Split-step Fourier Research Articles

ALL-OPTICAL LOGIC GATE OPERATION WITH A MULTI-CORE NONLINEAR PHOTONIC CRYSTAL FIBER

In this paper, we propose the design of a photonic crystal fiber with four central cores infiltrated by a high-index liquid to achieve highly efficient control of light guidance. We analyze the field distribution, effective mode area, and dispersion characteristics of the fundamental guided modes of the fiber. Within the coupled-mode theory, the pulse propagation in the fiber is governed by coupled nonlinear Schrödinger equations. We use the split-step Fourier method to simulate the propagation of pulses numerically. The results show three features of the dynamics: oscillation, switching, and self-trapping. We predict that the fiber could operate as a logic-gate device by introducing suitable input and control signals.

Mid-Infrared Supercontinuum Generation in Highly Nonlinear Chalcogenide Fibers

Interest in mid-infrared broadband laser light sources has surged due to applications in trace gas detection, free-space communications, and countermeasures. Progress in supercontinuum generation leverages fiber-based near-infrared and bulk-optic mid-infrared pump sources. In this paper, the Generalized Nonlinear Schrödinger Equation has been solved, using the Split Step Fourier Method, to simulate the pulse propagation and mid-infrared supercontinuum generation, inside a fiber composed of highly nonlinear As2Se3/As2S3 chalcogenide glass. The effect of various parameters, including fiber nonlinearity, Group Velocity Dispersion (GVD), input power and pulse-width, anomalous and normal dispersion pumping regime, etc. on the output supercontinuum bandwidth has been extensively studied. A tapered chalcogenide fiber is modeled to facilitate continuous simultaneous modification of the GVD and the Kerr nonlinearity parameter. Pumping the waveguides with 230-fs secant pulses at a peak power of 4.2-kW yields a mid-IR supercontinuum extending from [Formula: see text] to [Formula: see text] micrometers.

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In this article, we investigated the solitary wave solutions of the KdV–mKdV equation using Hirota’s bilinear method. Closed-form analytical single and multiple solitary wave solutions were obtained. Through qualitative methods and the analysis of solitary waveforms, we discovered that in addition to sech-type solitary waves, the system also contains Sech2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext{Sech}^{2}$$\\end{document}-type solitary waves. By employing the trial functions method, we obtained a single Sech2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext{Sech}^{2}$$\\end{document}-type solitary wave and verified its existence and stability using the split-Step Fourier Transform method. Furthermore, we use the collision of two Sech2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext{Sech}^{2}$$\\end{document}-type single solitary waves to excite a stable Sech2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext{Sech}^{2}$$\\end{document}-type double solitary wave. Similarly, we excite a stable triple solitary wave with three Sech2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext{Sech}^{2}$$\\end{document}-type single solitary waves. This method can also be used to excite stable multiple solitary waves. It is shown that these solitary wave solutions enrich the dynamic behavior of the KdV–mKdV equation and provide methods for solving Sech2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\ ext{Sech}^{2}$$\\end{document}-type solitary waves, which hold significant theoretical value.

Manipulating circular Airy beam dynamics with quadratic phase modulation in fractional systems under some diffraction modulations and potentials

Based on a split-step Fourier algorithm, the transmission of circular Airy beams with quadratic phase modulation (QPM) is investigated in the fractional Schrödinger equation (FSE) under diffraction modulations (periodic modulation, linear modulation and power function modulation) and external potentials (parabolic potential and linear potential). The results show that QPM is able to change the focusing position and intensity, as well as the transmission trajectory of the beam. In a periodic modulation, the circular Airy beam (CAB) exhibits periodic variation characteristics, and the beam splitting is retarded under the action of the QPM. The self-focusing distance of the beam is significantly reduced, and its transmission trajectory and beam width are altered by the QPM under the linear modulation. The CAB progressively evolves into a non-diffraction beam under the power function modulation, and the QPM is able to reduce the light intensity and increase the beam width as the Lévy index decreases. In a parabolic potential, CABs display autofocusing and defocusing behavior, and the QPM affects the intensity distribution and optical width of the beam. The CAB is deflected and evolves periodically in a linear potential. The beam width increases and gradually stabilizes with the addition of the QPM. The propagation of CABs controlled with QPM in parabolic and linear potentials is also analyzed in the frequency domain. The results demonstrate that we can control the transmission of CABs in an FSE optical system by rationally setting parameters such as QPM, modulation coefficients, and external potentials.

