## Abstract

We consider experimentally and theoretically a refined parameter space in a laser system near the transition to multi-pulse mode-locking. Near the transition, the onset of instability is initiated by a Hopf (periodic) bifurcation. As the cavity energy is increased, the band of unstable, oscillatory modes generates a chaotic behavior between single- and multi-pulse operation. Both theory and experiment are in good qualitative agreement and they suggest that the phenomenon is of a universal nature in mode-locked lasers at the onset of multi-pulsing from N to N+1 pulses per round trip. This is the first theoretical and experimental characterization of the transition behavior, made possible by a highly refined tuning of the gain pump level.

© 2009 Optical Society of America

## 1. Introduction

The onset of multi-pulsing as a function of increasing laser cavity energy is a ubiquitous phenomenon observed in mode-locking [1,2]. Indeed, the multi-pulsing dynamics has been demonstrated in a wide variety of theoretical and experimental configurations with both passive and active laser cavities [3–17]. Until recently, there was no satisfactory theoretical understanding of the instability mechanism that initiated the multi-pulsing process. However, theoretical progress on a mode-locked laser cavity based upon a waveguide array (WGA) architecture led to a significant theoretical advancement in quantifying the multi-pulsing process [18]. Indeed, the analysis associated with the multi-pulsing instability showed that the underlying transition from *N* to *N*+1 pulses per round trip occurred in a nontrivial manner. Specifically, a linear stability analysis of the mode-locked solution 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 values, before the two pulse per round trip state stabilizes at higher gain. Irrespective of the saturable absorption mechanism used for mode-locking, i.e. a passive polarizer, carbon nanotube, wave-guide array, quantum well, saturable absorber, etc., this transition phenomenon seems to be fairly universal in mode-locking models. In this manuscript, a highly-sensitive experimental study is performed near the multi-pulsing instability threshold in a laser mode-locked with single-walled carbon nanotube (SWCNT) saturable absorption. The experimental findings confirm qualitatively the theoretical multi-pulsing predictions [18] and suggests that the phenomenon is relevant to a broad range of mode-locked laser cavity configurations, regardless of the specific saturable absorption mechanism. To our knowledge, this is the first experimental study of its kind and shows the key role that the gain saturation plays in determining the nonlinear, dynamical transition in mode-locked laser cavities.

Theoretical progress on the multi-pulsing transition has often been hampered by the theoretical difficulty in characterizing the instabilities of mode-locked solutions in various laser cavity configurations [2]. Indeed, a full analytic treatment of the characteristic instabilities of mode-locked solutions in an averaged master mode-locking equation has only recently been given [19, 20]. Much of this difficulty is due to the fact that the gain saturation acts as a nonlocal term in the governing equations, making a standard linear stability analysis highly nontrivial [19,20]. And yet, the gain saturation term is critical in determining the overall stability of the system. Ignoring the term, or treating it as a constant, completely misses the multi-pulsing stability transition and leads to erroneous conclusions about the stability of the mode-locked pulses in general. Here we consider a mode-locking model where the necessary saturable absorber is achieved by the nonlinear mode-coupling of a wave-guide array. The primary advantage of the waveguide array based laser cavity [18, 21, 22] is that exact solutions and their stability can be explicitly calculated and as the gain is increased, robust multi-pulse operation occurs. This is unlike the qualitative master mode-locking equation which is not very robust and suffers from collapse and blowup of the mode-locked solution near the multi-pulsing transition point. The mode-locked solutions are first found to undergo a Hopf bifurcation to stable, periodic breathers as the gain is increased in the cavity. The resulting periodic breathers are then destabilized and a chaotic behavior is observed as the cavity energy intermitingly switches between one and two pulses. A slight increase in gain brings the pulses finally into the stable multi-pulsing state for which the inter-pulse dynamics becomes important [3, 15, 23]. The range of gain values for which this oscillatory and chaotic transition happens is quite small, suggesting that it could be easily missed in experiment if a refined study is not performed near the transition point. A modified gain parameter for N pulses per round trip allows this theory to be extended and to include the transition from *N* to *N*+1 pulses per round trip [18]. Although we focus on this wave-guide array based system, it should be emphasized that the waveguide array is not the mechanism responsible for generating the chaotic behavior and is simply the saturable absorption mechanism for initiating the mode-locking process.

The paper is outlined as follows: In Sec. 2, a brief overview is given of the experimental configuration used to generate mode-locked pulses and investigate the transition dynamics. Section 3 outlines the basic theoretical model used for predicting the transition phenomena at the multi-pulsing threshold. Section 4 highlights the computational and experimental findings and shows the qualitative agreement between theory and experiment in regards to the periodic and chaotic transition regions. A brief overview of the results and its implications are given in Section 5.

