Investigation of mode coupling in low and high NA step index plastic optical fibres using the Langevin equation

By employing the Langevin equation, we have examined a mode coupling in low and high NA step index plastic optical fibres. The numerical integration of the Langevin equation is based on the computer-simulated Langevin force. The solution matches the experimental data reported previously. We have shown that by solving the Langevin equation (stochastic differential equation) one can treat a mode coupling in multimode low and high NA step index plastic optical fibres, which is the result of fibre's intrinsic random perturbations.     

Optical power flow in plastic-clad silica fibers

Using the time-independent power-flow equation, we have examined the mode coupling caused by intrinsic perturbation effects of step-index plastic clad silica fiber carrying more than 10(5) modes. Result show that the equilibrium mode distribution for this fiber is achieved at a length of approximate 550 m, which is longer than reported previously. While this coupling length is much longer than that of plastic optical fibers, it is sorter than that of all-glass fibers.     

Mode coupling in strained and unstrained step-index plastic optical fibers

Using the power-flow equation, we have examined the state of mode coupling in strained and unstrained step-index plastic optical fibers. The strained fibers show much stronger mode coupling than unstrained fibers of the same types. As a result, the coupling lengths where equilibrium mode distribution is achieved and the lengths of fiber required for achieving a steady-state mode distribution for strained fibers are much shorter than the corresponding lengths for unstrained fibers.     

Invalidity of the spectral Fokker–Planck equation for Cauchy noise driven Langevin equation

The standard Langevin equation is a first order stochastic differential equation where the driving noise term is a Brownian motion. The marginal probability density is a solution to a linear partial differential equation called the Fokker–Planck equation. If the Brownian motion is replaced by so-called -stable noise (or Lévy noise) the Fokker–Planck equation no longer exists as a partial differential equation for the probability density because the property of finite variance is lost. Instead it has been attempted to formulate an equation for the characteristic function (the Fourier transform) corresponding to the density function. This equation is frequently called the spectral Fokker–Planck equation.

This paper raises doubt about the validity of the spectral Fokker–Planck equation in its standard formulation. The equation can be solved with respect to stationary solutions in the particular case where the noise is Cauchy noise and the drift function is a polynomial that allows the existence of a stationary probability density solution. The solution shows paradoxic properties by not being unique and only in particular cases having one of its solutions closely approximating the solutions to a corresponding Langevin difference equation. Similar doubt can be traced Grigoriu's work [Stochastic Calculus (2002)].     

Influence of numerical aperture on mode coupling in step-index plastic optical fibers

Using the power-flow equation, we have examined the state of mode coupling in step-index plastic optical fibers with different numerical apertures. Our results confirm that the coupling rates vary with the coupling coefficient of the fibers as the dominant parameter, especially in the early stage of coupling near the input fiber end. However, we show that the fiber's numerical aperture has a significant influence on later stages of this process. Consequently, equilibrium mode distribution and steady-state distribution are achieved at overall fiber lengths that depend on both of these factors. As one of our examples demonstrates, it is possible for the coupling length of a high-aperture fiber to be similar to that of a low-aperture fiber despite the three-times-larger coupling coefficient of the former.     

On a quasiclassical Langevin equation

It is shown that the motion of a quantum mechanical particle coupled to a dissipative environment can be described by a Langevin equation where the stochastic force is generalized such that its power spectrum is in accordance with the fluctuation-dissipation theorem. This generalized Langevin equation has an interesting range of applicability. It includes the quasiclassical regime provided that the damping, that is, the coupling of the particle to its environment, is sufficiently strong.     

Theoretical analysis of longevity testing on bubble memory devices

Theoretical results of magnetic bubble device long-term reliability testing are reported. The bubble during propagation along Permalloy tracks is represented by a simple, one-dimensional stochastic model. An equation to describe fluctuation in cylindrical bubble radius is approximated in the Langevin type stochastic differential equation, in which a set of small effects, such as interaction among bubbles and crystal nonuniformity, are considered as a white noise forcing term. Estimating the average time to bubble annihilation or runout (bubble memory mean time to failure) is reduced to a level-crossing problem for a random process. Calculated bias field margin degradation shows a qualitative agreement with experimental results for an actual bubble device. Bubble material parameters for obtaining maximum operation time are suggested.     

