THE NON-LINEAR SCHRODINGER EQUATION and SOLITONS James

Total Page：16

File Type：pdf, Size：1020Kb

THE NON-LINEAR SCHRODINGER EQUATION AND SOLITONS James P. Gordon 1 Soliton John Scott Russell ( Solitary Water Wave –1834) Korteweg and DeVries (KdeV equation – 1895) Zabusky and Kruskal (Solitons -- 1965) Zhakharov and Shabat (NLSE – 1971) Hasegawa and Tappert (Lightwave Solitons – 1973) Mollenauer et. al. (Observation in fiber – 1980) 2 THE NONLINEAR SCHRODINGER EQUATION 3 PHASE AND GROUP VELOCITIES Z=VT 4 INVERSE GROUP VELOCITY TIME DELAY = Z/V 5 FIBER NONLINEARITY 6 DERIVATION OF THE NLS EQUATION / 7 DERIVATION OF THE NLS EQUATION Step II: Shift to central frequency and retarded time 8 PULSE REPRESENTATION IN ORDINARY VS RETARDED TIME Pulses separated by δλ = 0.4 nm in fiber with D = 0.1 ps/nm-km Pulses at 0 km: Pulses at 5000 km: Pulse at ω0 Pulse at 0.4 nm longer wavelength 0 100 200 300 400 25,000,000,100 25,000,000,350 Ordinary Time (ps) 0 100 200 300 400 0 100 200 300 400 Retarded Time (ps) 9 DERIVATION OF THE NLS EQUATION Step III: Rescale the independent variables 10 FOURIER TRANSFORMS 11 THE NLS EQN: ACTION OF THE DISPERSIVE TERM 12 DISPERSIVE BROADENING OF A GAUSSIAN PULSE VS Z (Minimum temporal width at origin) 13 THE NLS EQN: ACTION OF THE NON-LINEAR TERM t 14 SPECTRAL BROADENING OF A GAUSSIAN PULSE AT ZERO DISPERSION (Peak non-linear phase shift indicated next to each spectrum.) 15 ORIGIN OF THE SOLITON 16 THE DISPERSIVE AND NON-LINEAR PHASE SHIFTS OF A SOLITON Note that the dispersive and non-linear phase shifts sum to a constant. 17 PATH-AVERAGE SOLITONS 18 PULSE ENERGY AND FIBER DISPERSION IN SAMPLE OF TRANSMISSION LINE USED FOR TEST OF “PATH-AVERAGE” SOLITONS 19 SIMULATED TRANSMISSION THRU SYSTEM WITH LUMPED AMPS AND PERIODICALLY VARYING DISPERSION 20 DISPERSION RELATION FOR SOLITONS AND LINEAR WAVES SOLITON SPECTRAL DENSITY ALSO SHOWN 21 NUMERICAL SOLUTION OF THE NLS EQN: THE SPLIT-STEP FOURIER TRANSFORM METHOD 22 CONSTANTS OF THE NLS ∂u 1 ∂2u − i = + | u |2 u ∂z 2 ∂u 2 W = ∫| u |2u (Pulse Energy) 1 ∂u H = ∫ dt(| |2 − | u |4 ) (Hamiltonian) 2 ∂t (More) 23 GAUSSIAN PULSE APPROXIMATION 1 ⎛η ⎞4 ⎡ 1 2 ⎤ u = W ⎜ ⎟ exp − (η + iβ )t + iφ ⎝π ⎠ ⎣⎢ 2 ⎦⎥ η , β , and φ may depend on z but not on t 1 ⎡η 2 + β 2 η ⎤ H = W ⎢ −W ⎥ 2 ⎣ 2η 2π ⎦ dH 1 ⎡⎛ β 2 W ⎞ dη 2β dβ ⎤ W ⎜1 ⎟ 0 = ⎢⎜ − 2 − ⎟ + ⎥ = dz 4 ⎣⎝ η 2πη ⎠ dz η dz ⎦ dβ dη This result gives a required relation between and dz dz 24 dη From the NLS: = 2βη dz dβ 2 2 W 1.5 From the equation for H: = β −η + η dz 2π These equations are very useful in picturing how solitons behave To make a soliton: β = 0 W = 2πη 25.

