Continuous-Wave Versus Pulse Regime in a Passively Mode-Locked Laser with a Fast Saturable Absorber
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234 J. Opt. Soc. Am. B/Vol. 19, No. 2/February 2002 Soto-Crespo et al. Continuous-wave versus pulse regime in a passively mode-locked laser with a fast saturable absorber J. M. Soto-Crespo Instituto de O´ ptica, Consejo Superior de Investigaciones Cie`ntifica` s, Serrano 121, 28006 Madrid, Spain N. Akhmediev Optical Sciences Centre, Research School of Physical Sciences and Engineering, Institute of Advanced Studies, Australian National University, Australian Capital Territory 0200, Australia G. Town School of Electrical and Information Engineering (J03), University of Sydney, New South Wales, 2006, Australia (Received February 27, 2001; revised manuscript received August 15, 2001) The phenomenon of modulation instability of continuous-wave (cw) solutions of the cubic–quintic complex Ginzburg–Landau equation is studied. It is shown that low-amplitude cw solutions are always unstable. For higher-amplitude cw solutions, there are regions of stability and regions where the cw solutions are modu- lationally unstable. It is found that there is an indirect relation between the stability of the soliton solutions and the modulation instability of the higher-amplitude cw solutions. However, there is no one-to-one corre- spondence between the two. We show that the evolution of modulationally unstable cw’s depends on the sys- tem parameters. © 2002 Optical Society of America OCIS codes: 190.5530, 190.5940, 140.3510, 320.5540. 1. INTRODUCTION loss (both linear and nonlinear). Some delicate balances between them give rise to the majority of the effects ob- Passive mode locking allows for the generation of self- served experimentally. On the other hand, we assume shaped ultrashort pulses in a laser system. It has been that relaxation times do not enter explicitly into the mas- demonstrated in a number of works that the pulses gen- erated by mode-locked fiber lasers are solitons.1–4 In ad- ter equation. This is possible when the pulse duration is dition to this very important feature, the mode-locked la- much shorter than the relaxation times in the system, ser is a dissipative nonlinear system that can have very relative to both the gain medium and the saturable ab- rich dynamics, including not only the generation of pulses sorber. This condition excludes, for example, lasers that use a semiconductor Bragg reflector as a saturable of accurate shape but also much more complicated behav- 4,21,22 ior. The approach that uses the complex Ginzburg absorber. We also exclude the influence of gain – 22,23 Landau equation (CGLE) in relation to passively mode- depletion on pulse generation. However, the CGLE locked lasers was pioneered by Haus.5 It is now used allows us to describe a large range of solid-state and fiber extensively to describe pulse behavior in solid-state or lasers and serves as a basic model in laser theory. fiber-based passively mode-locked lasers,6–9 optical para- One of the key issues in the theory of passively mode- metric oscillators,10 free electron laser oscillator,11 etc. locked lasers is the self-starting condition. There are 24–26 The CGLE has also been used to describe transverse soli- various approaches to the problem of self-starting. ton effects in wide-aperture lasers.12–17 The reason is They depend on the type of laser, and, in particular, on that the CGLE is the equation of minimum complexity the type of mode locking. One of the models for self- that nonetheless includes the most important effects that starting is the transition from continuous-wave (cw) op- are present in any active optical system. Various short- eration to steady-state mode-locked operation.26 It has pulse laser designs result in a similar master been suggested27 that such a transition occurs through equation.18,19 In the case of fast saturable absorbers the modulational instability of cw solutions. One of the with response times much faster than the pulse duration, motivations of the present research is to make a detailed the main features of pulse generation can be described by study of this possibility, based on solutions of the complex the cubic–quintic CGLE. The quintic terms in the equa- Ginzburg–Landau equation. In particular, we need to tion are essential to ensure that pulses are stable.20 know when cw solutions of the CGLE become unstable This continuous model takes into account the major and whether the soliton solutions are stable for the same physical effects occurring in a laser cavity, such as disper- set of parameters. The crucial question is, are the condi- sion, self-phase modulation, spectral filtering, and gain/ tions for stability of cw’s and solitons somehow related? 