Multiscale pulse dynamics in communication systems with strong dispersion management

The evolution of an optical pulse in a strongly dispersion-managed fiber-optic communication system is studied. The pulse is decomposed into a fast phase and a slowly evolving amplitude. The fast phase is calculated exactly, and a nonlocal equation for the evolution of the amplitude is derived. In the limit of weak dispersion management the equation reduces to the nonlinear Schr?dinger equation. A class of stationary solutions of this equation is obtained; they represent pulses with a Gaussian-like core and exponentially decaying oscillatory tails, and they agree with direct numerical solutions of the full system.     

Pulse propagation in a nonlinear dielectric slab waveguide

The evolution of an optical pulse in a single-mode, step index dielectric slab waveguide which is characterized by an intensity dependent dielectric function in the core and cladding regions is treated by means of differential equation techniques. A cubic order non-linearity is considered. The electromagnetic field distribution in the slab waveguide region satisfies a non-linear wave equation. This field can be represented in terms of even TE guided modes with a slowly varying envelope amplitude function.Then using the well known approximation, based on the slowly varying character of the amplitude function, a non linear partial differential equation is obtained for the amplitude function. As the coefficients of this equation depend on the distance across the transverse direction X, an averaging technique over x is applied to reduce the nonlinear partial differential equation into a form that is easily transformed to the so-called non-linear Scroedinger differential equation.This equation is then attacked by means of the well known Inverse Scattering method in the case of reflection less potentials. The single and double soliton solutions are obtained explicitly for a single-mode slab waveguide. Finally numerical results are presented in the time domain.     

Correction to: Generation of amplitude- and phase-modulated signal beam with a phase-only spatial light modulator and its detection by the transport of intensity equation method for holographic data storage

This study evaluates a novel holographic data storage (HDS) that uses a phase-only spatial light modulator (SLM) for the multilevel complex amplitude modulation of a signal beam and the transport of intensity equation (TIE) method to detect the signal beam without interferometry, to increase the capacity of the HDS, simplifying its optical system, and improving the stability of the signal beam modulation and detection. Both the amplitude and phase of the signal beam were modulated by a computer-generated hologram displayed in a phase-only SLM, a 4-f optical system, and a pinhole placed in the Fourier plane. The complex amplitude-modulated signal beam generated by this scheme does not always perfectly match the target complex amplitude, and deviations from the amplitude and phase of the target complex amplitude may exist. It is unclear whether the TIE method, which is sensitive to the state of the beam intensity and the phase distributions to be detected (such as zero-intensity points and phase discontinuities), can accurately detect a signal beam whose complex amplitude is modulated by the modulation scheme with a phase-only SLM. Here, we demonstrate via numerical simulations and experiments that several methods of complex amplitude generation using a phase-only SLM can achieve multilevel modulation of the amplitude and phase of a signal beam and are suitable for detection by the TIE method in HDS.

     

Computer-generated line holograms using an xy plotter

A new method is presented for constructing a computer-generated Fourier-transformed line hologram using an xy plotter. Phase and amplitude of the complex amplitude in the hologram plane are represented by varying, respectively, the position and height of triangles formed in the cells composing the hologram. An image reconstructed optically from the hologram can be moved arbitrarily from the optical axis since the hologram contains information about the phase and amplitude of the complex amplitude in the hologram plane. A three-dimensional hologram from which an image was successfully reconstructed is also presented.     

普适性的光学双稳及激光动力学方程和稳定性分析

本文从慢包络近似的麦克斯韦方程出发,在优质腔极限下,结合腔的边界条件与平均场极限,建立了适用于任何物质的光学双稳系统及相应的激光系统的动力学方程;并通过对方程的线性稳定性分析,得到了一个普适性的失稳判据。 关键词：     

Autophasing of solitons

The effect of an external wave perturbation with a slowly varying frequency on a soliton of the nonlinear Schrödinger equation is investigated. The equations that describe the time evolution of the perturbed-soliton parameters are derived. The necessary and sufficient soliton phase locking conditions that relate the rate of change in the frequency of the perturbation, its amplitude, wave number, and phase to the initial values of parameters for the soliton have been found.     

Observation of chirped soliton dynamics at lambda = 1.55 microm in a single-mode optical fiber with frequency-resolved optical gating

We present an experimental observation of the dynamics of an initially chirped optical soliton at 1.55microm that is propagating through a single-mode optical fiber, using frequency-resolved optical gating (FROG). FROG permits observation of both the amplitude and the phase profiles of ultrashort pulses, providing complete information on the pulse evolution. The features that are detected, which include what is believed to be the first experimental observation of phase slips, are in quantitative agreement with numerical simulations that employ the nonlinear Schr?dinger equation.     

