Optics Counications 223 (2003) 151 156 www.elsevier.co/locate/optco Phase transition theory of pulse foration in passively ode-locked lasers with dispersion and Kerr nonlinearity Ariel Gordon, Baruch Fischer * Departent of Electrical Engineering, Technion-Israel Institute of Technology, Haifa 32000, Israel Received 19February 2003; accepted 30 May 2003 Abstract We extend our statistical echanics approach (Phys. Rev. Lett. 89(2002) 103901) to passively ode-locked lasers to also include dispersion and the Kerr nonlinearity and in particular the soliton laser case. The theory predicts for these general cases a discontinuous first-order phase transition at the onset of pulses that explains the power threshold behavior of ode-locked lasers. Ó 2003 Elsevier Science B.V. All rights reserved. PACS: 42.65.Re; 42.60.Fc; 42.60.Mi; 42.55.Wd 1. Introduction In the course of the recent three decades, uch progress has been ade in the study of passive ode locking (PML) of lasers. The analytical part of the studies has led to a rather consensual theory [2,3]. There are any nuances of this theory, but in ost of the conteporary analytical descriptions of PML the core reains the sae: an equation of otion (the aster equation), reiniscent to the coplex Landau Ginzburg equation, for the evolution of optical pulses, odeling the action of gain dispersion, group velocity dispersion (GVD), a usually fast saturable absorber, the Kerr nonlinearity and * Corresponding author. Tel.: +97248294736; fax: +97248323041. E-ail address: fischer@ee.technion.ac.il (B. Fischer). slow saturable gain. This faily of theories has been used for extensive study of PML: the pulse shape [4], stability issues [5], the threshold behavior [6], noise characteristics [7] of the pulses and any ore. A recent review is given in [3]. Most of the existing theories of the PML threshold behavior [6,8 11] consider noise or soe sort of incoherence to be responsible for it. When the effect of noise on PML is studied through the Langevin aster equation, noise is often assued to be sall, which facilitates linearization of the Langevin aster equation [7]. This way, the lowest oents describing the statistics of the output signal of PML lasers can be calculated. As we find below, crucial features of the syste can be issed in this procedure when the noise is large. In a recent work [1] we presented a statistical echanics approach for a any interacting ode laser syste. We have shown that such systes 0030-4018/03/$ - see front atter Ó 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/s0030-4018(03)01622-5
152 A. Gordon, B. Fischer / Optics Counications 223 (2003) 151 156 exhibit phase transition of the first kind, a result of the interplay between entropy and (a generalization of) energy in a syste of any interacting odes, where the role of teperature is taken by noise. Contrary to sall noise in the laser, which causes noise in its output, increasing the noise (or equivalently decreasing the signal) to a certain critical value causes an abrupt breakdown of the pulses and a transition to a noisy CW operation. The pulse foration in lasers is shown to be siilar to elting freezing behavior in solid liquid and paraferroagnet transitions. We have recently experientally deonstrated the predicted first-order phase transition in a passively ode-locked fiber laser [12], showing the therodynaic elting freezing path. In this paper we extend the study of the threshold behavior of PML [1] to include GVD, the Kerr nonlinearity and a ore coon description of the optical saturable gain. We extend the Hailtonian foralis [1] to include gain saturation. We discuss the purely dissipative otion, the otion induced by the gain and losses (linear and nonlinear) alone, and present a dynaical picture of the balance between forces and noise, reflecting the balance between interaction and entropy. We then write the Fokker Planck equation associated with the Langevin equation of otion and look for steady-state solutions which are functions of the Hailtonians alone. We find such solutions when the soliton condition is satisfied [3], which extends our theory fro [1] to this iportant case. Beyond the soliton condition we investigate the Langevin equations of otion nuerically, for general values of dispersion and the Kerr nonlinearity. We find that the equations consistently exhibit the first-order phase transition behavior. We suggest this echanis as an explanation to the passive ode-locking threshold behavior. 