Symmetrized Split-step Fourier Method Research Articles

Embedded split-step methods optimized with a step size control for solving the femtosecond pulse propagation problem in the nonlinear fiber optics formalism

We present the suitability of two optimized split-step methods for validating the femtosecond pulse propagation problem in the nonlinear fiber optics formalism that is governed by an extended nonlinear Schrödinger equation. In particular, the embedded split-step Fourier method (embedded-SSFM) and the embedded symmetrized split-step Fourier method (embedded-SymSSFM), which are optimized by the implementation of a step size control algorithm, are tested in terms of the femtosecond soliton fission phenomenology to analyze the numerical stability. As a result, it is possible to validate that these numerical methods introduce the appropriate resolution as function of the accuracy order that is needed at different stages of the soliton fission dynamics, which involves very large spectral shifts driven by the interaction between the nonlinear response and the high-order dispersion contribution of the optical fiber. Thus, the presented numerical methods can be used to validate accurately complex processes such as the development of supercontinuum spectra in the current formalism.

Supercontinuum generation in organic liquid-liquid core-cladding photonic crystal fiber in visible and near-infrared regions

In this paper, we propose a liquid core-cladding photonic crystal fiber (PCF), which is engineered with different available organic optofluidics, to generate supercontinuum in the visible and near-infrared regimes by using the symmetrized split-step Fourier method. Simulations reveal that in response to launching 50 fs input pulses of 10 kW peak power, centered about λ0=1032 and 1560 nm, into a 10 mm long liquid core-cladding PCF, maximum 2 μm supercontinua from 500 to 2500 nm can be achieved. Our numerical study is important for the new field of visible and near-IR supercontinuum generation in liquid-core optical fibers.

Ultra-Wide Mid-Infrared Supercontinuum Generation in As40Se60Chalcogenide Fibers: Solid Core PCF Versus SIF

Due to the pronounced scientific and technical attention paid to the mid-infrared spectral region, we focus on ultra-wide mid-infrared supercontinuum generation in As40Se60 chalcogenide SIF and PCF—step index and photonic crystal fibers—using a symmetrized split step Fourier method (S-SSFM). Simulations reveal, in response to launching 100 fs input pulses of 50 kW peak powers, with near zero dispersion wavelengths centered about $\lambda _{0} = 8.5$ μm, into 50-mm long As40Se60 SIF and solid core PCF, 8.5 and 11-μm supercontinua can be achieved, respectively. Moreover, numerical results also show, when ultra-short pulses of center wavelengths $\lambda _{0} = 12$ μm are launched into those SIF and PCF, ripple free spectra as wide as 10 and 13 μm can be attained, respectively. These spectra are useful for safe spectroscopic medical imaging diagnosis such as early cancer diagnostics, and other spectroscopic applications such as gas sensing and food quality control.

A Tapered Chalcogenide Microstructured Optical Fiber for Mid-IR Parabolic Pulse Generation: Design and Performance Study

This paper presents a theoretical design of chalcogenide glass based tapered microstructured optical fiber (MOF) to generate high power parabolic pulses (PPs) at the mid-IR wavelength (~2 μm). We optimize fiber cross-section by the multipole method and studied pulse evolution by well-known symmetrized split-step Fourier Method. Our numerical investigation reveals the possibility of highly efficient PP generation within a very short length (~19 cm) of this MOF for a Gaussian input pulse of 60 W peak power and full width at half maximum (FWHM) of 3.5 ps. We examined quality of the generated PP by calculating the misfit parameter including the third order dispersion and fiber loss. Further, the effects of variations in input pulse power, pulse width and pulse energy on generated PP were also studied from the point of view of tolerances in fabrication of such a device.

An integral split-step fourier method for digital back propagation to compensate fiber nonlinearity

In optical fiber transmission systems, the split-step Fourier method (SSFM) has been widely used in digital back propagation (DBP) to compensate fiber nonlinearity. In this paper, by using the Lagrange's Integral Mean Value Theorem (LIMVT), we derive an analytical expression to calculate the optimal value of the nonlinearity calculation position (NLCP) for different systems and we propose an integral SSFM (I-SSFM) based on the expression. The I-SSFM can be performed more accurately and efficiently without parameter optimization. Simulations of various transmission links show that the I-SSFM outperforms the conventional asymmetric SSFM (A-SSFM) and the symmetric SSFM (S-SSFM) significantly, especially when we employ less amount of steps to ensure computation efficiency. The computation effort of the I-SSFM reaches as low as 50% of that of the S-SSFM.

