (a) below [ and ], (b) near [ and ], and (c) beyond [ and ] the Hopf bifurcation point. In (c), the temporal CS appears to be breathing with a period of about 11 roundtrips. The delay between subsequent pulses in the output sequences shown is equal to the 1.85 cavity roundtrip time (adapted from [33]).
Figure 2.15 Experimental (red) and numerical (blue) spectra of temporal CSs for three different values of average cavity dispersion as shown. In each case, the black dashed line indicates the position of the ZDW while the red dotted line highlights the observed DW wavelength (adapted from [35]; parameters listed therein).
Chapter 3: Dynamics and Interaction of Laser Cavity Solitonsin Broad-Area Semiconductor Lasers
Figure 3.1 (a) Scheme of VCSEL. (b) Microscope image of emission window of a VCSEL with 106 m circular aperture.
Figure 3.2 Experimental setup. VCSEL, vertical-cavity surface-emitting laser; BS, beam splitter; VBG, volume Bragg grating; HWP, half-wave plate; LP, linear polarizer; WB, external writing beam; A, aperture; M, mirror; PD, photo diode; CCD2, CCD camera in near field image plane of VCSEL; CCD1, CCD camera in far field image plane of VCSEL; FPI, Fabry–Perot interferometer; MMF, multimode fibre; FPD, fast photodetector; AMP, RF-amplifier; OSC, single-shot digitizing oscilloscope; RF, electrical spectrum analyzer for RF analysis.
Figure 3.3 Light–current characteristic and near field emission structures from the VCSEL. The dots in the LI-curve denote the points were the images were taken. The current values are also in insets. The images are displayed in a linear gray scale with white denoting high intensity. All images are obtained at the same gain and filter settings of CCD2, but each image is scaled to obtain maximum contrast within the image. Hence the images containing only spontaneous emission are dominated by RF-interference. Submount temperature, 44 C, VBG 2; displayed area, m.
Figure 3.4 LI-curves obtained for different submount temperatures (in legend). The temperature parameter for the particular LI-curves decreases from left (50 C) to right (38 C) via the steps given by the values in the legend. Other parameters: VBG 2.
Figure 3.5 (a) Switch-on (black squares) and switch-off thresholds (red circles) in dependence on submount temperature. The straight lines are linear fits to the data. (b) Illustration of mechanism for soliton-switch-on. The transverse wavenumber of the resonances is plotted against frequency. denotes the frequency of the VBG reflection peak. It is nearly independent of transverse wavenumber for the purpose of these considerations, hence the VBG characteristic is shown as a straight line. represents the longitudinal resonance frequency of the VCSEL for a certain set of parameters. All high-order VCSEL modes are at higher frequencies. For increasing current or increasing submount temperature the resonance red-shifts from to . Alternatively, one can think of the different lines as dispersion curves at different spatial locations in the VCSEL at a certain temperature and current, shifted by disorder. is the final LS frequency.
Figure 3.6 (a,c) Profile of the soliton field from Eq. (3.6) for the parameters listed at the end of the caption and (a) , (c) . Dots show the phase (left vertical axis); solid line shows the amplitude distribution (right vertical axis, semi-logarithmic scale). (b, d) “interaction plane” of two LCS (b) , (d) ). The arrows indicate the direction of motion of the system along any trajectory. The dark-gray shaded region is a dense spiral. Black dots near are foci, while the white dots (along and ) are saddle points. The vertical dash-dotted lines link subplots (a) and (b), and confirm that the saddles have the same spatial period as the phase of an isolated soliton. The shading is explained in the text. In (b), the red line crossing the horizontal axis around indicates the existence of two unstable foci above and below the horizontal axis. The two solitons finally merge into one. In (d), these two foci are stable, see the black and green lines. Parameters: , , , (a, b) (see [72]), (c, d) ((a, b) adapted from [73]).
Figure 3.7 Emission of (a) LS1 alone, (b) LS2 alone, and (c) both detected together. The left panel shows the near field, the right panel the far field (both in an inverse gray scale with black denoting highest intensity), the lower panel the optical spectrum. The lower right panel of (c) shows a cut through the far field distribution orthogonal to the fringes with a fit to Eq. (3.7). Other parameters: C, current mA, VBG 2.
Figure 3.8 (a) Fringe visibility (black) and fringe phase (blue curves) as a function of the tilt angle that changes the difference between the feedback phases for two LS, m apart. This difference is converted to a frequency scale by multiplying it by the free spectral range of the external cavity thus providing the change of the relative detuning between the two LCS in the external cavity. The zero of this detuning scale is arbitrary. The solid and dashed blue curves are obtained for scanning the tilt back and forth. The sigmoid red curve in the central, locked region is a superimposed Adler curve, Eq. (3.8), scaled and shifted appropriately. (b) Evolution of frequencies, the frequency distribution is obtained from the optical spectra recorded by the FPI. (c) Cut through far field intensity distribution orthogonal to fringe orientation. Other parameters: temperature C, current mA, VBG 1 (adapted from [73, 80]).
Figure 3.9 Locked phase differences of pinned LCS for different frequency detunings (controlled by the potential depths and ) from integration of Eq. (3.5) (dots, LCS separation of 5.3 soliton widths) and the Eqs. (3.1–3.3) (triangles, LCS separation of 4 soliton widths). The solid line refers to the Adler equation (3.8) (adapted from [73, 80]).
Figure 3.10 Far field images (a, d, g, and j) averaged over 2 s, optical spectra (b, e, h, and k) for a time window of 5 s and evolution of the phase difference (c, f, i, and l) for 0 (a–c), 0.99 (d–f), 2.0 (g–i), and 7.5 (j–l). In (b) and (e) the spectral peaks (dashed and solid lines) of the LS overlap. The simulations have been done using Eqs. (3.1)–(3.3) with LCS of initial separation of 4 soliton widths and an initial phase difference of (c)–(f), (i), and 0 (l).
