Nonlinear Coupled-mode Equations Research Articles

Gap solitons on an integrated CMOS chip

Abstract Nonlinear propagation in periodic media has been studied for decades, yielding demonstrations of numerous phenomena including strong temporal compression and slow light generation. Gap solitons, that propagate at frequencies inside the stopband, have been observed in optical fibres but have been elusive in photonic chips. In this manuscript, we investigate nonlinear pulse propagation in a chip-based nonlinear Bragg grating at frequencies inside the stopband and observe clear, unequivocal signatures of gap soliton propagation, including slow light, intensity-dependent transmission, intensity-dependent temporal delay and gap soliton compression. Our experiments which are performed in an on-chip ultra-silicon-rich nitride (USRN) Bragg grating with picosecond time scales, reveal slow light group velocity reduction to 35%–40% of the speed of light in vacuum, change in the temporal delay of 7 ps at low peak powers between 15.7 W–36.6 W, which is accompanied by up to 2.7× temporal compression of input pulses. Theoretical calculations using the nonlinear coupled mode equations confirm the observations of intensity-dependent temporal delay. Of fundamental importance, this demonstration opens up on-chip platforms for novel experimental studies of gap solitons as the basis of all-optical buffers, delay lines and optical storage.

Nonlinear switching of optical solitons in fiber Bragg gratings with non-instantaneous Kerr nonlinearity

Introduction:Optical soliton switching in nonlinear uniform fiber Bragg gratings with non-instantaneous Kerr nonlinearity is numerically investigated. Two cases of pulse propagation are considered: propagation through the reflection and transmission regions. Objectives:The purpose is to investigate the influence of a delayed response of the Kerr effect on the switching characteristics of uniform fiber Bragg gratings and compare to the instantaneous nonlinear regime. Methods:A modified set of nonlinear coupled mode equations are numerically resolved on a two-dimensional grid using an implicit four-order Runge–Kutta method. Results:It is found that the reflectivity of the gratings only depend on the relaxation time of the grating medium when the input pulse intensity reaches a certain level. In addition, marked differences in shape of the reflected and transmitted pulses are observed.

Switching dynamics of dissipative cnoidal waves in dual-coupled microresonators

Dual-coupled structure is typically used to actively change the local dispersion of microresonator through controllable avoided mode crossings (AMXs). In this paper, we investigate the switchability of dissipative cnoidal waves (DCWs) based on dual-coupled microresonators. The switching dynamics of DCWs are numerically simulated using two sets of nonlinear coupled-mode equations. It is found that the pulse number of DCWs can only be decreased (i.e. switched unidirectionally) when working as perfect soliton crystals and can either be decreased or increased (i.e. switched bidirectionally) when working as Turing rolls. Moreover, the stable regions of DCWs can be greatly expanded due to the existence of AMXs. The switchability of DCWs would further liberate the application potential of microcombs in a wide range of fields, including frequency metrology, optical communications, and signal-processing systems.

Theoretical solution of second-harmonic generation in periodically poled lithium niobate and chirped periodically poled lithium niobate thin film via quasi-phase-matching

We present an effective nonlinear coupled mode coefficient model to evaluate and analyze the second harmonic generation (SHG) process under the quasi-phase-matching (QPM) scheme in periodically poled lithium niobate (PPLN) thin film. The theoretical model derives directly from the nonlinear coupled mode equations and simplifies the QPM process in PPLN thin film. The simulated results agree excellently with the rigorous numerical calculations in a variety of situations. We also propose using the chirped PPLN (CPPLN) thin film to achieve the SHG of ultrashort laser pulse and develop a broadband nonlinear coupled mode theory to evaluate this complex nonlinear coupling interaction process. All the frequency components in fundamental wave (FW) and second harmonic wave (SHW) have been considered in the theory, and the spectral evolution of FW and SHW can be obtained by solving a series of nonlinear coupled mode equations. CPPLN thin film supports broadband up-conversion in telecommunications bands and can generate broadband entangled photons in infrared region, which shows great potential in optical communication and quantum optics. The developed theories provide an efficient tool to design and optimize the PPLN and CPPLN thin film for various applications based on nonlinear frequency conversion and help to achieve high-efficiency on-chip SHG of the ultrashort laser pulse.