Copy Link

Recommended publications

Real-Time Observation of Dissipative Soliton Formation in Nonlinear Polarization Rotation Mode- Locked fibre Lasers
ARTICLE DOI: 10.1038/s42005-018-0022-7 OPEN Real-time observation of dissipative soliton formation in nonlinear polarization rotation mode- locked fibre lasers Junsong Peng1, Mariia Sorokina2, Srikanth Sugavanam2, Nikita Tarasov2,3, Dmitry V. Churkin3, 1234567890():,; Sergei K. Turitsyn2,3 & Heping Zeng1 Formation of coherent structures and patterns from unstable uniform state or noise is a fundamental physical phenomenon that occurs in various areas of science ranging from biology to astrophysics. Understanding of the underlying mechanisms of such processes can both improve our general interdisciplinary knowledge about complex nonlinear systems and lead to new practical engineering techniques. Modern optics with its high precision mea- surements offers excellent test-beds for studying complex nonlinear dynamics, though capturing transient rapid formation of optical solitons is technically challenging. Here we unveil the build-up of dissipative soliton in mode-locked fibre lasers using dispersive Fourier transform to measure spectral dynamics and employing autocorrelation analysis to investi- gate temporal evolution. Numerical simulations corroborate experimental observations, and indicate an underlying universality in the pulse formation. Statistical analysis identifies cor- relations and dependencies during the build-up phase. Our study may open up possibilities for real-time observation of various nonlinear structures in photonic systems. 1 State Key Laboratory of Precision Spectroscopy, East China Normal University, 200062 Shanghai,
[Show full text]
Optical Event Horizons from the Collision of a Soliton and Its Own Dispersive Wave S
Optical event horizons from the collision of a soliton and its own dispersive wave S. Wang, Arnaud Mussot, Matteo Conforti, A. Bendahmane, X. Zeng, Alexandre Kudlinski To cite this version: S. Wang, Arnaud Mussot, Matteo Conforti, A. Bendahmane, X. Zeng, et al.. Optical event horizons from the collision of a soliton and its own dispersive wave. Physical Review A, American Physical Society, 2015, 92 (2), 10.1103/PhysRevA.92.023837. hal-02388991 HAL Id: hal-02388991 https://hal.archives-ouvertes.fr/hal-02388991 Submitted on 2 Dec 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. PHYSICAL REVIEW A 92, 023837 (2015) Optical event horizons from the collision of a soliton and its own dispersive wave S. F. Wang,1,2 A. Mussot,1 M. Conforti,1 A. Bendahmane,1 X. L. Zeng,2 and A. Kudlinski1,* 1Laboratoire PhLAM, UMR CNRS 8523, IRCICA, USR CNRS 3380, Universite´ Lille 1, 59655 Villeneuve d’Ascq, France 2Key Laboratory of Specialty Fiber Optics and Optical Access Network, Shanghai University, 200072 Shanghai, China (Received 22 May 2015; published 20 August 2015) We observe experimentally the spectral signature of the collision between a soliton and the dispersive wave initially emitted from the soliton itself.
[Show full text]
Integrable Nonlinear Schrödinger Systems and Their Soliton Dynamics
Dynamics of PDE, Vol.1, No.3, 239-299, 2004 Integrable Nonlinear Schrödinger Systems and their Soliton Dynamics M. J. Ablowitz, B. Prinari, and A. D. Trubatch Communicated by Charles Li, received June 16, 2004. Abstract. Nonlinear Schrödinger (NLS) systems are important examples of physically-significant nonlinear evolution equations that can be solved by the inverse scattering transform (IST) method. In fact, the IST for discrete and continuous, as well as scalar and vector, NLS systems all fit into the same framework, which is reviewed here. The parallel presentation of the IST for each of these systems not only clarifies the common structure of the IST, but also highlights the key variations. Importantly, these variations manifest themselves in the dynamics of the solutions. With the IST approach, one can explicitly construct the soliton solutions of each of these systems, as well as formulas from which one can determine the dynamics of soliton interaction. In particular, vector solitons, both continuous and discrete, are partially charac- terized by a polarization vector, which is shifted by soliton interaction. Here, we give a complete account of the nature of this polarization shift. The po- larization vector can be used to encode the value of a binary digit (“bit”) and the soliton interaction arranged so as to effect logical computations. Contents 1. Introduction 240 2. Scalar Nonlinear Schrödinger equation (NLS) 244 3. Integrable Discrete Nonlinear Schrödinger equation (IDNLS) 253 4. Vector Nonlinear Schrödinger (VNLS) equation 266 5. Integrable Discrete Vector Nonlinear Schrödinger equation (IDVNLS) 274 6. Soliton Interactions 288 7.
