Timing of laser pulses produced by combined passive and active mode-locking†

Abstract

The equations of combined active and passive mode-locking are solved approximately in the limit when the saturable absorber determines the width of the pulse, and the active modulator the timing. An equation identical with the Adler equation of oscillator phase locking is obtained for the phase of the pulse with respect to the active modulation. In the case of synchronization between the mode-locked pulse-train and the active modulation (time independent solution), the phase of the pulse with respect to the modulation maximum is determined as a function of detuning and modulation (depth. Beyond a critical detuning of the round-trip time T_n from the modulation period T_M, synchronisation ceases and the pulses' slip through ' the active modulation with a phase that has a periodic variation superimposed on the uniform slippage rate. Finally, we investigate the effect of noise on the pulse timing. The spectrum of the pulse phase is the same as the spectrum of the phase of an injection-locked single-mode oscillator in the presence of noise.