IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL. 32, NO. 6, JUNE 1996 94 I Supermodes of High-Repetition-Rate Passively Mode-Locked Semiconductor Lasers Randal A. Salvatore, Steve Sanders, Thomas Schrans, and Amnon Yariv, L~ Fellow, IEEE Abstract-We present a steady-state analysis of high-repetition- rate passively mode-locked semiconductor lasers. The analysis includes effects of amplitude-to-phase coupling in both gain and absorber sections. A many-mode eigenvalue approach is pre- sented to obtain supermode solutions. Using a nearest-neighbor mode coupling approximation, chirp-free pulse generation and electrically chirp-controlled operation are explained for the first time. The presence of a nonzero alpha parameter is found to change the symmetry of the supermode and significantly reduce the mode-locking range over which the lowest order supermode remains the minimum gain solution. An increase in absorber strength tends to lead to downchirped pulses. The effects of individual laser parameters are considered, and agreement with recent experimental results is discussed. I. INTRODUCTION REVIOUSLY, the theory of passive modelocking has P been analyzed thoroughly in the time domain [ 11. Haus’ analysis has provided a clear picture of the evolution of pulses through gain, absorptive, and bandwidth-limiting elements within a cavity. A steady-state solution was found when these effects are included. Certain approximations were deemed necessary in order to present an analytic solution. For example, in the steady-state solution, a symmetric and unchirped pulse envelope is assumed as limited by the approximation of all time-domain effects only up to the quadratic term. The model has been extended to include chirped pulses due to self- phase modulation (SPM) yet only for a fast absorber [2], [3], and still restricts the analysis to exponents quadratic in time and achieves symmetric pulses. No recovery is assumed to occur during pulses. Additionally, both models include an approximation of the discrete-mode spectrum by a continuous spectrum. Although the latter approximation works well for mode-locked lasers having many closely-spaced modes, and a slightly-varying gain with frequency, it, along with the assumption of no material recovery during the pulse, is not adequate for the case of high-repetition-rate passively mode- locked lasers (250 GHz). In this case, the difference in gain Manuscript received August I, 199.5; revised January 30, 1996. This work was supported by the National Science Foundation under Grant ECS-9001272 and by ARPA and the Office of Naval Research under Grant N00014-91-J- 119.5. R. A. Salvatore is with the Electrical and Computer Engineering Depart- ment, University of California, Santa Barbara, Santa Barbara, CA 93106 USA. S. Sandcrs is with SDL Inc., 80 Rose Orchard Way, San Jose, CA 95134 USA. T. Schrans is with the Ortel Corp., 201.5 West Chestnut Street. Alhambra, CA 91803 USA. A. Yariv is with the Department of Applied Physics 128-95, California Institute of Technology, Pasadena, CA 91125 USA. Publisher ltem Identifier S 0018-9197(96)04131-0. between neighboring modes can be significant, and typically only a small number of rnodes (around 3-10) dominate. Active modelocking, on the other hand, has been analyzed thoroughly in both the time domain and the frequency domain [4]-[6]. It has been suggested that passive mode-locking should be analyzed in the time domain since simple products in the time-domain analysis result in cumbersome convolutions in the frequency domain analysis 171, however, in the case of high-repetition-rate passive modelocking, where few modes are involved and the induced carrier modulation is much closer to a sinusoid [8], the frequency domain approach becomes more appropriate. In this paper, we present a steady- state analysis of passive modelocking directed toward high- repetition-rate semiconductor lasers. The analysis is done in the frequency domain extending that presented in [8]. For the first time, passive mode-locking supermodes are found while amplitude-to-phase coupling from slow saturation is permitted. Section I1 describes the model and arrives at an equation for each mode in the supermode. It incorporates dispersive effects through the common semiconductor laser parameters and unlike previous frequency domain calculations, does not force all modes beyond (the minimum) three modes to contribute zero coupling. Section I11 describes the eigen- value formulation used to arrive at a self-consistent solution of the coupled nonlinear equations. Section IV presents an approximate analytical expression based on (the minimum) three modes in order to reduce the complexity and allow one to build physical intuition about the gain requirements and amplitudes and phases of the supermode spectrum. Section V presents results for the full calculation. Section VI compares the results with experiments for high-repetition-rate passively mode-locked lasers. Finally, Section VI1 includes conclusions. 11. THE MODEL High-repetition-rate modelocking (250 GHz) was first demonstrated by Vasil’ev [9] and by Sanders et al. [lo]. To date, semiconductor lasers are the only mode-locked lasers that have been able to generate repetition rates of hundreds of GHz. Due to their large material gain coefficients, fast recovery times, and the ability to be made into short monolithic cavities, high-repetition-rate pulse trains can be generated easily. Typically, high-repetition-rate lasers involve a monolithic semiconductor laser structure, meaning no external cavity is used. The model presented is intended to analyze the monolithic multisection laser, and no intention of including an external cavity is made here although one could easily modify 0018-9197/96$05.00 0 1996 IEEE