Split-step Method Research Articles

A Fast Wavelet-to-Wavelet Propagation Method for the Simulation of Long-Range Propagation in Low Troposphere

As an alternative method to the discrete split-step Fourier (DSSF) method, the split-step wavelet method has recently been derived. This method based on a wavelet decomposition of the field is efficient and accurate in simulating the long-range propagation of electromagnetic waves in the low troposphere. In this article, to further improve the computational efficiency and take full advantage of wavelet characteristics, a wavelet-to-wavelet propagation (WWP) method is proposed. The propagation, variable refractivity, and apodization are accounted in the wavelet domain. The computational complexity is reduced, since no transforms are performed between wave fields and wavelet coefficients. This method works well for the propagation over a planar ground with a constant impedance. When a variable impedance ground condition and/or an irregular relief are considered, a hybridization with the split-step wavelet method is proposed. These methods are tested and validated by means of numerical experiments, showing very good efficiency and accuracy.

Algorithm for Solving a System of Coupled Nonlinear Schrödinger Equations by the Split-Step Method to Describe the Evolution of a High-Power Femtosecond Optical Pulse in an Optical Polarization Maintaining Fiber

This article proposes an advanced algorithm for the numerical solution of a coupled nonlinear Schrödinger equations system describing the evolution of a high-power femtosecond optical pulse in a single-mode polarization-maintaining optical fiber. We use the algorithm based on a variant of the split-step method with the Madelung transform to calculate the complex amplitude when executing a nonlinear operator. In contrast to the known solution, the proposed algorithm eliminates the need to numerically solve differential equations directly, concerning the phase of complex amplitude when executing the nonlinear operator. This made it possible, other things being equal, to reduce the computation time by more than four times.

Modulation instability in fractional Schrödinger equation with cubic–quintic nonlinearity

In this paper, we investigate the properties of modulation instability (MI) in the fractional Schrödinger equation with cubic–quintic nonlinearity (SECN), particularly focusing on some novel dynamical behaviors different from those in the conventional SECN. First, we apply the linear-stability theory to derive an analytical expression for instability gain in such novel fractional physical system. Just like the case of the conventional SECN, our analysis shows that the dynamical behavior of MI in the fractional SECN still depends on the various combinations of the signs of cubic and quintic nonlinearities. In particular, MI is found to still occur even in the fractional Schrödinger equation (SE) with self-defocusing cubic nonlinearity if the quintic nonlinearity is positive. Moreover, we further find that the Lévy indices related to the fractional effects mainly act to increase not only the fastest growth frequency but also the bandwidth, when compared to the case in the conventional SECN. However, the corresponding maximum gain is completely independent on such Lévy indices. The theoretical predictions are confirmed by the numerical results from the split-step Fourier method. Our findings suggest that the novel fractional systems with cubic–quintic nonlinearity provide some possibilities to control the dynamical behavior of MI and new solitary waves.

The local wave phenomenon in the quintic nonlinear Schrödinger equation by numerical methods

The nonlinear Schrödinger hierarchy has a wide range of applications in modeling the propagation of light pulses in optical fibers. In this paper, we focus on the integrable nonlinear Schrödinger (NLS) equation with quintic terms, which play a prominent role when the pulse duration is very short. First, we investigate the spectral signatures of the spatial Lax pair with distinct analytical solutions and their periodized wavetrains by Fourier oscillatory method. Then, we numerically simulate the wave evolution of the quintic NLS equation from different initial conditions through the symmetrical split-step Fourier method. We find many localized high-peak structures whose profiles are very similar to the analytical solutions, and we analyze the formation of rouge waves (RWs) in different cases. These results can be helpful to understand the excitation of nonlinear waves in some nonlinear fields, such as the Heisenberg ferromagnetic spin system in condensed matter physics, ultrashort pulses in nonlinear optical fibers, and so on.