2. Hailtonian forulation of the aster equation The coon equation of otion [3] of PML (the aster equation ), which is usually written in the tie doain, can be written in the spectral doain as follows: _a ¼ðg l þ i/ Þa þðic k c s Þ X a j a k a l þ C : j kþl ¼0 ð1þ The boundaries of the cavity adit only discrete axial odes into the laser, the slowly varying aplitudes of which are denoted by a. _a stands for da =ds, where s is a diensionless variable easuring tie in units of the cavity roundtrip tie. g denotes the net gain at the wavelength corresponding to and is signal dependent. / represents the phase acquired by the th ode after a roundtrip, which in particular describes dispersion. l represents the wavelength-dependent linear loss. c s and c k represent the nonlinear self-aplitude (saturable absorber) and self-phase odulation (Kerr nonlinearity) coefficients, and C is a white Gaussian Langevin additive noise ter which in particular describes quantu noise [7]. We assue the CÕs tobe Gaussian, white and uncorrelated C ðt 1 ÞC n ðt 2Þ ¼ 2T dn dðt 1 t 2 Þ; ð2þ hc ðt 1 ÞC n ðt 2 Þi ¼ 0 with hi denoting enseble averaging. T is the spectral power of the noise, which functions as teperature. It is well known that lasers and in particular PML lasers are stable due to gain saturation [5]. When the response of the laser aplifier and the evolution of the wavefor are uch slower than the cavity roundtrip tie, the gain at a given oent is deterined by the whole wavefor all over the cavity. The ost coon odel for gain saturation in that case is g 0 g ¼ ; ð3þ 1 þ P=P sat where P ¼ X a a ð4þ and P sat is the saturation power of the aplifier. Eq. (3) describes a wavelength-independent aplifier: the filtering action of the aplifier is ebedded in l and its signal dependence has been neglected. This coon approxiation is often satisfying [5], but wavelength-dependent saturation behavior is easily incorporated in our foralis by defining
A. Gordon, B. Fischer / Optics Counications 223 (2003) 151 156 153 g 0 g ¼ ; ð5þ 1 þ G=G sat where G ¼ X g 0 a a : ð6þ Eq. (5) reflects the fact that in a hoogeneously broadened aplifier the gain saturation is deterined by the overlap between the optical spectru and the gain profile [13]. G sat is the analog of the saturation power for that case, and can be related to its fundaental properties [13]. Defining H R ¼ X / a a c X k a j a k 2 a la ð7þ j kþl ¼0 and H I ¼ X l a a c X k a j a k 2 a la j kþl ¼0 G sat lnðg sat þ GÞ; ð8þ it is straight forward to verify that (1) can be rewritten as _a ¼ oðh R ih I Þ þ C ioa : ð9þ Changing variables to the real and iaginary parts of a (a ¼ a R þ iai ) we have _a R ¼ oh R oa I _a I ¼ oh R oa R oh I oa R oh I oa I þ C R ; þ C I ; ð10þ C R and CI are the real and iaginary parts of C. Observing Eq. (10), we see clearly that H R generates a Hailtonian otion and H I generates a purely dissipative otion (a gradient flow ). Eqs. (10) are Langevin equations describing a rando process, which can be equivalently described by the associated Fokker Planck equation [14] _q ¼ X þ X o oa R þ T X oh R o oa I oa I oh R o 2 oa R2 oh I q oa R oa R þ o2 oa I2 oh I q oa I! q; ð11þ where q is the statistical distribution of the odes a and T is half the spectral power of the Langevin force. As shown in [1], when H R ¼ 0 it is straight forward to see that the distribution qða 1 ;...; a N Þ/e H I=T ð12þ is the exact solution of Eq. (11) with _q ¼ 0 (the steady-state solution). The solution (12) has far reaching consequences, since it is identical by structure to the statistical distribution of a canonical enseble. We elaborate this case in the next section (Section 3). The ore general cases that include the Kerr nonlinearity and dispersion (H R 6¼ 0) are discussed afterwards (in Sections 4 and 5). 3. Purely dissipative otion (H R =0) For the case of zero GVD and no Kerr nonlinearity (H R ¼ 0) the dynaics of the syste is gradient flow (_a R ¼ oh I=oa R, _ai ¼ oh I=oa I ). This is useful for additional intuitive understanding of the dynaics induced by the saturable absorber, gain saturation and filtering. Let us exaine the evolution of H I without noise and for H R ¼ 0. We have then " 2 þ dh I ds ¼ X oh I a R which eans that H I oh # 2 I a I ; ð13þ keeps decreasing until a stationary point of H I is reached. If the stationary point is not a iniu, then a sall perturbation will push H I down the hill again. If it is a local iniu, then the point is locally stable: after a sall perturbation the syste will return to the iniu, but a large one will throw the syste away without necessarily going back. Only a global iniu of H I will be a truly stable solution to Eq. (1). It follows therefore that a truly stable solution to H I will exist if and only if H I is bounded fro below. This provides a convenient tool for stability analysis, and a atheatical ebodient to the coonly used intuitive arguents on what lasers like and dislike : lasers tend to iniize H I.