A theoretical study on deep ultraviolet supercontinuum generation in subwavelength optical fiber

In order to obtain supercontinuum (SC) spectra in the deep ultraviolet (UV) region, a generation method of UV SC is proposed based on subwavelength fiber (SWF) composed of background material MgF2. The nonlinear Schrödinger equation is solved by the split-step Fourier method, the optical fiber transmission characteristics are analyzed by the full-vector finite element, and the evolution of the pumping light into SC is numerically simulated in detail. The results of the numerical analysis show that the SC spectra have a spectral width extending from 229.11 nm to 2366.68 nm when 350 fs input pulses of 6 kW peak power, centered about λ0 = 400 nm, are launched into a 4 cm long MgF2 SWF. Such a deep UV SC spectrum has many applications, especially in telecommunication and military fields.

Nonlinear impairment compensation in multi-channel communication systems based on correlated digital backpropagation with separation of walk-off effect

Digital backpropagation (DBP) is a commonly used method to compensate for the nonlinear impairments of optical fiber channels in the electrical domain, offering a new approach to enhancing the data capacity without affecting the system performance. However, due to the repeated Fourier transform and inverse transform of the split-step Fourier method (SSFM), the complexity of DBP algorithm has increased significantly. In this paper, a correlated backpropagation algorithm with the separation of the walk-off effect is developed, which improves the computational step and significantly reduces the complexity of the algorithm. Simulation results demonstrate that in a 6-channel PDM-16QAM system, the proposed hybrid algorithm reduces the computational complexity by 73.96 % compared to the traditional DBP algorithm for an 800 km transmission.

CUDA-based focused Gaussian beams second-harmonic generation efficiency calculator

We present an object-oriented programming (OOP) CUDA-based package for fast and accurate simulation of second-harmonic generation (SHG) efficiency using focused Gaussian beams. The model includes linear as well as two-photon absorption that can ultimately lead to thermal lensing due to self-heating effects. Our approach speeds up calculations by nearly 40x (11x) without (with) temperature profiles with respect to an equivalent implementation using CPU. The package offers a valuable tool for experimental design and study of 3D field propagation in nonlinear three-wave interactions. It is useful for optimization of SHG-based experiments and mitigates undesired thermal effects, enabling improved oven designs and advanced device architectures, leading to stable, efficient high-power SHG. Program summaryProgram Title:cuSHGCPC Library link to program files:https://doi.org/10.17632/hn76s7x848.1Developer's repository link:https://github.com/alfredos84/cuSHGLicensing provisions: MITProgramming language:▪, CUDANature of problem: The problem which is solved in this work is that of second-harmonic generation (SHG) performance degradation in a nonlinear crystal with focused Gaussian beams due to thermal effects. By placing the nonlinear crystal in an oven that controls temperature, the package computes the involved electric fields along the medium. The implemented model includes the linear and nonlinear absorption which occasionally lead to self-heating effect, degrading the performance of the SHG.Solution method: The coupled differential equations for three-wave interactions, which describe the field evolution along the crystal, are solved using the well-known Split-Step Fourier method. The temperature profiles are estimated using the finite-elements method. The field evolution and thermal effects are embedded in a self-consistent algorithm that sequentially and separately solves the electromagnetic and thermal problems until the system reaches the steady state. Due to the eventual computational demand that some problems may have, we chose to implement the coupled equations in the ▪/CUDA programming language. This allows us to significantly speed up simulations, thanks to the computing power provided by a graphics processing unit (GPU) card. The output files obtained are the interacting electric fields and the temperature profile, which have to be analyzed during post-processing.

