  Complete chaotic synchronization of end lasers has been observed in a line of
mutually coupled, time-delayed system of three lasers, with no direct
communication between the end lasers. The present paper uses ideas from
generalized synchronization to explain the complete synchronization in the
presence of long coupling delays, applied to a model of mutually coupled
semiconductor lasers in a line. These ideas significantly simplify the analysis
by casting the stability in terms of the local dynamics of each laser. The
variational equations near the synchronization manifold are analyzed, and used
to derive the synchronization condition that is a function of the parameters.
The results explain and predict the dependence of synchronization on various
parameters, such as time-delays, strength of coupling and dissipation. The
ideas can be applied to understand complete synchronization in other chaotic
systems with coupling delays and no direct communication between synchronized
sub-systems.
