An exact analysis is presented of the steady-state stability of a semiconductor laser subjected to feedback from a phase-conjugate mirror. Reduced stability occurs at low feedback whenever the effective external delay time is an integer multiple of the relaxation oscillation period. The role of a finite response time of the mirror is to enhance drastically the steady-state stability. © 1998 Optical Society of America OCIS codes: 140.2020, 140.3430, 140.3570, 140.5960, 190.5040. Optical feedback is known to affect severely the properties of a semiconductor laser. For stabilization purposes feedback from a phase-conjugate mirror is preferred over conventional optical feedback (COF), since in the latter case the laser suffers from extreme sensitivity to mirror-distance variations within an optical wavelength. This sensitivity is due to the fact that with an ordinary mirror the phase of the returning light depends strongly on the mirror position, whereas in phase conjugation there is no such dependence. 2 However, in the case of phase-conjugate feedback (PCF) one is always confronted with a certain sluggishness of the ref lector owing to the f inite response time, which should be taken into account when one is analyzing the stability of the laser operation with a phase-conjugate mirror. Most of the previous stability analyses disregarded this sluggishness. 3,4 In this Letter we present a linear stability analysis of the steady state of single-frequency operation of a single-mode diode laser with PCF, including the finite response-time effect. Owing to the time-delay term in the rate equations an exponential appears in the characteristic equation D͑s͒, the roots of which determine the stability of the laser. This exponential, which also shows up in the case of COF, complicates the analysis, and several kinds of approximation have been made in stability analysis.
1,4,5We report on a stability analysis without any such approximation, the results of which are valid for arbitrary laser parameters. This method was also used by Cohen et al. 6 in analyzing a diode laser with COF.When multiple external round trips can be ignored, the rate equations for a single-mode semiconductor laser with sluggish PCF are given bywhere E is the slowly varying amplitude of the optical field with respect to the optical carrier exp͑iv 0 t͒, with v 0 as the emission frequency of the solitary laser (i.e., the same laser without feedback) at threshold. E is normalized such that jEj 2 P equals the number of photons inside the cavity. N N th 1 DN is the number of electron -hole pairs (inversion) in the active layer, and N th is the inversion at threshold of the solitary laser. j is the differential gain, e is the nonlinear gain parameter, G 0 is the photon decay rate, a is the linewidth-enhancement factor, g p is the feedback rate, t m is the response time of the mirror, d 0 is the detuning of the mirror pump beam with respect to v 0 , t is the external-cavity round-trip time, J is the number of carriers injected into the active layer per un...