Thermally activated magnetisation reversal

A model of thermally activated magnetisation reversal is presented. The approach is based on the numerical solution of the Langevin equation of the problem. It is shown that the random thermal perturbations give rise to correlated magnetisation fluctuations (spin waves) in coupled systems. Simulations of longitudinal thin films are presented which show an effective non-local damping arising from spin waves.     

Stochastic model of crack paths in composites based on the Langevin equation

A stochastic model based on the Langevin equation was used to describe crack paths in composites. A single crack path in a fiberglass reinforced epoxy matrix composite was used to predict Langevin parameters quantifying architectural drift and material variability. The predicted Langevin parameters were consistent with moderate architectural drift (Langevin drift parameter of 0.04 ± 0.02) and moderate material variability (Langevin variability parameter of 0.05 ± 0.03). These Langevin parameters were then used to predict a family of crack paths exhibiting the same stochastic characteristics as the original crack path. Architectural drift was qualitatively related to the orientation of the reinforcing phase. A strong correlation between the predicted and experimental crack path suggests the utility of the Langevin model to quantify crack paths.     

平面应变轴对称压电圆柱壳的随机响应

提出求解随机激励轴对称压电圆柱壳响应的一种方法，并导出相应的解析表达式。首先给出压电圆柱壳在边界随机激励下的基本方程；然后通过位移与电势的变换，将随机激励变换到运动方程中；再利用Legendre多项式展开位移，应用Galerkin法化偏微分的运动方程为常微分方程组；最后根据随机振动理论，得到压电圆柱壳位移与加速度响应的均方值，由此可计算随机响应、分析有关因素的影响与机电耦合关系等。分析说明了存在的机电耦合项，及由此产生广义刚度的非对称性。     

Supersensitivity due to uncertain boundary conditions

We study the viscous Burgers' equation subject to perturbations on the boundary conditions. Two kinds of perturbations are considered: deterministic and random. For deterministic perturbations, we show that small perturbations can result in O(1) changes in the location of the transition layer. For random perturbations, we solve the stochastic Burgers' equation using different approaches. First, we employ the Jacobi‐polynomial‐chaos, which is a subset of the generalized polynomial chaos for stochastic modeling. Converged numerical results are reported (up to seven significant digits), and we observe similar ‘stochastic supersensitivity’ for the mean location of the transition layer. Subsequently, we employ up to fourth‐order perturbation expansions. We show that even with small random inputs, the resolution of the perturbation method is relatively poor due to the larger stochastic responses in the output. Two types of distributions are considered: uniform distribution and a ‘truncated’ Gaussian distribution with no tails. Various solution statistics, including the spatial evolution of probability density function at steady state, are studied. Copyright © 2004 John Wiley & Sons, Ltd.     

Influence of bending on power distribution in step-index plastic optical fibers and the calculation of bending loss

A means of calculating optical power distribution in bent multimode optical fibers is proposed. It employs the power-flow equation approximated by the Fokker-Planck equation that is solved by the explicit finite-difference method. Conceptually important steps of this procedure include (i) dividing the full length of the bent optical fiber into a finite number of short, straight segments; (ii) solving the power equation for each segment sequentially to find its output distribution; and (iii) expressing that output distribution in rotated coordinates of the subsequent segment along the curved fiber to determine the input distribution for that subsequent segment and thus enable the calculation of the power flow and output distribution for it. The segment length and bend-induced perturbation of output angles are determined by geometric optics. The resulting power distributions are given at different cross sections along the curved fiber axis. They vary with the radius of fiber curvature and launch conditions. Results are compared to those for straight fiber. Bending loss is calculated as well.     

Maximum fiber coupling efficiency and optimum beam size in the presence of random angular jitter for free-space laser systems and their applications

Free-space laser communication systems use optical-fiber-based technology such as optical amplifiers, receivers, and high-speed modulators. In these systems using single-mode fibers, the fiber coupling efficiency is one of the most significant issues to be solved. Optimum relationships between a focused optical beam and mode field size of the optical fiber in the presence of random angular jitter are discussed in relation to fiber-coupled optical systems. Maximum fiber coupling efficiency is analytically derived with the optimum Airy disk radius normalized by the mode field radius as a function of random angular jitter. The fade level of fiber-coupled signals at desired fade probability is investigated. It is shown that the average bit error ratio significantly degrades with the random angular jitter normalized by the mode field radius larger than about 0.3 when the Airy disk size is optimally selected.     

