1. Introduction

Harmonic mode-locking in passive or active laser cavities has been achieved in a wide variety of theoretical and experimental configurations [1]–[14]. Indeed, this multi-pulse per round trip laser operation is a ubiquitous phenomena in mode-locking [15, 16]. To date, there is no satisfactory way to construct a general theoretical characterization of the instability which results in harmonic mode-locking. Regardless, some theoretical progress has been made in specific models which characterize the harmonic mode-locking inter-pulse dynamics [1, 13, 17], the energy quantization [14], or active laser dynamics [6]. In this manuscript, a complete theoretical description is given of the instability and bifurcation structure of a laser cavity mode-locked with a waveguide array [18, 19, 20]. Specifically, we characterize the instability of the single-pulse per round trip configuration and show that this solution is subject to both a radiation mode-instability and a Hopf bifurcation. The theory also allows for a treatment of the instability from M to M+1 pulses per round trip. This is the first quantitative theory of its kind to describe the instability process responsible for harmonic mode-locking. Further, the method presented can be extended to a broader range of harmonically mode-locked systems.

A common qualitative understanding of the harmonic mode-locking process can be understood from energy considerations [14]. The critical idea is that there is a basic competition between the bandwidth limited gain and the intensity discrimination element in the laser cavity [15, 16]. The intensity discrimination provides the mechanism ultimately responsible for the mode-locking. Often this mode-locking mechanism is only a small perturbation to the governing laser cavity equations, and it has been successfully achieved in a wide variety of experimental configurations including: a fiber ring laser with a linear polarizer, the figure-eight laser with nonlinear interferometry, a linear-cavity configuration with a semi-conductor saturable absorber, a ring or linear cavity configuration via nonlinear mode-coupling, and a laser cavity with an acousto-optic modulator. In its simplest form, intensity discrimination preferentially attenuates weaker intensity portions of individual pulses or electromagnetic energy. This attenuation is compensated by the gain medium (e.g. an Erbium-doped fiber) which acts to preserve the total cavity energy. Thus pulse shaping occurs since the peak of a pulse, for instance, experiences a higher net gain per round trip than its lower intensity wings. This time-domain narrowing of a propagating pulse is limited, however, by the bandwidth of the gain medium (typically≈20–40 nm [15]) or the chromatic dispersion. Thus the mode-locked pulse stream achieves a balance between the intensity discrimination and gain dynamics. By increasing the gain, the mode-locked pulses are made narrower in the time-domain, thereby increasing their bandwidth. However, it is energetically unfavorable for the bandwidth to increase significantly beyond that of the gain. Thus a more energetically favorable situation is created in the laser cavity whereby two mode-locked pulses are generated which are broader in the time-domain (narrower in the frequency domain) and therefore can extract a maximal amount of gain. This simple premise serves to qualitatively identify the underlying physical mechanism responsible for the harmonic mode-locked dynamics.

We theoretically characterize the instability process responsible for the resulting harmonic mode-locking by considering a laser cavity mode-locked with waveguide arrays [18, 19, 20]. Here, the nonlinear mode-coupling in the waveguide arrays is responsible for the intensity discrimination necessary for the mode-locking. By considering an averaged model for this laser cavity [21], exact pulse solutions can be constructed and their linear stability examined. Indeed, this is one of the great strengths of this quantitative model: it admits exact solutions for which the stability can be explicitly determined. We show that as the gain increases the pulse solution undergoes a Hopf bifurcation which leads to a stable mode-locked breather solution. Increasing the gain further splits the pulse into two pulses. Further increasing the gain leads to another Hopf bifurcation and eventual splitting. This process, which can be completely characterized, repeats itself and is the basic mechanism responsible for the harmonic mode-locking phenomenon.

The paper is outlined as follows: Section 2 introduces the governing equations of the wave-guide array based mode-locked cavity. Various approximations are considered in this section in order to construct the analytic solution presented in Sec. 3. Section 3 is devoted to calculating the linear stability of this pulse solution and determining the harmonic mode-locking behavior. Section 4 provides a collection of numerical simulations which verify the analytic findings of the previous section. The basic phenomena associated with the Hopf bifurcation and onset of harmonic mode-locking are exhibited. We conclude in Sec. 5 with a brief review of our findings and the implications of this model to the broader issue of harmonic mode-locking.

