a ., __ , __ l!iiB ELSEVIER 15 March 1996 Optics Communications124 (1996) 475-480 OPTICS COMMUNICATIONS Simulation of passively mode locked lasers, using natural boundary conditions: multi pulse evolution and ordering M. Margalit, M. Orenstein zyxwvutsrqponmlkjihgfedcbaZYXWVUTS Department of Electrical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel Received 29 June 1995;revised version received6 November 1995;accepted 10 November 1995 zyxwvutsrqponmlkjihgfed Abstract We present a method for simulating a passively mode locked laser. By selecting the simulation time step as the cavity round trip ti.me, we achieved a proper tracking of the temporal evolution of the individual cavity modes of the laser. Employing the split step Fourier algorithm using this natural modes base, an efficient simulation method was obtained, while retaining at each propagation step the direct physical relations between the calculated phase terms in the time and frequency domains to the actual optical field and individual cavity modes, respectively. Using this method, we tracked the formation of multiple pulses in a passively mode locked fiber laser cavity, and explored the effect of delayed optical feedback as a means for pulse ordering in fiber lasers. Passive mode locking in fiber lasers has drawn much attention. Many of the fiber lasers, utilize either a non linear amplified loop mirror or a non-linear polarization rotation mechanism, as effective fast saturable absorb- ers for achieving the passive mode locking mode of operation [ 1,2]. Because of the relatively low satura- tion power of the absorbers, these lasers tend to exhibit multiple pulse formation in,the laser cavity [ 31. This mode of operation is a reflection of an intriguing dynamic system, which is not yet fully explored. Par- ticularly, the inter pulse interactions in the cavity may cause, as observed in previously reported experimental results, pulse bunching and giant pulse formation [ 31. Many of the numerical schemes employed for the analysis of passively mode locked fiber lasers, utilized the split step Fourier algorithm [4]. This method, which is an elaboration on closely related computa- tional schemes commonly used for the analysis of long distance soliton transmission [ 51, suffers, in the con- text of laser simulation, from two drawbacks. The first is a result of the fundamental difference between soliton lasers and soliton transmission. In soliton transmission the pulse propagation is a transient process for which the timing and the phase of the soliton, in the absence of noise, are determined by their initial values. In fiber lasers, on the other hand, the phase and the timing of the soliton are determined inherently by the cavity round-trip criterion. The second drawback is related to the numerical constraints, which limits the computed optical field to a region of interest, localized in time around the center of a coordinate system moving with the soliton. This temporal localization, limits the sim- ulation of the pulse evolution process, transient changes of the laser parameters and the analysis of complex schemes such as high order harmonic mode locking [61. We describe a scheme for simulating passively mode locked lasers which overcomes the drawbacks of pre- vious numerical schemes, by properly addressing the physical interpretation of the frequency domain terms 0030.4018/96/$12.00 0 1996Elsevier Science B.V. All rights reserved SSDIOO30-4018(95)00696-6