Split-step Method Research Articles

A Closed-Form Expression for the Gaussian Noise Model in the Presence of Inter-Channel Stimulated Raman Scattering Extended for Arbitrary Loss and Fibre Length

A closed-form formula for the nonlinear interference (NLI) estimation using the Gaussian noise (GN) model in the presence of inter-channel stimulated Raman scattering (ISRS) is derived. The formula enables accurate estimation of the NLI evolution along any portion of the fibre span together with arbitrary values of optical fibre losses. The formula also accounts for wavelength-dependent fibre parameters, variable modulation formats and launch power profiles. The formula is suitable for ultra-wideband (UWB) optical transmission systems and its accuracy is assessed for a system with 20 THz optical bandwidth over the entire S-, C-, and L- band through comparison with numerical integration of the ISRS GN model and split-step Fourier method (SSFM) simulations in point-to-point transmission and inline NLI estimation scenarios.

The balanced split step theta approximations of stochastic neutral Hopfield neural networks with time delay and Poisson jumps

The balanced numerical approximations of the stochastic neutral Hopfield neural networks (SNHNN) with time delay and Poisson jumps are examined to ascertain the nature of the mathematical model. The numerical approximations of the balanced split-step theta methods for the SNHNN with time delay and Poisson jumps are taken into consideration primarily because they maintain almost surely (a.s.) exponential stability property of numerical methods and produce negligible mean square error when compared to other approaches. Furthermore, in the recent development of numerical approximations for SNHNN with time delay, we note that balanced split-step theta-approximations are a more stable scheme. We showed that the numerical approximations of balanced split-step theta methods of SNHNN with time delay and Poisson jumps have strong convergence order 1/2 and are numerically almost exponentially stable by applying some theoretical significance criteria.Moreover, our main research tools are Lipschitz conditions, linear growth conditions, and the discrete semi martingale convergence theorem. Through numerical experiments, we try to demonstrate the theoretical results obtained in this paper. Finally, we got the confirmation about the theoretical results of the split-step theta-approximations of SNHNN with time delay and Poisson jumps via particular numerical example.

Limitations of homogeneous and segmented single-crystal compact TEM00-mode Ho3+:YAG laser resonators

We present investigations on output power limitations of near-diffraction-limited compact hbox {Ho}^{3+}:YAG laser resonators employing a homogeneous or segmented laser crystal. An approach for designing a segmented crystal is presented. Maximum output powers of {57.6},hbox {W} and {51.9},hbox {W} are reached with the homogeneous and segmented crystal, respectively, resulting in pulse energies of {1.14},hbox {mJ} and {1.04},hbox {mJ} at a repetition rate of {50},hbox {kHz} in Q-switched operation. Interferometric experiments are conducted to derive the radial temperature profile for both crystals. Simulations based on a split-step beam propagation method are used to model the longitudinal temperature gradient in both crystals.

Reducing the numerical dispersion of the one-way Helmholtz equation via the differential evolution method

This study is devoted to increasing the performance of the numerical methods for solving the one-way Helmholtz equation in large-scale domains. The higher-order rational approximation of the propagation operator was taken as a basis. Computation of an appropriate approximation coefficients and grid sizes is formulated as the problem of minimizing the discrete dispersion relation error. Keeping in mind the complexity of the developed optimization problem, the differential evolution method was used to tackle it. The proposed approach does not require manual selection of the artificial parameters of the numerical scheme. The stability of the scheme is provided by an additional constraint on the optimization problem. Various configurations of the differential evolution method are studied as applied to the considered problem. The advantages of the proposed method has been demonstrated on several classical diffraction problems. Comparisons with the Padé approximation method, rational interpolation and split-step Fourier method are carried out. Theoretical considerations and numerical experiments prove that the proposed method requires much less computational resources than the existing ones. Python 3 implementation of the proposed method is freely available. This work is an extended version of the ICCS-2022 conference paper (Lytaev, 2022).