2. Nonlinear mode-coupling laser model

Mode-locking is achieved by applying intensity discrimination in conjunction with bandwidth-limited gain. The intensity discrimination in the specific laser considered here is provided by nonlinear mode-coupling (NLMC) in waveguide arrays [18, 19, 20, 21]. NLMC as the basis for a mode-locked laser was originally considered in a dual-core fiber configuration [22, 23] and later extended to long-period fiber gratings [24] and fiber arrays [20]. In all cases, the NLMC provides the necessary intensity-discrimination to theoretically generate stable and robust mode-locking. In what follows, the theoretical description of the harmonic mode-locking via NLMC is considered for waveguide arrays only. Figure 1 demonstrates a typical laser cavity configuration which is, in theory, capable of producing stable mode-locked pulse trains. Included in the cavity is a section of optical fiber which integrates a small section of Erbium-doped fiber for the bandwidth limited gain. The intensity discrimination is accomplished by inserting (butt-coupling) a waveguide array into the cavity. The governing equations for this laser cavity system are considered in what follows.

 

Fig. 1. Possible experimental laser configuration for a mode-locked laser cavity with a waveguide array responsible for the NLMC.

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Fig. 2. Spatial diffraction of a CW beam propagating in a waveguide array governed by Eq. (1). The left figure shows the low-intensity evolution and associated discrete spatial diffration with initial condition A ₀=1, while the right figure shows the high-intensity evolution and associated self-focusing for A ₀=3.

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2.1. Waveguide arrays

It has now been well established that the equations governing the nearest-neighbor coupling of electromagnetic energy in the waveguide array are given by [25, 26, 27, 28, 29]

where A_n represents the normalized [19, 20] electric field amplitude in the n^th waveguide (n=-N,…,-1,0,1,…,N and there are 2N+1 waveguides). The periodic waveguide spacing is fixed so that the nearest-neighbor linear coupling dominates the interaction between wave-guides. Over the distances ξ of propagation considered here (e.g. 6 mm), chromatic dispersion and linear attenuation can be ignored in the waveguide array. Note that chromatic dispersion effects have been considered elsewhere [30].

Christodoulides and Joseph [25] were the first to explore the connection between Eq. (1) and the second-order finite difference discretization of the nonlinear Schrödinger equation. Indeed, the fundamental insight drawn from this work was the description of the spatial focusing which occurs and allows for a new form of discrete spatial soliton [26, 27, 28, 29]. In the mode-locking application considered here, the NLMC is used as an intensity-dependent pulse shaping mechanism [18]. It is, in fact, equivalent to a discrete Kerr lens effect [31, 32]. To see the effects of the intensity dependence, the evolution of a plane wave subject to Eq. (1) is simulated for the physical values given in Ref. [26]. Figure 2 demonstrates the classic phenomena [25] of intensity dependent, discrete spatial diffraction which occurs in the waveguide array. From this figure, we can conjecture that low intensities are preferentially attenuated, since they couple to neighboring waveguides which eject light from the cavity, thereby giving rise to the pulse-narrowing (intensity discrimination) necessary for mode-locking.

2.2. Averaged model

One method capable of describing the laser cavity is direct simulation of the constitutive components in the laser cavity [19, 20]. A second method follows the idea of Haus in developing a master mode-locking model for the laser cavity [15]. This method effectively averages over all the physical effects which occur per round trip in the laser cavity. A master mode-locking model has also been recently developed and justified for the waveguide array based mode-locking [21]. The governing equations in this case are given by

where again A_n represents the normalized [21] electric field amplitude in the n^th waveguide (n=-N,…,-1,0,1,…,N and there are 2N+1 waveguides). The parameters γ_n, β_n, and τ_n determine the loss, nonlinear self-phase modulation and gain bandwidth in each waveguide. The gain G_n(Z) is the gain in each waveguide. Since we only amplify the center waveguide in the laser cavity configuration (See Fig. 1), we have G_n(Z)=0 for n≠0 and

where g ₀ is the gain parameter and e ₀ is the saturated gain in the cavity. The parameters β_i and C̄ are the average self-phase modulation and average coupling per round trip of the laser cavity.

The governing equations (2) with the gain model Eq. (3) give a complete and quantitatively accurate description of the laser cavity dynamics [21]. Indeed, these equations are the basis for the analysis which follows. Few, if any, analytic methods for characterizing the dynamics in these governing equations are available. However, approximations can be made which render the system analytically tractable. These approximations are pursued in the following subsections.

