Lambert function methods for laser dynamics with time-delayed feedback

TL;DR

This paper demonstrates that certain laser rate equations with time-delayed feedback can be analytically solved using the Lambert W function, offering a new approach to understanding laser dynamics with delay effects.

Contribution

It introduces an analytical solution method for laser delay differential equations using the Lambert W function, which was traditionally solved numerically.

Findings

Analytical solutions for laser rate equations with delay using Lambert W function.

Conditions identified under which coupled equations reduce to a single solvable equation.

Application of the method to coupled laser systems with time-delayed feedback.

Abstract

Time-delayed differential equations arise frequently in the study of nonlinear dynamics of lasers with optical feedback. Traditionally, one has resorted to numerical methods because the analytical solution of such equations are intractable. 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 intra-cavity electric field and excess 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. We conclude the manuscript with a similar discussion for two lasers coupled via time-delayed feedbacks.