Numerical Simulations for the Gross-Pitaevskii Equation

In this study the split-step Fourier method for the numerical simulation of the Gross-Pitaevskii equation. Approximate numerical solutions of the GrossPitaevskii equation are obtained by using Matlab software. It is shown that the proposed method improves the computational effort significantly. This improvement becomes more significant especially for large time evolutions.

Computer Simulation of Multichannel Beam Focusing System in Turbulent Atmosphere

The results of a computer simulation of laser beams focused by a combined multichannel array system in the atmosphere are presented. Beam propagation on atmospheric paths is considered taking into account the influence of turbulence and time-dependent thermal blooming interactions. The increase in the maximum intensity of the focal spot due to the optimization of the initial power density of the emitting aperture is estimated. It is shown that the scaling of the optimal initial power in multichannel arrays preserves the proportional dependence of the maximum intensity in focus with an increase in the number of channels. For wave-optics simulation, custom made software was developed based on the split-step method and modified for parallel algorithms using the functions of the Intel ® Math Kernel Library.

Effect of Some Parameters on Optical Soliton Pulses in Photonic Crystal Fibers

Fiber optics have been greatly enhanced by photonic crystal fibers based on microstructured air-glass designs. On the one hand, such fibers enable very tight light confines in a small mode shape region, resulting in significantly improved alternative options between light and dielectric medium. Photonic crystal fibers, on the other hand, allow light to be guided via air core instead of glass. As a result, the latter form of fiber decreases optical nonlinearities in ways that classic fiber designs cannot. The chirp effect and dispersion of photonic crystal fibers with super-Gaussian pulses during various pulses durations are examined in this paper in both normal and anomalous dispersion patterns. The chirp effect and fiber dispersive nonlinear effects are investigated. For this study, a mathematical model of the solution of a nonlinear equation involved the split-step Fourier method. Peak power was reduced for broad pulses. When the magnitude of the super-Gaussian pulse increased proportionally, pulse constriction was also noticeable, Furthermore, the results reveal that an anomalous dispersion system was superior to a regular dispersion system for pulse . The experimental findings will facilitate further research into more understanding of photonic crystal fibers, and to improve data speeds in modern optical communication systems

Fast Predicting the Complex Nonlinear Dynamics of Mode‐Locked Fiber Laser by a Recurrent Neural Network with Prior Information Feeding

AbstractAs an imperative method of investigating the internal mechanism of femtosecond lasers, traditional femtosecond laser modeling relies on the split‐step Fourier method (SSFM) to iteratively resolve the nonlinear Schrödinger equation suffering from the large computation complexity. To realize inverse design and optimization of femtosecond lasers, numerous simulations of mode‐locked fiber lasers with different cavity settings are required, further highlighting the time‐consuming problem induced by the large computation complexity. Here, a recurrent neural network is proposed to realize fast and accurate femtosecond mode‐locked fiber laser modeling. The generalization over different cavity settings is achieved via the proposed prior information feeding method. With the acceleration of GPU, the mean time of the artificial intelligence (AI) model inferring 500 roundtrips is less than 0.1 s, which is ≈146 times faster than the SSFM running on a CPU. The proposed AI‐enabled method is promising to become a standard approach to femtosecond laser modeling.

Spin-polarised wave-packets transport through one-dimensional nonlinear multiple barriers: Rashba and Dresselhaus Spin–Orbit interactions

ABSTRACT The time evolution of a Gaussian wave packet travelling through a one- dimensional semiconductor based on multiple potential barriers has been explored in order to obtain the spin-based transmission, reflection, and trapping coefficients. We have used a split-step finite difference method to solve the resulting nonlinear coupled Schrodinger equations. The related behaviour of scattering properties of the system as a function of Rashba and Dresselhaus Spin–Orbit interactions, wave-packet velocity, the geometry of the barriers, and the number of neighbouring barriers have been discussed. Based on the results, it is found that by tuning the Rashba and the Dresselhaus coupling strengths, and the characteristics of the system, one can control the spin polarisation of the wave-packet and its propagation coefficients.