Optimal power allocation in nonlinear MDM‐WDM systems using Gaussian noise model

Mode-division multiplexing (MDM) using few-mode fibre (FMF) has received increasing attention to address the exponential growth of data traffic in long-haul optical communication systems. Also, combining the MDM with wavelength-division multiplexing (WDM) is a promising approach for dramatically growing the transmission capacity in such systems. However, a major barrier in this regard is the FMF nonlinear effects, which can significantly reduce the link performance. In this paper, in order to alleviate the FMF nonlinear effects, we focus on power allocation in FMF links by optimizing the input power of each optical WDM channel of each spatial mode, which leads to maximizing the total capacity transmission and also the minimum signal to noise ratio (SNR) margin. The FMF nonlinearity has been already modelled as the Gaussian noise (GN) for which no closed-form formulation has been developed so far. Here, we derive a closed-form GN model for this problem and verify it by comparing with the integral-form GN model and split-step Fourier method. In this approach, an optimal power is independently determined for each channel of each mode by optimizing a capacity maximization and a minimum SNR margin maximization problem in convex forms. The performance of different links including the single mode fibre-WDM, MDM-single channel, and MDM-WDM are compared using computer simulations. These systems are comprehensively investigated in equal/non-equal required SNR as well as flat/non-flat amplifier gain scenarios. It is shown that optimized power allocation to each channel of each mode has a significant enhancement in the minimum SNR margin maximization scheme compared to the best equal power allocation. Furthermore, this improvement is much more in non-equal required SNR and the non-flat amplifier gain scenarios, showing the efficiency of the established approach in practical communication links.

Synergy between the negative absolute temperature and the external trap for a Bose-Einstein condensate under optical lattices

We provide a generic but exact analytical model for realizing negative absolute temperature in ultracold atoms under a quasi-periodic optical lattice (bi-chromatic) and an expulsive trap. We keep freedom for maximum tunability for both kinds of trap profile, where bi-chromatic lattice is quasi-periodic except two tunable limits, from a pure optical lattice to another doubly periodic optical lattice. The expulsive trap can be tuned from zero to a stronger one. All the required conditions for achieving negative temperature domain have been explored by identifying appropriate trap parameters. Condensate density in each lattice site is controlled and Anderson-like localization is emphasized. A thorough study of a quantitative estimate of the temperature is performed in the negative domain by controlling the trap, where the roles of all the trap parameters are precisely identified. A numerical stability analysis by Fourier split-step method makes the study more useful and the proposed family of solution are found quite stable. This is also connected to the merit of negative temperature through mean deviation data with varied trap parameters.

Influence of cubic nonlinearity effect on quadratic solitons in boundary-constrained self-focusing oscillatory response function system

In this paper, we theoretically study the influence of cubic nonlinearity effect on quadratic solitons in the boundary-constrained self-focusing oscillatory response function system. Based on the Newton iteration approach, we numerically solve the nonlinear coupled-wave equations with both quadratic and cubic nonlinearity. Moreover, a family of quadratic solitons is obtained. By comparing the quadratic solitons with both quadratic and cubic nonlinearity with those with only quadratic nonlinearity, we find that the cubic nonlinearity changes the transverse distribution of the soliton profiles only slightly. However, because of the existence of the cubic nonlinearity, quadratic solitons can be found only in the strongly nonlocal case and general nonlocal case, and they cannot be found in the weakly nonlocal case, in which the quadratic solitons with only quadratic nonlinearity can be found. In addition, the existence of cubic nonlinearity reduces the number of extended half-periods of the quadratic solitons. Through the linear stability analysis of the obtained soliton solutions, it is found that the stability intervals of solitons are also shrunk due to the existence of the cubic nonlinearity. The results of the linear stability analysis are verified by the numerical simulations of soliton propagations through using the split-step Fourier method.

Numerical Study of High-Order Soliton Generation in Photonic Crystal Fibers: Effect of the Pulse Shape and Its Duration

High-order soliton generation in photonic crystal fibers was simulated using split-step Fourier method (SSFM) and MATLAB. The input pulse shape was studied for various types, super-Gaussian, chirped and pulses obtained from a train of individual Gaussian pulses. The study shows that the order of soliton generated depends on pulse shape, input power and dispersion and nonlinear parameters of the photonic crystal fiber.

