Averaged models for passive mode-locking using nonlinear mode-coupling

Mathematics and Computers in Simulation 74 (2007) 333–342

Averaged models for passive mode-locking using nonlinear mode-coupling J. Proctor, J. Nathan Kutz∗ Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA Available online 1 December 2006

Abstract An averaged mode-locking model is presented in which the nonlinear mode-coupling behavior in a wave-guide array, dual-core fiber, and/or fiber array is used to achieve stable and robust passive mode-locking. By using the discrete, nearest-neighbor spatial coupling of these nonlinear mode-coupling devices, low-intensity light can be transferred to the neighboring wave-guides and attenuated. In contrast, higher intensity light is self-focused in the launch wave-guide and remains largely unaffected. This nonlinear effect, which is described by linearly coupled nonlinear Schr¨odinger equations, leads to the temporal intensity discrimination required in the laser cavity for mode-locking. Computations of this pulse shaping mechanism show that stable and robust modelocked soliton-like pulses can be produced in this master mode-locking model. © 2006 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Mode-locked lasers; Solitons; Averaging; Wave-guide arrays; Nonlinear mode-coupling

1. Introduction Nonlinear mode-coupling (NLMC) is a well-established phenomenon which has been both experimentally verified [9,6,1,7,21] and theoretically characterized [15,26,3]. NLMC has been an area of active research in all-optical switching and signal processing applications using wave-guide arrays [6,1,7,21], dual-core fibers [9,15,26], and fiber arrays [27,20]. Recently, the temporal pulse shaping associated with NLMC has been theoretically proposed for the passive intensity-discrimination element in a mode-locked fiber laser [16,22,23,28,19,14]. The models derived to characterize the mode-locking consist of two governing equations: one for the fiber cavity and a second for the NLMC element [16,22,23,14] (see Fig. 1). Although the two discrete components provide accurate physical models for the laser cavity, analytic methods for characterizing the underlying laser stability and dynamics is often rendered intractable. Thus, it is often helpful to construct an averaged approximation to the discrete components model in order to approximate and better understand the mode-locking behavior. Indeed, this is the essence of Haus’ master mode-locking theory [12]. In this manuscript, we develop an averaged approximation to the discrete laser cavity system based upon NLMC and characterize the resulting laser cavity dynamics. The resulting averaged equations are the equivalent of a master mode-locking theory for a laser cavity based upon nonlinear mode-coupling. ∗

Corresponding author. E-mail address: [email protected](J. Nathan Kutz).

0378-4754/$32.00 © 2006 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.matcom.2006.10.030

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Fig. 1. Possible experimental configuration of a mode-locked laser cavity based upon a wave-guide array. The wave-guide array is butt-coupled into the fiber laser cavity (a) as demonstrated in (b).

