IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 43, NO. 11, NOVEMBER 2007 1109 Relaxation Oscillations and Pulse Stability in Harmonically Mode-Locked Semiconductor Lasers Farhan Rana and Paul George Abstract—In this paper, we discuss pulse dynamics in harmoni- cally mode-locked semiconductor lasers and present the conditions necessary for stability. In a laser mode-locked at the th har- monic, the pulse energy fluctuations have different modes of relaxation oscillations. Different modes correspond to different patterns for the energy fluctuations in the different pulses inside the laser cavity. In the higher order relaxation oscillation modes, the energy fluctuations are negatively correlated in different pulses inside the laser cavity, and these modes can cause instability. Gain saturation on time scales of the order of the pulse width (dynamic gain saturation) stabilizes pulse energy fluctuations with respect to relaxation oscillations. The precise limits on the stable operating regime depend on the gain dynamics at both slow and fast time scales. We also discuss harmonic mode-locking in the presence of a slow saturable absorber. Dynamic loss saturation in a saturable absorber can work against dynamic gain saturation and limit the stability range for harmonic mode-locking. Index Terms—Laser stability, optical pulses, semiconductor lasers. I. INTRODUCTION H ARMONICALLY mode-locked lasers are attractive as sources of high-repetition-rate optical pulses that can be used in electrooptic sampling, optical analog-to-digital conver- sion, optical telecommunication systems, and ultrafast optical measurements [1]–[5]. Stability of the pulses in harmonically mode-locked lasers is important for most of these applications. A laser mode-locked at the th harmonic has optical pulses propagating inside the laser cavity. The requirements for pulse stability in fiber lasers were analyzed in [6] and [7]. It was shown that pulse stability results from the combination of Kerr nonlinearity and optical filtering. For soliton pulses, the pulse width is inversely proportional to the pulse energy. If the pulse energy increases, the pulse width decreases and the pulse expe- riences less loss from the active modulator. On the other hand, since the pulse bandwidth also increases with decrease in the pulse width, the pulse experiences more loss from the optical filter. The pulse energy fluctuations are damped if the increase in loss from the optical filter is more than the increase in gain Manuscript received March 27, 2007; revised July 11, 2007. This work was supported in part by a National Science Foundation Faculty CAREER Award, in part by ILX Lightwave Corporation, in part by Intel Inc., and in part by Info- tonics Inc. The authors are with the School of Electrical and Computer Engineering, Cor- nell University, Ithaca, NY 14853 USA (e-mail: fr37@cornell.edu; pag25@cor- nell.edu). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/JQE.2007.907261 from the active modulator. This condition was used to obtain a minimum value for the pulse energy for stable operation. The stability of soliton pulses in fiber lasers was also analyzed numerically in [8], and the stability requirements predicted in [6] and [7] were verified. It was also shown in [8] that, when the pulse power is smaller than the minimum value required for stable operation, instabilities can lead to pulse dropouts. In the case of harmonically mode-locked semiconductor lasers, the following questions arise: 1) what stabilizes the pulses and 2) what are the limits on the stable operating regime. In this paper, we present a theoretical model to answer these questions. In the analytical treatments of [6] and [7], it was assumed that every pulse in the laser cavity is the same. This assumption makes it impossible to study the slow gain dynamics and find the precise limits on the stable operating regime. In harmon- ically mode-locked semiconductor lasers, the gain relaxation times can be much longer than the pulse repetition times. Pulse energy fluctuations that are negatively correlated in different pulses inside the laser cavity do not change the average power much and are therefore almost invisible to the gain medium on slow time scales. These negatively correlated pulse energy fluctuations can grow, causing instability and pulse dropouts. The energy fluctuations in all of the pulses in the laser cavity as well as the gain dynamics need to be taken into account to find the precise limits on the stable operating regime. Har- monically mode-locked semiconductor lasers can have more than 100 different pulses propagating inside the laser cavity [3], [9]–[11], and it seems like a daunting task to numerically model the fluctuations in all of the pulses. The theoretical technique presented in this paper takes into account the energy fluctua- tions in all of the pulses in the laser cavity as well as the gain dynamics via nonlinear finite-difference equations, which are then linearized to obtain finite-difference equations for the pulse photon-number fluctuations. A similar method was used by the authors earlier to characterize the pulse timing fluctuations in harmonically mode-locked lasers [12]. Pulse stability in fundamentally and harmonically mode-locked semiconductor lasers can be affected by a number of different processes. For example, in [13], it was shown that, in actively mode-locked lasers, dynamic gain saturation in the gain medium causes the pulse to move off the gain maximum (in time) of the modulator, and the excess gain just behind the pulse results in instabilities that limit the maximum pulse energy. In lasers, the oscillatory dynamics of small perturbations in laser gain and photon number around an operating point are called relaxation oscillations. The operating point is considered stable if the perturbations are damped in time. The operating point is unstable if small fluctuations 0018-9197/$25.00 © 2007 IEEE