Pulsating intensity mode solutions for a semiconductor laser with phase-conjugate feedback

This work analyzes the phase conjugate feedback (PCF) laser equations in the singular limit T large. T = /spl tau//sub e///spl tau//sub p/ is defined as the ratio of the carrier and photon lifetimes and is typically an O(10/sup 3/) large quantity for semiconductor lasers. The authors show analytically and verify numerically that the ECMs of the PCF laser are in first approximation given by Y /spl sim/ A/sub 1/exp(iw/sub 1/t) + A/sub 2/exp(iw/sub 2/t) where Y is the complex electric field. The frequencies /spl omega//sub 1/ and /spl omega//sub 2/ satisfy a resonance condition. In the simplest case of zero phase and frequency (DFWM) shift at the phase conjugate mirror, /spl omega//sub 1/ = -/spl omega//sub 2/ = /spl omega/ and the intensity |Y|/sup 2/ is a /spl pi///spl omega/-periodic function of t. Equations for the amplitudes A/sub 1/, A/sub 2/ and /spl omega/ are derived allowing a simple description of the PCF bifurcation diagram. In addition to the successive branches of external-cavity modes (ECMs), a nearly vertical Hopf bifurcation is found connecting steady state and ECM branches. These connections represent different bridges between isolated solutions compared to the bridges in the conventional optical feedback laser.