154 A. Gordon, B. Fischer / Optics Counications 223 (2003) 151 156 An iediate result of this stability analysis technique is that the aster equation adits no globally stable solutions: it is easy to see fro Eq. (8) that H I is not bounded fro below. The reason for this is that in the siplest odel of absorption saturation we used, which takes into account only its the lowest nonlinear ter, increasing the optical power enough will eventually turn the absorber into an aplifier, which is physically incorrect. This proble can be fixed by adding to the aster equation a stabilizing ter which will turn the local iniu of the odelocked configuration into a global one. An exaple for such a ter is the next one in the expansion of the saturable absorber. When noise is present, along with sliding down, the syste will be randoly thrown fro state to state. This rando otion is likely to draw the syste away fro rare states (like the iniu of H I ), or in other words, noise tends to increase the entropy of the syste. As entioned above, just like in statistical echanics, this copetition between iniizing H I and axiizing entropy leads to the Gibbs distribution, with the noise power taking the role of teperature. Contrary to linearization techniques, which can provide inforation on the few lowest oents only, Eq. (12) is an exact solution for the entire distribution qða 1 ;...; a N Þ/e H I=T of the wavefors (or odes) in the laser. Needless to say equipped with an exact solution for the distribution one can thoroughly study the noise properties of PML lasers. More interesting is that the threshold behavior of PML is ebedded in Eq. (12): order (ode locking) eerges in the laser upon cooling. This can be shown analytically too, through the ean field approxiation [1]. Eq. (12) establishes an intiate relation between PML lasers and equilibriu statistical echanics and phase transition theory, and indeed it can be shown that the occurrence of ode locking upon reaching threshold is a phase transition of the first kind. The results provided by statistical echanics, both through nuerical siulation (a Monte-Carlo siulation) and the ean field approxiation are shown in Fig. 1. The connection to phase transition theory and statistical echanics is not accidental. In fact, the situation in the PML laser is very siilar to the one in traditional systes exhibiting order disorder transitions like elting solids, ferroagnets and any ore. Just like the latter, the laser is a syste of any interacting degrees of freedo the odes, with an ordering interaction the nonlinear saturable absorber, and with teperature (which in our case is replaced by the noise variance) encouraging gain of entropy which opposes the ordering interaction. The theory of phase transitions is a ature theory with any well-established powerful results, which can be now directly applied to PML. One of the interesting consequences of our analysis is that whenever a saturable absorber is present (for zero GVD and no Kerr nonlinearity) CW is not a stable solution of Eq. (1): it is easy to show that a CW solution is a axiu or a saddle point of H I. As we have seen, what stabilizes a CW solution is noise alone. This reinds other situations in nature, for exaple the liquid phase, which reains liquid due to entropy considerations, although its energy would be lower had it solidified, and nevertheless it reains liquid. 4. Analytical solution for a case with H R 6¼ 0 including the soliton laser case Let us now proceed to H R 6¼ 0, that is, a laser with nonzero dispersion and Kerr coefficient. Inspired by equilibriu statistical echanics, where a very fundaental stateent is that the statistical distribution is a function of the Hailtonian only, and even ore by the solution (12), we can try to seek for a solution to the steady-state distribution which is again a function of the Hailtonians only. We therefore substitute the solution qða 1 ;...; a N Þ/fðH R ða 1 ;...; a N Þ; H I ða 1 ;...; a N ÞÞ ð14þ into Eq. (11) and equating to zero leads via a lengthy but trivial calculation to the result that Eq. (12) is the only possible solution of the for (14), and it solves Eq. (11) as long as
A. Gordon, B. Fischer / Optics Counications 223 (2003) 151 156 155 Fig. 1. The order paraeter Q, inversely proportional to the pulse duty cycle, as function of the generalized teperature T for several choices of the different paraeters. Graphs 1 4, which alost coincide, are results of nuerical siulations: A Monte-Carlo siulation of Eq. (12) and a direct siulation of the Langevin equation (10), with (graphs 3 and 4) or without (graphs 1 and 2) the Kerr nonlinearity. The thick lines are their analytical ean field based approxiations [12]. Graphs 5 10 siulations of Eq. (10) with GVD, and graphs 11 and 12 are the sae siulations with both GVD and the Kerr effect. Qualitatively the behavior of graphs 5 12 is very siilar to that of graphs 1 4, all exhibiting an abrupt transition to a ode-locked phase. In the siulations we used 100 discrete odes (a 1 ; a 2...; a 100 ), with equal spectral spacing. The stabilization we used is a constraint of constant power P ¼ P ja k j 2. T is in units of c s P 2. Graphs 3 and 4 are with c k =c s ¼ 2 and 3. In graphs 5 12 we took a parabolic dispersion relation