Numerical demonstration of generation of bound solitons in figure of eight microstructured fiber laser in normal dispersion regime

We have numerically studied passive mode locking in figure of eight laser containing microstructured optical fiber. We demonstrate for the first time to the best of our knowledge the generation of bound states in such configuration in the normal dispersion regime. The numerical simulations are based on extended nonlinear Schrödinger equation using the symmetrized split-step Fourier method. The numerical results show that single pulse or bound solitons can be obtained when the parameters of the cavity are suitably chosen. We identify the small signal gain and the microstructured optical fiber's nonlinear coefficient as key parameters for the generation of bound solitons in figure of eight fiber laser.

A Discrete-Time Model for Uncompensated Single-Channel Fiber-Optical Links

An analytical discrete-time model is introduced for single-wavelength polarization multiplexed nonlinear fiber-optical channels based on the symmetrized split-step Fourier method (SSFM). According to this model, for high enough symbol rates, a fiber-optic link can be described as a linear dispersive channel with additive white Gaussian noise (AWGN) and a complex scaling. The variance of this AWGN noise and the attenuation are computed analytically as a function of input power and channel parameters. The results illustrate a cubic growth of the noise variance with input power. Moreover, the cross effect between the two polarizations and the interaction of amplifier noise and the transmitted signal due to the nonlinear Kerr effect are described. In particular, it is found that the channel noise variance in one polarization is affected twice as much by the transmitted power in that polarization than by the transmitted power in the orthogonal polarization. The effect of pulse shaping is also investigated through numerical simulations. Finally, it is shown that the analytical performance results based on the new model are in close agreement with numerical results obtained using the SSFM for a symbol rate of 28 Gbaud and above.

Effect of Fiber Self Phase Modulation on the Splitting Error using the Strang Formulas

The generalized nonlinear Schrodinger equation describes the different physical phenomena encountered when ultrashort pulses propagate through dispersive and nonlinear fibers. If the pulse duration is of picoseconds order, the nonlinear Schrodinger equation can be simp lified. However the analyt ical solution remains inaccessible except for some special cases like soliton. The symmetric split-step Fourier method (S-SSFM) which is derived fro m the St rang formulas, subdivides the global propagation distance into small steps of length h to calculate the numerical solution of this equation. By using only the fact that the dispersive and nonlinear operators do not commute the Baker-Campbell-Hausdorff formu la shows that the global relative error of this method is O(h 2 ). Our numerical simulation results show that this error depends also on the self phase modulation nonlinear term. For this purpose, we emp loy in this work an exp licit representation of the nonlinear operator and we present four imp lementations: the S-SSFM 1, S-SSFM 2, T-SM1 and T-SM 2 obtained respectively fro m

Numerical analysis of aperiodic photonic crystal fiber structures for supercontinuum generation

An unconventional photonic crystal fiber (PCF) is designed in order to obtain supercontinuum (SC) generation. The distance of the air hole rings is not constant, but it follows the Thue-Morse (ThMo) aperiodic sequence. The spectral broadening of high power femtosecond (fs) input pulses in anomalous dispersion regime is investigated by means of a homemade computer code, based on the generalized nonlinear Schrodinger equation (GNLSE) and implemented via the symmetrized split-step Fourier method (S-SSF). A lot of ThMo PCF transversal sections are simulated, and the one providing the flattest and the widest spectrum is identified and further refined by varying the operating conditions. Three different peak powers (5, 10, 15 kW) and three different pulse durations (40, 50, 100 fs) are investigated at the pump wavelength λP = 1064 nm. The novel ThMo PCF proposed in this paper exhibits a SC figure of merit Φ = 16, very high with respect to the state of the art pertaining to fs input pulses.