[Show full text]
Soliton Crystals in Kerr Resonators
Soliton crystals in Kerr resonators Daniel C. Cole1,2, Erin S. Lamb1, Pascal Del’Haye1,†, Scott A. Diddams1, and Scott B. Papp1 1National Institute of Standards and Technology (NIST), Boulder, CO 80305, USA 2Department of Physics, University of Colorado, Boulder, CO 80309, USA †Present address: National Physical Laboratory (NPL), Teddington, TW11 0LW, United Kingdom Corresponding author: [email protected] Strongly interacting solitons confined to an optical resonator would offer unique capabilities for experiments in communication, computation, and sensing with light. Here we report on the discovery of soliton crystals in monolithic Kerr microresonators—spontaneously and collectively ordered ensembles of co-propagating solitons whose interactions discretize their allowed temporal separations. We unambiguously identify and characterize soliton crystals through analysis of their ‘fingerprint’ optical spectra, which arise from spectral interference between the solitons. We identify a rich space of soliton crystals exhibiting crystallographic defects, and time-domain measurements directly confirm our inference of their crystal structure. The crystallization we observe is explained by long-range soliton interactions mediated by resonator mode degeneracies, and we probe the qualitative difference between soliton crystals and a soliton liquid that forms in the absence of these interactions. Our work explores the rich physics of monolithic Kerr resonators in a new regime of dense soliton occupation and offers a way to greatly increase the efficiency of Kerr combs; further, the extreme degeneracy of the configuration space of soliton crystals suggests an implementation for a robust on-chip optical buffer. Optical solitons have recently found a new realization in frequency combs generated in passive, monolithic Kerr- nonlinear resonators1 (microcombs).
[Show full text]
Plasmon-Soliton
Plasmon-Soliton Eyal Feigenbaum and Meir Orenstein Department of Electrical Engineering, Technion, Haifa 32000, Israel [email protected] Abstract : Formation of a novel hybrid-vector spatial plasmon-soliton in a Kerr slab embedded in-between metal plates is predicted and analyzed with a modified NLSE, encompassing hybrid vector field characteristics. Assisted by the transverse plasmonic effect, the self trapping dimension of the plasmon-soliton was substantially compressed (compared to the dielectrically cladded slab case) when reducing the slab width. The practical limitation of the Plasmon-soliton size reduction is determined by available nonlinear materials and metal loss. For the extreme reported values of nonlinear index change, we predict soliton with a cross section of 300nm×30nm (average dimension of 100nm). 1 The downscaling of conventional photonics is halted when approaching a transverse dimension of ~ λ/2 ( λ is the wavelength). This limitation can be alleviated by the incorporation of metals, giving rise to surface plasmon polaritons (SPP) [1,2]. Here we analyze a novel configuration where light is transversely tightly guided between two metal layers, and laterally self trapped by a nonlinear Kerr effect to yield a plasmon-soliton. The tight field confinement of this scheme is important for potential excitation of plasmon-solitons by a low power continuous wave optical source, which is further assisted by the expected small group velocity of the plasmon-soliton. In this letter we present for the first time both an analysis of TM spatial solitons, using the Non- linear Schrödinger Equation (NLSE), where the full vectorial field is considered, as well as the prediction and characteristics of SPP based solitons.