Figure 3.11 (a) Minimum peak pulse power required to switch on a LCS for varying pulse lengths. Insets: time trace of switch-on events for pulse length of (b) 15 ns (AOM limit), (c) 100 ns, and (d) 300 ns pulse. The red line corresponds to the WB pulse and the black line shows the response of the LCS. The LCS always switches to the same power. (e) Minimum energy required to switch a CS on for varying pulse lengths. The lower linear fit (red) is for all pulses shorter than and including 100 ns, the upper linear fit (blue) is for all pulses longer than and including 100 ns. Cavity round trip time 0.606 ns, VBG 1, bias current 3 mA below the spontaneous switch-on threshold. Data taken with APD ((a, c, d) adapted from [87], (e) Figure 4 of [87], with kind permission from Springer Science and Business Media).
Figure 3.12 (a, b) Switch-on transient of an LS initiated by a 50 ns pulse of 4.8 times minimum threshold power monitored with an AC-coupled fast pin-photo detector. Cavity round-trip time: 1.05 ns, VBG 1. (c, d) Numerical simulation of a switch-on transient of a CS initiated by a 20 ns pulse of 2.3 times minimum threshold power. (a, c) Initial transient regime with strong pulsing. (b, d) Time-frequency spectrogram of intensity time series. Cavity round-trip time: 0.41 ns. ((b) adapted from [87], (c,d) adapted from [88]).
Figure 3.13 Examples for possible dynamical states characterized by time averaged optical spectra (left column) and RF-spectra (right column) for three different LCS and currents. Parameters: VBG 1, cavity round-trip time 1.05 ns.
Figure 3.14 (a) Snapshot of temporal dynamics. (b) Time–frequency spectrogram of the intensity time series (note the change of scale compared to (a)!). (c) Time-averaged optical spectrum. (d) Time-averaged RF spectrum. Parameters: VBG 1, cavity round-trip time 1.05 ns.
Figure 3.15 Permutation entropy versus delay for pattern length of time series in Figure 3.14.
Chapter 4: Localized States in Semiconductor Microcavities, from Transverse to Longitudinal Structures and Delayed Systems
Figure 4.1 Experimental setup: two semiconductor lasers ( and ) are imaged onto each other. A reflection beam splitter is inserted in the center of the cavity to extract two output beams from the system. Two output beams are directed on CCD cameras and to a detector monitoring a small portion of the transverse section of . A narrow (<m) tunable beam is used as an external optical perturbation for LS addressing (from [33]).
Figure 4.2 (a) Instantaneous field intensity of a chaotic laser LS observed in the model of [37]: (i) view of the full spatial extension of the system; (ii–iv) snapshots of the spatial region containing the chaotic LS. (b) Temporal evolution of a cut along the horizontal axis across two chaotic LS. Each LS is in a chaotic regime (from [76]).
Figure 4.3 Panels (a–c) depict the bifurcation scenario as a function of the gain for different values of the delay: (a) , (b) , and (c) . The fundamental PML solution is plotted with the color line and its stability domain is indicated with a thicker line. The CW solution is plotted with the thin black line and its stability domain is not indicated. Panel (d) shows, for , the folding of several PML solutions having a different number of equally separated pulses per round-trip, while the folding point is represented by a circle (from [88]).
Figure 4.4 Evolution over round-trips of a bit pattern written optically by injecting ps light pulses in the cavity (a) and detail over a single period (b). Parameters as in Figure 4.3 with . The bit sequence is 10101010010001001110011000111001100 (from [88]).
Figure 4.5 (a)–(d) Coexisting time output traces ( mA). (e) Experimentally obtained bifurcation diagram for the number of pulses per round-trip. The stability of each solution is indicated by the solid horizontal lines (from [88]).
Figure 4.6 Several forms of localized states may be formed within a delayed retroaction loop applied to a nonlinear element that is either bistable or excitable, in the absence of any actual spatial degree of freedom.
Figure 4.7 Laser intensity through the polarizer (dark green is high intensity). In a spatiotemporal representation, the horizontal scale is a fast time scale related to the delay and the vertical scale is discrete time in units of delay (see [53, 96]). In that comoving reference frame, fronts move apart at constant speed until “spatial” forcing is applied, which pins the fronts (from [96]).
Figure 4.8 Many localized states can be formed in a bistable system with delayed feedback. (a) Evolution from the initial condition (a tiny spatial region in the “green” state) to a stable localized state. Inset: many localized states enclosing each a single maximum can coexist. (b, c) Different kinds of localized states can coexist, the size of each of them is a multiple of the period of the forcing in pseudospace (from [96]).
Figure 4.9 The pinning region (in which the front velocity is zero) is delimited by saddle-node bifurcations leading to oscillatory velocity of the front around a nonzero value. When the asymmetry between the states is increased the
left
front unpins first, while the right front is still locked to the forcing (a). Upon further increase, the
right
front finally also unpins (from [96]).
Figure 4.10 Response to external perturbations of an injection-locked semiconductor laser close to the unlocking transition in the excitable regime. For any perturbation beyond some threshold (here about 60 phase jumps), all responses are identical (insets). Each phase rotation translates into a small intensity pulse (10–20% of the continuous level) due to interference with the forcing beam (from [111]).
Figure 4.11 An injection locked semiconductor laser close to the unlocking transition with the addition of delayed feedback can produce periodic emission of relative phase rotations. The use of the delay term as a spatial-like variable suggests the use of the available “space” to store information in the configuration of mutually independent phase bits. (a, b) A second phase bit has been nucleated by a phase perturbation, without impacting the pre-existing phase bit. (a, c) Spatiotemporal representation; (b, d) corresponding time traces (from [115]).
Figure 4.12 Analysis of model (4.1). Temporal traces (a,c,e,g) for the output intensity and phase and histogram of the Floquet multipliers (b,d,f,h) in the cases of 1,2,3, and 6 -bits. One notices that the number of neutral modes located in the vicinity of increases linearly with the number of -bits. After [115].
Figure 4.13 A multimode ring laser with coherent forcing can host nondispersive wave packets consisting of phase rotations embedded in a uniformly locked domain (from [101]).
Chapter 5: Dynamics of Dissipative Solitons in Presence of Inhomogeneities and Drift
Figure 5.1 (a) Bifurcation diagram showing the maximum of the steady state as a function of for . (b) Bifurcation diagram when the reflection symmetry is weakly broken, for . (c) Zoom of (b) showing the reconnection of the branches. (d) Main solutions corresponding to the labels in (a).