Operation of a mode-locked laser [12,5] is achieved using an intensity discrimination element in a laser cavity with bandwidth limited gain [12]. The intensity discrimination preferentially attenuates weaker intensity portions of individual pulses or electromagnetic energy. This attenuation is compensated by the saturable gain medium (e.g. Erbium-doped fiber). Pulse narrowing occurs since the peak of a pulse, for instance, experiences a higher net gain per round trip than its lower intensity wings. This pulse compression is limited by the bandwidth of the gain medium (typically ≈ 20–40 nm [12,5]). This intensity discrimination, which is often only a small perturbation to the laser cavity dynamics, can be achieved with NLMC due to the well-known discrete self-focusing properties of the NLMC element. Indeed, the NLMC dynamics in wave-guide arrays is well-documented experimentally and provides the motivation for the current work. An overview of the techniques and methods which are capable of producing intensity discrimination and mode-locking are reviewed in Refs. [12,17]. The paper is outlined as follows: In Section 2 the governing equations for the optical fiber cavity are given along with the NLMC equations which arise for a wave-guide array. The wave-guide array will serve as the standard example of the NLMC element. The resulting mode-locking dynamics in the discrete component system are briefly summarized. In Section 3 we derive the averaged evolution equations which merge the two laser cavity components, i.e. the optical fiber and the wave-guide array. The evolution dynamics are explored with emphasis given to practical mode-locking and the limitations of the averaged NLMC model. A summary of the averaged evolution model along with its strengths, weaknesses, and connection to the discrete model is discussed in the final section. 2. Governing equations In addition to the cavity (fiber) propagation equations, theoretical models are required to describe the NLMC element. Although nonlinear mode-coupling can be achieved in at least three ways [23] (wave-guide arrays, dual-core fibers, and fiber arrays), we will consider only wave-guide arrays since they illustrate all the basic properties of NLMC based mode-locking. The NLMC models are fundamentally the same, the only difference being in the number of modes coupled together. It should be noted that the NLMC theory presented here is an idealization of the dynamics of the full Maxwell’s equations. For very short temporal pulses (i.e. tens of femtoseconds or less), modifications and corrections to the theory may be necessary. 2.1. Fiber propagation The theoretical model for the dynamic evolution of electromagnetic energy in the laser cavity is composed of two components: the optical fiber and the NLMC element. The pulse propagation in a laser cavity is governed by the interaction of chromatic dispersion, self-phase modulation, linear attenuation, and bandwidth limited gain. The propagation is given by [12]   ∂Q 1 ∂2 Q ∂2 2 i (1) |Q| Q + iγQ − ig(Z) 1 + τ 2 Q = 0, + ∂Z 2 ∂T 2 ∂T where g(Z) =

2g0 , 1 + Q2 /e0

(2)

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∞ Q represents the electric field envelope normalized by the peak field power |Q0 |2 , and Q2 = −∞ |Q|2 dT . Here the variable T represents the physical time in the rest frame of the pulse normalized by T0 /1.76 where T0 = 200 fs is the typical full-width at half-maximum of the pulse. The variable Z is scaled on the dispersion length Z0 = ¯ 0 /1.76)2 corresponding to an average cavity dispersion D ¯ ≈ 12 ps/km nm. This gives the one-soliton (2πc)/(λ20 D)(T 2 −16 cm2 W−1 is the nonlinear coefficient in the fiber, peak field power |Q0 | = λ0 Aeff /(4πn2 Z0 ). Further, n2 = 2.6 × 10 2 Aeff = 60 ␮ m is the effective cross-sectional area, λ0 = 1.55 ␮m is the free-space wavelength, c is the speed of light, and γ = Z0 ( = 0.2 dB/km) is the fiber loss. The bandwidth limited gain in the fiber is incorporated through the dimensionless parameters g and τ = (1/ 2 )(1.76/T0 )2 . For a gain bandwidth which can vary from λ = 20–40 nm,  = (2πc/λ20 ) λ so that τ ≈ 0.08–0.32. The parameter τ controls the spectral gain bandwidth of the mode-locking process, limiting the pulse width. It should be noted that a solid-state configuration can also be used to construct the laser cavity. As with optical fibers, the solid state components of the laser can be engineered to control the various physical effects associated with (1). Given the robustness of the mode-locking observed, the theoretical and computational predictions considered here are expected to hold for the solid-state setup. Indeed, the NLMC acts as an ideal saturable absorber and even large perturbations in the cavity parameters (e.g. dispersion-management, attenuation, polarization rotation, higher-order dispersion, etc.) do not destabilize the mode-locking. 2.2. Nonlinear mode-coupling equations The leading-order equations governing the nearest-neighbor coupling of electromagnetic energy in the wave-guide array is given by [6,1,7,21,3] i

dan + C(an−1 + an+1 ) + β|an |2 an = 0, dξ

(3)