[Show full text]
Discrete Diffraction Managed Spatial Solitons
VOLUME 87, NUMBER 25 PHYSICAL REVIEW LETTERS 17DECEMBER 2001 Discrete Diffraction Managed Spatial Solitons Mark J. Ablowitz and Ziad H. Musslimani Department of Applied Mathematics, University of Colorado, Campus Box 526, Boulder, Colorado 80309-0526 (Received 25 June 2001; published 30 November 2001) Motivated by recent experimental observations of anomalous diffraction in linear waveguide array, we propose a novel model governing the propagation of an optical beam in a diffraction managed nonlinear waveguide array. This model supports discrete spatial solitons whose beamwidth and peak amplitude evolve periodically. A nonlocal integral equation governing the slow evolution of the soliton amplitude is derived and its stationary soliton solutions are obtained. DOI: 10.1103/PhysRevLett.87.254102 PACS numbers: 05.45.Yv Discrete spatial solitons in nonlinear media have at- Kerr media in the presence of both normal and anomalous tracted considerable attention in the scientiﬁc community diffraction. Importantly, even though optical diffraction for many years [1]. They have been demonstrated to exist and chromatic dispersion originate from different physics in a wide range of physical systems such as atomic chains (the former being geometric and the latter being media de- [2,3], molecular crystals [4], biophysical systems [5], elec- pendent), nevertheless, discrete diffraction spatial solitons trical lattices [6], and recently in arrays of coupled nonlin- and dispersion managed solitons share many properties ear optical waveguides [7,8]. Such discrete excitations can which highlight the universality and diversity of solitons. form when discrete diffraction is balanced by nonlinearity. A nonlocal integral equation governing the slow evolution In an optical waveguide array this can be achieved by using of the beam amplitude is derived, and its stationary soli- an intense laser beam which locally changes the nonlinear ton solutions are obtained.
[Show full text]
Soliton-Guided Quantum Information Processing
Chapter 1 Soliton-Guided Quantum Information Processing Ken Steiglitz Computer Science Department Princeton University Princeton NJ 08544 Abstract We describe applications of solitons and soliton collisions to the transport, transfer, and beam-splitting of qubits carried by optical photons. The transport and transfer realize the “flying qubits” necessary for quantum information processing, and the beam-splitting leads, in theory, to an implementation of quantum computing using linear optics. These proposed applications are embedded in a uniform optical fiber and require no special device fabrication, no cooling, and no vacuum. The pioneering papers of Feynman [1] and Deutsch [2] in the 1980s sparked the rapid development of the field of quantum information processing. The theoretical and experimentalprogress has been remarkable, with the development, for example, of quantum error correction and a fast algorithm for factoring, and the exploration of a wide variety of physical implementations. In the latter category, the optical photon as the carrier of a qubit has played an important role in the experimental demonstra- tion of quantum cryptography and other important applications to communications and information processing. At the same time there has been tremendous progress in our understanding of classical nonlinear waves, and, in particular, solitons in op- tical fibers. In this chapter we will explore what role solitons and soliton collisions might play in the development of quantum information processing with optical pho- tons. For a more detailed account, the reader is referred to [3, 4, 5], from which the material in this chapter was drawn. 1 2 Steiglitz 1.1 Photon trapping A pulse traveling down a fiber forms a soliton when the dispersion, which tends to widen the pulse, is counterbalanced by the nonlinear Kerr effect, whereby the elec- tric field changes the index of refraction of the material.
[Show full text]
Kerr Microresonator Soliton Frequency Combs at Cryogenic Temperatures
Kerr Microresonator Soliton Frequency Combs at Cryogenic Temperatures Gregory Moille,1, 2, ∗ Xiyuan Lu,1, 2 Ashutosh Rao,1, 2 Qing Li,1, 2, 3 Daron A. Westly,1 Leonardo Ranzani,4 Scott B. Papp,5, 6 Mohammad Soltani,4 and Kartik Srinivasan1, 7, y 1Microsystems and Nanotechnology Division, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA 2Institute for Research in Electronics and Applied Physics and Maryland Nanocenter, University of Maryland,College Park, Maryland 20742, USA 3Electrical and Computer Engineering, Carnegie Mellon University, Pittsburgh, PA 15213, USA 4Raytheon BBN Technologies, 10 Moulton Street, Cambridge, MA, 02138, USA 5Time and Frequency Division, National Institute of Standards and Technology, 385 Broadway, Boulder, CO 80305, USA 6Department of Physics, University of Colorado, Boulder, Colorado, 80309, USA 7Joint Quantum Institute, NIST/University of Maryland, College Park, Maryland 20742, USA (Dated: February 19, 2020) We investigate the accessibility and projected low-noise performance of single soliton Kerr fre- quency combs in silicon nitride microresonators enabled by operating at cryogenic temperatures as low as 7 K. The resulting two orders of magnitude reduction in the thermo-refractive coefficient relative to room-temperature enables direct access to single bright Kerr soliton states through adia- batic frequency tuning of the pump laser while remaining in thermal equilibrium. Our experimental results, supported by theoretical modeling, show that single solitons are easily accessible at tem- peratures below 60 K for the microresonator device under study. We further demonstrate that the cryogenic temperature primarily impacts the thermo-refractive coefficient. Other parameters critical to the generation of solitons, such as quality factor, dispersion, and effective nonlinearity, are unaltered.
