CONSTANT PHASE, PHASE DRIFT, AND PHASE ENTRAINMENT IN LASERS WITH AN INJECTED SIGNAL

In a laser with an injected signal (LIS), the phase of the laser field can be constant (phase locking) or grow unbounded (phase drift) or vary periodically (phase entrainment). We analyze these three responses by studying the bifurcation diagram of the time-periodic solutions in the limit of small values of the population relaxation rate (small), small values of the detuning parameters (small and), and small values of the injection field (small y). Our bifurcation analysis has led to the following results: (1) We determine analytically the conditions for a Hopf bifurcation to stable time-periodic solutions. This bifurcation appears at y=yH and is possible only if the detuning parameters are sufficiently large compared to [specifically, and must be O(1/2) quantities]. (2) We construct the periodic solutions in the vicinity of y=yH. We show that the phase of the laser field varies periodically. We then follow numerically this branch of periodic solutions from y=yH to y=0. Near y=0, the phase becomes unbounded. (3) A transition between bounded and unbounded phase time-periodic solutions appears at y=yc (0<yc<yH). This transition does not appear at a bifurcation point. We characterize this transition by analyzing the behavior of the phase as yyc+ and as yyc-. (4) We find conditions for a secondary bifurcation from the periodic solutions to quasiperiodic solutions. The secondary branch of solutions is then investigated numerically and is shown to terminate at the limit point of the steady states. (5) We develop a singular perturbation analysis of the LIS equations valid as 0. This analysis allows the determination of a complete branch of bounded time-periodic solutions. We show that the amplitude of these time-periodic solutions becomes unbounded as y0 provided that and are sufficiently small compared to [specifically, and are zero or O() quantities]. © 1990 The American Physical Society.