Symbol error rate minimization using deep learning approaches for short-reach optical communication networks

Short-reach optical communication networks have various applications in areas where high-speed connectivity is needed, for example, inter- and intra-data center links, optical access networks, and indoor and in-building communication systems. Machine learning (ML) approaches provide a key solution for numerous challenging situations due to their robust decision-making, problem-solving, and pattern-recognition abilities. In this work, our focus is on utilizing deep learning models to minimize symbol error rates in short-reach optical communication setups. Various channel impairments, such as nonlinearity, chromatic dispersion (CD), and attenuation, are accurately modeled. Initially, we address the challenge of modeling a nonlinear channel. Consequently, we harness a deep learning model called autoencoders (AEs) to facilitate communication over nonlinear channels. Furthermore, we investigate how the inclusion of a nonlinear channel within an autoencoder influences the received constellation as the optical fiber length increases. Another facet of our work involves the deployment of a deep neural network-based receiver utilizing a channel influenced by chromatic dispersion. By gradually extending the optical length, we explore its impact on the received constellation and, consequently, the symbol error rate. Finally, we incorporate the split-step Fourier method (SSFM) to emulate the combined effects of nonlinearities, chromatic dispersion, and attenuation in the optical channel. This is accomplished through a neural network-based receiver. The outcome of this work is an evaluation and reduction of the symbol error rate as the length of the optical fiber is varied.

High-order rogue wave and mixed interaction patterns for the three-component Gross-Pitaevskii equationsin F=1 spinor Bose-Einstein condensates.

Under investigation are the three-component Gross-Pitaevskii equationsin F=1 spinor Bose-Einstein condensates. Various localized waves' generation mechanisms have been derived from plane wave solutions using modulation instability. The perturbed continuous waves produce a large number of rogue wave structures through the split-step Fourier numerical method. Based on the known Lax pair, we construct the generalized iterative (n,N-n)-fold Darboux transformation to generate various high-order solutions, including the bright-dark-bright structure of rogue waves, periodic waves, and their mixed interaction structures. Numerical simulations show that rogue waves with a two-peaked structure have robust noise resistance and stable dynamical behavior. The asymptotic states of high-order rogue waves as the parameter approaches infinity are also predicted using the large parameter asymptotic technique. In addition, the position of these localized wave patterns can be controlled by some special parameters. These results may help us understand the dynamic behavior of spinor condensates for the mean-field approximation.

Supercontinuum Generation by Controlling Pitch in Photonic Crystal Fibers

The influence of varying the distance between air holes (pitch) on the geography of solitone propagation through a photonic crystal fiber has been tested. The study depends on the Split-Step Fourier method and the results quantified using MATLAB. The first-order solitone was tested with the change in pitch, and it was found that there is a clear decay in the amplitude of the resulting pulse with an increase in pitch. When increasing the pitch in the case of second-order solitons, it was noticed that the pulse would split into multiple-order solitons down to higher-order solitons with the increase in pitch. In the case of third-order solitons, solitonic fission leads to the supercontinuum generation with increasing the pitch, where the supercontinuum generation was reached in this way depending on a very small energy source compared to the high energies used to generate this type of spectrum by the previous methods. In this study, I observed that when the pitch values increased in the third-order soliton, this result led to the use of supercontinuum generation (ScG) which has many applications such as medical and industrial applications and has an important role in modern communication systems. Keywords: Photonic Crystal Fiber, Split-Step Fourier, Soliton Pulse, Pitch, Supercontinuum Generation.

Two-dimensional modulation instability and soliton clusters in nonlocal media with competing cubic–quintic nonlinearities

Using linear-stability analysis, two-dimensional modulation instability (MI) of plane wave is studied in nonlocal media with competing cubic–quintic (CQ) nonlinearities, which shows that MI can be effectively eliminated by strong nonlocality and competing quintic nonlinearity. Furthermore, propagation properties of higher-order soliton clusters, i.e., Hermite–Gaussian (HG) and Laguerre–Gaussian (LG) solitons are also investigated. Bifurcated solutions of these solitons are obtained analytically with variational approach. We also demonstrate in detail the propagation dynamics of the HG and LG solitons with split-step Fourier transform.

Modulation instability in nonlinear media with sine-oscillatory nonlocal response function and pure quartic diffraction.