where an represents the normalized amplitude in the nth wave-guide (n = −N, . . . , −1, 0, 1, . . . , N and there are 2N + 1 wave-guides). The peak field power is again normalized by |Q0 |2 as in Eq. (1). Here, the variable ξ is scaled by the typical wave-guide array length [7] of Z0∗ = 6 mm. This gives C = cZ0∗ and β = (γ ∗ Z0∗ /γZ0 ). To make connection with a physically realizable wave-guide array [21], we take the linear coupling coefficient to be c = 0.82 mm−1 and the nonlinear self-phase modulation parameter to be γ ∗ = 3.6 m−1 W−1 . Note that for the fiber parameters considered, the nonlinear fiber parameter is γ = 2πn2 (λ0 Aeff ) = 0.0017 m−1 W−1 . These physical values give C = 4.92 and β = 15.1. The periodic wave-guide spacing is fixed so that the nearest-neighbor linear coupling dominates the interaction between wave-guides. Over the distances of propagation considered here (e.g. Z0∗ = 6 mm), dispersion and linear attenuation can be ignored in the wave-guide array. The values of the linear and nonlinear coupling parameters are based upon recent experiment [7]. For alternative NLMC devices such as dual-core fibers or fiber arrays, these parameters can be changed substantially. Further, in the dual-core fiber case, only two wave-guides are coupled together so that the n = 0 and n = 1 are the only two modes present in the dynamic interaction. For fiber arrays, the hexagonal structure of the wave-guides couples an individual wave-guide to six of its nearest neighbors. Regardless of these model modifications, the basic NLMC dynamics remains qualitatively the same. 2.3. Mode-locking via NLMC The self-focusing property of the wave-guide array is what allows the mode-locking to occur. The proto-typical example of the NLMC self-focusing as a function of input intensity is illustrated in Fig. 2(a) which is simulated with 41 (N = 20) wave-guides [21] for two different launch powers. For this simulation, light was launched in the center wave-guide with initial amplitude a0 (0) = 1 (top) and a0 (0) = 3 (bottom). Lower intensities are clearly diffracted via nearest-neighbor coupling whereas the higher intensities remain spatially localized due to self-focusing. The spatial self-focusing can be intuitively understood as a consequence of (3) being a second-order accurate, finite-difference discretization of the focusing nonlinear Schr¨odinger equation [3]. This fundamental behavior has been extensively verified experimentally [6,1,7,21]. When placed within an optical fiber cavity, the pulse shaping associated with Fig. 2(a) leads to robust and stable modelocking behavior [16,22,23]. The computational model considered in this subsection evolves (1) while periodically

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Fig. 2. (a) The classic representation of spatial diffraction and confinement of electromagnetic energy in a wave-guide array considered by Peschel et al. [21]. In the top figure, the intensity is not strong enough to produce self-focusing and confinement in the center wave-guide, whereas the bottom figure shows the self-focusing due to the NLMC. Note that light was launched in the center wave-guide with initial amplitude a0 (0) = 1 (top) and a0 (0) = 3 (bottom). (b) Stable mode-locking using a wave-guide array with g0 = 0.7. The mode-locking is robust to the specific gain level, cavity parameter changes, and cavity perturbations. Here is is assumed that a 20% coupling loss occurs at the input and output of the wave-guide array due to butt-coupling (see Fig. 1(b)).