Joint Power and Gain Allocation in MDM-WDM Optical Communication Networks Based on Enhanced Gaussian Noise Model

Achieving reliable communication over different wavelength channels and modes is one of the main goals of Mode Division Multiplexing-Wavelength Division Multiplexing (MDM-WDM) transmission. The reliability can be described by the minimum Signal to Noise Ratio (SNR) margin which depends on launch power, the gain of Few-Mode Erbium-Doped Fiber Amplifiers (FM-EDFA), and the nonlinear impairments of Few-Mode Fiber (FMF). In this paper, we develop the Enhanced Gaussian Noise (EGN) nonlinear model for FMF, which can be used in both weak and strong coupling regimes. We validate the model by comparing simulation results with those obtained through the Split-Step Fourier Method. Based on our proposed EGN model, we address the problem of joint optimized power and gain allocation based on minimum SNR margin maximization when accounting for practical FM-EDFA constraints such as saturation power and maximum gain. The problem is solved using a convex optimization approach and considering different scenarios such as the best equal power, optimized power, and joint optimized power and gain. Results demonstrate that the minimum SNR margin improvement for the joint optimized power and gain allocation compared to the best equal power allocation is <inline-formula> <tex-math notation="LaTeX">$1.4~dB$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$1.7~dB$ </tex-math></inline-formula> for MDM-single channel and single-mode fiber-WDM systems, respectively.

Real-Time Dense Field Phase-to-Space Simulation of Imaging Through Atmospheric Turbulence

Numerical simulation of atmospheric turbulence is one of the biggest bottlenecks in developing computational techniques for solving the inverse problem in long-range imaging. The classical split-step method is based upon numerical wave propagation which splits the propagation path into many segments and propagates every pixel in each segment individually via the Fresnel integral. This repeated evaluation becomes increasingly time-consuming for larger images. As a result, the split-step simulation is often done only on a sparse grid of points followed by an interpolation to the other pixels. Even so, the computation is expensive for real-time applications. In this paper, we present a new simulation method that enables <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">real-time</i> processing over a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">dense</i> grid of points. Building upon the recently developed multi-aperture model and the phase-to-space transform, we overcome the memory bottleneck in drawing random samples from the Zernike correlation tensor. We show that the cross-correlation of the Zernike modes has an insignificant contribution to the statistics of the random samples. By approximating these cross-correlation blocks in the Zernike tensor, we restore the homogeneity of the tensor which then enables Fourier-based random sampling. On a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$512\times 512$</tex-math></inline-formula> image, the new simulator achieves 0.025 seconds per frame over a dense field. On a <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$3840 \times 2160$</tex-math></inline-formula> image which would have taken 13 hours to simulate using the split-step method, the new simulator can run at approximately 60 seconds per frame.

Numerical Treatment of Boundary Conditions to Reduce High-Frequency Artifacts in Simulations of Distributed-Feedback Lasers

High-frequency artifacts may occur when the coupled-mode equations describing distributed-feedback lasers are solved by a variety of numerical methods, such as method-of-characteristics schemes and the split-step method. We propose a simple technique to suppress this artifact. Its idea is to modify the numerical implementation of the boundary conditions to promote the leakage of highest-frequency modes out of the medium. At the same time, this implementation only minimally affects the physical part of the solution. We demonstrate the effectiveness of this technique for first- and second-order schemes.

Simulation of spatial coherence of laser echo light field from random rough surface in turbulent atmosphere

According to the generalized Huygens-Fresnel principle, we derive the analytical formula for the complex degree of coherence of the echo light field under the von Karman atmospheric turbulence spectrum condition. Based on split-step beam propagation method of the turbulent phase screen and the target surface model, the fold pass propagation simulation of the laser in the turbulent atmosphere is realized. The dynamic speckle characteristics on the image plane are consistent with the experimental phenomenon. Firstly, the simulation results of the complex degree of coherence and phase structure function of the mirrored reflection echo light field are compared with the theoretical values, which verifies the correctness of the algorithm. Based on this, the complex degree of coherence of the echo light field reflected by the optical rough surface is calculated and analyzed. The results show that on a double-path turbulent flow path of 1.1 km, in other words, it transmits 2.2 km in unfolded mode, the spatial coherence of the echo light field is very sensitive to the root mean square value of height. When the root-mean-square value of height is close to the wavelength, the coherence is seriously degraded. When the correlation length of the target surface is much larger than the atmospheric coherence length, the coherence length of the echo light field is relatively close to the set spatial coherence length. When the correlation length of the target surface is close to the atmospheric coherence length, the influence of the rough surface of the target on the beam coherence cannot be ignored. When the correlation length of the target surface is much smaller than the atmospheric coherence length, the target surface characteristics have a dominant influence on the echo coherence, the spatial coherence of the light field is seriously degraded, and the echo is close to incoherent light. Considering the smooth target reflection surface, the greater the strength of turbulence, the faster the complex coherence decreases with space. The atmospheric coherence diameter &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}${r_0}$\end{document}&lt;/tex-math&gt;&lt;alternatives&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20212420_M1.jpg"/&gt;&lt;graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="14-20212420_M1.png"/&gt;&lt;/alternatives&gt;&lt;/inline-formula&gt; can be calculated further according to the complex degree of coherence. For the Pearson correlation coefficient the simulation value and theoretical value are both 0.998, which indicates that the atmospheric coherence diameter calculated by the complex degree of coherence has a high correlation with the theoretical value. This research provides a theoretical basis for the coherent detection scheme of echoes from rough surfaces in the turbulent atmosphere. The simulation algorithm extracts the target surface features by analyzing the variation of the complex coherence of laser echo signals in the turbulent atmosphere with the spatial distance, and also provides a method of using the known target surface to obtain path turbulence information.