Modulation instability of one-dimensional plane wave is demonstrated in nonlinear Kerr media with sine-oscillatory nonlocal response function and pure quartic diffraction. The growth rate of modulation instability, which depends on the degree of nonlocality, coefficient of quartic diffraction, type of the nonlinearity and the power of plane wave, is analytically obtained with linear-stability analysis. Different from other nonlocal response functions, the maximum of the growth rate in media with sine-oscillatory nonlocal response function occurs always at a particular wave number. Theoretical results of modulation instability are confirmed numerically with split-step Fourier transform. Modulation instability can be controlled flexibly by adjusting the degree of nonlocality and quartic diffraction.

Optimizing ultrashort pulse in fiber laser based on artificial intelligence algorithm

Ultrashort pulses, characterized by their short pulse duration, diverse spectral content, and high peak power, are widely used in fields including laser processing, optical storage, biomedical sciences, and laser imaging. The complex, highly-nonlinear process of ultrashort pulse evolution within fiber lasers is influenced by numerous aspects such as dispersion, loss, gain, and nonlinear effects. Traditionally, the split-step Fourier transforms method is employed for simulating ultrashort pulses in fiber lasers, which involves traversing multiple parameters within the fiber to attain the pulse’s optimal state. The simulation is a significantly time-consuming process. Here, we use a neural network model to fit and predict the impact of multiple parameters on the pulse characteristics within fiber lasers, enabling parameter optimization through genetic algorithms to determine the optimal pulse duration, pulse energy, and peak power. Integrating artificial intelligence algorithms simplifies the acquisition of optimal pulse parameters and enhances our understanding of multiple parameters’ impact on the pulse characteristics. The investigation of ultrashort pulse optimization based on artificial intelligence holds immense potential for laser design.

Co-LSTM-based fiber link modeling with ASE noise tracking for long-haul coherent optical transmission.

In this Letter, a novel, to the best of our knowledge, channel modeling scheme based on cascading chromatic dispersion-nonlinearity feature decoupling modules is proposed with the center-oriented long short-term memory (Co-LSTM) network structure adopted for modeling nonlinearity of each optical fiber span. By tracking the amplified spontaneous emission (ASE) noise at the output of each fiber span, the Co-LSTM-based channel modeling scheme achieves high waveform accuracy for long-haul coherent optical transmission compared with the conventional split-step Fourier transform method (SSFM) while saving calculation time by almost one order of magnitude.

Synergic action of linear dispersion, second-order nonlinearity, and third-order nonlinearity in shaping the spectral profile of a femtosecond pulse transporting in a lithium niobate crystal

We present a detailed theoretical and numerical analysis on the temporal-spectral-spatial evolution of a high-peak-power femtosecond laser pulse in two sets of systems: a pure lithium niobate (LN) plate and a periodically poled lithium niobate (PPLN) plate. We develop a modified unidimensional pulse propagation model that considers all the prominent linear and nonlinear processes and carried out the simulation process based on an improved split-step Fourier transformation method. We theoretically analyze the synergic action of the linear dispersion effect, the second-order nonlinearity (2nd-NL) second-harmonic generation (SHG) effect, and the third-order nonlinearity (3rd-NL) self-phase modulation (SPM) effect, and clarify the physical mechanism underlying the peculiar and diverse spectral broadening patterns previously reported in LN and PPLN thin plate experiments. Such analysis and discussion provides a deeper insight into the synergetic contribution of these linear and nonlinear effects brought about by the interaction of a femtosecond laser pulse with the LN nonlinear crystal and helps to draw a picture to fully understand these fruitful optical physical processes, phenomena, and laws.

Dark solitons in liquid crystals with competing re-orientational and thermal nonlocal nonlinearities

The properties of dark solitons in liquid crystals with competing nonlinearities are investigated by the model proposed by Jung. Under the condition of the rectangular response function, the relationship between the width of the dark soliton and the parameters of the medium is obtained by the variational method. It is found that the width of the dark solitons first decreases and then increases with the increment of the orientational nonlocality, the width of the soliton increases monotonously with the increase of the thermal nonlocality, and the width of the soliton decreases monotonously with the increase of the thermal nonlinearity coefficient. It is also found that the width of dark solitons is more sensitive to the orientational nonlocality than the other parameters. In addition, the width of the soliton increases monotonously with the grayness of the solitons. The analytical results are consistent with the numerical simulations that are obtained by the split-step Fourier algorithm.