Figure 5.2 Bifurcation diagram as Figure 5.1 for (a) , (b) , and (c) . In (c), the crosses indicate the maximum and minimum values of the oscillatory DS at a given spatial location.
Figure 5.3 Train of solitons for and , corresponding to the oscillatory region in Figure 5.2b.
Figure 5.4 Excitable excursions of DSs. (a) A type-I excitable excursion for system parameters in region IV close to the SNIC bifurcation, and (see Figure 5.2b). (b) A type-II excitable excursion for system parameters in region III close to , and (see Figure 5.2c). The parameters of the perturbation were and .
Figure 5.5 Scaling of the period for type II (a) at , and type I (b) at excitability.
Figure 5.6 Excitable excursion of the fundamental solution in region I, close to the FC ().
Figure 5.7 (a) A ring fiber cavity pumped by a continuous wave (CW) of frequency . (b) The same cavity but now pumped synchronously by pulses with a frequency . Here, and are the reflection and transmission coefficients of the beam splitter, respectively. is the length of the fiber and is the round-trip time of the cavity.
Figure 5.8 Bifurcation diagram of the different pinned states in function of . The solid (dashed) lines represent the energy of the stable (unstable) states.
Figure 5.9 Bifurcation diagrams as in Figure 5.8 for (a) , (b) , and (c) .
Figure 5.10 Train of solitons taking over the whole -domain for and .
Figure 5.11 Evolution of the CS after a suitable perturbation of the steady state in the LLE model for a ring cavity that brings the system momentarily into region II. For panels on (a) an initial perturbation with is applied over a time to the fundamental solution and (region I). Panels on (b) correspond to the application of a perturbation over to the pinned CS for parameter values and (region III).
Figure 5.12 Spatiotemporal evolution after a perturbation of the fundamental steady state (a) and the pinned CS (b) for the same parameter values as in Figure 5.11.
Chapter 6: Dissipative Kerr Solitons in Optical Microresonators
Figure 6.1 Time- and frequency-domain picture of mode-locked laser-based frequency combs. A periodic train of pulses with a pulse repetition rate (a) corresponds to a comb spectrum of equidistant lines in the frequency domain (b). The line spacing is given by . The offset of the frequency-comb spectrum relates to the carrier–envelope phase shift between two consecutive pulses via . The two parameters and fully define all comb frequencies .
Figure 6.2 Microresonator platforms. (a) Diamond-turned magnesium fluoride resonator containing two protrusions that confine high- optical whispering-gallery modes. (b) Manually shaped and polished resonator. (c) Polishing of a diamond-turned preform. (d) Scanning electron micrograph of a silicon nitride microresonator before applying the fused silica cladding.
Figure 6.3 Universal Kerr-comb formation processes. (a) Formation of primary sidebands. (b) Formation of subcombs. (c) Overlap between inconsistent subcombs can lead to multiple lines per cavity resonance and explains noise phenomena in Kerr-combs.
Figure 6.4 Kerr-comb formation and noise. (a) Formation of the optical spectrum as the laser is tuned into resonance and the intracavity power increases. (b) Comb-line spacing measured as the radio frequency (RF) beat note between neighboring comb lines. Multiple and broad beat notes indicate multiple and inconsistent line spacings present in the comb spectrum.
Figure 6.5 Subcomb synchronization in an optical microresonator achieved by changing the pump laser wavelength.
Figure 6.6 Crystalline resonator for temporal dissipative Kerr soliton generation. (a) Crystalline resonator and simulated mode profile. (b) Measured anomalous group velocity dispersion for the resonator shown in panel (a). The dispersion of a resonator can be quantified in terms of the deviation of its resonance frequencies from an equidistant frequency grid , where is the FSR at the pump wavelength. An anomalous group velocity dispersion corresponds to a parabolic curve as shown in (b).
Figure 6.7 Signatures of soliton formation. (a) A staircase-like step structure in the pump laser transmission indicates the formation of several DKS in the microresonator. High intensity noise is apparent in the transmission signal before soliton formation. (b) The formation of the solitons is associated with a transition to low noise. This is evidenced here by the transition of a broad RF beat note (cf. Figure 6.4) to a single narrow-band signal. CF, center frequency.
Figure 6.8 Stability of soliton states: considering only the Kerr-nonlinear resonance shift the intracavity power can be described by bistability curves where the upper branch solution corresponds to high and the lower branch solution to low intracavity power. When tuning into the resonance with decreasing optical frequency (increasing wavelength) the intracavity power follows the upper branch of the Kerr-bistability curve. After the transition to a soliton state, the major fraction of the pump light is described by the lower branch of the bistability curve. The fraction of the pump light that propagates with the soliton inside the microresonator experiences a larger phase shift and is effectively blue detuned on the upper branch of another bistability curve. The extend of the “soliton bistability curve” toward longer wavelength depends on the peak power of the solitons (i.e., the maximal nonlinear phase shift), the relative height of the curve depends on the relative fraction of the pump light that is affected by the high-peak-power soliton. The overall intracavity power can be inferred by adding the bistability curves resulting in the black curve.
Figure 6.9 Laser detuning and soliton formation. (a) The transmitted power and a series of steps associated with multiple temporal cavity solitons. The background shading indicates the laser detuning that is derived from a Pound–Drever-Hall (PDH) error signal shown in (b). The soliton formation coincides with the transition to red detuning (indicated by a sign change in the PDH error signal). Note that the additional Kerr-frequency shift due to the high-peak-power soliton does not significantly impact the PDH error signal as its relative contribution is small.
Figure 6.10 Generation of stable solitons via laser tuning. (a) Illustration of laser transmission, scan control voltage (corresponding to pump wavelength), and resonator temperature for slow, ideal, and fast laser tuning speed. In the ideal case, the resonator temperature does not change once the desired soliton state is reached and the laser scan can be stopped for stable operation. (b) Regular scan of the pump laser over a resonance showing a “soliton step.” (c) The laser tuning method allows to stop the laser scan once the soliton is generated. Once generated in this manner, the soliton circulates stably inside the microresonator.