Design of Partially Suspended Silicon Nitride Slot Waveguides for Efficient Forward Stimulated Brillouin Scattering

We design a partially suspended silicon nitride (Si <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> ) slot waveguide on silica to realize efficient on-chip forward stimulated Brillouin scattering (FSBS). The slot waveguide can intensify radiation pressure close to the slot and simultaneously confine both optical and acoustic modes, resulting in an enhanced FSBS gain on the order of 300 W <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> m <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">-1</sup> . The acoustic radiation loss is significantly suppressed due to the strong acoustic coupling in the narrow slot waveguides. We analyze the impacts of structural parameters on the SBS properties, which produces the optimal parameters for maximum SBS gain. Based on the nonlinear coupled-mode equations, we compare the performance of Stokes amplification between the Si <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> slot waveguides and the silicon nanowires. Simulation results indicate that the Si <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> slot waveguides with low linear loss can offer a larger net Stokes amplification than silicon waveguides under moderate pump power. Such approach enables efficient on-chip SBS devices in CMOS-compatible Si <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sub> N <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sub> platform.

Energy Exchange at Degenerate Mixing of Four Fast TE2 Modes of a Thin Left-Handed Film on a Nonlinear Substrate

Frequency-degenerate mixing of forward and backward propagating fast modes is considered in a planar optical waveguide based on a thin left-handed film and a right-handed substrate exhibiting the Kerr effect. Nonlinear coupled-mode equations are solved for the intensities of these modes in the neglect of the contributions of elementary processes that fail to satisfy the phase-matching condition. It is shown that the intramode self-action of four TE modes is accompanied by energy exchange between these modes. The efficiency of energy exchange varies periodically along the propagation direction of modes, thus leading to the inversion of the intensities of forward and backward propagating light fields on the interaction half-period.

Spatial-Dependent Hamiltonian Formulation of Cross-Mode Modulation

In the absence of random mode mixing (RMM), linearization of the cross-mode modulation (XMM) under pump-probe configuration is in general complicated because the evolution of the pump light depends on the local mode decomposition of itself. In this work, general derivation of the Hamiltonian of the XMM system without RMM is carried out and discussed under two interesting scenarios where the pump evolution can be effectively linearized, leading to spatial dependent Hamiltonian of the probe light. Representative evolutions of pump and probe light over transmission are investigated with the assistance of Poincare sphere based on the Hamiltonian approach. Our results are benchmarked against the results given by precession equations examined by Lin and Agrawal and numerical simulations using nonlinear coupled mode equations (CMEs). By highlighting the eigenstates as well as eigenvalues of the system, our approach provides an intuitive yet powerful approach to understand the XMM nonlinear problems and to dynamically manipulate the spatial profiles of the probe light.

Super Gaussian pulse generation in a passively mode-locked fiber laser

The generation of self-similar super Gaussian-like pulses from a passively mode-locked fiber laser is numerically demonstrated. The laser cavity utilizes a long period fiber grating as a pulse-shaping element. The model is based on the normalized complex Ginzburg-Landau equation and the nonlinear coupled mode equations of the grating. The numerical results show stable super Gaussian pulses with a linear frequency chirp, a 1.5 ps pulse duration and 2 nJ energy. The generated pulses are scalable and exhibits no bifurcation or instability as the gain is increased.

Intensity dependent spectral properties of long-period fiber grating for applications in lightwave communication

Using nonlinear coupled mode equations, the spectral properties of long-period fiber grating are studied analytically taking into account the Kerr type optical nonlinearity in the medium. The nonlinear coupled-mode equations were solved for co-propagating core and cladding mode amplitudes and expressions for transmittivity and phase factor at high excitation intensity are obtained. It is observed that the resonance wavelength of grating shifts towards higher wavelength side with increasing excitation intensity. Also, the minimum transmittivity of the grating shows oscillating behavior with increasing grating length, whereas the phase of the transmitted wave increases monotonically with length at high input intensities.