Numerical investigation of a wideband supercontinuum source based on a large-mode-area photonic crystal fiber pumped at 1.3 μm

A numerical investigation was conducted to obtain a supercontinuum spanning about two octaves using a large mode area photonic crystal fiber (PCF) pumped at 1.3 μm. In our study, a 1.3 μm femtosecond laser and a silica-based large mode area PCF were selected as the pump source and nonlinear medium, respectively. The nonlinear Schrodinger equation was solved with split-step Fourier method to simulate the evolution of pulse and the broadening of spectrum. The effect of several parameters including the length of PCF, the pulse width, and the average pump power on characteristics of the output spectrum was studied. The simulation results revealed that the supercontinuum extended from near 600 nm to over 2450 nm at 20 dB with length of 30 cm, pulse width of 100 fs and average power of 12 W, respectively. This work proved this large mode area PCF a potentially excellent medium for supercontinuum source and provided some theoretical guidance for future experiments.

Tunable Three-Wavelength Fiber Laser and Transient Switching between Three-Wavelength Soliton and Q-Switched Mode-Locked States

We report a passive mode-locked fiber laser that can realize single-wavelength tuning and multi-wavelength spacing tuning simultaneously. The tuning range is from 1528 nm–1560 nm, and up to three bands of soliton states can be output at the same time. These results are confirmed by a nonlinear Schrödinger equation model based on the split-step Fourier method. In addition, we reveal a way to transform the multi-wavelength soliton state into the Q-switched mode-locked state, which is period doubling. These results will promote the development of optical communication, optical sensing and multi-signal pulse emission.

Design of three layered step index fiber for femtosecond pulse delivery at 1550 nm wavelength

This paper presents the design and numerical analysis of a step index three-layered fiber optimized for singlemode operation at a wavelength of 1550 nm. The fiber exhibits an extraordinary mode area, measuring 2020 μm², enabling the delivery of ultra-short pulses without distortion. We have used the Transfer Matrix Method for analysis and the Split-Step Fourier Method to solve the nonlinear Schrödinger equation. The study also incorporates considerations for Raman scattering response and self-steepening phenomena in the analysis, enhancing the comprehensive understanding of the proposed step index three-layered fiber for single-mode operation. Demonstrating the propagation of 100-fs pulses with a peak power of 57.7 kW over a 3.4 m length, the fiber maintains a dispersion length-to-nonlinear length ratio of 1 to ensure pulse fidelity. This research offers insights into designing high-performance optical fibers for efficient and distortion-free pulse transmission in ultra-fast laser applications.

Temporal evolution of the light emitted by a thin, laser-ionized plasma source

We present an experimental and simulation-based investigation of the temporal evolution of light emission from a thin, laser-ionized helium plasma source. We demonstrate an analytic model to calculate the approximate scaling of the time-integrated, on-axis light emission with the initial plasma density and temperature, supported by the experiment, which enhances the understanding of plasma light measurement for plasma wakefield accelerator (PWFA) plasma sources. Our model simulates the plasma density and temperature using a split-step Fourier code and a particle-in-cell code. A fluid simulation is then used to model the plasma and neutral density, and the electron temperature as a function of time and position. We then show the numerical results of the space-and-time-resolved light emission and that collisional excitation is the dominant source of light emission. We validate our model by measuring the light emitted by a laser-ionized plasma using a novel statistical method capable of resolving the nanosecond-scale temporal dynamics of the plasma light using a cost-effective camera with microsecond-scale timing jitter. This method is ideal for deployment in the high radiation environment of a particle accelerator that precludes the use of expensive nanosecond-gated cameras. Our results show that our models can effectively simulate the dynamics of a thin, laser-ionized plasma source. In addition, this work provides a detailed understanding of the plasma light measurement, which is one of the few diagnostic signals available for the direct measurement of PWFA plasma sources.