[Show full text]
Microresonator Kerr Frequency Combs with High Conversion Efficiency Xiaoxiao Xue,1,2* Pei-Hsun Wang,2 Yi Xuan,2,3 Minghao Qi,2,3 and Andrew M
Microresonator Kerr frequency combs with high conversion efficiency Xiaoxiao Xue,1,2* Pei-Hsun Wang,2 Yi Xuan,2,3 Minghao Qi,2,3 and Andrew M. Weiner2,3 1Department of Electronic Engineering, Tsinghua University, Beijing 100084, China 2School of Electrical and Computer Engineering, Purdue University, 465 Northwestern Avenue, West Lafayette, Indiana 47907-2035, USA 3Birck Nanotechnology Center, Purdue University, 1205 West State Street, West Lafayette, Indiana 47907, USA *Corresponding author: [email protected] ABSTRACT: Microresonator-based Kerr frequency comb 1 shows the energy flow chart in microcomb generation. Considering the (microcomb) generation can potentially revolutionize a optical field circulating in the cavity, part of the pump and comb power is variety of applications ranging from telecommunications absorbed (or scattered) due to the intrinsic cavity loss, and part is to optical frequency synthesis. However, phase-locked coupled out of the cavity into the waveguide; meanwhile, a fraction of the microcombs have generally had low conversion efficiency pump power is converted to the other comb lines, effectively resulting in limited to a few percent. Here we report experimental a nonlinear loss to the pump. At the output side of the waveguide, the results that achieve ~30% conversion efficiency (~200 mW residual pump line is the coherent summation of the pump component on-chip comb power excluding the pump) in the fiber coupled from the cavity and the directly transmitted pump; the power present in the other comb lines excluding the pump constitutes the telecommunication band with broadband mode-locked usable comb power. Here we omit any other nonlinear losses possibly dark-pulse combs.
[Show full text]
Formation of Soliton Trains in Bose–Einstein Condensates As a Nonlinear Fresnel Diffraction of Matter Waves
Physics Letters A 319 (2003) 406–412 www.elsevier.com/locate/pla Formation of soliton trains in Bose–Einstein condensates as a nonlinear Fresnel diffraction of matter waves A.M. Kamchatnov a,∗,A.Gammalb,c, F.Kh. Abdullaev d,R.A.Kraenkele a Institute of Spectroscopy, Russian Academy of Sciences, Troitsk 142190, Moscow region, Russia b Instituto de Física, Universidade de São Paulo, C.P. 66318, 05315-970 São Paulo, Brazil c Department of Physics and Astronomy and Rice Quantum Institute, Rice University, Houston, TX 77251, USA d Physical-Technical Institute, Uzbek Academy of Sciences, G. Mavlyanov str. 2-b, 70084 Tashkent-84, Uzbekistan e Instituto de Física Teórica, Universidade Estadual Paulista, UNESP, Rua Pamplona 145, 01405-900 São Paulo, Brazil Received 6 October 2003; accepted 14 October 2003 Communicated by V.M. Agranovich Abstract The problem of generation of atomic soliton trains in elongated Bose–Einstein condensates is considered in framework of Whitham theory of modulations of nonlinear waves. Complete analytical solution is presented for the case when the initial density distribution has sharp enough boundaries. In this case the process of soliton train formation can be viewed as a nonlinear Fresnel diffraction of matter waves. Theoretical predictions are compared with results of numerical simulations of one- and three-dimensional Gross–Pitaevskii equation and with experimental data on formation of Bose–Einstein bright solitons in cigar- shaped traps. 2003 Elsevier B.V. All rights reserved. Realization of Bose–Einstein condensate (BEC) mation of bright soliton trains in nonlinear wave sys- [1–3] has created new active ﬁeld of research of quan- tems is often explained as a result of modulational in- tum macroscopical behavior of matter.