Digital Postcompensation Using Volterra Series Transfer Function

We propose a noniterative digital backward propagation technique, based on an inverse modified Volterra series transfer function to postcompensate transmission linear and nonlinear impairments in the presence of optical noise. Using a single-channel 40-Gb/s nonreturn-to-zero quadrature phase-shift-keying optical signal propagated over 20 × 80 km of standard single-mode fiber, and performing digital postcompensation around the Nyquist rate, our compensation algorithm is able to surpass the maximum accuracy obtained with a symmetric split-step Fourier method, enabling us to increase the nonlinear tolerance by approximately 2 dB.

Soliton interaction in birefringent optical fibers: Effects of nonlinear gain devices

This paper presents the influences of polarization mode dispersion (PMD) on the performance of soliton transmission system in birefringent fibers. Dispersive waves generated in single mode fibers due to PMD degrade the soliton transmission system in two aspects. First, solitons continuously lose their energy, thus cause enhancement in pulse width. Second, the dispersive waves interact with neighboring pulses and cause distortion in a sequence of pulses. Both these effects reduce the effective bit-rate and degrade the performance of high-speed optical transmission systems. Optical fibers with large group velocity dispersion (GVD) have less dispersive waves and are relatively robust to pulse broadening, but it enhances the interaction between the adjacent pulses. In this paper, we analyzed these effects of PMD on soliton propagation in birefringent fibers and introduced nonlinear gain devices with perturbation terms proportional to second and fourth power of amplitudes to reduce these effects. We proposed Symmetric Split-Step Fourier Method to solve the coupled nonlinear Schrödinger equations (CNLSE); which yields better results over the existing Split-Step Fourier Method.

Optimized digital backward propagation for phase modulated signals in mixed-optical fiber transmission link

The parametric optimization of Digital Backward Propagation (DBP) algorithm for mitigating fiber transmission impairments is proposed and numerically demonstrated for phase modulated signals in mixed-optical fiber transmission link. The optimization of parameters i.e. dispersion (D) and non-linear coefficient (γ) offer improved eye-opening (EO). We investigate the optimization of iterative and non-iterative symmetric split-step Fourier method (S-SSFM) for solving the inverse non-linear Schrödinger equation (NLSE). Optimized DBP algorithm, with step-size equal to fiber module length i.e. one calculation step per fiber span for obtaining higher computational efficiency, is implemented at the receiver as a digital signal processing (DSP) module. The system performance is evaluated by EO-improvement for diverse in-line compensation schemes. Using computationally efficient non-iterative symmetric split-step Fourier method (NIS-SSFM) upto 3.6 dB referenced EO-improvement can be obtained at 6 dBm signal launch power by optimizing and modifying DBP algorithm parameters, based on the characterization of the individual fiber types, in mixed-optical fiber transmission link.

Methods of Step-Size Distribution Optimization Used in S-SSFM Simulations of WDM Systems

Brief review of methods used for simulation of signal propagation in wavelength division multiplexed (WDM)links is presented. We propose two novel methods of stepsize distribution optimization used to improve symmetrized split step Fourier method (S-SSFM) numerical efficiency: presimulated local error S-SSFM (PsLE S-SSFM) and modified logarithmic (ML S-SSFM). The PsLE S-SSFM contains two stages: in the initial stage step-size distribution optimization is carried out by combining local error method and presimulation with signal spectrum averaging; in the second stage conventional SSFM is used by applying optimal step-size distribution obtained in the initial stage. The ML S-SSFMis generalization of logarithmic method proposed to suppress spurious FWM tones, in which a slope of logarithmic step-size distribution is optimized. Overall time savings exceed 50%, depending of a simulated system scenario.

Adaptive higher-order split-step Fourier algorithm for simulating lightwave propagation in optical fiber

In this paper, we propose an adaptive step-size control algorithm for solving nonlinear Schrödinger’s equations. The proposed algorithm has a fourth-order local accuracy and is a system-independent rule for adjusting the step sizes. This algorithm has two potential advantages, an automatic step adjustment mechanism and higher-order accuracy. A test example shows that, by comparing to the fixed step-size method and the local-error method, our method decreases the computational time by about 10 and 70 times, respectively. The performance of the proposed method is validated and compared to commonly used step-size selection methods by simulating the evolution of a third-order soliton, the collision of a fundamental soliton pair, and the supercontinuum generation. Numerical simulations show that the proposed method can increase the computational efficiency by more than one and two orders of magnitude in comparison with the symmetrized split-step Fourier method. In addition, the computational efficiency is improved with the increase of the accuracy of solutions.