## 2. Experimental Setup

Although the transition from single to multi-pulse operation is a ubiquitous phenomena in pulsed laser systems, here we investigate this phenomena using a soliton Erbium-doped fiber laser [24] shown in Fig. 1(a). It contains ~2 m of Er-doped fiber with a 4 *µ*m core diameter and ~11 dB/m absorption at around 980 nm. The active fiber was pumped by a single-mode telecom-grade diode laser operating at λ=980 nm via a fused 980/1550 WDM fiber coupler. An all-fiber optical isolator was added to the cavity to ensure unidirectional operation. The saturable absorber was based on SWCNTs, which was fabricated using the technique reported previously [24]. We have found that the transition region is stretched over a broader pump power range for larger output coupling. For that reason we used a rather large output coupler (95 percent of the power is extracted from the cavity) in this experiment to facilitate the observation of the transition from *N* to *N*+1 pulses.

The change of the mode of operation of the laser with increasing pump power is shown in Fig. 1(b). Note that similar operation modes as a function of pump power have been observed in various fiber lasers (see, for instance [17]). In contrast to the work done in Ref. [17], here we focus on the pulse dynamics within the transition region from *N* to *N*+1 pulses [shown in red in Fig. 1(b)]. Mode-locking occurs when the pump current is increased to ~71 mA. The laser is quite stable and produces pulses with duration of 650 fs and a 4.5 nm bandwidth. As the pump level is increased from 71 mA, the single pulsing mode undergoes an instability and transitions into a breather at around 83 mA [see Fig. 2(a) (Media 1)]. Further increasing the pump level leads to a transition regime in which the cavity energy oscillates between single and two-pulse operation. Figure 2 shows the pulse dynamics for four different gain pumping values within the transition region. To characterize the dynamics, we measure the pulse separation of the two pulses (when separation is zero, it is a single pulse) from the video footage. Since each frame in the experimental movies is on the order of a fraction of a second, several million round trips of the cavity will have elapsed. Thus the measured quantities miss the fast timescale dynamics but give an overall representation of the periodicity of the pulse separation. Figure 3(a) shows the time series of the separation at the pump level 84 mA as shown in Fig. 2(b) (Media 2). The Fourier transform of the time series highlights that two dominant modes exist [Fig. 3(c)], typical of periodic behavior. Increasing the pump level to 85 mA [see Fig. 2(c) (Media 3)], the oscillation period becomes more irregular as seen in the time series in Fig. 3(b). Taking the Fourier transform shows that indeed there are more than two modes present in the oscillations [Fig. 3(d)]. Further, the dense, rapidly varying modal distribution shown here is characteristic of chaotic behavior.

In general, the transition from one- to two-pulse operation consists of an interesting regime where the pulse is in-between the two states. Within this region, as the pump level is increased the oscillations between the two states becomes more irregular, until finally stable two-pulse operation is obtained. Although we have focused on the one to two pulse transition, we have experimentally observed the same dynamics between any *N* to *N*+1 transition [see Fig. 1(b) for the transition from two to three pulses, for instance].

## 3. Governing Equations

There are a large number of theoretical models [2] that have been developed for quantifying the energy equilibration and mode-locking exhibited by numerous experimental configurations [1]. To make progress on quantifying the transition dynamics observed in a mode-locked laser cavity, we consider a specific theoretical model where the intensity discrimination is provided by the nonlinear mode-coupling in waveguide arrays [18,21,22]. We emphasize that this is for convenience only in illustrating the transition phenomena. Indeed, the experimental configuration considered is mode-locked not by waveguide arrays, but by the saturable absorption action of the SWCNTs described in the previous section. Regardless, the transition phenomenon considered holds in both cavity configurations and exhibits the universal nature of the multi-pulsing transition behavior.