2.3. Simplified model

The first approximation to be made to the system given by Eq. (2) concerns the number of interacting waveguides. The total number of waveguides is 2N+1 where there are typically N=20 (i.e. 41 total waveguides) in the experimentally considered waveguide array [26]. Proctor and Kutz [21] established that the fundamental behavior (qualitatively and quantitatively) in the laser cavity as governed by Eq. (2) and (3) does not change for N≥2. Thus we can take N=2 (5 waveguides) as the fundamental set of interacting waveguides. Additionally, for initial data confined to the center waveguide A ₀, which is the case of physical interest, the coupling to the neighboring waveguides A _±1 and A _±2 is symmetric. Thus, we reduce from five to three the number of waveguides of importance in determining the dynamics of the system. The governing equations are then given by

where C=2C̄ which accounts for double the coupling strength which would be present in the five waveguide model. This retains all the features of Eq. (2) with N=2, but now we only have to consider three coupled partial differential equations. Provided the initial conditions are only in A ₀, this is a valid assumption since there is symmetry between the pairs A _±1 and A _±2.

Although Eq. (4) greatly simplifies the analysis from the general model of Eq. (2), it remains analytically intractable. Regardless, computations are greatly simplified since now on three partial differential equations are simulated. But given the fact that the pulse shapes in A ₁ and A ₂ largely inherit their structure from A ₀ due to the linear coupling, we make two further approximations in order to extract an analytically tractable waveguide array laser model. One assumption comes from observing numerically that the amplitudes in the neighboring waveguides remain small, thus allowing us to neglect the self-phase modulation in the neighboring waveguides. And finally, we ignore the effects of the chromatic dispersion in the neighboring waveguides. This is a relatively good approximation given that the chromatic dispersion per round trip in the cavity is nearly zero, i.e. the propagation in the neighboring channels do not experience the anomalous dispersion given by the optical fiber that is part of the center waveguide A ₀ [21]. The resulting approximate evolution dynamics is given by

where now we have only ordinary differential equations for A ₁ and A ₂. It is this approximate system which will be the basis for our analytic findings. In fact, Eq. (5) provides a great deal of analytic insight into the bifurcation structure of the harmonic mode-locking process. Further, the approximation is shown to be in good agreement with the system Eq. (4).

3. Mode-locked solutions

Steady-state solutions to the simplified equations (5) can be constructed by performing an amplitude-phase decomposition:

where i=0,1, 2. When inserted into Eq. (5c), a relation is found for Q ₂ in terms of Q ₁:

A similar result is obtained when applying the amplitude-phase decomposition to Eq. (5b). Specifically, by using Eq. (7) Q ₁ can be found in terms of of Q ₀ alone:

where P is a complex constant which depends upon the solution variables Θ₁ and Θ₂. This leaves only Eq. (5a) as a single differential equation for Q ₀:

 

Fig. 3. Plotted are the amplitude η versus the fixed-gain parameter g_f [left curve], from solving Eq. (14), and the corresponding solutions [right curve] for variable gain with g ₀ computed via Eq. (15) from g_f.

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In this calculation, the gain from Eq. (3) is a function of Q ₀ alone. Specifically, we find

Equation (9) can be solved with a solution of the form [15]

Any solution of this form has ‖Q ₀‖²=2η ²/ω, and we may therefore define the fixed-gain parameter

so that the gain from Eq. (3) is given by

Denoting the real and imaginary parts of the coefficient P=P_r+iP_i leads to the center waveguide phase ${\Theta }_0=-\frac{{\omega }^2}{2}\left(1-A^2\right)-{\mathrm{CP}}_r-4g_f{\mathrm{\tau \omega }}^{2}A$ , and the 3×3 system

which, in theory, we can solve for η, ω and A. To revert back to the original variable-gain from Eq. (3), we only need to recover g ₀ via

The chirped (A≠0) localized hyperbolic secant solution is the basis for understanding the harmonic mode-locking structure in the laser cavity. In particular, the consistency equations (14) can be solved numerically to yield a localized steady-state mode-locked solution: all computations presented here are for

and we refer to Fig. 3 for the existence region of secant solutions. Of critical importance is that a linear stability analysis can now be performed and the instabilities of the steady-state branch of solutions characterized. This generates the bifurcation structure of the harmonic mode-locking.