Figure 6.11 Numerical simulations of dissipative Kerr soliton formation in a crystalline microresonator. (a) Intracavity power (corresponding to the transmission signal in Figure 6.7a when mirrored horizontally) during a simulated laser scan (101 simulated modes) over a resonance in a microresonator. The step features are clearly visible. The light gray lines trace out all possible states of the system during the scan. The unshaded area corresponds to the area where DKSs can exist, the light shaded area allows for breather solitons with a time variable, oscillating envelope; no solitons can exist in the dark shaded area. (b) Optical spectrum and intracavity intensity for different detuning values (1–5) in the laser scan.
Figure 6.12 Characteristics of a Dissipative Kerr soliton in a crystalline resonator. (a) Optical spectrum showing the characteristic sech-squared envelope. (b) Magnified part of the spectrum, resolving the individual comb lines of which it is composed. (c) The low-noise radio frequency beat note at 14.09 GHz corresponds to the comb line spacing and the soliton pulse repetition rate. (d) SHG-FROG trace revealing femtosecond pulse duration. The pulse to pulse separation of 71 ps corresponds to the pulse repetition rate of 14.09 GHz and the pulse duration of 194 fs can be inferred (in agreement with the spectral width). RBW, resolution bandwidth; CF, center frequency).
Figure 6.13 Spectral and temporal characterization of multisoliton states in a crystalline resonator. (a) Optical spectra of a single soliton (top) and two multisoliton states (middle, bottom). (b) SHG-FROG traces corresponding to (a).
Figure 6.14 Transmission spectrum of an crystalline microresonator with an FSR of approximately 14.09 . The upward transmission spikes (values ) result from cavity-ringdown. Frequency-comb-assisted diode laser spectroscopy ensures precise calibration of the laser detuning (megahertz level).
Figure 6.15 Mode structure of an resonator with an FSR of 14.09 GHz. (a) Two-dimensional Echelle-type representation where for all measured mode families the deviation of the resonance frequency from an equidistant -spaced frequency grid ( is an approximate average FSR of all modes) is shown (plus some offset) in function of the mode number . Dots forming a continuous line represent a particular mode family. Different free spectral ranges correspond to different slopes of the lines, whereas dispersion and variation of the FSR show as curvature and bending of the lines. The dispersion can be strongly affected by mode crossings. (b) Two mode families have been extracted from the data set shown in (a). The upper one is characterized by an anomalous dispersion, the lower one exhibits two avoided mode crossings that induce deviations from the anomalous dispersion.
Figure 6.16 Mode structure and spectral envelope. (a) Typical sech-squared envelope for the case of weak (a) and strong anomalous dispersion (b). (c) Higher-order dispersion (such as non zero ) leads to an asymmetric spectrum, dispersive wave emission (cf. Section 6.11) and a shift of the spectral soliton peak intensity away from the pump laser (soliton recoil). (d) Avoided mode crossings manifest themselves in a characteristic spectrally local variation of the spectrum.
Figure 6.17 Illustration of – self-referencing. If the frequency-comb spectrum spans more than two-thirds of an octave, the second and third harmonics of blue and red wings of the spectrum can be overlapped. The difference frequency beat note between the two harmonics yields the comb's offset frequency .
Figure 6.18 Counting the cycles of light by self-referencing a soliton-based frequency comb. (a) Soliton spectrum and nonlinearly broadened spectrum. The broadened spectrum spans more than two-thirds of an octave and allows for self-referencing. (b) Magnified part of the broadened spectrum. The line spacing is the same as for the soliton spectrum (14.09 GHz). (c) Pulse repetition rate beat note measured via direct photodetection of the comb spectrum. (d) Offset frequency signal measured via a modified – self-referencing technique, where two transfer lasers are used for signal enhancement (see [89] for details). RBW, resolution bandwidth; CF, center frequency.
Figure 6.19 Single temporal soliton generation in an microresonator. The upper panel shows the single soliton spectrum that covers a spectral bandwidth of 2/3 of an octave. The black line in the background is the spectral envelope obtained by the simulation described in Section 6.7. The lower panel shows the measured dispersion (dots) free of avoided mode crossing and the dispersion over the full-spectral span as obtained through finite element simulation. The dispersive wave forms at the wavelength of approximately 1.9 micron where the phase-matching condition is fulfilled. The zero dispersion point (ZDP) is marked by a vertical, dashed line. The inset in the upper panel shows a heterodyne beat note between the dispersive wave and an external laser. Its narrow width proves the coherence of the dispersive wave.
Figure 6.20 Generation of multisoliton states and soliton-induced Cherenkov radiation in an microresonator. The characteristic spectral modulations occur due to the interference of solitons at different positions in the microresonators as indicated in the insets.
Chapter 7: Dynamical Regimes in Kerr Optical Frequency Combs: Theory and Experiments
Figure 7.1 Typical experimental setup for the generation of Kerr frequency combs in WGM resonators. The light of a CW-laser is amplified and coupled to the high- resonator via the evanescent field of a tapered fiber. The output of the same fiber is used to monitor the optical spectrum or to further process the generated comb.
Figure 7.2 Examples of WGM resonators manufactured at the FEMTO-ST Institute. (a) Photograph of a crystalline whispering-gallery disk coupled with a tapered fiber. (b) Scanning electron microscope image of a planar, integrated resonator, and tapered coupling waveguide.
Figure 7.3 Stability map of the LLE in the (a) anomalous () and (b) normal () dispersion regimes. The dashed line in both cases corresponds to the threshold value . Between the two thick lines, the Lugiato–Lefever equation has three steady-state solutions, two of which are stable. In the anomalous case, Turing patterns can be found above threshold when the detuning is smaller than , and the lines indicate the iso-values of the roll pattern. Numerical simulations show that bright solitons (breathers) can be excited in the light gray (dark gray) area. Since it can be generated for pump powers below the bifurcation line, the soliton is a subcritical structure. In the normal dispersion regime, dark solitons (breathers) can also be generated in the light gray (dark gray) area. The critical value separates supercritical (or soft) from subcritical (or hard) excitation of the comb [24]. The bifurcation lines , , and correspond to a structural change of the Jacobian eigenvalues, as explained in Table 7.1.
Figure 7.4 (a) Stationary spatiotemporal distribution of the optical power and (b) optical spectrum of a supercritical Turing patterns (or “primary comb”) generated from small amplitude noise. The parameters used in this simulation are , , and .
