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Transport Theory and Statistical Physics

Discrete ordinate solution of a Fokker-Planck equation in laser physics

Robert Blackmore, Ulrich Weinert, Bernie D. Shizgal

Published Feb 1, 1986• 11 citations
Publisher

Abstract

A newly developed discrete ordinate method of solution of differential equations is applied to the solution of the Fokker-Plank equation employed in the a nalysis is of optical bistability. The solution is written in terms of the eigenfunctions of the Fokker-PlancK operator and the time evolution of the probability density function is studied. The present analysis includes a study of the convergence of the eigenualues and eigen functions for different choices of the quadrature rule upon which the discrete ordinate method is based.

DOI
10.1080/00411458608210450

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A newly developed discrete ordinate method of solution of differential equations is applied to the solution of the Fokker-Plank equation employed in the a nalysis is of optical bistability. The solution is written in terms of the eigenfunctions of the Fokker-PlancK operator and the time evolution of the probability density function is studied. The present analysis includes a study of the convergence of the eigenualues and eigen functions for different choices of the quadrature rule upon which the discrete ordinate method is based.
Based on the well-known relation between Fokker-Planck equations and Schrödinger equations of quantum mechanics (QM), we propose new deformed Fokker-Planck (FP) equations associated with the Schrödinger equations of “discrete” QM. The latter is a natural discretization of QM and its Schrödinger equations are difference instead of differential equations. Exactly solvable FP equations are obtained corresponding to exactly solvable “discrete”QM, whose eigenfunctions include various deformations of the classical orthogonal polynomials.
The one-dimensional Fokker-Planck equation is solved by the Asymptotic Taylor Expansion Method for the time-dependent probability density of a particle. Using an ansatz wave function, one obtains the series expansion of the solution for the Schrödinger and it allows one to find out the eigen functions and eigen energies of the states to the evaluation of the probability. The eigen energies of some certain kind of Bistable potentials are calculated for some certain potential parameters. The probability function is determined and graphed for potential parameters. The numerical results are compared with existing literature, and a conclusion about the advantages and disadvantages on the method is given.
A dynamic correspondence is set up between the density matrix of the electromagnetic field and an associated classical function. Means of one and two time quantum operators are shown to be obtainable from the corresponding averages of the associated classical random problem. When all atomic response rates are fast compared to photon rates, the equation of motion is obtained for the field density matrix. The associated classical problem is shown to obey a Fokker-Planck equation. Over a broad region that includes the threshold region, the Fokker-Planck equation is reduced to that for the rotating-wave van der Pol oscillator for which exact numerical solutions have been previously obtained by Hempstead and Lax.
A novel numerical method to solve Fokker-Planck equations in quantum optics is presented, based on a Monte Carlo simulation of the probability diffusion process. The method is especially useful for multimode analysis and hence for studying realistic models of nonlinear optical systems. Two simple examples are given: the first is a 1D Fokker-Planck equation in the number representation, which describes a simple model of optical amplifier; the second is a 2D equation in the P-function representation-the quasi-probability for normal-ordered averages-which corresponds to the customary Van der Pol model of the laser threshold. In this case also the field correlation function and spectrum are numerically simulated.