Order Soliton Research Articles

Transforming higher order bright and dark solitons to the first order solitons in Kerr medium: A review

A higher order soliton is a soliton where the initial energy is higher than the fundamental one by a specific coefficient. The value of the coefficient is square of an integer number. The intensity profile of higher order soliton (spatial or temporal) is not constant but varies periodically during propagation. The period of their evolution, called soliton-period. However, when the coefficient is not the square of an integer, the complicated intensity profile by propagation occurs. In this work, it is demonstrated and discussed the reason why higher order soliton is made, and how it modifies a factor in Nonlinear Schrödinger Equation (NLS), in positive and negative third order nonlinear (Kerr) medium. In this modified NLS equation, the exact analytical fundamental soliton solution which doesn’t show the soliton-period is obtained. Followed by numerical simulation by MATLAB, with Split-Step method, and comparison between modified initial beam-width which produces fundamental soliton and nonmodified initial beam-width that gives the other order of solitons for bright and dark solitons.

Efficient Precoding Scheme for Dual-Polarization Multi-Soliton Spectral Amplitude Modulation

Soliton pulses are special waveforms to account for nonlinearity in fiber optical communication. They can be represented in a nonlinear spectrum by eigenvalues and spectral amplitudes which have simple transformation equations along the ideal link. This motivates to encode data in the nonlinear spectrum. In this paper, we consider dual-polarization modulation of spectral amplitudes, and show that they become highly correlated during propagation along a noisy fiber link. Thus, joint equalization is generally needed for detection at the receiver. We propose a simple precoding scheme that almost removes these correlations. This allows to significantly improve the detection performance even without any complex equalization. The spectral amplitudes are transformed into pairs of common and differential information part. We show that the differential part is almost preserved along the link, even in the presence of noise. Thus, it can be directly detected from the received spectral amplitudes with high reliability. Exploiting the differential gain of the precoding allows much higher bit rates at comparable error rates. We analyze our precoding scheme and verify its performance gain in split-step-Fourier simulations by comparing it to the conventional independent modulation of spectral amplitudes for first and second order solitons.

Broadband supercontinuum generation from 600 to 5400 nm in a tapered fluorotellurite fiber pumped by a 2010 nm femtosecond fiber laser

We demonstrate broadband supercontinuum (SC) generation from 600 to 5400 nm in a tapered fluorotellurite fiber pumped by a 2010 nm femtosecond fiber laser. All-solid fluorotellurite fibers with a core diameter of ∼6 μm are fabricated by using a rod-in-tube method. Tapered fluorotellurite fibers with an untapered region length of ∼2 cm and a tapered transition region length of ∼1.05 cm are prepared by employing an elongation machine. By using the tapered fiber as the nonlinear medium and a 2010 nm femtosecond fiber laser as the pump source, SC generation from 600 to 5400 nm is obtained, the 30 dB bandwidth of the generated SC light is about 3600 nm, and the corresponding output power is about 0.85 W for a launched average pump power of ∼1.57 W. The spectral broadening in the tapered fiber is caused by higher order soliton compression, Raman soliton, and blue-shifted and red-shifted dispersive wave generation. Our results show that fluorotellurite fibers are promising nonlinear media for generating broadband SC light expanding from visible to 5400 nm.

The soliton solutions for the Wadati–Konno–Ichikawa equation

By taking the Wadati–Konno–Ichikawa (WKI) equation as an example, this paper refines the Olmedilla’s technique of solving the Gelfand–Levitan–Marchenko (GLM) equation with one higher-order pole. In order to solve the GLM equation, the Laurent’s series and residue theorem are used, and the formulae of the Nth order soliton solution and Nth order bound-state soliton solution are expressed in a unified determinant form. Moreover, these formulae are simpler and more explicit to calculate solutions of the nonlinear integrable equations, compared with the results of Olmedilla.

Riemann–Hilbert method for the Wadati–Konno–Ichikawa equation: N simple poles and one higher-order pole

Abstract We present a Riemann–Hilbert (RH) method directly by using the inverse scattering method (ISM) for Wadati–Konno–Ichikawa equation (WKIE) i q t + q 1 + | q | 2 x x = 0 . The RH problem is related to two cases of scattering data: N simple poles and one N th order pole. Under the reflection-less situation, we solve the RH problem and obtain the formulae of N th order soliton and positon solutions in the form of determinants. As applications, the first-order soliton and the second-order positon solutions are displayed in analytical and graphical ways. For the first-order soliton, it displays analytic form, bursting form, and loop form. For the second-order positon solution, an interesting inelastic phenomenon is observed that a singular positon solution including an analytic soliton and a loop soliton is split into an analytic soliton and a bursting soliton after the collision.

