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Journal of the Optical Society of America B

Fokker-Planck and Langevin analyses of noise accompanying the amplification of optical pulses in semiconductor optical amplifiers

A. Bilenca, G. Eisenstein

Published Aug 1, 2005• 8 citations
Publisher

Abstract

The Fokker-Planck approach is used to study the statistics of an optical pulse propagating in semiconductor optical amplifiers in which the amplified spontaneous emission noise dominates. The probability density functions (pdfs) of the pulse peak sample are calculated for different saturation levels. Analytical results are possible for an ideal linear amplifier and when the simplest first-order approximation for nonlinear amplification is used. In other cases, we solve numerically the Fokker-Planck equation and the appropriate Langevin equation. Multicanonical Monte Carlo simulations ensure efficient calculations of the pdfs whose high- and low-power tails deviate from noncentral chi-square statistics for moderate and deep saturation levels, thus implying that the electric field process is not Gaussian.

DOI
10.1364/josab.22.001632

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The Fokker-Planck approach is used to study the statistics of an optical pulse propagating in semiconductor optical amplifiers in which the amplified spontaneous emission noise dominates. The probability density functions (pdfs) of the pulse peak sample are calculated for different saturation levels. Analytical results are possible for an ideal linear amplifier and when the simplest first-order approximation for nonlinear amplification is used. In other cases, we solve numerically the Fokker-Planck equation and the appropriate Langevin equation. Multicanonical Monte Carlo simulations ensure efficient calculations of the pdfs whose high- and low-power tails deviate from noncentral chi-square statistics for moderate and deep saturation levels, thus implying that the electric field process is not Gaussian.
We present a theoretical analysis and an experimental study of the statistical properties of the noise accompanying an optical pulse propagating in a nonlinear semiconductor optical amplifier. Several degrees of gain saturation corresponding to different levels of optical signal-to-noise ratios (OSNRs) are examined. We employ the Heun numerical procedure to ensure proper convergence to the Stratonovich solution of the multiplicative propagation equation. This algorithm takes also into account the effect of gain saturation due to the amplified spontaneous emission noise. Moreover, the multicanonical Monte Carlo algorithm is used to efficiently calculate the probability density functions (pdfs), including the tails, of the peak of a pulse which emerges at the output of a saturated semiconductor optical amplifier. The results are compared to the corresponding pdfs obtained in a linear amplification system (where the optical noise is additive and Gaussian) having the same gain and under identical OSNR levels. We demonstrate that the pdf of the saturated amplifier is shifted toward lower power levels and is narrower, or equivalently, the mean and variance for the saturated amplifier case are smaller. Also, the difference between the two configurations increases with the degree of gain saturation. The theoretical predictions are confirmed by a series of experiments in which the pdfs at the output of the linear amplification scheme and saturated semiconductor optical amplifier are measured for an optical pulse of /spl sim/ 70-ps duration.
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.
This paper derives Langevin equations for an optical cavity where the dominant nonlinearity arises from free-carrier dispersion. We define a generalized Wigner function, compute a Fokker-Planck equation that approximates the master equation, and convert this to a system of stochastic differential equations. These equations are similar to the Wigner equations for an optical Kerr cavity, but have additional noise terms due to the incoherent carrier excitation and decay processes. We use these equations to simulate a phase-sensitive amplifier and latch and compare the results to a Kerr model.
In the work, analytic expressions describing the behaviour of a strongly saturated nonlinear optical amplifier are deduced. These expressions make possible the calculation of the mean and variance of the amplified light photon number. Results are compared with those obtained by the numerical resolution of the photon density matrix equation, and good accordance is observed in the strong saturation regime. The results numerically obtained enable us to make conclusions about the statistical fluctuations of the optical power at the nonlinear optical amplifer output.
A study is carried out of the evolution of the classical phase portrait of the density operator of nonlinearly amplified light. A single-mode optical amplifier is considered, with quantumstatistical fluctuations of the two-level active medium and the light being amplified. The Heisenberg-Langevin equations for the field operators are obtained by adiabatic elimination of the atomic variables in the van der Pol approximation. The Fokker-Planck equation for the P-function, corresponding to the Heisenberg-Langevin equations, was solved numerically using the polynomial-expansion method. The initial quasiprobability distribution function corresponded to a pure coherent state. The Wigner quasiprobability distribution was calculated with the help of an integral transform of the P-function. Such quantumstatistical parameters of the amplifier radiation as the mean photon number and its variance, the Mandel parameter, and the mean field and the variance of the phase-amplitude quadratures of the field were calculated. O 1996 American Institute of Physics. [S1063-7761(96)00405-21