Figure 7.5 (a, c, e) Three experimental primary combs corresponding to Turing patterns with different mode spacings. (b, d, f) Corresponding numerical simulations. The parameters used are (b) , , and ; (d) , , and ; (f) , , and .
Figure 7.6 (a, c, e) Time evolution of the relative phases of the first three modes of a primary comb. After a delay increasing with the mode number, the relative phases reach a constant value, the Kerr comb becoming phase-locked. (b) Evolution of the relative phase of the first excited modes for three different initial conditions. The asymptotic value depends on the initial condition. Evidence of triplet phase locking (d) , and global phase locking (f) .
Figure 7.7 (a) Spatiotemporal distribution of the intracavity field and (b) corresponding optical spectrum for a bright soliton in the anomalous regime of dispersion. The parameters used are , , , and the initial intracavity field envelope is .
Figure 7.8 (a) Intracavity intensity profiles of a bright breather in the anomalous regime of dispersion, at two different evolution times. (b) Spatiotemporal distribution of the field intensity of the bright breather. The parameters used are , , and , and the initial intracavity field envelope is .
Figure 7.9 Spatiotemporal distribution of the (a,c) intracavity field and (b,d) corresponding optical spectrum for two dark solitons in the normal regime of dispersion. The parameters for the first soliton are , , , and the initial intracavity field envelope is . For the second soliton, the parameters are unchanged except for and .
Figure 7.10 (a) Experimental frequency comb obtained in a resonator with -factor around and FSR 6 GHz. (b) Numerical simulation of the spectrum of a dark soliton.
Figure 7.11 (a) Intensity profiles of a dark breather at two different evolution times where it reaches its maximum and minimum amplitudes. (b) Spatiotemporal distribution of the optical intensity of the dark breather, in the normal regime of dispersion. , , , and the initial intracavity field envelope is .
Figure 7.12 Anomalous dispersion bifurcation diagrams for two different detunings (a) and (b) , when the pump power is increased. For , the first bifurcation leads to the formation of Turing patterns with an integer number of rolls (22). For higher pump powers, this structure evolves to a different number of rolls, 23 in this case. For higher gains, the amplitude of each roll oscillates, and finally, a chaotic regime is reached. In the case , the initial condition is a Gaussian pulse leading to the formation of a soliton just above the bifurcation (dashed gray line). This soliton is stable at excitations lower than the bifurcation limit, revealing its subcritical nature. With higher pump power, the soliton evolves into a bright breather, and ultimately, the system becomes chaotic in the vicinity of the bifurcation.
Figure 7.13 (first column) Spatial distribution of the optical intensity when the highest wave occurs for different pump powers: (a) , (c) 10, and (e) 20. The second column represents the number of events recorded for each wave height bin during the simulation time. (g) Spatiotemporal distribution of the field when the highest wave occurs at .
Figure 7.14 Comparison between experiments and simulations for several regimes of Kerr combs. In the experiments (first column), the pump power is kept constant while the laser frequency is progressively decreased ( is increased). For the numerical simulations (second column), the excitation is kept constant at , the dispersion is and the detuning is progressively increased. The excellent agreement between the experimental results and numerical simulations over different regimes proves the relevance of the LLE model to describe the generation of Kerr frequency combs.
Chapter 8: Nonlinear Effects in Microfibers and Microcoil Resonators
Figure 8.1 Uniform microcoil formed by wrapping microfiber around a rod. Inset shows the local
x–y
fiber axes (from: [9]).
Figure 8.2 Mode dispersion curves showing the effective index () of various modes with changing OMN diameter for a wavelength of µm.
Figure 8.3 Mode field intensity profiles of the fundamental mode for a silica OMN with a core diameter of (a) 5 µm and (b) 1 µm. µm.
Figure 8.4 (a) Effective nonlinearity of OMNs and (b) effective modal area versus microfiber diameter for the fundamental mode at µm and µm.
Figure 8.5 (a) Schematic of a microcoil resonator taken from [8]. (b) Transmission spectrum for a lossy OMR with eight coils. The loss is while the coupling strength is . The green line shows the expected transmission for a straight microfiber with the same length and loss (from [28]).
Figure 8.6 Nonlinear response of a three turn microcoil resonator. (a) Full solution (both stable and unstable branches) as a function of the input power. (b) Hysteresis curves for the resonator for a range of wavelengths (from [28]).
Figure 8.7 (a) Linear transmission spectrum for a broken OMR (red line) and unbroken OMR (green line). (b) The nonlinear response for a broken OMR for both increasing (green line) and decreasing (red line) input power (from [32]).
Figure 8.8 (a) The microcoil geometry and (b) a cross section of three adjacent turns and the local axes when the birefringent fiber is twisted, shown here for a linear rotation , which permits cross-polarization coupling (from [33]).
Figure 8.9 (a) Linear transmission spectrum for a coil with a weak birefringence of . (b) The energy stored in the OMR against the input power, showing the double nonlinear hysteresis loops for a pump wavelength red-detuned by pm from the resonance. (c) The nonlinear transmission characteristic (from [33]).
Figure 8.10 Effective index curves for the fundamental mode at µm (red dashed line) and third-harmonic hybrid modes for azimuthal order (solid blue and dotted green lines). Phase-matching points occur where the pump and harmonic curves intersect.
Figure 8.11 The effect of detuning on THG conversion efficiency in an OMN. The insets detail the evolution of efficiency along the taper for different . Here, µm and kW.
Figure 8.12 THG efficiency dependence on the random surface fluctuations, for four different surface tension values . Here, C, and the RMS radius is 383.2 nm, with µm (adapted from [38]).
Figure 8.13 (a) Schematic of the OMN loop resonator and (b) the experimental implementation used to generate and detect the third harmonic (from [39]).
Figure 8.14 (a) Simulated output-pump power and (b) third-harmonic power from a silica loop resonator against pump detuning from resonance (solid lines). The dashed line indicates the linear resonance spectrum, and dotted lines represent the output from an equivalent nonlinear length of straight OMN. W (from [39]).
Figure 8.15 Theoretical enhancement of the THG efficiency from a loop resonator against pump detuning from resonant wavelength and proximity to critical coupling (adapted from [41]).