A split-step Fourier pseudo-spectral method for solving the space fractional coupled nonlinear Schrödinger equations

In this paper, we propose a split-step Fourier pseudo-spectral method for solving the space fractional coupled nonlinear Schrödinger (CNLS) equations. The space fractional CNLS equations can be split into two subproblems such that one of them is linear. The solution of the nonlinear subproblem is computed exactly. The Riesz space fractional derivative is approximated by a Fourier pseudo-spectral method. The unconditional stability, convergence, discrete charge and multi-symplectic preserving properties for the proposed method are investigated. Then, the proposed method is extended for solving the two-dimensional problem. Finally, some numerical experiments are performed to confirm our theoretical analysis and illustrate the efficiency of the proposed scheme.

Split-step birefringence simulation method for optical Kerr gates

A split-step birefringence simulation method is proposed to investigate the gating efficiency and intensity distribution of the Kerr signal field considering the evolution of the switch beam and probe beam in their path. Using this simulation method, we investigated the switch-beam power-dependent gating efficiency and conducted an experiment to prove its reliability. Furthermore, we analyzed the optical intensity distribution of the Kerr signal exiting the Kerr medium under different switch-beam powers. This study provides an effective theoretical tool for the design and optimization of optical Kerr gates.

Analytical and numerical study of chirped optical solitons in a spatially inhomogeneous polynomial law fiber with parity-time symmetry potential

This work studies the dynamical transmission of chirped optical solitons in a spatially inhomogeneous nonlinear fiber with cubic-quintic-septic nonlinearity, weak nonlocal nonlinearity, self-frequency shift and parity-time () symmetry potential. A generalized variable-coefficient nonlinear Schrödinger equation that models the dynamical evolution of solitons has been investigated by the analytical method of similarity transformation and the numerical mixed method of split-step Fourier method and Runge–Kutta method. The analytical self-similar bright and kink solitons, as well as their associated frequency chirps, are derived for the first time. We found that the amplitude of the bright and kink solitons can be controlled by adjusting the imaginary part of the -symmetric potential. Moreover, the influence of the initial chirp parameter on the soliton pulse widths is quantitatively analyzed. It is worth emphasizing that we could control the chirp whether it is linear or nonlinear by adjusting optical fiber parameters. The simulation results of bright and kink solitons fit perfectly with the analytical ones, and the stabilities of these soliton solutions against noises are checked by numerical simulation.

Periodic and solitary waves generating in optical fiber amplifiers and fiber lasers with distributed parameters

We study self-similar dynamics of picosecond light pulses generating in optical fiber amplifiers and fiber lasers with distributed parameters. A rich variety of periodic and solitary wave solutions are derived for the governing generalized nonlinear Schrödinger equation with varying coefficients in the presence of gain effect. The constraint on distributed optical fiber parameters for the existence of these wave solutions is presented. The dynamical behaviour of those self-similar waves is discussed in a periodic distributed amplification system. The stability of periodic and solitary wave solutions is also studied numerically by adding white noise. It is proved by using the numerical split-step Fourier method that the profile of these nonlinear self-similar waves remains unchanged during evolution.

Balanced implicit methods with strong order 1.5 for solving stochastic differential equations

The aim of this article is to propose a family of balanced implicit methods and split-step balanced implicit methods for solving Itô stochastic differential equations. These numerical methods represent a class of highly efficient linear-implicit schemes which covered the implicitness in the stochastic terms. The convergence in mean-square with strong order 1.5 and the asymptotic mean-square stability properties of these methods are analysed. What is more, the linear mean-square stability regions of some proposed methods are given. Several numerical examples are carried out to illustrate the theoretic findings.