Equilibrium mode distribution and steady-state distribution in 100-400 μm core step-index silica optical fibers

Using the power flow equation, the state of mode coupling in 100-400 μm core step-index silica optical fibers is investigated in this article. Results show the coupling length L(c) at which the equilibrium mode distribution is achieved and the length z(s) of the fiber required for achieving the steady-state mode distribution. Functional dependences of these lengths on the core radius and wavelength are also given. Results agree well with those obtained using a long-established calculation method. Since large core silica optical fibers are used at short distances (usually at lengths of up to 10 m), the light they transmit is at the stage of coupling that is far from the equilibrium and steady-state mode distributions.     

Asymptotic stability of a stochastic delay equation

The investigation of a nonlinear stochastic delay equation with structural tool and regenerative time delays is presented. The conditions of Hopf bifurcation are computed in order to describe the regions of stability and instability. Explicit expressions characterizing the influence of nonlinear and stochastic perturbations, valid in the first order centre manifold approximation, are derived. In addition to this, we describe the underlying mathematical ideas of the centre manifold reduction of delay differential equations to ordinary differential equations for fixed time delays.     

On Solution Regularity of Linear Hyperbolic Stochastic PDE Using the Method of Characteristics

The generalized Polynomial Chaos (gPC) method is one of the most widely used numerical methods for solving stochastic differential equations. Recently, attempts have been made to extend the the gPC to solve hyperbolic stochastic partial differential equations (SPDE). The convergence rate of the gPC depends on the regularity of the solution. It is shown that the characteristics technique can be used to derive general conditions for regularity of linear hyperbolic PDE, in a detailed case study of a linear wave equation with a random variable coefficient and random initial and boundary data.     

Rigorous integral equation analysis of nonsymmetric coupled grating slab waveguides

A rigorous integral equation formulation in conjunction with Green's function theory is used to analyze the waveguiding and coupling phenomena in nonsymmetric (composed of dissimilar slabs) optical couplers with gratings etched on both slabs. The resulting integral equation is solved by applying an entire-domain Galerkin technique based on a Fourier series expansion of the unknown electric field on the grating regions. The proposed analysis actually constitutes a special type of the method of moments and provides high numerical stability and controllable accuracy. The singular points of the system's matrix accurately determine the complex propagation constants of the guided waves. The results obtained improve on those derived by coupled-mode methods in the cases of large grating perturbations and highly dissimilar slabs. Numerical results referring to the evolution of the propagation constants as a function of the grating's characteristics are presented. Optimal grating parameters with respect to minimum coupling length and maximum coupling efficiency are reported. The coupler's efficient operation as an optical bandpass filter is thoroughly investigated.     

Stochastic analysis of fatigue crack growth for metallic structures

In order to estimate the statistical variability of fatigue crack growth in metallic structures, a stochastic model is proposed by combining stochastic theory with experimental results. A stochastic differential equation is derived from the stochastic model for fatigue crack growth. By using the solution of the stochastic differential equation, some distribution functions related to fatigue crack growth were derived. Sample functions of fatigue crack growth time histories have been simulated as random processes.     

Impulse dispersion in a multimode optical fiber from its far-field radiation pattern

A simple geometric-optical theory predicts impulse broadening in multimode fibers using the far-field power distribution of an incoherently excited waveguide with arbitrary refractive-index profile. The accuracy is comparable with that of scalar optical methods if they do not incorporate mode coupling and a differential mode attenuation model.     

Exponential ergodicity of some Markov dynamical systems with application to a Poisson-driven stochastic differential equation

We are concerned with the asymptotics of the Markov chain given by the post-jump locations of a certain piecewise-deterministic Markov process with a state-dependent jump intensity. We provide sufficient conditions for such a model to possess a unique invariant distribution, which is exponentially attracting in the dual-bounded Lipschitz distance. Having established this, we generalize a result of J. Kazak on the jump process defined by a Poisson-driven stochastic differential equation with a solution-dependent intensity of perturbations.