0740-3224/2002/020234-09$15.00 © 2002 Optical Society of America Soto-Crespo et al. Vol. 19, No. 2/February 2002/J. Opt. Soc. Am. B 235 Of course, the modulation instability is not the only way we obtain an equation for two coupled functions, a and . of self-starting.24,25 However, a clear understanding of Separating real and imaginary parts, we get the following the self-starting phenomenon requires a careful study of set of two ODEs: each model. Hence here we concentrate on the question raised above. D D In particular, we carefully consider both cw and soliton ͫ Ϫ Ј2 ϩ  Љ ϩ v Јͬa ϩ 2 ЈaЈ ϩ aЉ regimes of passively mode-locked lasers. To be specific, 2 2 in studying the stability properties of cw and soliton solu- tions of the cubic–quintic CGLE, we have found that, al- ϩ a3 ϩ a5 ϭ 0, though the stability properties of the two are related, there is no one-to-one correspondence between them. Modulation instability of the cw’s does not necessarily D ͩ Ϫ␦ ϩ  Ј2 ϩ Љͪ a ϩ ͑D Ј Ϫ v͒aЈ lead to pulse generation. Various possibilities of cw evo- 2 lution are possible, depending on the parameter values of the laser system. Only at certain values of the param- Ϫ aЉ Ϫ ⑀a3 Ϫ a5 ϭ 0, (3) eters does modulation instability of a cw solution lead to the generation of solitons. In the optical context the cubic–quintic CGLE has the where each prime denotes a derivative with respect to . following form28: It can be transformed into D D D i ϩ ϩ ͉ ͉2 ϩ ͉ ͉4 ͩ Ϫ 2 ϩ  Ј ϩ ͪ ϩ  Ј ϩ Љ z tt M M vM a 2 Ma a 2 2 2 ϭ i␦ ϩ i⑀͉ ͉2 ϩ i ϩ i͉ ͉4, (1) tt ϩ a3 ϩ a5 ϭ 0, where z is the cavity round-trip number, t is the retarded time, is the normalized envelope of the field, D is the D group-velocity dispersion coefficient with D ϭϮ1 de- ͩ Ϫ␦ ϩ M2 ϩ MЈͪ a ϩ ͑DM Ϫ v͒aЈ pending on whether the group-velocity dispersion is 2 anomalous or normal, respectively, ␦ is the linear gain–  3 5 loss coefficient, i tt account for spectral filtering or lin- Ϫ aЉ Ϫ ⑀a Ϫ a ϭ 0, (4) ear parabolic gain ( Ͼ 0), ⑀͉ ͉2 represents the nonlin- ear gain (which arises, e.g., from saturable absorption), where M ϭ Ј is the instantaneous frequency. the term with represents, if negative, the saturation of Separating derivatives, we obtain the nonlinear gain, and the one with corresponds, also if negative to the saturation of the nonlinear refractive in- 2 ϩ Ϫ dex. y͑8 M 2M 2Dv͒ MЈ ϭϪ An important question is that of the correspondence be- a͑42 ϩ 1͒ tween the equation and laser parameters. As we do not specify a particular type of laser, our theory allows us to 4 ϩ 4Mv Ϫ 2␦D find dimensionless parameters in Eq. (1), where cw and Ϫ ϩ 2 soliton solutions exist and can be compared. We have 1 4 made this comparison over a wide range of the parameter a2͑2D⑀ Ϫ 4͒ a4͑2D Ϫ 4͒ values. To apply the theory to any particular experimen- ϩ ϩ , tal case would require certain adjustments. However, 1 ϩ 42 1 ϩ 42 the latter is beyond our aims. 2͑D ϩ 2␦͒ 2͑D ϩ 2⑀͒ yЈ ϭ M2a Ϫ a Ϫ a3 2. STATIONARY SOLUTIONS 1 ϩ 42 1 ϩ 42 Equation (1) has both soliton and cw stationary solutions. 2͑D ϩ 2͒ 4v 2Dv We ignore fronts, sinks, and sources that are beyond our Ϫ 5 Ϫ Ϫ 2 a 2 y Ma, interest here. To study the interrelation between the 1 ϩ 4 1 ϩ 4 1 ϩ 42 cw’s and solitons, we carry out the following analysis. We reduce Eq. (1) to a set of ordinary differential equa- aЈ ϭ y. (5) tions (ODEs). Namely, we look for solutions in the form ͑t, z͒ ϭ ͑͒exp͑Ϫiz͒ ϭ a͑͒exp͓i͑͒ Ϫ iz͔, This set contains all stationary and uniformly translating 0 (2) solutions. The parameters v and are the eigenvalues of Eq. (5). Pulse solutions exist only at certain values of where a and are real functions of ϭ t Ϫ vz, v is the v and . pulse inverse velocity, and is the nonlinear shift of the If we are only interested in zero-velocity (v ϭ 0) solu- propagation constant. Substituting Eq. (2) into Eq. (1), tions, Eqs. (5) can be further simplified: 236 J. Opt. Soc. Am. B/Vol. 19, No. 2/February 2002 Soto-Crespo et al. y͑82M ϩ 2M͒ 4 Ϫ 2␦D a2͑2D⑀ Ϫ 4͒ MЈ ϭ Ϫ ϩ a͑42 ϩ 1͒ 1 ϩ 42 1 ϩ 42 a4͑2D Ϫ 4͒ ϩ , 1 ϩ 42 2͑D ϩ 2␦͒ 2͑D ϩ 2⑀͒ yЈ ϭ M2a Ϫ a Ϫ a3 1 ϩ 42 1 ϩ 42 2͑D ϩ 2͒ Ϫ a5, 1 ϩ 42 aЈ ϭ y.