On coordinate transformations associated with the optical 1 + 1 lossless cubic–quintic Schrödinger equation

The mathematical structure of the transformations of coordinates involved in the optical 1 + 1 lossless cubic–quintic Schrödinger equation is investigated in relation to the amplitude and phase of the solitary-wave functions which are solutions to the equation in question. In fact, the above coordinate transformations are characterized algebraically by using matrices.     

Amplitude and phase of the solutions to the optical lossless cubic-quintic Schrödinger equation for some relevant values of dynamical parameters

The amplitude and phase of the wavefunctions obeying the optical lossless cubic-quintic Schrödinger equation for relatively weak nonlinearity relative to fifth degree are determined for relevant values of two fundamental dynamical parameters involved in the aforementioned wavefunctions, one of them being the solitary-wave velocity which is assumed to be positive and small enough.     

用线化和校正法近似求解一种非谐振动

利用拉格朗日方程建立了单质点弹簧振子非线性振动方程,作出了回复力、势能随坐标的变化曲线以及相图.应用第一类完全椭圆积分求出了非线性振动周期的精确解.应用线化和校正法对单质点弹簧振子的周期和近似解进行了求解.利用Maple计算机绘图,分别作出了它们的周期近似解与周期精确解随振幅的变化曲线以及近似解与数值解的变化曲线.利用线化和校正法所求得的近似解与数值解比较,具有简单实用、精度高、相对误差低等优点,在求解非线性振动中具有一定的实用价值.     

Novel accurate computer algorithm for modeling light propagation and diffraction in inhomogeneous, anisotropic medium - Applied to the acousto-optic interaction

There is a high demand for a computational model that calculates effectively the phase and amplitude distribution of the beams emerging from an acousto-optic cell. We present a model based on a new algorithm that is capable to solve the vectorial optical wave equation on consecutive planes in an optically anisotropic medium with an arbitrary refractive index distribution with limited refractive index amplitude. Strength of the presented method is that it does not require the paraxial approximation. We used the model successfully to calculate the amplitude and phase distribution of the diffracted and undiffracted beams generated in optically anisotropic and isotropic acousto-optic interaction.     

Control of the energy exchange upon double two-wave mixing in photorefractive crystals

The problem of double two-wave mixing in photorefractive crystals is analytically solved in the framework of the two-level model of optical transitions. Relationships are derived for calculating the intensities of all four interacting waves and the amplitude of the holographic grating. It is demonstrated that the magnitude and direction of energy exchange between the interacting beams can be controlled by a purely optical method, namely, by varying the intensity ratio and the initial phase shift between the interference patterns.     

单模光纤中超快受激拉曼散射的反斯托克斯波

由光纤中光的基本传输方程出发 ,利用慢变振幅近似 ,给出了包含反斯托克斯波的光纤超快受激拉曼散射的耦合波方程。以此为基础讨论连续、超快受激拉曼散射中泵浦波、斯托克斯和反斯托克斯波的耦合 ,分析了单模光纤相位匹配和群速匹配对光纤超快受激拉曼散射反斯托克斯波产生的影响     

Application of higher-order KdV——mKdV model with higher-degree nonlinear terms to gravity waves in atmosphere

Higher-order Korteweg-de Vries (KdV)-modified KdV (mKdV) equations with a higher-degree of nonlinear terms are derived from a simple incompressible non-hydrostatic Boussinesq equation set in atmosphere and are used to investigate gravity waves in atmosphere. By taking advantage of the auxiliary nonlinear ordinary differential equation, periodic wave and solitary wave solutions of the fifth-order KdV--mKdV models with higher-degree nonlinear terms are obtained under some constraint conditions. The analysis shows that the propagation and the periodic structures of gravity waves depend on the properties of the slope of line of constant phase and atmospheric stability. The Jacobi elliptic function wave and solitary wave solutions with slowly varying amplitude are transformed into triangular waves with the abruptly varying amplitude and breaking gravity waves under the effect of atmospheric instability.     

Comparison of Solutions of the General Nonlinear Amplitude Equation and a Modified Schrӧdinger Equation

We review the dynamics of narrow and broad-band optical pulses in nonlinear dispersive media. A major problem that arises during the development of theoretical models, which describe accurately and correctly the behavior of these pulses, is the limited application of the nonlinear Schr?dinger equation. It describes very well the evolution of nanosecond and picosecond laser pulses. However, when we investigate the propagation of femtosecond and attosecond light pulses, it is necessary to use the more general nonlinear amplitude equation. We show that in this equation two additional terms are included and they have a significant impact on the phase of the pulse. We perform numerical simulations and show the temporal shift of the position of fundamental solitons. This effect depends on the initial duration of the laser pulses. To clarify the influence of the additional terms on the parameters of the optical pulses, we consider the nonlinear amplitude equation, which is a modified nonlinear Schr?dinger equation.     

