  We consider a model for two lasers that are mutually coupled
optoelectronically by modulating the pump of one laser with the intensity
deviations of the other. Signal propagation time in the optoelectronic loop
causes a significant delay leading to the onset of oscillatory output.
Multiscale perturbation methods are used to describe the amplitude and period
of oscillations as a function of the coupling strength and delay time. For weak
coupling the oscillations have the laser's relaxation period, and the amplitude
varies as the one-fourth power of the parameter deviations from the bifurcation
point. For order-one coupling strength the period is determined as multiples of
the delay time, and the amplitude varies with a square-root power law. Because
we allow for independent control of the individual coupling constants, for
certain parameter values there is an atypical amplitude-resonance phenomena.
Finally, our theoretical results are consistent with recent experimental
observations when the inclusion of a low-pass filter in the coupling loop is
taken into account.