Spatio-temporal enhancement of Raman-induced frequency shifts in graded-index multimode fibers

We investigate the impact of intrapulse Raman scattering and third-order dispersion on the propagation of a pulsed optical beam inside graded-index (GRIN) fibers by solving an effective nonlinear Schrodinger equation that includes the spatial self-imaging effects through a periodically varying effective mode area. Numerical simulations are used to show that the Raman-induced frequency shift of the shortest fundamental soliton, created after the fission process, is enhanced considerably inside GRIN fibers compared to single-mode fibers for the same value of the soliton order. We also discuss the role of spatial-width contraction during each self-imaging cycle on the Raman-induced frequency shifts.

Analyses of Supercontinuum Generation in Photonic Crystal Fibers

In this work, the solid PCF are studied, where the solid core was surrounded by a number of holes with a diameter away from each other by distance. The strange behavior of dispersion with the presence of nonlinear phenomenon such as: self-phase modulation, self-steepening and stimulated Raman scattering allows the formation of higher order solitons in a short distance range . The study of SCG has been done by the numerical solution for the generalized propagation equation, where the Fourier transform method is used to solve the linear part and Runge-Kutta method to solve the nonlinear part. Higher order soliton means the possibility of soliton fission to the fundamental solitons and emissions the dispersive wave, this means the formation SCG. The SCG has been affected by each of: width, power and wavelength of the input pulses. On the other side, it is also affected by the fiber parameters and that in turn controls the zero dispersion wavelength , where the conditions and form bright soliton and dark soliton, respectively.

Transmission of Higher Order Solitons Created by Optical Multiplexing

The nonlinear Fourier transform (NFT) is a promising tool to linearize the inherently nonlinear optical fiber channel. The NFT transforms a time-domain signal into the continuous and the discrete spectrum. The discrete spectrum is composed of an arbitrary number of complex valued discrete eigenvalues and their associated amplitudes. These discrete eigenvalues relate to solitons, which maintain their shape or return to it in an oscillating manner, while passing through the optical channel. Higher order solitons consisting of multiple eigenvalues are complex pulses, which are created and demodulated by sophisticated digital signal processing (DSP) leading to demanding hardware requirements. This paper shows a way to work with higher order solitons in a wavelength division multiplexing such as fashion by using optical-electrical signal processing and presents boundaries of this method. Optical-electrical signal processing decreases the required electrical and electro-optical hardware specifications substantially and enables to use a simplified DSP. The proposed creation method is subsequently employed to transmit higher order solitons consisting of five QPSK modulated eigenvalues. Furthermore, the optical-electrical processing is benchmarked against the Darboux transformation, which creates higher order solitons purely numerically. The results show that for a fifth-order soliton transmission the proposed method can significantly reduce the hardware requirements and DSP complexity.

Time-Bandwidth Product Perspective for Nonlinear Fourier Transform-Based Multi-Eigenvalue Soliton Transmission

Multi-soliton pulses are potential candidates for fiber optical transmission where the information is modulated and recovered in the so-called nonlinear Fourier domain. While this is an elegant technique to account for the channel nonlinearity, the obtained spectral efficiency, so far, is not competitive with classic Nyquist-based schemes. This is especially due to the observation that soliton pulses generally exhibit a large time-bandwidth product. We consider the phase modulation of spectral amplitudes of higher order solitons, taking into account their varying spectral and temporal behavior when propagating along the fiber. For second and third order solitons, we numerically optimize the pulse shapes to minimize the time-bandwidth product. We study the behavior of multi-soliton pulse duration and bandwidth and generally observe two corner cases where we approximate them analytically. We use these results to give an estimate on the minimal achievable time-bandwidth product per eigenvalue.

Ultra high transmission capacity based on optical first order soliton propagation systems

Abstract This study presents the first order optical soliton propagation systems for ultra-high transmission distance that reaches up to 20,000 Km with transmission data rates of 80 Gb/s and its corresponding maximum Q-factor and Soliton peak power. The simulation has been performed for the distance up to 20000 Km at the operating wavelength of 1300 and 1550 nm.

Vortex and multipole soliton modes in the (2 + 1)-dimensional cubic-quintic-septimal nonlinear media with the spatially modulated distributions

A (2 + 1)-dimensional cubic-quintic-septimal nonlinear Schrodinger equation with the spatially modulated distributions is investigated. When it is linked to the stationary cubic-quintic-septimal nonlinear Schrodinger equation, constraint conditions exist connecting inhomogeneous cubic, quintic and septimal nonlinearities and the amplitude of soliton mode. Under these conditions, analytical vortex and multipole soliton mode solutions are derived. If the value of the modulation depth is set as 1 and 0, vortex and multipole soliton modes are formed respectively. If the value of the soliton order number n grows, the layer structures of vortex and multipole soliton modes are added and are determined by the value of n. The stability of multipole and vortex soliton modes is tested by direct simulations with initial white noise.