 

Fig. 4. The left plot shows the existence curve of secant solutions for variable gain Eq. (10). The modelocking solution is stable along the solid curve; it destabilizes on the lower branch at g ₀=1.4 due to unstable counter-propagating radiation modes as shown in the lower-right panel, while the instability on the upper branch at g ₀=2.3 is due to a Hopf bifurcation which will be discussed later. The upper-right panel contains the linear dispersion relation λ(k) associated with radiation modes. The instability at g ₀=1.4 for low-amplitude solitons is due to the upper branch of radiation modes crossing the imaginary axis, resulting in the instability caused by the counter-propagating waves shown in the lower-right panel.

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3.1. Stability of mode-locked solutions

We now discuss the stability of the mode-locked solutions computed in Eq. (11). A convenient way for assessing stability is to calculate the spectrum of the linearization of Eq. (4) about the underlying solitary wave. The spectrum comes in two parts, namely radiation modes, which are determined by the asymptotic background state (A ₀,A ₁,A ₂)=0, and eigenvalues, which are linked to the profile and shape of the underlying soliton.

We begin with radiation modes which are computed by linearizing Eq. (4) about the background state (A ₀,A ₁,A ₂)=0 for g(Z)=2g_f, where g_f is given by Eq. (12) in the case of variable gain. Seeking solutions of the form

where (B ₀,B ₁,B ₂) are constant complex amplitudes, for the resulting linearized equation, we arrive at the equation

Solutions λ for real k correspond to radiation modes. We plot the resulting stability boundaries in Fig. 4 together with a representative plot of the linear dispersion relation.

Next, we discuss the linearization of Eq. (4) about the modelocking solution (11) in the case of variable gain, where g(Z) is given by Eq. (10). The resulting operator depends on T, through the profile of the secant solution, which makes is difficult to compute its eigenvalues explicitly. Instead, we discretize the operator using centered finite differences in T on the interval (-ℓ, ℓ) with Neumann boundary conditions and find the eigenvalues, and eigenfunctions, of the resulting matrix numerically. For parameters as in Eq. (16), we find that the mode-locking solution on the upper branch destabilizes in a Hopf bifurcation at g ₀=2.3; see Fig. 5. The associated eigenfunction, also shown in Fig. 4, is clearly localized. We believe that it emerged from the continuous spectrum in an edge bifurcation [33]. Direct simulations, presented in more detail below, indicate that the Hopf bifurcation leads to stable mode-locked breather solutions.

 

Fig. 5. The upper-right panel contains the complete spectrum for the mode-locking solution at the instability point g ₀=2.3 shown in the left panel. Eigenvalues are plotted for the linearization on the interval (-4,4) [blue bullets] and (-10,10) [red crosses] with 300 equidistant mesh points in T. The instability is caused by a localized Hopf eigenmode, whose real and imaginary part we plot in the lower-right figure, with temporal frequency 12.06.

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3.2. Harmonic mode-locking

The mode-locked solutions we computed from Eq. (14) are chirped. Thus, their tails in T are oscillatory, and we can therefore glue well-separated copies of these localized solutions together to create harmonic mode-locked solutions with several pulses per round trip. As discussed above, if the mode-locked secant exists for a fixed gain g_f, its power is given by 2η ²/ω and it exists when the variable gain coefficient g ₀ is g ₀=g_f(1+2η ²/ωe ₀). Thus, a harmonic mode-locked solution with M pulses per round trip has power approximately equal to 2Mη ²/ω and exists for g ₀=g_f(1+2Mη ²/ωe ₀).

The above argument allows us to predict the existence region of harmonic mode-locked solutions using only the mode-locked secant solution and the relation between fixed and variable gain coefficients. Since the background states of primary and harmonic mode-locked solution are identical, their radiation-mode instabilities coincide. Similarly, we expect that each of the individual pulses in a harmonic mode-locked solution will undergo the Hopf bifurcation that the primary secant solution experiences. The theoretical predictions are summarized in Fig. 6 which gives the existence curves of harmonic mode-locked solutions and their stability boundaries. Figure 6 further illustrates that multiple branches of solutions are stable simultaneously. For instance, for g ₀=2, the one-, two-, three- and four-pulse solutions are all simultaneously stable. However, the one-pulse mode-locked solution is the most energetically favorable for this value of g ₀ and is the solution which is formed from white-noise initial data.