Impact of the input OSNR on data-driven optical fiber channel modeling

The data-driven approach is promising for computationally efficient modeling of optical fiber channels. Here, for the first time to our knowledge, we investigate the impact of the input optical signal-to-noise ratio (OSNR) on the accuracy of data-driven modeling for optical fiber channels, based on a conditional generative adversarial network (cGAN). Initially, considering a single span of 80 km of standard single mode fiber (SSMF), we vary the launched optical power for the ease of emulating both linear and nonlinear transmission scenarios. When the input OSNR of a root raised cosine (RRC)-shaped 16 quadrature amplitude modulation signal with a roll-off factor of 0.1 varies, we identify that there occurs an OSNR threshold for accurately modeling a single span of 80 km of SSMF in order to reach the normalized mean square error of less than 0.02 in relevance to the traditional split-step Fourier method. The OSNR thresholds are 19 dB and 21 dB, respectively, under scenarios of linear and nonlinear transmissions. Moreover, we verify that the OSNR threshold of accurate modeling is insensitive to both the modulation format and the RRC shaping of the input signal. Furthermore, we find that the cascaded use of cGAN well-trained for the single SSMF span is capable of modeling the multiple-span SSMF transmission in the case that the input OSNR threshold of 22.6 dB is satisfied. We believe that the identification of the OSNR threshold for data-driven optical fiber channel modeling is useful for the accurate and efficient emulation of long-haul fiber optical transmission.

Propagation characteristics of rotating beams under combined effect of atmospheric turbulence and thermal blooming

The rotating beam is a beam that is generated by the heterodyne interference of vortex beams with different topological charges, whose intensity and wavefront rotate rapidly with time. Under Taylor’s frozen air hypothesis, such a rotating beam, whose rotating period is much shorter than the characteristic time of atmospheric turbulence and the thermal blooming, can travel through all the inhomogeneities due to its rapid rotation during its propagation in the air. The rotation can contribute to the smooth of the wavefront distortion caused by the atmospheric turbulence and the thermal blooming, and thus improve the beam quality and reduce the centroid drift of the rotating beam in far field. The physical model of rotating beam is established by the heterodyne interference of two vortex beams. Taking the atmospheric turbulence and the thermal blooming effect into consideration, the propagation model of the rotating beam in the air is established by using the split-step Fourier method. For simplicity without loss of generality, the influences of the atmospheric turbulence and the thermal blooming effect on the propagation of rotating beam, are both treated as wavefront screen. The difference is that the wavefront screen of atmospheric turbulence is generated by the Kolmogorov power spectrum, while that of thermal blooming is generated by the fluid mechanics equation. The physical mechanism of how the rotating beam mitigates the atmospheric turbulence and the thermal blooming effect is analyzed in detail, that is, when the rotating speed of laser beam is faster than the airflow, the laser beam can pass through all the inhomogeneous and anisotropic atmosphere in the azimuthal direction within the time interval of airflow. After the laser propagates through the frozen air, the total wavefront distortion in the azimuthal direction becomes centrosymmetric with lower PV value, reducing the beam quality degradation. On the basis, the influence of rotation frequency, power ratio of the sub beams, strength of turbulence and thermal blooming on the propagation characteristics of the rotating beam in atmosphere are analyzed. Within a certain range, as the beam rotation frequency increases, the mitigation effect of the rotating beam on atmospheric turbulence and thermal blooming effects are enhanced. With the increase of turbulence intensity and thermal blooming intensity, the mitigation effect of the rotating beam is weakened but still can maintain well, which can provide reference for the engineering applications of laser beam in atmosphere.

Split-step quintic B-spline collocation methods for nonlinear Schrödinger equations

<abstract><p>Split-step quintic B-spline collocation (SS5BC) methods are constructed for nonlinear Schrödinger equations in one, two and three dimensions in this paper. For high dimensions, new notations are introduced, which make the schemes more concise and achievable. The solvability, conservation and linear stability are discussed for the proposed methods. Numerical tests are carried out, and the present schemes are numerically verified to be convergent with second-order in time and fourth-order in space. The conserved quantity is also computed which agrees with the exact one. And solitary waves in one, two and three dimensions are simulated numerically which coincide with the exact ones. The SS5BC scheme is compared with the split-step cubic B-spline collocation (SS3BC) method in the numerical tests, and the former scheme is more efficient than the later one. Finally, the SS5BC scheme is also applied to compute Bose-Einstein condensates.</p></abstract>