Generation of low power and ultrashort laser pulses at 800 nm through soliton compression in chloroform-infiltrated cascaded photonic crystal fibers

A unique modest approach to obtain low energy, few-cycle laser pulses at 800 nm through chloroform filled cascaded photonic crystal fiber (PCF) is theoretically presented. Temperature is considered as the control parameter in designing the cascaded PCF to compress femtosecond pulses using higher order soliton compression scheme. The temperature-dependent parameters of both fiber and the infiltrated chloroform liquid are utilized to solve the modified nonlinear Schrödinger equation that facilitates the engineering of ultrashort pulses. Through this method, cascaded PCF could be designed using a single fiber for the generation of short pulses with minimal pedestal energy deprived of any other supplementary modules. We numerically prove that the compression of pulses from an input pulse of 140 fs up to 6.98 fs is possible by using temperature-controlled PCF. The designed fiber compressor is estimated to provide the possible compression factor of 20.05, quality factor of 0.49, output energy of 4.81 pJ, with 50.81% of pedestal energy, through 2.4 cm of entire length.

Vortex and multipole coupled solitons in the spatially modulated cubic–quintic–septimal nonlinear material

A (2+1)-dimensional N-coupled cubic–quintic–septimal nonlinear Schrödinger equation with spatial distribution of nonlinearity and transverse modulation describes beam propagation of mutually incoherent components with different frequencies in cubic–quintic–septimal nonlinear material. From this equation with two-component case, coupled soliton solutions are analytically obtained. When the value of the modulation depth q is fixed as 0 and 1, vector multipole–multipole soliton, vector vortex–vortex soliton and scalar vortex soliton are found. When the soliton order number p adds, vector solitons all show the layer structures. The number of layer is decided by the value of p. The “petals” number of multipole solitons is determined by the value of the topological charge m.

Time-Domain Neural Network Receiver for Nonlinear Frequency Division Multiplexed Systems

The nonlinear Fourier transform is a new approach for addressing the capacity limiting Kerr nonlinearities in optical communication systems. It exploits the property of integrability of the lossless nonlinear Schrodinger equation and thus incorporates nonlinearities as an element of the transmission. However, practical links employing erbium-doped fiber amplifiers include losses/gains and introduce noise which breaks the integrability of the nonlinear Schrodinger equation. Although the lossless path average approximation proposes an integrable model, its imprecision still leads to unintended distortions and thus performance degradation. We propose an alternative receiver for nonlinear frequency division multiplexing optical communication systems using techniques from machine learning. It is highly adaptive as it learns from previously transmitted pulses and thus holds no assumptions on the system and noise distribution. The detection method presented is fully applied in time-domain and omits the nonlinear Fourier transform. The numerical results provide a benchmark for nonlinear Fourier transform based detection of high order solitons for fiber links with losses and noise present.

Generation of single or double parallel breathing soliton pairs, bound breathing solitons, moving breathing solitons, and diverse composite breathing solitons in optical fibers.

Interactions of two truncated Airy pulses with arbitrarily initial relative phases, initial pulse intervals, and different soliton order are numerically investigated in optical fibers. When the soliton order is 1, depending on different initial pulse intervals, the initial in-phase Airy pulses may evolve to single breathing solitons, bound breathing solitons, and single parallel breathing solitons. While the out-of-phase Airy pulses may evolve to parallel or repulsive soliton pairs with breathing or weak breathing, after radiating away some dispersive waves. When the initial relative phases take arbitrary values except 0 and π, moving single breathing solitons and repulsive or parallel soliton pairs will form. Moreover, the whole temporal profiles may become asymmetric. The repulsive soliton pairs consist of two moving breathing solitons with different intensities, moving velocities, and breathing periods. The most interestingly is that, when the soliton order is larger than one, we observe double bound breathing solitons, double parallel breathing soliton pairs, and diverse composite breathing solitons which consist of two or more different breathing solitons. one can effectively manipulate and select the soliton expected and its evolution dynamics by adjusting the soliton order, initial pulse intervals, and initial relative phases.

Wavelength Separation Tunable 2-Color Soliton Generation and Its Application to 2-Color Fluorescence Multiphoton Microscopy

Optical solitons generated through soliton self-frequency shift in optical waveguides, has found the widespread applications in multiphoton microscopy (MPM). The multicolor fluorescence MPM, involving the simultaneous imaging of multiple structures labeled by different fluorophores, typically necessitates multiple-color pulses whose wavelengths match the peak excitation wavelengths of the fluorophores. Here we demonstrate experimentally an efficient method of generating wavelength separation tunable 2-color solitons, based on our recently proposed theoretical scheme of prechirping to manipulate the soliton order of the excitation pulse. Using this method, we reach a maximum soliton separation tuning range of 190 nm, with 1550-nm excitation. As an experimental demonstration of the applicability of this technique, we further show the MPM results using this 2-color soliton source for 2-color, 3-photon fluorescence imaging of fluorescent beads.