applying (3) every round trip of the laser cavity (see Fig. 1). The simulations assume a cavity length of 5 m and a gain bandwidth of 25 nm (τ ≈ 0.1). The loss parameter is taken to be γ = 0.1 which accounts for losses due to the output coupler and fiber attenuation. To account for the significant butt-coupling losses between the wave-guide array and the optical fiber, an additional loss is taken at the beginning and end of the wave-guide array. Fig. 2(b) demonstrates the stable mode-locked pulse formation over 40 round trips of the laser cavity starting from noisy initial conditions with a coupling loss in and out of the wave-guide array of 20% and with a constant gain g0 = 0.7. Due to the excellent intensity discrimination properties of the wave-guide array, the mode-locked laser converges extremely rapidly to the steady-state mode-locked solution. It is this generation of a stabilized mode-locked soliton pulse which the averaged model needs to reproduce. Note that the gain level g0 has been chosen so that only a single pulse per round trip is supported. Further, in Fig. 2(b) the initial condition is chosen for convenience only. 3. Averaged evolution models The principle concept behind the averaging method presented here is to derive a single, self-consistent, and asymptotically correct representation of the dynamics in the laser cavity. In order to do so, we require an equation of evolution for each individual wave-guide which accounts for both the fiber propagation and wave-guide array coupling. Thus the term averaged equations refers to the governing set of equations which account for the average effect of dispersion, self-phase modulation, mode-coupling, attenuation, and bandwidth-limited gain in the wave-guide array based laser cavity configuration of Fig. 1. The following important guidelines must be met: • Individual wave-guides are subject to chromatic dispersion and self-phase modulation. • Coupling between neighboring wave-guides is a linear process with coupling coefficient C. • The central wave-guide A0 is subject to bandwidth-limited gain given in (1) and (2) since this wave-guide is coupled back into the fiber laser cavity. No other wave-guides experience gain due to amplification. • The wave-guides neighboring the central wave-guide experience large attenuation due to the fact that they do not couple back into the laser cavity. These simple guidelines, along with the governing Eqs. (1)–(3), allow for an asymptotically correct averaged description of the laser cavity dynamics. Fig. 3 shows a schematic of the averaging process which includes five wave-guides. Specifically, each wave-guide Ai is subject to two distinct physical propagation regions: the optical fiber region and the wave-guide array region. The period L of the laser cavity depicted theoretically in Fig. 3 is established with mirrors as demonstrated in Fig. 1. In the averaging process, only the center wave-guide A0 experiences bandwidth-limited gain as given by (1) with (2) since this wave-guide contains the only optical fiber which has an Erbium doped section of fiber and physically butt-couples

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Fig. 3. Schematic of averaging process. Each wave-guide Ai is subject to two distinct physical propagation regions: the optical fiber region and the wave-guide array region. Here the period of the laser cavity L is determined by the mirror locations and fiber lengths in Fig. 1. The averaging procedure used is equivalent to the split-step method in reverse [25] which holds asymptotically for L  1, i.e. a short cavity length.

in and out of the wave-guide array (see Fig. 1). The optical fibers A±1 and A±2 representing the connection between wave-guide arrays are fictitious and only for averaging purposes. Indeed, as demonstrated in Fig. 1 the energy in the neighboring wave-guide arrays are allowed to escape the cavity into free-space. Put another way, one can think of the optical fiber propagation links in A±1 and A±2 in Fig. 3 as being governed by (1) with large attenuation but no gain, no dispersion, and no self-phase modulation. It should be noted that the attenuation in the neighboring wave-guides A±1 may not be too large since the optical fiber radius is significantly larger than the wave-guide array diameters. Thus the butt-coupling process illustrated in Fig. 1(b) can transfer significant energy in A±1 from one round-trip to the next. The averaging is then accomplished by applying the principles of the split-step method, or Strang splitting, in reverse [25], i.e. we take the evolution for the two components of the laser cavity and fuse them into a single governing equation. In its simplest form, the split-step method decomposes a partial differential equation into two principle operators: ∂A (4) = N1 (A) + N2 (A), ∂Z where N1 and N2 are in general nonlinear operators which characterize two fundamentally different behaviors or phenomena [25]. Here, N1 and N2 would represent the optical fiber propagation (1) and wave-guide array evolution (3), respectively. The split-step method then solves (4) numerically by decomposing it into two pieces over a single forward-step Z  1: ∂A = N1 (A), (5a) ∂Z ∂A = N2 (A). (5b) ∂Z Thus over each step Z, the evolution is separated into two distinct evolution equations. Thus to advance the solution, (5a) would be solved for a Z forward-step. The final solution of this step would be the initial data for (5b) which would also be advanced Z. The two step process (5) is asymptotically equivalent to (4) provided the cavity period L, which is effectively Z, is sufficiently small [25]. The details of the split-step method and its asymptotic validity are outlined by Strang [25] and will not be considered here. In essence, the averaged equations account for the average dispersion, self-phase modulation, attenuation, gain and coupling which occurs over a single round trip of the laser cavity. The only remaining modeling issue is the choice in the number of wave-guides (n = 2N + 1, see below (3)) to be considered in the averaged equations. From a practical viewpoint, each additional wave-guide considered implies the coupling of the system to another partial differential equation. Thus it is beneficial in the model to consider the minimal set of coupled equations which allow for the correct mode-locking dynamics. From a physical standpoint, the amount of energy in the wave-guides neighboring the central wave-guide is only a small fraction of the total cavity energy [23]. This suggests that a small number of wave-guides can be considered. In what follows, two models are investigated. The first model has the minimum number of wave-guides possible, i.e. the central wave-guide and its two neighbors. This model is shown to encounter significant difficulties in reproducing mode-locking in the laser cavity. The second model couples to two neighboring wave-guides on each side. This model is shown to lead to robust mode-locking characteristic of the discrete mode-locking cavity model of the last section.