Multichannel Wavelength Conversion Using Fourth-Order Soliton Decay

In this paper, we present the decaying behavior of a fourth-order temporal soliton, which propagates through optical fibers with various dispersion profiles, using the symmetrized split-step Fourier method. We use these behaviors to design two types of 1 times 4 and on 1 times 5 channel wavelength converters using three different types of dispersion profiles along the optical fibers. We also demonstrate that under certain conditions one can use one of the 1 times 4 converters for a pulse train with duration of 30 ps.

Effective Algorithms and Their Applications in Fiber Transmission Systems

A class of fast and accurate algorithms for solving differential equations are proposed and derived. The proposed methods are considerably more accurate and more stable than classical numerical methods, and they can obtain a significantly faster speed, a higher accuracy and a larger stability region in parallel. Two kinds of basic equations, the fiber amplifier propagation equations and nonlinear Schrödinger equations, in the fiber transmission systems are effectively solved by use of these methods. Numerical simulations show that the proposed methods can increase the accuracy by more than an order of magnitude in comparison with the classical fast methods for solving Raman amplifier propagation equations, and shorten the central processing unit (CPU) time by a factor of ∼2 to ∼6 compared with the symmetrized split-step Fourier method for solving nonlinear Schrödinger equations.

A fast method for nonlinear Schrodinger equation

The predictor-corrector split-step Fourier method (SSFM) is proposed for solving a nonlinear Schrodinger equation. In comparison with the symmetrized SSFM, the proposed method greatly decreases the relative error and increases the computing speed by /spl sim/2.8 to /spl sim/5.5 times at the same accuracy.

Self- and cross-phase modulation of high-intensity laser beams emerging from a diamond-turned KDP wedge

The Laboratory for Laser Energetics plans to install KDP wedges on each beam line of the OMEGA laser system in order to improve the on-target laser uniformity by decreasing the instantaneous speckle through spatial averaging of the two resultant orthogonally polarized beams. The proposed wedge-production method, diamond turning, produces small residual scratch marks, causing each beam to acquire a pseudorandom phase perturbation. In addition, the orthogonally polarized beams interfere such that their combined polarization state continuously cycles through all elliptical states along any transverse plane. Since the nonlinear refractive index depends on the polarization state, intense beams accumulate a periodic phase perturbation that is greater for linear polarization. Propagation of both types of phase perturbation yields an intensity modulation that tends to be larger in the neighborhood of linear polarization, through a combination of diffraction and self- and cross-phase modulation. However, one- and two-dimensional calculations using a symmetrized split-step Fourier method demonstrate that diamond-turned KDP wedges are not a significant source of intensity modulation under OMEGA laser conditions. Installation of diamond-turned rather than polished wedges will reduce costs without adversely affecting the system performance.

Symmetric Daubechies' wavelets and numerical solution of NLS equations

Recent work has shown that wavelet-based numerical schemes are at least as effective and accurate as standard methods and may allow an 'easy' implementation of a spacetime adaptive grid. Up to now, wavelets which have been used for such studies are the 'classical' ones (real Daubechies' wavelets, splines, Shannon and Meyer wavelets, etc) and were applied to diffusion-type equations. The present work differs in two points. Firstly, for the first time we use a new set of complex symmetric wavelets which have been found recently. The advantage of this set is that, unlike classical wavelets, they are simultaneously orthogonal, compactly supported and symmetric. Secondly, we apply these wavelets to the physically meaningful cubic and quintic nonlinear Schrodinger equations. The most common method to simulate these models numerically is the symmetrized split-step Fourier method. For the first time, we propose and study a new way of implementing a global spacetime adaptive discretization in this numerical scheme, based on the interpolation properties of complex-symmetric scaling functions. Second, we propose a locally adaptive 'split-step wavelet' method.