Figure 8.16 Experimental demonstration of enhanced third-harmonic generation in a loop resonator. The different colors show the effect of tightening the loop radius and thus increasing the coupling (from [39]).
Figure 8.17 Experimentally measured second-harmonic spectra from a loop resonator recorded as its diameter was tightened from 4 mm (Loop A) to 1 mm (Loop B), showing a resonant efficiency enhancement compared with the original straight OMN (from [18]).
Chapter 9: Harmonic Laser Mode-Locking Based on Nonlinear Microresonators
Figure 9.1 Sketch of the FD-FWM fiber laser: a nonlinear microcavity, here a microring resonator, is inserted in an active fiber loop.
Figure 9.2 (a) Sketch of the propagation geometry used in the simulations. The field
a
(
z
R
,
t
) [
f
(
z
F
,
t
)] propagates in the microresonator (fiber) over a length
L
R
[
L
F
]. The fields and
a
(
z
R
,
t
) and
f
(
z
F
,
t
) are coupled at the microresonator ports. (b) Summary of the numerical parameters used in the simulations, in the frequency space
ω
. The gain bandwidth (blue line) is controlled by the constant Ω. The position of the microcavity modes set (red continuous lines) is controlled by the parameter
ϕ
RC
. The position of the main cavity modes (inset, yellow lines) with respect to the microcavity resonances (red dashed curves) is regulated via the parameter
ϕ
MC
. The bandwidth of the microcavity resonances is controlled by adjusting the coefficients
T
and
R
.
Figure 9.3 Simulations for an even symmetry of the microcavity modes with respect to the gain band.
ϕ
RC
=
π
/2 in all cases;
ϕ
MC
=
π
/8 for (a, b, e, f, i, j); and
ϕ
MC
= 7
π
/8 for (c, d, g, h, k, l). The lasing is investigated for different ratios of the ring lines FWHM over the main cavity FSR, FWHM
RC
/FSR
MC
= 0.25, 1.25, and 2.5 for the top, central, and bottom panels, respectively. The top–bottom curve for each panel is for growing cavity saturation powers (
P
0
, 2
P
0
, 3
P
0
, and 4
P
0
, respectively). The main cavity has FSR
RC
= 12 GHz. Here, (a, c, e, g, i, k) report the power spectral densities and (b, d, f, h, j, l) report the temporal evolution of the intensity.
Figure 9.4 Simulation for an even symmetry of the microcavity modes with respect to the gain band. The parameter
ϕ
RC
is
π
/2 in all cases, while
ϕ
MC
varies as reported in the
x
-axis. The lasing is investigated for different ratios of the ring lines FWHM over the main cavity FSR, FWHM
RC
/FSF
MC
= 0.25, 1.25, and 2.5, top to down panels, respectively. Increasingly darker gray curves are for growing cavity saturation powers (
P
0
, 2
P
0
, 3
P
0
, and 4
P
0
, respectively). The main cavity has FSR = 12 GHz. The RF spectrum bandwidth is reported on the left side
y
-axis (a, c, e), the optical spectrum bandwidth is reported on the right side
y
-axis (b, d, f).
Figure 9.5 (a) Schematic of the central component – a monolithically integrated four-port high-
Q
(
Q
= 1.2 million) microring resonator (fiber pigtails not shown). (b) SEM picture of the ring cross-section before depositing the upper cladding of SiO
2
. The waveguide core is made of high index (1.7) doped silica glass.
Figure 9.6 High-repetition-rate laser: sketch of the experimental setup.
Figure 9.7 Experimental optical spectra and autocorrelation traces of the laser output, for different FSRs of the main cavity. (a,b) Laser emission for increasing (top to bottom) pump powers, achieved through the use of a long EYDFA, for 5.5, 28, 40, and 68 mW average powers at the ring input, respectively. In (b), the autocorrelation traces calculated starting from the experimental optical spectra for a fully stable-coherent and transform-limited system are also shown in green. These profiles are calculated considering each line of the experimental optical spectra perfectly monochromatic and in-phase with the others, thus yielding to a soliton output pulse with a width (FWHM) of 730 fs (duty cycle
ρ
= 0.15) for the highest excitation condition. The measured autocorrelation shows a considerable higher background (the peak-to-background ratio is 2.5 : 1 for the 68 mW case) than the expected autocorrelation (50 : 1). (c,d) Unstable oscillation condition for a main cavity FSR = 6 MHz. The average power in the ring was 68 mW (c) temporal modulation of the output intensity measured with a slow photodetector and (d) RF noise.
Figure 9.8 Unstable regime for the short-length cavity: optical (a,b) and radio-frequency (c,d) characterization of the laser output: (a) experimental autocorrelation trace (black) and (b) experimental optical spectrum. The autocorrelation trace for a fully coherent transform-limited system calculated from the spectrum in (b) is shown (dashed line) in (a). RF signal of the laser output in time (c) and spectrum (power spectral density (PSD)) (d). The RF signal also shows a pulsation due to the beating of the MC modes, as visible in the spectral components at 65 MHz in (d).
Figure 9.9 Experimental spectra and autocorrelation traces of the filter-driven mode-locked laser. (a,b) Same as Figure 9.7a and b for short-cavity laser emission. The average powers in the ring were, respectively, 7, 11.4, 15.2, and 15.4 mW (top–bottom).
Figure 9.10 Stable regime for the short-length cavity: optical (a, b) and radio frequency (c, d) characterization of the laser output: (a) experimental autocorrelation trace (black) and (b) experimental optical spectrum. The autocorrelation trace for a fully coherent transform-limited system calculated from the spectrum in (b) is shown in dashed in (a). RF signal of the laser output in time (c) and spectrum (d).
Figure 9.11 Dual-line regime for the short-length cavity: optical (a, b) and radio frequency (c, d) characterization of the laser output: (a) experimental autocorrelation trace (black) and (b) experimental optical spectrum. The autocorrelation trace for a fully coherent transform-limited system calculated from the spectrum in (b) is shown (dashed line) in (a). RF signal of the laser output in time (c) (inset in (e)) and spectrum (d).
Figure 9.12 Dual-line regime for the short-length cavity: (a) experimental autocorrelation trace (black) and (a) RF signal output, over shorter time scale than Figure 9.11c. (b) RF spectrum showing the first and second harmonic at the main cavity FSR = 65 MHz. (c) RF spectrum around the 65 MHz oscillation.