Analysis of Mode Coupling Assisted Kerr Comb Generation in Normal Dispersion System

By using nonlinear coupled mode equations, we numerically investigated the generation of a Kerr comb in a normal dispersion microcavity system, where mode coupling between two cavity modes is present. In contrast to previous studies, our model is rigorous in which we fully considered the mode coupling between two modes. We investigated the phase matching condition to obtain the suitable parameters needed to form a free-spectral-range (FSR)-selectable comb. Our calculations are in good agreement with existing experimental results, and enable us to obtain a good understanding of the phenomenon; moreover our model also allows us to reveal new phenomena. We investigated 1-FSR comb generation in detail and found that randomly oscillating behavior will appear when detuning is scanned, which has never before been clearly observed. This modeling approach will be a powerful tool that will assist dispersion engineering for Kerr comb generation and the frequency tuning needed to generate a deterministic mode-locked comb.

A numerical investigation of enhanced backward second-harmonic generation in one-dimensional PIM/NIM structure

In this paper, we have demonstrated a numerical investigation of an enhanced backward second-harmonic generation (BSHG) effect in a one-dimensional positive-index material/negative-index material (1D-PIM/NIM) structure with nonlinear deep grating. The 1D-PIM/NIM structure composed of common linear PIM layers and NIM layers, whose electric permittivity and magnetic permeability are described by Drude model for allowing negative refractive index behavior, embedded in nonlinear χ(2) material in periodically arrangement. To model BSHG phenomenon, we have developed a completed set of nonlinear coupled-mode equations (NCMEs) by perturbing nonlinear wave equation by a small factor with appropriate scale following a way of multiple-scale approach (MSA). Then, we have numerically solved the NCMEs to achieve the second-harmonic frequency output and conversion efficiency of backward field. We have also discussed a backward phase-matching (BWPM) condition, which has been satisfied by tuning a fundamental frequency (FF) at ω in a negative refractive index region and second-harmonic (SH) frequency at 2ω in a positive refractive index region, and band-edge local field enhancement condition, which has been created by rearranging PIM and NIM layers in optimal periodic fashion. By using both conditions, a conversion efficiency of BSHG can be dramatically enhanced.

Principle and noise performance of optical phase arithmetic devices using four wave mixing

The existing theoretical equations cannot provide an excellent guidance for developing four-wave mixing (FWM)-based optical logic devices, though the experiments have been done in several researches. The optimization of noise figure performances of such devices should be further investigated. In the paper, the universal analytic expressions for the amplitude and phase of the idler in degenerate or non-degenerate FWM process under pump depletion are derived in detail from the nonlinear coupled-mode equations for guiding optical waves propagation in highly nonlinear fiber. The universal analytic expressions are obtained by the first-and the third-kind of elliptic integrals. By using equivalent infinitesimal to calculate the limit of phase sensitive amplification, we find out the initial phase relationship between the idler and the input guided wave for phase-independent amplification, which is crucially important for explaining the operating principles of the FWM-based adder and subtracter. As an example, the configuration of non-degenerate FWM-based hybrid arithmetic device with three logic functions of A+B-C, A+C-B, and B+C-A for QPSK signals is presented, and then the noise transfer characteristics in terms of signal-to-noise ratio (SNR) and error vector magnitude (EVM) are taken into account by adjusting the fiber length, input wavelength, and optical power. The calculation results show as follows. 1) This kind of arithmetic device has a noise figure of about 1.1 dB and an input SNR of more than 24 dB is necessary for the symbol error rate of 10-3 without forward error correction, corresponding to an output EVM of 23.2%. 2) The length of highly nonlinear fiber used in the hybrid arithmetic device may be taken flexibly, provided that the variation of FWM conversion efficiency is controlled in a range of 1 dB relative to the maximum, with an EVM fluctuation of less than for the idlers. 3) The hybrid arithmetic device has an operating optical bandwidth of about 16 nm for the SNR degradation of 1.3 dB. 4) The output EVM increases with the increase of input power, and the allowable input power should be no more than 100 mW for an input SNR of 28 dB, noting that the larger the input SNR, the higher the allowable input power is.

Solitary wave solutions of the cubic-quintic-septic nonlinear Schrödinger equation in fiber Bragg gratings

We consider the nonlinear coupled mode equations with competing cubic-quintic-septic nonlinearities, describing propagation of femtosecond optical pulses through fiber Bragg grating structure. By means of the multiple scale analysis, the model is reduced into the perturbed nonlinear Schrödinger equation with non-Kerr terms and higher-order dispersions. The complex envelope function ansatz of Li is adopted to investigate general analytic solitary wave solutions and derive explicit bright and dark solitons for the resulting nonlinear Schrödinger type equation. The physical parameters of the shape preserving solutions such as power and pulse width are calculated in terms of the dependent model coefficients. The constraint conditions for the existence of exact solutions are presented.