[Show full text]
Fully Integrated Ultra-Low Power Kerr Comb Generation
Fully integrated ultra-low power Kerr comb generation Brian Stern1,2, Xingchen Ji1,2, Yoshitomo Okawachi3, Alexander L. Gaeta3, and Michal Lipson2 1School of Electrical and Computer Engineering, Cornell University, Ithaca, NY 14853, USA 2Department of Electrical Engineering, Columbia University, New York, NY 10027, USA 3Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY 10027, USA Corresponding Author: [email protected] Optical frequency combs are broadband sources that offer mutually-coherent, equidistant spectral lines with unprecedented precision in frequency and timing for an array of applications1•9. Kerr frequency combs in microresonators require a single-frequency pump laser and have offered the promise of highly compact, scalable, and power efficient devices. Here, we realize this promise by demonstrating the first fully integrated Kerr frequency comb source through use of extremely low-loss silicon nitride waveguides that form both the microresonator and an integrated laser cavity. Our device generates low-noise soliton-modelocked combs spanning over 100 nm using only 98 mW of electrical pump power. Our design is based on a novel dual-cavity configuration that demonstrates the flexibility afforded by full integration. The realization of a fully integrated Kerr comb source with ultra-low power consumption brings the possibility of highly portable and robust frequency and timing references, sensors, and signal sources. It also enables new tools to investigate the dynamics of comb and soliton generation through close chip-based integration of microresonators and lasers. Frequency combs based on chip-scale microresonators offer the potential for high- precision photonic devices for time and frequency applications in a highly compact and robust platform.
[Show full text]
Optical Solitons
Lisa Larrimore Physics 116 Optical Solitons An optical soliton is a pulse that travels without distortion due to dispersion or other effects. They are a nonlinear phenomenon caused by self-phase modulation (SPM), which means that the electric field of the wave changes the index of refraction seen by the wave (Kerr effect). SPM causes a red shift at the leading edge of the pulse. Solitons occur when this shift is canceled due to the blue shift at the leading edge of a pulse in a region of anomalous dispersion, resulting in a pulse that maintains its shape in both frequency and time. Solitons are therefore an important development in the field of optical communications. Following closely from A. Ghatak and K. Thyagarajan's Introduction to Fiber Op- tics, we will consider the theoretical description of solitary waves. First, let us consider the nonlinear effect of SPM. Recall from Hecht Ch. 13 that for nonlinear media, the electric polarization of a wave is described by (2) 2 (3) 3 P = 0χE + 0χ E + 0χ e + ; (1) ··· where χ is the dielectric susceptibility. For an optical fiber, χ(2) = 0, so we keep only the first and third terms (also neglecting weaker higher-order terms). By considering the 2 polarization of a plane wave, E = E0 cos(!t kz), whose intensity is I = c0n0E =2, − 0 and remembering that the general relationship between polarization and refractive 2 index is P = 0(n 1)E, we find ::: [Space for you to take notes!] − n = n0 + n2I; (2) 1 where 3 χ(3) n2 = 2 : (3) 4 c0n0 Now, let us consider a pulse in a nonlinear dispersive medium, which is approxi- mately described by the so-called non-linear Schrödingerequation (NLSE), 2 @f 1 @f α @ f 2 i + 2 + Γ f f = 0; (4) − @z vg @t − 2 @t j j where 1 dk = (5) vg d! !=!0 d2k α = (6) d!2 !=!0 1 Γ = ! n n (7) 2 0 0 0 2 and f(z; t) is the envelope of the pulse: E(z; t) = exp [i (!0t k0z)] f(z; t): (8) − It is often easier to write Eq.
[Show full text]