Chapter 10: Collective Dissipative Soliton Dynamics in Passively Mode-Locked Fiber Lasers
Figure 10.1 Experimental high-power fiber laser setup. PC, polarization controller; PBS, polarization beam-splitter; DSF, dispersion-shifted fiber; DCF, double-clad fiber; VSP, V-groove side pumping; SMF, standard single-mode fiber (from [67]).
Figure 10.2 Multihysteresis of the lasing regime and number of mode-locked pulses: (a) Experimental evolution of the number of pulses versus the pump power, in the normal dispersion regime with 0.0572 (from [68]). (b) Numerical observation of the number of pulses in steady-state operation versus the pumping level in the normal dispersion regime (from [68]). (c) Dependence of the intracavity energy on pumping . Digits indicate the number of pulses in the cavity for the corresponding branch.
Figure 10.3 Experimental characterization of a soliton crystal: (a) direct laser output recording with a 12-GHz oscilloscope, (b) optical autocorrelation trace, (c) optical spectrum, and (d) spectral fringes of high contrast (from [42]).
Figure 10.4 (a) Numerical simulation of the soliton crystal buildup: (a) optical intensity, and (b) optical spectrum, versus the round-trip number.
Figure 10.5 Schematics of the gain evolution during a cavity round-trip. Each of the pulses takes off an amount of gain , and after the pulse train, pumping restores the amount (from [35]).
Figure 10.6 Harmonic soliton crystal mode-locking, obtained after dislocation of a long soliton crystal at a pump power of 25 W. Each square pulse of the inset is in fact a minicrystal of pulses (from [61]).
Figure 10.7 Initial soliton distribution (a) temporal oscilloscope recording, (b) optical autocorrelation trace. After applying the external injection, an HML regime is obtained: (c) temporal oscilloscope recording, and (d) RF spectrum (from [53]).
Figure 10.8 Snapshot of the temporal soliton distribution in (a) a soliton gas and (b) a soliton liquid (from [59]). These distributions change from one round-trip to the next.
Figure 10.9 Soliton polycrystal. (a) Temporal distribution, (b) optical spectrum, and (c) optical autocorrelation trace (from [59]).
Figure 10.10 Soliton rain dynamics: (a) fiber laser setup. EDF, erbium-doped fiber; WDM, wavelength division multiplexer; ISO, polarization insensitive optical isolator; PBS, polarization beam splitter; PC, polarization controller; OC, output coupler; DCF, dispersion–compensation fiber. (b) Typical soliton rain optical spectrum. (c) Oscilloscope stroboscopic recording of the laser output intensity. The three “soliton rain” components, namely the noisy background, drifting solitons, and the large peak representing the liquid soliton phase, clearly appear (from [58]).
Figure 10.11 (a–d) Successive snapshots, separated by 40 ms, of the temporal close-up view of the drifting solitons impinging on the liquid soliton phase (from [97]).
Figure 10.12 Numerical simulation of a noise-like pulse behavior in a fiber ring laser. The intracavity optical intensity is plotted, in dimensionless units, versus the cavity time , for successive round-trip numbers . The chaotic bunch of pulses, through nonlinear collisions, produces transient pulses of extreme intensity, akin to optical rogue waves (from [106]).
Figure 10.13 (a) Example of the stroboscopic recording of a rogue wave event at the laser output, with a 20-GHz electronic bandwidth. (b) Probability distribution function of the output optical intensity maxima, recorded as voltage amplitudes (in millivolt) by the 20-GHz oscilloscope (from [109]).
Figure 10.14 Characterization of a noise-like pulse regime obtained with the setup of Figure 10.10a, in the anomalous dispersion regime under 700-mW pumping. (a) Averaged (i.e., multishot) optical autocorrelation trace, and (b) series of successive single-shot optical spectra obtained with the dispersive Fourier-transform measurement technique, in color scale. The statistical distribution of the spectral maxima, recorded in millivolt by a 6-GHz oscilloscope, is displayed (c), revealing a heavy-tailed distribution that is a feature of rogue wave dynamics (from [113]).
Chapter 11: Exploding Solitons and Rogue Waves in Optical Cavities
Figure 11.1 Model of the mode-locked laser used in the numerical simulations.
Figure 11.2 The recorded (top) peak power and (bottom, logarithmic scale) pulse profiles over 5000 round-trips showing nine consecutive explosions. The simulation parameters are given in Table 11.1. The gray-shaded area with two explosions is selected for detailed presentation in Figure 11.3.
Figure 11.3 Two explosions selected from a large set of data. This selection is shown as gray area in Figure 11.2. The explosions are presented in (a) temporal and (b) spectral domains. We use the same logarithmic scale color map as in Figure 11.2. The first explosion occurs on one side of the soliton (“asymmetric”), whereas the second explosion occurs on two sides (“symmetric”). Consequently, the second explosion reaches higher peak power than the first one.
Figure 11.4 Historgram (log scale) of recorded pulse peak power after each round-trip for 10 million round-trips with uniform 0.02 kW intervals. The distribution shows a flattened tail at high peak powers. The scale on the right-hand side vertical axis shows the data in probability density given in the units of . The dashed vertical line indicates four times the significant pulse peak power.
Chapter 12: SRS-Driven Evolution of Dissipative Solitons in Fiber Lasers
Figure 12.1 (a) Block scheme of a fiber laser with ring cavity configuration, (b) typical scheme of the real experimental setup.
Figure 12.2 Solution existence areas in the plane (): positive () branch of the analytic solution exists in the area IV, negative () branch—in the areas III and IV. Coordinates of the example numerical solutions are marked by points. Dotted lines illustrate the paths with self-similar pulse shape corresponding to = const [27].
Figure 12.3 (a) Comparison between numerics (color) and positive branch of the analytical solution (solid black) for the spectral shape. (b) Time-domain shape of the positive branch (solid) and fitting (dashed) by parabola and . Additional black curve shows analytics in the limit ().
Figure 12.4 (a) Energy of the output pulses depends on cavity length; inset –dimensionless chirp parameter defined by Eq. (12.1); (b) nonlinearity and dispersion phase shift ratio over one round-trip in the cavity [28].