Magnetically-controllable optical multi-stability in magneto-optic fiber Bragg gratings with potential applications to multi-level all-optical regeneration

Starting with the nonlinear coupled-mode equations of guided optical waves in the magneto-optic fiber Bragg grating (MFBG), the amplitude transfer curve of the transmitted light is numerically calculated for the incident right-circularly polarized wave, and the multi-stability is analyzed by introducing the parameter of jitter suppression. It is shown that, (i) the performance of amplitude jitter suppression in the stable states of high level is better than that of low level; (ii) the jitter suppression in the multi-stable regions can be enhanced when the magnetic field is applied to the MFBG in the opposite direction of the incident wave; and (iii) by adjusting the applied magnetic field, the multi-stable levels can be tuned flexibly, which is helpful for developing the intelligent all-optical devices for multilevel regeneration.

Mode Spectrum and Temporal Soliton Formation in Optical Microresonators

The formation of temporal dissipative solitons in optical microresonators enables compact, high-repetition rate sources of ultrashort pulses as well as low noise, broadband optical frequency combs with smooth spectral envelopes. Here we study the influence of the microresonator mode spectrum on temporal soliton formation in a crystalline MgF2 microresonator. While an overall anomalous group velocity dispersion is required, it is found that higher order dispersion can be tolerated as long as it does not dominate the resonator's mode structure. Avoided mode crossings induced by linear mode coupling in the resonator mode spectrum are found to prevent soliton formation when affecting resonator modes close to the pump laser frequency. The experimental observations are in excellent agreement with numerical simulations based on the nonlinear coupled mode equations. The presented results provide for the first time design criteria for the generation of temporal solitons in optical microresonators.

Optical parametric amplification in one-dimensional photonic bandgap structures

In this paper, optical parametric amplification based on the degenerate four-wave mixing principle in a one-dimensional photonic bandgap (PBG) structure has been numerically studied. First, the multiple scale method was introduced to derive a complete set of nonlinear coupled-mode equations for a finite structure with different inhomogeneous nonlinear coefficients than those used in previous works. This finite structure is composed of 680 dielectric layers, which are alternating half-wave/eight-wave films. The wavelengths of the pump, signal, and idler pulses have been determined from the transmission spectrum, which was illustrated by using the transfer matrix method. The parametric interaction of the pump, signal, and idler pulses inside PBG structure has been numerically simulated by using the split-step Fourier transform method. The results of the simulation have shown that the intensities of the signal and idler have exponential growth with respect to the number of layers in the medium. Meanwhile, pump wavevector detuning directly affects the intensities of both pulses due to a band-edge phase-matching condition that might be achieved from only one optimal detuning parameter. Moreover, both the amplification gain and the conversion efficiency of the idler pulse have been shown to be dependent on the bandwidth of the pump pulse spectrum. A very narrow pulse, with a bandwidth much less than the relevant transmission peak, enables the highest amplification and conversion efficiency in this medium because the most efficient phase-matched condition occurs in this situation. Finally, the conversion efficiency grows exponentially with input pump intensity for several input signal intensities. Furthermore, the maximum conversion efficiencies directly vary with input signal intensity.

Investigation of crosstalk suppression techniques for multi-wavelength regeneration based on data-pump FWM

Multi-wavelength regenerators free of inter-channel crosstalk are desirable for wavelength-division-multiplexing systems from the cost-effectiveness point of view. Three crosstalk suppression methods, including polarization multiplexing, time interleaving and bidirectional transmission, are experimentally investigated in a data-pump four-wave mixing (FWM)-based dual-wavelength regeneration system. The results show that: (1) polarization multiplexing can suppress inter-channel degenerate FWM, but it cannot suppress the non-degenerate FWM unless the fiber has a sufficiently high birefringence; (2) the optimal relative delay time is half a bit period for input signals with 50% duty cycle in the time-interleaving system; (3) in the bidirectional system, if the regenerated signal and the opposite input signal have the same wavelength, the influence of the scattering wave should be considered. Furthermore, an eight-wavelength regeneration scheme is put forward and the inter-modulation characteristics of the regenerated channels are calculated from the nonlinear coupled-mode equations. The Q factor of each regenerated channel can be more than doubled when the duty cycle is 0.2 and the fiber DGD is 40ps/km.

