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 = τ_e/τ_p is defined as the ratio of the carrier and photon lifetimes and is typically an O(10³) 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 ∼ A₁exp(iw₁t) + A₂exp(iw₂t) where Y is the complex electric field. The frequencies ω₁ and ω₂ satisfy a resonance condition. In the simplest case of zero phase and frequency (DFWM) shift at the phase conjugate mirror, ω₁ = -ω₂ = ω and the intensity |Y|² is a π/ω-periodic function of t. Equations for the amplitudes A₁, A₂ and ω 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.