4. Mode-locking dynamics

The dynamical properties and bifurcation structure of the solution in Eq. (11) to the evolution equations (5) are analytically characterized in the previous section. This section aims to verify the analytic predictions and to demonstrate the stability of the mode-locked solution and the onset of harmonic mode-locking for increasing values of the gain parameter g ₀.

 

Fig. 6. Plotted are the amplitude η versus variable gain g ₀ for the harmonic modelocking solutions with M=1,…,4 and the fixed-gain modelocking solutions [left]. Solutions are stable for amplitudes η in the range indicated to the right; they are unstable to oscillatory modes above the upper horizontal dotted line and to counter-propagating waves below the lower horizontal dotted line.

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Fig. 7. Demonstration of mode-locking (A ₀, A ₁ and A ₂) in the variable gain model with parameters as in Eq. (16) and initial white-noise. The L ² norm is also depicted for A ₀, A ₁ and A ₂. Note that the dynamics here and the settling to a steady-state solution is tremendously robust with what appears to be an infinite basin of attraction.

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To begin the exploration of the dynamics associated with mode-locking, Eq. (5) is simulated with a gain parameter g ₀ for which only a single pulse per round trip is generated. Figure 7 demonstrates the mode-locking process. The center cavity A ₀ is initially seeded with a small amount of white-noise as shown in the top left panel of Fig. 7. From this white-noise, the solution (11) acts as an attractor to the governing system Eq. (5). Numerical simulations suggest that all initial data will eventually settle to the localized mode-locked solution (11), which leads to the conjecture that there is an infinite ball of attraction for this mode-locked solution. This should be contrasted to the master mode-locked equation [15, 33] for which there is only a small region of attraction for the mode-locked solution, i.e. initial conditions must be carefully prepared in simulations.

Figure 7 also shows in the bottom two panels the evolution of electromagnetic energy in the neighboring waveguides A ₁ and A ₂. Their form of solution mimics that in the center waveguide A ₀ as is clear from the reduction of solution Eqs. (7) and (8) in Sec. 3. A critical aspect of the mode-locking behavior is the energy which saturates the individual waveguides. The top right panel of Fig. 7 shows the evolution L ² norm as a function of distance Z. It is clear from this figure that the pulse mode-locks to the steady-state solution after Z≈150. Note that there is a significant difference in energy between the different waveguides. This is expected since only A ₀ experiences gain through Eq. 3, whereas A ₁ and A ₂ only receive energy from the linear coupling. Additionally, A ₂ is strongly attenuated so that its energy never builds up.

 

Fig. 8. Dynamic evolution and associated bifurcation structure of the transition from one pulse per round trip to two pulses per round trip. The corresponding values of gain are g ₀=2.3,2.35,2.5,2.55,2.7, and 2.75. For the lowest gain value only a single pulse is present. The pulse then becomes a periodic breather before undergoing a “chaotic” transition between a breather and a two-pulse solution. Above a critical value (g ₀≈2.75), the two-pulse solution is stabilized. The corresponding gain dynamics is given in Fig. 9.

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4.1. One- to two-pulse transition

The basic mode-locking dynamics illustrated in Fig. 7 is altered once the gain parameter g ₀ is increased. In particular, the analysis of the last section suggests that the steady-state pulse solution of Fig. 7 first undergoes a Hopf bifurcation before settling to a two pulse per round trip configuration. However, between the Hopf bifurcation and the stable two-pulse configuration there is a region of chaotic dynamics. Figure 8 shows a series of mode-locking behaviors which occur between the steady-state one pulse per round trip and the two pulses per round trip configurations. The gain values in this case are progressively increased from g ₀=2.3 to g ₀=2.75. As the dynamics change from one to two pulses per round trip steady-state, oscillatory and chaotic behaviors are observed. To characterize this behavior, we consider the gain dynamics g(Z) of Eq. (3) in Fig. 9 which correspond to the evolution dynamics shown in Fig. 8. The gain dynamics provides a more easily quantifiable way of observing the transition phenomena.

