Small noise asymptotic of the timing jitter in soliton transmission

Abstract

We consider the problem of the error in soliton transmission in long-haul optical fibers caused by the spontaneous emission of noise inherent to amplification. We study two types of noises driving the stochastic focusing cubic one dimensional nonlinear Schr\"{o}dinger equation which appears in physics in that context. We focus on the fluctuations of the mass and arrival time or timing jitter. We give the small noise asymptotic of the tails of these two quantities for the two types of noises. We are then able to prove several results from physics among which the Gordon--Haus effect which states that the fluctuation of the arrival time is a much more limiting factor than the fluctuation of the mass. The physical results had been obtained with arguments difficult to fully justify mathematically.Comment: Published in at http://dx.doi.org/10.1214/07-AAP449 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org