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Fig. 4. Evolution of electromagnetic energy in the central wave-guide A0 in the three wave-guide model with the average cavity parameters C = 6, β = 15, τ = 0.1, e0 = 1, γ0 = 0.1 and γ±1 = 175. The gain parameter is set to (a) g0 = 0.32 and (b) g0 = 0.4. For (a) the initial condition is close to the one-soliton solution of (6b) in the absence of loss, gain and linear coupling. In (b), a white-noise seed initializes the evolution process. Neither is capable of producing mode-locked soliton pulses. Note that we have assumed a round trip cavity distance of Z = 1.

3.1. Averaged evolution with three wave-guides (N = 1) The case for which three wave-guides are coupled together, i.e. a central wave-guide (A0 ) and two neighboring wave-guides (A±1 ), gives the following governing equations: i

∂A−1 1 ∂2 A−1 + β|A−1 |2 A−1 + CA0 + iγ−1 A−1 = 0, + ∂Z 2 ∂T 2

  ∂A0 1 ∂ 2 A0 ∂2 2 i + + β|A0 | A0 + C(A1 + A−1 ) + iγ0 A0 − ig(Z) 1 + τ 2 A0 = 0, ∂Z 2 ∂T 2 ∂T

(6a) (6b)

∂A1 1 ∂ 2 A1 + β|A1 |2 A1 + CA0 + iγ1 A1 = 0. (6c) + ∂Z 2 ∂T 2 The parameter γi is the loss in each wave-guide and the gain g(Z) is given by (2) where A0 = Q. The parameters D, β and C are the average cavity dispersion, self-phase modulation, and coupling per round trip of the laser cavity, respectively. The three wave-guide interaction is incapable of producing stabilized mode-locked pulses. Fig. 4 demonstrates the characteristic behavior in the laser cavity for this model. In Fig. 4(a), the initial condition considered is reasonably close to the one-soliton solution of (6b) in the absence of loss, gain and linear coupling. Initially, the pulse looks like it will stabilize. However, after approximately 100 round trips the pulse destabilizes and quasi-periodic behavior ensues. Fig. 4(b) illustrates the more generic behavior of the laser cavity since the initial condition is a white-noise seeding expected in experiments. In this case, the electromagnetic energy forms localized structures which continue to evolve and change in a chaotic manner. This is clearly not the mode-locking behavior required to model the solitons produced in Fig. 2(b) of the discrete model. The unstable behavior in the three wave-guide interaction can be characterized by considering a model for which the dispersion in the neighboring wave-guides A±1 is neglected. In this case, localized steady-state solutions of the hyperbolic secant form can be constructed and their stability characterized [18]. It is found that these solutions undergo a radiation-mode instability. Details of these stability calculations, which are beyond the scope of the current manuscript, can be found in Ref. [18]. i

3.2. Averaged evolution with five or more wave-guides (N = 2) Given the inability of the three wave-guide model to effectively mode-lock the laser, additional wave-guides are assumed in the averaged model. The easiest case to consider is that of five coupled wave-guides: the central-wave-

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guide (A0 ) coupled to two neighboring wave-guides on each side (A±1 and A±2 ). This model allows for the robust mode-locking operation of the laser cavity. The governing equations are given by: i

