Abstract
<p><!--StartFragment-->Time-delayed differential equations arise frequently in the study of nonlinear dynamics of lasers with optical feedback and because the analytical solution of such equations can be intractable, one resorts to numerical methods. In this manuscript, we show that under some conditions, the rate equations model that is used to model semiconductor lasers with feedback can be analytically solved by using the Lambert W function. In particular, we discuss the conditions under which the coupled rate equations for the intracavity electric field and carrier inversion can be reduced to a single equation for the field, and how this single rate equation can be cast in a form that is amenable to the use of the Lambert W function.<!--EndFragment--></p>
Highlights
Time-delayed differential equations arise naturally in a wide variety of physical phenomena where one or more system parameters are fed back into the system after a certain amount of time
The mathematical model for a time-delayed feedback system often reduces to a first order differential equation with a time-delayed term, and the analytical solution of such differential equations can be difficult because one has to deal with an infinite-dimensional equation
We demonstrate that the Lambert W -function [8] can be invoked in some situations to obtain analytical solutions to time-delayed equations of physical interest, and explore some of the consequences of using this method
Summary
Time-delayed differential equations arise naturally in a wide variety of physical phenomena where one or more system parameters are fed back into the system after a certain amount of time. There are combinations of delay and feedback strengths that produce single-tone oscillations in the optical frequency of the laser, period doubling routes to chaos, and coherence collapse and line-narrowing Each of these dynamical responses have been studied for a variety of applications such as the development of stable, all-optical microwave frequency oscillators, chaotic synchronization for all-optical encryption, and stable, narrow line-width lasers [11]. Another system that has been of immense interest to the semiconductor laser community is the coupling of two lasers by mutual injection of light from each laser into the other [13]. These systems have a natural time-delay built into them due to the finite amount of time it takes for the light from one laser to reach the other laser due to the physical separation between the lasers
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