Figure 12.5 Experimental setup of an all-fiber highly chirped DS fiber oscillator.
Figure 12.6 The measured optical pulse spectrum (in –inside the cavity at 1% port, out –out of the PM splitter): spectrum for a maximum DS output energy for 30 m and 90 m cavity length is present on (a) and (b) correspondingly.
Figure 12.7 Autocorrelation functions: (a) by intensity for chirped pulse at different length of cavity and (b) interferometric for compressed pulse at 30 m and 90 m cavity length.
Figure 12.8 Stability thresholds () of HCDS solutions of CQGLE without noise and SRS (solid black curve), with noise and no SRS (red dashed curve), as well as with noise and SRS (blue dashed-dotted curve). A sole HCDS exists below and left of the corresponding curves. The parameters of CQGLE are and . The spectral filtering parameter corresponds to an approximately 40 nm bandwidth. Other details can be found in [76].
Figure 12.9 The calculated shapes of the generated pulses in corresponding points (A,B,C,D,E) of the scheme without and with the feedback loop shown in cloud-inset of Figure 12.5. The RP in box C (right) present before it attenuated by factor .
Figure 12.10 Evolution of the pulse shape (a) and spectrum (b) along PMF of m in the scheme of Figure 12.5 without feedback loop.
Figure 12.11 Calculated and measured pulse shapes (a) and spectra (b) and at the output port E in scheme of Figure 12.5 without feedback loop. Calculated and measured ACF and CCF traces are present in inset.
Figure 12.12 Energy of the DS in the experiment (green triangles) and simulation (green solid line). Calculated total energy for the DS and RP (dashed line). The experimental point obtained for m ( nJ) is not added because the pump power was reduced due to lower stability at this length.
Figure 12.13 Evolution of the pulse shapes and spectra along the PMF and SMF (with the active Yb
3+
-doped part) sections in a scheme of Figure 12.5 with the Raman feedback loop comprising the delay line (DL) shown as the white box, with parameters similar to that in the experiment. Points A, B, C, D, E, and PBS (polarization beam splitter) of the scheme of Figure 12.5 are marked at the corresponding distances [31].
Figure 12.14 Pulse shapes: (a) the simulated feedback-defined evolution of the intracavity pulses in the scheme of Figure 12.5 (point B) versus the round-trip number. The zero time offset is bound to the position of the main dissipative soliton. (b) The calculated DS and RDS pulse shapes inside the cavity (point B in Figure 12.5) and their instant frequencies. Inset: the dechirped DS–RDS complex with 70-ps delay compensation in the Raman feedback loop [31].
Figure 12.15 Output spectra: calculated (a) and measured (b) at the DS (blue) and RDS output (red) ports for the 40-m long cavity with the feedback coefficient and without feedback (gray). Note that the (i) the spectral division between output ports in the experiment ( dB) is not as high as in the simulations ( dB), and (ii) the RDS port has significant spectral ripples around 1010 nm [31].
Figure 12.16 Autocorrelation traces and radio frequency beating spectra of the realized pulses. The measured (points) autocorrelation traces of the DS (a) and RDS (b), and the calculated (lines) for the 40-m cavity oscillator with the feedback coefficient . Inset: the corresponding FROG traces; Interferometric ACF traces shown for DS (c) abd RDS (d) after a double-pass compressor consisting of a grating pair. The compressor introduces one-pass group delay dispersion of . Insets: Radio frequency spectra measured at a repetition rate of 5 MHz with 1 Hz bandwidth [31].
Figure 12.17 (a) Transmission spectra of the and couplers of the experimental scheme (Figure 12.5 of the paper). (b) Simulation results for the DS (blue) and RDS (red) spectra at different coupler parameters: 1010/1055 nm (solid) and 1015/1060 nm (dashed) at the fixed parameters (the cutoff wavelength at 1005 nm), for 70 ps Raman delay and 1025 (1030) nm gain maximum.
Figure 12.18 Intracavity spectra in experiment and simulation corresponding to different feedback coefficients. The spectra are measured at point B of the scheme of Figure 12.5.
Figure 12.19 Evolution of the pulse shapes and spectra along the PMF and SMF sections in a scheme of Figure 12.5 in the case of two feedback loops providing delay and filtering for the first and second Stokes waves with the additional output port .
Figure 12.20 Three-color bound solitons. Results of simulation for the generated spectra in an 80-m-long PM-fiber cavity oscillator with the Raman feedbacks for the first-order Stokes () and the second-order Stokes () waves, in the presence of a band-stop filter at 1025 nm (10 nm width) and the following parameters of the th feedback loop: delay time 140 ps () and 280 ps (), stepwise couplers with cutoff wavelengths at 1035 nm () and 1085 nm ().
Chapter 13: Synchronization in Vectorial Solid-State Lasers
Figure 13.1 (a) Anisotropic laser cavity principle. is the active medium gain and the birefringence. (b) Schematic of the two sets of eigenfrequencies.
Figure 13.2 (a) Cavity with crossed phase and loss anisotropies. (b) Beat frequency versus birefringence. Dashed lines indicate the lossless case of Eq. (13.2 (adapted from [11]).
Figure 13.3 Self-pulsed regime in a vectorial laser (adapted from [11].) (a) Simulation. is the intensity of the field after a polarizer at of the eigenstate directions, is the phase difference between the two components of the field. (b) Corresponding experimental result.
Figure 13.4 Laser cavity containing a quarter-wave plate crossed with an anisotropic gain.
Figure 13.5 Laser eigenstates at different locations inside the cavity of Figure 13.4.
Figure 13.6 Polarization self-modulation in a few-mode Nd : YAG laser containing a QWP. (a) Single-eigenstate intensity. (b) Interference between the two oscillating eigenstates.
Figure 13.7 (a) Experimental laser and detection setup. H, half-wave plate; PBS, polarization beamsplitter; D1,D2, photodiodes. (b) Schematic polarization sequences emitted when and .
Figure 13.8 Experimental eigenstates beats (red), and (blue), for (a) , (b) , (c) , and (d) .
Figure 13.9 Experimental FFT spectrum of the laser output, when . (a) outside the locking range. (b) inside the locking range, and (c) resulting -periodic spectrum.