for GVD / ¼ c d ðn 50:5Þ 2 with c d ¼ 0:002; 0:003; 0:004; 0:006; 0:009; 0:02 for graphs 5; 6; 7; 8; 9; 10; respectively. In graphs 11 and 12 c k ¼ 3c s and c d is )0.005 and 0.005, respectively. X oh R oa I oh I oa R oh R oa R oh I ¼ 0; oa I ð15þ that is, H I is a constant of otion under the evolution generated by H R alone. Putting the coon parabolic profiles for dispersion and gain dispersion and restricting ourselves to Eq. (3) for odeling gain saturation, it is straight forward to show that Eq. (15) holds in particular for solitons [3]. The right balance between dispersion and the Kerr nonlinearity is kept. The conclusion of the above calculation is twofold. First, we have seen that Eq. (12) and hence all the subsequent analysis [1] holds as it is in a soliton laser. Second, we have seen that the steady-state distribution is not a function of the Hailtonians alone when dispersion and Kerr are nonzero and the soliton condition does not hold. It is therefore an explicit function of the aõs, which is difficult to find and presuably also to analyze. We have therefore turned to nuerical studies. 5. Nuerical study of the general H R 6¼ 0 case The ain question we were addressing in the nuerical study is whether the sharp breakdown of pulses upon increasing noise occurs with the ost general choice of the paraeters in Eq. (1). We have therefore conducted tens of nuerical experients, where we nuerically solved Eq. (1) and calculated the tie average of the noralized order paraeter P j kþl ¼0 ~Q ¼ a ja k a la 2 ð16þ P ja j 2
156 A. Gordon, B. Fischer / Optics Counications 223 (2003) 151 156 as function of the noise level T. This paraeter is inversely proportional to the pulsewidth and directly proportional to the nuber of correlated odes: In the ode-locked phase it is of order of the total nuber of odes in the band, whereas in the CW phase it is of order one. All the siulations presented in this paper were perfored with 100 odes and with the constraint of constant optical power P 0 as explained above. All the nuerical experients exhibited the sae sharp breakdown of the pulses, however the critical T depends on the paraeters. It sees that pulses with higher chirp paraeter are less stable against noise, which is siilar to the behavior of stability against sall perturbations [3]. Additionally the general effect of the unbalanced GVD and Kerr nonlinearity is to reduce the effective nuber of locked odes. Soe results are presented in Fig. 1. With nonzero GVD and Kerr nonlinearity, the behavior of a PML laser with noise is qualitatively the sae as PML in the purely dissipative case (H R ¼ 0), and in particular the phase transition occurs, at least for a wide variety of paraeters. This is not very surprising, since a phase transition of the first kind is a draatic singularity, and it is not expected to be reoved by slight odifications of the dynaics. 6. Conclusion We have extended our statistical echanics approach for any interacting ode systes to general PML lasers that include dispersion and Kerr nonlinearity. Such systes were shown to exhibit a first-order transition. For a specific balance between dispersion and Kerr nonlinearity, that includes the iportant case of soliton lasers, the proof was given by an exact analysis. For the broader case we used a nuerical study. We also ention again our recent [12] experiental deonstration of the phase transition with a variety of therodynaic properties of pulse foration in a fiber PML laser. We have shown the elting freezing transition as the teperature noise was varied. The present paper can provide a further understanding of such systes when solitons are incorporated, as often happens in fiber lasers. We end with an experiental reark. We have analyzed the behavior of PML lasers as the noise power is altered. In an experient, the odelocking threshold is usually observed as the power in the cavity is raised. The interaction energy responsible for the ode locking is quadratic with the power in the cavity (see the last ter in Eq. (8)), while the noise power is close to be linear with it. This explains the existence of a threshold power. Acknowledgeents We thank Shuel Fishan for fruitful discussions. We acknowledge the support of the Division of Research Funds of the Israeli Ministry of Science, and the Fund for Prootion of Research at the Technion. References [1] A. Gordon, B. Fischer, Phys. Rev. Lett. 89(2002) 103901. [2] H.A. Haus, J. Appl. Phys. 46 (1975) 3049. [3] H.A. Haus, IEEE J. Sel. Top. Quant. 6 (2000) 1173. [4] H.A. Haus, C.V. Shank, E.P. Ippen, Opt. Coun. 15 (1) (1975) 29. [5] C.J. Chen, P.K. Wai, C.R. Menyuk, Opt. Lett. 19(3) (1994) 198. [6] C.J. Chen, P.K. Wai, C.R. Menyuk, Opt. Lett. 20 (4) (1995) 350. [7] H.A. Haus, A. Mecozzi, IEEE J. Quantu Electron. 29(3) (1993) 983. [8] F. Krausz, T. Brabec, Ch. Spilann, Opt. Lett. 16 (1991) 235. [9] J. Herrann, Opt. Coun. 98 (1993) 111. [10] E.P. Ippen, L.Y. Liu, H.A. Haus, Opt. Lett. 15 (1990) 737. [11] H.A. Haus, E.P. Ippen, Opt. Lett. 16 (1991) 1331. [12] A. Gordon, B. Vodonos, V. Sulakovski, B. Fischer, unpublished. [13] M. Horowitz, C.R. Menyuk, S. Keren, IEEE Photon Technol. Lett. 11 (10) (1999) 1235. [14] H. Risken, The Fokker Planck Equation, second ed., Springer, Berlin, 1989.