EFFECTS OF NONLINEARITY ON WAVE PROPAGATION IN FIBER BRAGG GRATINGS

In this paper, we show that the solution of linear coupled mode equations (LCME) is a good approximation to that of the nonlinear coupled mode equations (NLCME) for small times, provided that the nonlinearity is weak. We bound the difierence between the two solutions using energy estimates. We illustrate our flndings in numerical examples. This work is motivated by the need to examine the behavior of a flber Bragg grating when large amplitude light is propagated in the flber. We start with the assumption that the grating has been designed for operation in the linear regime, while the flber itself has a mild nonlinearity. A flber Bragg grating, in its simplest form, is a length of flber whose index of refraction is periodic in the direction of light propagation. It can be made by exposing a treated flber to ultra- violet radiation which changes the index of refraction. It has many uses, for instance, flltering and dispersion compensation (6). It works by coupling the forward- and backward-moving waves in the flber. The periodicity and the wavelength of light determines the coupling strength. While it is not the scope of this work to examine the rich phenomena that result from nonlinearity and periodicity, it should be mentioned that some work exploring such aspects as solitons, gap solitons, and light-stopping, exists (3,4,9). In this work, we start with the nonlinear coupled mode equations (NLCME), whose derivation can be found in (9). It has been pointed out that NLCME is related

Polychromatic Solitary Waves in a Periodic and Nonlinear Maxwell System

We consider the one-dimensional Maxwell equations with low contrast periodic linear refractive index and weak Kerr nonlinearity. In this context, wave packet initial conditions with a single carrier frequency excite infinitely many resonances. On large but finite time-scales, the coupled evolution of backward and forward waves is governed by nonlocal equations of resonant nonlinear geometrical optics. For the special class of solutions which are periodic in the fast phase, these equations are equivalent to an infinite system of nonlinear coupled mode equations, the so-called extended nonlinear coupled mode equations, or xNLCME. Numerical studies support the existence of long-lived spatially localized coherent structures, featuring a slowly varying envelope and a train of carrier shocks. In this paper we explore, by analytical, asymptotic, and numerical methods, the existence and properties of spatially localized structures of the xNLCME system for the case where the refractive index profile consists of a periodic array of Dirac delta functions. We consider, in particular, the limit of small amplitude solutions with frequencies near a spectral band edge. In this case, stationary xNLCME is well approximated by an infinite system of coupled, stationary, nonlinear Schrödinger (NLS) equations, the extended nonlinear Schrödinger system, xNLS. We embed xNLS in a one-parameter family of equations, xNLS$^\epsilon$, which interpolates between infinitely many decoupled NLS equations ($\epsilon=0$) and xNLS ($\epsilon=1$). Using bifurcation methods we show existence of solutions for a range of $\epsilon\in(-\epsilon_0,\epsilon_0)$ and, by a numerical continuation method, establish the continuation of certain branches all the way to $\epsilon=1$. Finally, we perform time-dependent simulations of a truncated xNLCME and find the small-amplitude near–band edge gap solitons to be robust to both numerical errors and the NLS approximation.

Modeling Self-Similar Optical Pulse Compression in Nonlinear Fiber Bragg Grating Using Coupled-Mode Equations

Nearly pedestal-free optical pulse compression using self-similar chirped optical solitons near the photonic bandgap (PBG) structure of nonlinear fiber Bragg gratings (NFBGs) with exponentially decreasing dispersion is investigated using the generalized nonlinear coupled-mode equations (NLCMEs). The full dispersion characteristics and the effect of PBG are included. We find that the ratio of the frequency detune of the pulse's center frequency from the Bragg frequency of the NFBG to coupling coefficient of the NFBG is an important design parameter of the NFBG optical pulse compressor. We carried out a comprehensive study on the effect of the ratio of frequency detune to coupling coefficient, grating length, initial chirp, initial dispersion, and initial pulsewidth on the self-similar optical pulse compression. We also studied the compression of both the hyperbolic secant and Gaussian-shaped pulses and the effect of variation in the initial pulsewidth on the optical pulse compression.