Numerical study of self-focusing and filament formation of intense vortex beams in plasmas

We have studied the self-focusing and filamentation of vortex laser beams propagating in underdense plasmas with different values of the topological charge and initial laser powers. The self-focusing dynamics of vortex laser beams is closely related to the topological charge, which has attracted widespread attention. Based on the paraxial approximation of the Helmholtz equation, the steady-state solution of vortex beams propagating in underdense plasmas is deduced, and the expression for critical power of vortex laser beams is obtained. Furthermore, using the split-step Fourier method to solve the wave equation, we analyze the numerical images of vortex laser beams propagating in underdense plasma under different parameters. The simulation results show that the critical power for self-focusing is positively correlated with the topological charge of vortex laser beams. When the initial laser power is high enough, the vortex laser beam will first be focused into a thin ring, and then the modulation amplitude increases continuously, which eventually leads to the ring structure breaking into filaments. The number of filaments has an integer multiple relationship with the topological charge. In the process of filamentation, the radius and the maximum light intensity of vortex laser beam both change drastically. After the filamentation process is completed, the vortex laser beam continues to propagate with a new topological type. We further increase the incident laser power and find that the number of filaments of the vortex laser beam increases. The increased number of filaments is the value of its topological charge at each time. With the development of filament instability, higher-order modulation instability can be excited in the later stage, and the intensity of filaments will exhibit angular modulation. Our results show that in compared with the standard Gaussian beam, the propagation behavior of vortex laser beams in underdense plasmas is much more stable under the same power, wavelength and plasma parameters. The propagation characteristics of vortex laser beams are helpful to the theoretical and experimental study of stimulated backward Raman amplification of ultra-strong vortex beam in underdense plasmas.

Multi-span long-haul fiber transmission model based on cascaded neural networks with multi-head attention mechanism

In this paper, a novelty fiber transmission model consisting of cascaded neural networks with the multi-head attention mechanism has been put forward to solve signal transmission prediction problems in multi-span long-haul fiber link. After appropriately training the model with collected data by the gradient descent method, it can gradually handle the rules of signals&#x0027; changes over each span and predict the signal transmission results with notably low time cost compared with traditional split-step Fourier method based long-haul model. Through numerical demonstration, this new model can predict 1000km fiber link with symbol rate up to 40GBaud in the QAM modulation with extremely low predicting error.

Comparison principles and applications to mathematical modelling of vegetal meta-communities

&lt;abstract&gt;&lt;p&gt;This article partakes of the PEGASE project the goal of which is a better understanding of the mechanisms explaining the behaviour of species living in a network of forest patches linked by ecological corridors (hedges for instance). Actually we plan to study the effect of the &lt;italic&gt;fragmentation&lt;/italic&gt; of the habitat on biodiversity. A simple neutral model for the evolution of abundances in a vegetal metacommunity is introduced. Migration between the communities is explicitely modelized in a deterministic way, while the reproduction process is dealt with using Wright-Fisher models, independently within each community. The large population limit of the model is considered. The hydrodynamic limit of this split-step method is proved to be the solution of a partial differential equation with a deterministic part coming from the migration process and a diffusion part due to the Wright-Fisher process. Finally, the diversity of the metacommunity is adressed through one of its indicators, the mean extinction time of a species. At the limit, using classical comparison principles, the exchange process between the communities is proved to slow down extinction. This shows that the existence of corridors seems to be good for the biodiversity.&lt;/p&gt;&lt;/abstract&gt;

