  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.