Modulational instability in the cubic-quintic nonlinear Schrödinger equation through the variational approach

The dynamics of nonlinear pulse propagation in an average dispersion-managed soliton system is governed by a constant coefficient nonlinear Schrödinger (NLS) equation. For a special set of parameters the constant coefficient NLS equation is completely integrable. The same constant coefficient NLS equation is also applicable to optical fiber systems with phase modulation or pulse compression. We also investigate MI arising in the cubic-quintic nonlinear Schrödinger equation for ultrashort pulse propagation. Within this framework, we derive ordinary differential equations (ODE’s) for the time evolution of the amplitude and phase of modulation perturbations. Analyzing the ensuing ODE’s, we derive the classical modulational instability criterion and identify it numerically. We show that the quintic nonlinearity can be essential for the stability of solutions. The evolutions of modulational instability are numerically investigated and the effects of the quintic nonlinearity on the evolutions are examined. Numerical simulations demonstrate the validity of the analytical predictions.     

Mode Structure and the Envelope of a Short Pulse in a Graded-Index Optical Fiber with a Longitudinal Inhomogeneity and a Spatial Bending

We consider propagation of a short optical pulse in an optical fiber whose refractive index is strongly dependent on the radial coordinate and is weakly dependent on the longitudinal coordinate with allowance for the possible weak spatial bending of the fiber axis. The three-dimensional nonlinear wave equation modelling the pulse propagation is solved asymptotically with respect to a small parameter specifying the order of magnitude of the pulse amplitude. A relationship between the propagating modes and the eigenvalues and eigenfunctions of the singular Sturm-Liouville problem is established. The pulse propagation is shown to have three scales: the high-frequency carrier is modulated by the envelope which evolves in a two-scale manner and is described by a nonlinear Schrödinger equation whose coefficients depend on the longitudinal coordinate. The transverse distribution of the wave field and the envelope soliton are obtained in terms of elementary functions for several types of transverse and longitudinal inhomogeneities of the fiber. The possibility of controlling the pulse parameters by varying the transverse and longitudinal inhomogeneities of the fiber is pointed out.     

Parametrically forced pattern formation

Pattern formation in a nonlinear damped Mathieu-type partial differential equation defined on one space variable is analyzed. A bifurcation analysis of an averaged equation is performed and compared to full numerical simulations. Parametric resonance leads to periodically varying patterns whose spatial structure is determined by amplitude and detuning of the periodic forcing. At onset, patterns appear subcritically and attractor crowding is observed for large detuning. The evolution of patterns under the increase of the forcing amplitude is studied. It is found that spatially homogeneous and temporally periodic solutions occur for all detuning at a certain amplitude of the forcing. Although the system is dissipative, spatial solitons are found representing domain walls creating a phase jump of the solutions. Qualitative comparisons with experiments in vertically vibrating granular media are made. (c) 2001 American Institute of Physics.     

Nonlinear pulse propagation: a time-transformation approach

We present a time-transformation approach for studying the propagation of optical pulses inside a nonlinear medium. Unlike the conventional way of solving for the slowly varying amplitude of an optical pulse, our new approach maps directly the input electric field to the output one, without making the slowly varying envelope approximation. Conceptually, the time-transformation approach shows that the effect of propagation through a nonlinear medium is to change the relative spacing and duration of various temporal slices of the pulse. These temporal changes manifest as self-phase modulation in the spectral domain and self-steepening in the temporal domain. Our approach agrees with the generalized nonlinear Schr?dinger equation for 100 fs pulses and the finite-difference time-domain solution of Maxwell's equations for two-cycle pulses, while producing results 20 and 50 times faster, respectively.     

On the motion of non-linear oscillators with a fractional-order restoring force and time variable parameters

An analytical approach to determine the approximate solution for the periodic motion of non-conservative oscillators with a fractional-order restoring force and slowly varying parameters is presented. The solution has the form of the first-order differential equation for the amplitude and phase of motion. The method used is based on the combination of the Krylov-Bogoliubov method with Hamilton's variational principle with the uncommutative rule for the variation of velocity. The conservative systems with slowly varying parameters are also considered. The corresponding adiabatic invariant is obtained. Two examples are given to illustrate derived theoretical results.