Effect of system parameters on power scintillation of Ince Gaussian beam in turbulent atmosphere

The power scintillation of Ince Gaussian (IG) beam is investigated numerically based on the split step method. The effect of the IG beam modes on mitigating the effect of atmospheric turbulence is studied. Furthermore, the power scintillation indices of IG beams with different modes are compared with the fundamental Gaussian beam power scintillation indices. Moreover, the ability of different parameters, namely the beam waist, operating wavelength, and the receiver aperture, to reduce the scintillation is examined. The obtained results show that IG beams with even modes have the advantage over Gaussian beams for links longer than 2 km. Moreover, IG beams with even modes are able to reduce the scintillation more than the ones with odd modes. Yet, increasing the beam waist and the receiver radius reduces the scintillation of the beams with odd mode more than the beams with even mode. Our results contribute to the improvement of free-space optical communication systems operating in turbulent atmosphere.

Improving MDM–WDM optical network performance via optimized power allocation using Gaussian noise model

The few-mode fiber nonlinearity has been well modeled by an additive Gaussian Noise (GN) source. The available GN models in Mode Division Multiplexing–Wavelength Division Multiplexing (MDM–WDM) systems are incoherent and neglect the dispersion slope. The incoherent GN model assumes that the Nonlinear Interference (NLI) noise created at each span is summed up in power incoherently at the receiver and leads to underestimating the NLI noise power. In the first part of this paper, a coherent GN model is derived by taking into account the dispersion slope and verified through Split-Step Fourier Method (SSFM) simulations. In the second part of this paper, the total capacity maximization, as well as the minimum Signal to Noise Ratio (SNR) margin maximization problems are presented and solved based on the obtained GN model. MDM–WDM multi-node nonlinear network under different scenarios including Equal/Non-Equal Required SNR (ERS/NERS), and Flat/Non-Flat Amplifier Gain (FG/NFG) are considered in the numerical simulations. Considering 200 different sets of randomized traffic demands (random channel-mode distributions) the cumulative distribution function of minimum SNR margin improvement in optimized power allocation compared with the best flat power allocation is in 99% more than 1 dB, 1.2 dB, 2.4 dB, and 2.8 dB, for ERS-FG, ERS-NFG, NERS-FG, and NERS-NFG scenarios, respectively.

Transformer-Based Long Distance Fiber Channel Modeling for Optical OFDM Systems

The fiber channel model plays an essential role in the simulation and design of optical fiber communication systems. However, it is difficult for conventional model-driven modeling to balance accuracy and efficiency, especially in optical orthogonal frequency division multiplexing (OFDM) systems with complex and long-haul transmission. We introduce the simplified Transformer into optical OFDM systems and combine it with the feature decoupled distributed (FDD) scheme for fast and accurate fiber channel modeling. Unlike the popular Transformer architectures, we remove the Decoder part and cancel the self-attention with quadratic complexity, significantly reducing the computational cost. The modeling performance is investigated from the nonlinear fitting capability, accuracy, and generalization ability. The transmission distance ranges from 80 km to 1600 km. The highly matched four-wave mixing (FWM) power, low error vector magnitudes (EVMs), and similar signal-noise ratios (SNRs) demonstrate the high precision and robustness of the model. Furthermore, the modeling is studied under different transmission rates and is proved to be reliable over a wide transmission bandwidth. Compared to the bidirectional long short-term memory (Bi-LSTM), the Transformer performs better in accuracy and has lower computational and memory costs. For modeling under the same conditions, the required running time of the Transformer is about 60% of Bi-LSTM, less than 1% of the split-step Fourier method (SSFM). The Transformer-based method achieves high precision modeling of the fiber channel in the long-distance and high-rate optical OFDM system and makes a significant breakthrough in complexity.