∂A−2 1 ∂2 A−2 + + β|A−2 |2 A−2 + CA−1 + iγ−2 A−2 = 0, ∂Z 2 ∂T 2

1 ∂2 A−1 ∂A−1 + β|A−1 |2 A−1 + C(A−2 + A0 ) + iγ−1 A−1 = 0, + ∂Z 2 ∂T 2   ∂2 1 ∂ 2 A0 ∂A0 2 + β|A0 | A0 + C(A1 + A−1 ) + iγ0 A0 − ig(Z) 1 + τ 2 A0 = 0, + i ∂Z 2 ∂T 2 ∂T i

(7a) (7b) (7c)

i

∂A1 1 ∂ 2 A1 + + β|A1 |2 A1 + C(A2 + A0 ) + iγ1 A1 = 0, ∂Z 2 ∂T 2

(7d)

i

1 ∂ 2 A2 ∂A2 + + β|A2 |2 A2 + CA1 + iγ2 A2 = 0. ∂Z 2 ∂T 2

(7e)

As with (6), the parameter γi determines the loss in each wave-guide and the gain g(Z) is given by (2). The ideal mode-locking behavior in this five wave-guide scenario is shown in Fig. 5. For a gain value of g0 = 0.21, a single, stable soliton-like pulse forms from initial random noise after Z ≈ 1000 in the central wave-guide (A0 ) of interest. The intensity-discrimination in this case is sufficient to give rise to a mode-locked pulse attractor in the system. The coupling to the neighboring wave-guides is illustrated in Fig. 6. This shows the evolution of energy in wave-guide (a) A1 and (b) A2 . The evolution in wave-guides A−1 and A−2 resemble that in A1 and A2 , respectively. Fig. 6(c) shows the effect of the nonlinear mode-coupling. Specifically, the steady-state mode-locked profiles in the three wave-guides A0 , A1 and A2 are aligned and normalized to height unity. It is observed that the mode-locked pulse in the central wave-guide is narrowed in comparison to the neighboring wave-guides. This specifically illustrates that there is more energy transferred in the low-intensity portions of the pulse versus the high-intensity portions, i.e. a slight amount of intensity-discrimination is achieved. Interestingly enough, the five mode interaction is asymptotically equivalent to the three mode interaction for γ1  1. In this asymptotic regime, the majority of energy transferred to the neighboring wave-guides A±1 is strongly attenuated so that the wave-guides A±2 essentially play no role in the dynamics. As with the three wave-guide interaction, localized steady-state solutions of the hyperbolic secant form can be constructed and their stability characterized [18] in an appropriate parameter regime. It is found that for γ1  1, these solutions undergo a radiation-mode instability. Details of these stability calculations, which are beyond the scope of the current manuscript, can again be found in Ref. [18].

Fig. 5. Stable mode-locking dynamics in the central wave-guide A0 in the five wave-guide array model with the average cavity parameters g0 = 0.21, C = 6, β = 15, τ = 0.1, e0 = 1, γ0 = γ±1 = 0.1 and γ±2 = 175. A white-noise seed initializes the evolution process which mode-locks to a stable soliton-like solution after 700 round trips where we have assumed a round trip distance of Z = 1.

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Fig. 6. Mode-locking dynamics in the neighboring wave-guides (a) A1 and (b) A2 in the five wave-guide array model with the cavity parameters and initial conditions given in Fig. 5. The intensity discrimination which occurs in the wave-guide array via nonlinear mode-coupling is demonstrated in (c). We normalize the peak amplitude of the steady-state pulse solutions to unity and consider the pulse width in time T. The narrowing of the pulse in the temporal domain is demonstrated by comparing the central wave-guide temporal width (A0 ) with the neighboring wave-guide temporal widths (A1 and A2 ). The narrowing is due to the self-focusing of the highly intense portions of the central wave-guide pulse (see Fig. 2(a)). Note that the pulse-widths of A1 and A2 are effectively identical since the coupling between these two low-intensity pulses is linear.