Polarization Rotation Dynamics in Harmonically Mode-Locked Vector Soliton Fiber Lasers

Polarization rotation dynamics for group velocity locked vector solitons in higher order harmonically mode-locked soliton fiber lasers are studied for the first time, in a thulium/holmium fiber laser. In a linear ultrafast fiber laser with net anomalous dispersion, harmonically mode-locked states are generated continuously by increasing the pump power for specific net cavity birefringence settings. The polarization evolution frequency remains the same for all harmonically mode-locked pulse trains, converging toward a singular solution close to the maximum pulse intensity modulation depth for each respective harmonically mode-locked state. With a polarization resolved output, an intensity modulation of the ultrafast pulse train with varying pulse binning is recorded, corresponding to the polarization rotation of the vector solitons. In addition, a polarization rotation vector soliton state with two different sets of RF sidebands symmetric to the repetition rate is observed, which leads to a more complex modulated pulse train. The experimental results of the polarization evolution also matche well with the presented theoretical modeling. Similar relative intensity noise <0.4% for all harmonically mode-locked states is measured. This method to modulate the intensity of an ultrafast pulse train and tune its pulse spacing by integer fractions of the initial round trip time shows a strong potential for applications in optical communications, polarization de multiplexing schemes, and sensing.

Carrier-envelope offset frequency stabilization of a 100 fs-scale Ti:sapphire mode-locked laser for quantum frequency comb generation

The carrier-envelope offset (CEO) frequency stabilization of a Ti:sapphire mode-locked laser with a Fourier-transform-limited pulse width of 100 fs-scale has been successfully realized. With the home-built f-2f interferometer, the CEO beat signal has been measured with a signal-to-noise ratio (SNR) as high as 40 dB under a resolution bandwidth of 100 kHz, which means that the coherence of the octave-spanning supercontinuum (SC) is well retained, although the evaluated pulse soliton order (N ≈ 23) completely violate the previous theoretical limitation that N should be smaller than 16. Furthermore, a stabilized CEO frequency is achieved with residual phase noise suppressed to 660 mrad (integrated from 1 Hz to 100 kHz) and an Allan deviation relative to the optical carrier of 7.6 × 10−18 at 1 s integration time. This realization constructs a solid foundation for further generation of quantum frequency comb needed for ultra-sensitive optical timing measurement applications.

General high–order breathers and rogue waves in the [formula omitted]-dimensional KP–Boussinesq equation

In this work, we investigate the (3+1)-dimensional KP–Boussinesq equation, which can be used to describe the nonlinear dynamic behavior in scientific and engineering applications. We derive general high–order soliton solutions by using the Hirota’s bilinear method combined with the perturbation expansion technique. We also obtain periodic solutions comprising of high-order breathers, periodic line waves, and mixed solutions consisting of breathers and periodic line waves upon selecting particular parameter constraints of the obtained soliton solutions. Furthermore, smooth rational solutions are generated by taking a long wave limit of the soliton solutions. These smooth rational solutions include high-order rogue waves, high–order lumps, and hybrid solutions consisting of lumps and line rogue waves. To better understand the dynamical behaviors of these solutions, we discuss some illustrative graphical analyses. It is expected that our results can enrich the dynamical behavior of the (3+1)-dimensional nonlinear evolution equations of other forms.

Generation of bright-dark soliton trains with a central wide dip in optical fibers

The nonlinear propagation dynamics of the dark soliton pulse with hyperbolic tangent profile in the anomalous dispersion regime of an optical fiber is investigated numerically. The corresponding temporal evolutions are provided for different integral or non-integral soliton order and loss perturbations. The results show that, irrespective of the integral or non-integral soliton order, with increase of distance, the width of the dark soliton broadens and more and more bright and dark pulses appear symmetrically on the two sides of the central black soliton. Correspondingly, bright-dark soliton trains with a central wide black soliton can gradually form. The higher the soliton order, the more the bright-dark pulses at the same distance. With increase of distance, the peak values of the bright pulses exhibit oscillation behavior. The loss perturbation only influences the evolutions slightly, which means that the generated soliton trains can resist loss perturbations to some extent. Moreover, this work provides us an alternative approach to generate bright-dark soliton trains by using the dark soliton pulse propagating in the anomalous dispersion regime of the passive optical fiber besides directly using the fiber laser.