At a gain value of g ₀=2.3, the stable one-pulse configuration is observed in the top left panel of Fig. 8. The detailed evolution of this steady-state mode-locking process is shown in Fig. 7. The top right panel and middle left panel of Fig. 8 show the dynamics for gain values of g ₀=2.35 and g ₀=2.5 which are above the predicted threshold for a Hopf bifurcation. The resulting mode-locked pulse settles to a breather. Specifically, the amplitude and width oscillate in a periodic fashion. The oscillatory behavior is more precisely captured in Fig. 9 which clearly show the period and strength of oscillations generated in the gain g(Z). Note that as the gain is increased further, the oscillations become stronger in amplitude and longer in period. To further demonstrate the behavior near the Hopf bifurcation, we compute in Fig. 10 the Fourier spectrum of the oscillatory gain dynamics for g ₀=2.35,2.5 and 2.55. The dominant wavenumber of the Fourier modes for g ₀=2.35 near onset is 10.07, which is in very good agreement with the theoretical prediction of 12.06 derived in Sec. 3.3 for the Hopf bifurcation.

 

Fig. 9. Gain dynamics associated with the transition from one pulse per round trip to two pulses per round trip for the temporal dynamics given in Fig. 8. The left column is the full gain dynamics for Z∊[0,4000], while the right column is a detail over Z=10 or Z=50 units, for values of gain equal to g ₀=2.3,2.35,2.5,2.55,2.7, and 2.75. Initially a single pulse is present (top panel), which becomes a periodic breather (following two panels) before undergoing a “chaotic” transition between a breather and a two-pulse solution (following two panels) until the two-pulse is stabilized (bottom panel) at g ₀≈2.75.

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Increasing the gain further leads to an instability of the breather solution. The middle right and bottom left panels of Fig. 8, which have gain values of g ₀=2.55 and g ₀=2.7, illustrate the possible ensuing chaotic dynamics. Specifically, for a gain of g ₀=2.55, the mode-locking behavior alternates between the breather and a two pulse per round trip state. The alternating between these two states occurs over thousands of units in Z. As the gain is further increased, the cavity is largely in the two-pulse per round trip operation with an occasional, and brief, switch back to a one-pulse per round trip configuration. Figure 9 illustrates the two chaotic behaviors in this case. Note the long periods of chaotic behavior for g ₀=2.5 and the short bursts of chaotic behavior for g ₀=2.7. Above g ₀=2.75, the solution settles quickly to the two pulse per round trip configuration as shown in the bottom right panel of Fig. 8, which is therefore the new steady-state for the system. Thus the theoretical predictions of Sec. 3 capture the majority of the transition aside from the small window of parameter space for which the chaotic behavior is observed.

The bistability between the one- and two-pulse solutions in the laser cavity is demonstrated in Fig. 11. The numerical simulations performed for this figure involve first increasing and then decreasing the bifurcation parameter g ₀. Specifically, the initial value of g ₀=0.9 is chosen so that only the one-pulse solution exists and is stable. The value of g ₀ is then increased to g ₀=2.3 where the one-pulse solution is still stable. Increasing further to g ₀=2.55 excites the Hopf bifurcation demonstrated in Figs. 8–10. Increasing to g ₀=2.75 shows the two-pulse solution to be stable. The parameter g ₀ is then systematically decreased to g ₀=0.9,2.3 and 2.55. Figure 11 demonstrates the bistability by showing that at g ₀=2.3 (middle panels) and g ₀=2.55 (bottom panels) both a one-pulse and two-pulse solution are stable. Dropping the gain back to g ₀=0.9 reproduces the one-pulse solution shown in the top left panel. The top right panel shows the location on the solution curves (circles) where the one- and two-pulse solutions are both stable. It should be noted that the harmonic mode-locking is not just bistable. Rather, for a given value of the gain parameter g ₀, it may be possible to have one-, two-, three-, four- or more pulse solutions all simultaneously stable. The most energetically favorable of these solution branches is the global-attractor of white-noise initial data.

 

Fig. 10. Fourier spectrum of gain dynamics oscillations as a function of wavenumber. The last Z=51.15 units (which gives 1024 data points) are selected to construct the time series of the gain dynamics and its Fourier transform minus its average. This is done for the data in Figs. 8 and 9 for g ₀=2.35,2.5 and 2.55.

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4.2. Two- to three-pulse transition

The transition from two to three pulses per round trip follows much of the pattern of the transition from one to two pulses. As before, we expect a Hopf bifurcation to occur along the branch of solutions for the two pulses per round trip configuration. This is followed by a brief region of chaotic behavior before the three pulses per round trip configuration becomes the stable steady-state. Figure 12 illustrates the onset of the Hopf bifurcation, transition to chaotic behavior, and finally the stabilization of a three-pulse steady-state. The gain dynamics are much like those displayed in Fig. 9.