In addition to the steady-state, one pulse per round trip scenario shown in Figs. 5 and 6, a number of other characteristic cavity behaviors can be observed. In particular, the cavity gain parameter g0 acts like a bifurcation parameter which determines not only the number of pulses in the laser cavity, but their dynamical properties. The multi-pulse per round trip phenomena is called harmonic mode-locking [10,11,8,4,13]. Fig. 7 shows the behavior in the central wave-guide of the laser cavity as the gain parameter g0 is increased from that of Fig. 5. In Fig. 7(a), the evolution is shown to settle to a quasi-periodic breather-like solution. The oscillatory nature of the solution, which again arises from white-noise initial data, persists indefinitely even under perturbation. Such periodic amplitude fluctuations in the laser cavity can lead to the phenomena of Q-switching [24,2]. Once the cavity energy is increased, the formation of two mode-locked pulses in the cavity occurs. For lower values of g0 in this regime, the scenario depicted in Fig. 7(b) occurs in which a short, broad pulse and a tall, narrow pulse are mode-locked simultaneously. In this case, the oscillatory structure observed in Fig. 7(a) persists. A slight increase in the bifurcation parameter g0 allows for the formation of two steady-state mode-locked soliton pulses as shown in Fig. 7(c). Further increasing g0 gives fundamentally the same behavior with an increasing number of pulses in the laser cavity. As an example, Fig. 7(d) illustrates the formation of three pulses. Before closing this section, two important observations concerning the model are related: • The generic features of the mode-locking demonstrated for the five wave-guide (N = 2) hold for a larger number of wave-guides (N > 2). • A one-sided version of the model, i.e. one that would include only coupling to A1 and A2 , also demonstrates the same generic features of the current model. Thus a three wave-guide model (A0 , A1 and A2 ) represents the minimum number of wave-guides necessary for the averaged model to hold. A five wave-guide model (A0 , A±1 and A±2 ) is the minimum number necessary for a symmetric mode-coupling model.

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Fig. 7. Mode-locking dynamics in the central wave-guide A0 in the five wave-guide array model as a function of the bifurcation parameter g0 with the average cavity parameters as given in Fig. 5. A white-noise seed initializes the evolution process which mode-locks to (a) an oscillatory (breather) mode-locked pulse with g0 = 0.23, (b) two oscillatory (breather) mode-locked pulses of differing heights and widths with g0 = 0.25, (c) two identical, steady-state mode-locked pulses with g0 = 0.28, and (d) three mode-locked pulses with g0 = 0.36. Note that we have assumed a round trip cavity distance of Z = 1.

4. Conclusions We have developed an asymptotically valid averaged model for characterizing the mode-locking dynamics in a laser cavity mode-locked via nonlinear mode-coupling. The resulting evolution in the laser cavity is given by a set of linearly coupled nonlinear Schr¨odinger equations which include attenuation and bandwidth-limited gain. The specific averaged equations are developed using a wave-guide array as the nonlinear mode-coupling element, but the equations are qualitatively correct for a broader class of nonlinear mode coupling devices. Simulations show that nearest-neighbor coupling can indeed lead to robust and stable mode-locking behavior provided a sufficient number of wave-guides are coupled together. In particular, the coupling of a central wave-guide with only one attenuated neighboring wave-guide (A0 and A1 ) does not support mode-locked pulse solutions due to a radiation mode instability [18]. Thus the averaged evolution does not describe a laser mode-locked with a dual-core NLMC element since only two modes are present. In contrast, the coupling of a chain of at least three neighboring wave-guides (A0 , A1 and A2 ) gives rise to the spontaneous formation of a mode-locked soliton pulse, i.e. in this model the mode-locked pulse is an attractor to the system. With increased cavity energy, the onset of oscillatory breather solutions and multi-pulse solutions is observed; a result consistent with experiments. Although the need for at least two neighboring wave-guides is difficult to explain from a physical point of view, it seems to be related to the fact that if the neighboring wave-guides A±1 are strongly attenuated, then the system is, in some sense, over-damped and no stable localized steady-states exist. Explanation of this phenomena is of current [18] and continued interest. Given the ability of the averaged model to generate and stabilize mode-locked pulses from noise, it provides a potential framework for the analytic study of laser stability. In particular, methods from linear stability analysis can be applied to the numerically generated steady-state solutions produced in the laser. This can aid in understanding the

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