QTRAJ 1.0: A Lindblad equation solver for heavy-quarkonium dynamics

We introduce an open-source package called QTraj that solves the Lindblad equation for heavy-quarkonium dynamics using the quantum trajectories algorithm. The package allows users to simulate the suppression of heavy-quarkonium states using externally-supplied input from 3+1D hydrodynamics simulations. The code uses a split-step pseudo-spectral method for updating the wave-function between jumps, which is implemented using the open-source multi-threaded FFTW3 package. This allows one to have manifestly unitary evolution when using real-valued potentials. In this paper, we provide detailed documentation of QTraj 1.0, installation instructions, and present various tests and benchmarks of the code. Program summaryProgram Title: QTraj 1.0CPC Library link to program files:https://doi.org/10.17632/34hbsb6pzy.1Developer's repository link::https://bitbucket.org/kentphysics/qtraj-fftwLicensing provisions:: GPLv3Programming language:: CNature of problem:: Solution of the Lindblad equation for evolution of the heavy-quarkonium reduced density matrix in a quark-gluon plasma.Solution method:: We use the quantum trajectories algorithm to solve the Lindblad equation.

Impact of two-photon absorption on two-dimensional solitons in silicon waveguides with a nonstationary potential*

We study the effects of weak two-photon absorption on two-dimensional (2D) solitons interaction in a silicon-based waveguide under a framework of coupled perturbed D nonlinear Schrödinger equations with a pure Kerr nonlinearity and a nonstationary potential. We extend the perturbative analysis for collisions of two one-dimensional solitons in silicon waveguides to derive the theoretical expressions for the amplitude evolution of a single perturbed soliton and for the abrupt amplitude downshift of a colliding soliton due to a fast interaction between two 2D solitons in silicon lattices. The theoretical calculations are then validated by the extensive simulations of the coupled perturbed D nonlinear Schrödinger model with periodic and non-periodic potentials. The simulations are implemented by using the accelerated imaginary-time evolution scheme and the split-step Fourier method. These results allow us to accurately capture the abrupt energy downshift of a colliding soliton in silicon-based two-dimensional waveguides.

Spatio-spectral dynamics of soliton pulsation with breathing behavior in the anomalous dispersion fiber laser

We have numerically and experimentally observed the soliton pulsation with obvious breathing behavior in the anomalous fiber laser mode-locked by a nonlinear polarization rotation technique. The numerical study of the soliton pulsation with breathing behavior was analyzed through the split-step Fourier method at first, and it was found that the phase difference caused by the polarization controller would affect the breathing characteristics. Then, taking advantage of the dispersive Fourier transform technique, we confirmed the breathing characteristic of soliton pulsation in the same fiber laser as the simulation model experimentally. These results complement the research on the breathing characteristic of soliton pulsation.

A Phase Correction for Long-Distance Propagation Using Split-Step Methods in Nonrectangular Domains

In this letter, an additional phase correction to Donohue and Kuttlers shift map theories is proposed. This correction enables a fast and accurate evaluation of the electromagnetic field at the receiver, both in terms of field amplitude and phase. Numerical tests are led in the troposphere, and the observed difference between the field computed with the corrected shift map and the field computed with a classic split-step method is less than -35 dBV/m. The corrected shift map is also applied to a Radio Occultation configuration for which the phase information is important. The split-step wavelet method is applied in a reduced non-rectangular domain and shows good computational times.

Solitons propagation dynamics in a saturable PT-symmetric fractional medium

In the current research, the propagation of solitons in a saturable PT-symmetric fractional system is studied by solving nonlinear fractional Schrödinger equation. Three numerical methods are employed for this purpose, namely Monte Carlo based Euler–Lagrange variational schema, split-step method, and extrapolation approach. The results show good agreement and accuracy. The effect of different parameters such as potential depth, Levy indices, and saturation parameter, on the physical properties of the systems such as maximum intensity and soliton width oscillations are considered.