Beginning with the Hopf transition, we see from the top two panels of Fig. 12 the onset of oscillations. Interestingly though, only one of the two pulses becomes a breather. The second pulse remains a steady-state mode-locked solution. Of interest is the Hopf bifurcation calculation of Sec. 3.3 which holds here despite the two pulse per round trip configuration. In particular, the wavenumber found numerically agrees well with the analytic prediction of 12.06.

The onset of chaotic behavior observed for g ₀=4.1 is more complicated than for the one to two pulse transition. Still, the fundamental interplay and competition of the pulses in the cavity reflects similar chaotic behavior. Once above g ₀≈4.3, the evolution quickly settles to the three pulses per round trip configuration in the cavity.

 

Fig. 11. Demonstration of bi-stability of the one and two pulse per round trip configurations in the harmonic mode-locking process. The gain is first increased from g ₀=0.9 to g ₀=2.75 and then decreased back. Shown are stable single pulse per round trip (g ₀=0.9), one and two pulse per round trip (g ₀=2.3 in middle panels), and breathing one pulse and two pulse per round trip (g ₀=2.55 in bottom panels) configurations. The top right panel shows the one- and two-pulse branch of solutions along with the stable solutions from the remaining panels indicated by circles.

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4.3. M to M+1 pulse transitions

The behavior exhibited for one to two and two to three pulse transitions is generic for this system. Indeed, the basic behavior persists for transitions from M to M+1 pulses in the laser cavity. As the bifurcation parameter g ₀ is increased, one of the M pulses undergoes a Hopf bifurcation to a breathing state. Further increase of the bifurcation parameter leads to chaotic oscillations in the breathing pulse. This is followed by a chaotic region where the solutions intermittently switch from M to M+1 pulses. Increasing the gain further gives a stabilized M+1 pulse configuration. Figure 13 demonstrates the formation of four and five pulse per round trip configurations for g ₀=5.7 and 7.1, respectively. Increasing g ₀ further allows for the addition of a sixth and seventh pulse, and so on. In practice, it should be noted that the value of g ₀ is limited by the cavity losses and amplifier pumping in the laser cavity. Thus, for a given cavity, only a limited number of pulses can be produced.

5. Conclusions

We considered a theoretical model for a laser cavity which is mode-locked through the nonlinear mode-coupling in a butt-coupled waveguide array. Using this model, we characterized the instability process responsible for harmonic mode-locking by constructing exact one pulse per round trip solutions. A linear stability analysis of these solutions shows that, as the gain increases, the stable pulse solution undergoes an oscillatory Hopf bifurcation which leads to a stable mode-locked breather solution. Increasing the gain further leads to a transition from the breather to a two pulse per round trip steady-state configuration. The transition process is chaotic in nature, with the solution dynamically switching back and forth between the breather and the two-pulse state over a range of gain parameters, before the two pulse per round trip state stabilizes for higher gain parameters, leading to harmonic mode-locking. This process repeats itself, leading to M+1 pulse per round trip configurations for sufficiently large gain through Hopf bifurcations of one of the pulses in the M pulses per round trip state and subsequent chaotic transitions to M+1 pulses per round trip. Our analysis showed that harmonic mode-locking can be predicted by the gain model

 

Fig. 12. Dynamic evolution and associated bifurcation structure of the transition from two pulses per round trip to three pulses per round trip. The corresponding values of gain are g ₀=3.7,3.9,4.1, and 4.3. For the lowest gain, two pulses are present. One pulse then becomes a periodic breather before undergoing a “chaotic” transition between a breather and a two-pulse solution. Above a critical value, three pulses are stabilized.

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Fig. 13. Dynamic evolution to four and five pulses per round trip for values of gain equal to g ₀=5.7 and 7.1, respectively. The dynamical bifurcations to these steady-state configurations follow patterns similar to the transitions from one to two and two to three pulses.

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which gives the gain parameters for which M pulses per round trip configurations exist, assuming that g ₀ corresponds to a stable one-pulse solution. The theory also predicts bi-stability between several stable M pulse per round trip solutions. The theoretically calculated instability mechanism and the predicted bi-stability are in excellent agreement with numerical simulations.