When placed within an optical fiber cavity, the pulse shaping mechanism of a waveguide array leads to stable and robust mode-locking [18, 21, 22]. In its most simple form, the nonlinear mode-coupling is averaged into the laser cavity dynamics [18] to generate a master equation for the WGA-driven mode-locking dynamics. Symmetry considerations in the WGA and computational studies of the mode-locking dynamics predicts that the evolution dynamics in the WGA mode-locked cavity can be accurately represented by [18, 21]

where the saturated gain (energy equilibration) behavior [1] is given by

with ‖*u*‖=∫^{∞}
_{-∞}|*u*|^{2}
*dt*. Here the *u*(*z*, *t*), *v*(*z*, *t*) and *w*(*z*, *t*) represent the normalized electromagnetic fields in the center waveguide (*u*) and two neighboring waveguides (*v* and *w*). The variables *z* and *t* present the normalized propagation distance and time respectively. Note that the equations governing the neighboring WGA fields are ordinary differential equations. All fiber propagation and gain effects occur in the central waveguide (*u*) since this is the only waveguide that experiences saturated gain. This model accounts for all the leading-order physical effects in the cavity including the chromatic dispersion in the central waveguide, self-phase modulation (*γ*), linear attenuation in the three waveguides (*δ*
_{0},*δ*
_{1} and *δ*
_{2} respectively), WGA evanescent linear coupling (*C*), linear gain (*g*
_{0}), bandwidth-limit on the gain (*τ*), and energy saturation (*e*
_{0}). A detailed analysis of the scalings can be found in Refs. [18, 21]. Note that we are only considering the anomalous dispersion laser cavity in order to be consistent with the experimental findings. However, multi-pulsing is also observed in normal dispersion laser cavities as well.

It is this approximate system which will be the basis for our analytic findings and transition phenomenon studies. The primary reason for considering this model is the the fact that Eq. (1) provides a great deal of analytic insight due to its hyperbolic secant solutions

where the solution amplitude *η*, width *ω*, chirp parameter A, and phase *θ* satisfy a set of nonlinear equations [18]. Note that although this chirped pulse solution is the same as that found in the master mode-locking equation [1], the mode-locked solution here acts like a global attractor, is not susceptible to blow-up and is highly robust under perturbation. Indeed, stability of this solution, its Hopf bifurcation, chaotic transition, and corresponding solutions with N pulses per round trip can all be explored in a semi-analytic fashion [18].

To understand the stability of the multi-pulsing solutions, we consider a laser cavity with *N* mode-locked pulses of the form (3). Assuming the pulses are well-separated so that any pulse-to-pulse overlap can be neglected, the multi-pulse solution can be constructed by assuming a solution of the form (3) with the gain (2) modified to be [18]

The simple step of including the number of pulses *N* in the saturating gain gives the *N* pulse mode-locked state. Figure 4 demonstrates the stability and solution branches for *N*=1,2 and 3. The gray region in this figure denotes the area of stable mode-locked solutions. Stability is determined by linearizing about the steady-state solution (3) and considering the associated eigenvalue problem. Linear instability occurs for any eigenvalue whose ℜ{*λ*}>0 [18]. The insets on the 1-pulse solution branch (*N*=1) illustrate the Hopf instability mechanism computed previously. Further, the value of the gain parameter at which the instability occurs is within 5 percent of that obtained from numerical simulation of (1) [18]. Note that there is an entire band of unstable modes if one proceeds far enough beyond the Hopf bifurcation point. It is conjectured that this large number of unstable modes is ultimately responsible for the chaotic behavior observed in the multi-pulsing transition. Linear stability no longer holds in this case since the solutions are now time-periodic and full Floquet-type analysis is required. This analysis shows that the model (1) provides an ideal analytic framework and model for characterizing the multi-pulse transition.

## 4. Multi-pulse Transition: Theory and Experiment

To make a qualitative comparison with the experimental findings of Sec. 2, simulations are performed and evaluated near the theoretically predicted Hopf transition point of (1). Unlike the previously analysis of Kutz and Sandstede [18], emphasis is placed on the pulse-to-pulse interaction of the transition to demonstrate and confirm the experimental findings. Although the focus here will be on the transition from one to two pulses, a similar transition is predicted to occur from *N* to *N*+1 pulses [18].

Figure 5 demonstrates the mode-locking dynamics as a function of the gain strength parameter *g*
_{0}. Over the range of values *g*
_{0}=2.3,2.52,2.53,2.68,2.72 and 2.75, the mode-locking is observed to go from 1-pulse per round trip to 2-pulses per round trip. During the transition, the Hopf (periodic) bifurcation is clearly observed (top right panel of Fig. 5) to preceed the onset of chaotic dynamics. Specifically, the breather solution begins to breath erratically and drift while intermittingly forming two pulses and then one again. For sufficiently high gain, two stable mode-locked pulses are formed. To make connection with experimental findings of this process, the distance between neighboring pulses is computed for the simulations shown in Fig. 5 with *g*
_{0}=2.68 and *g*
_{0}=2.72. For these two gain values, Fig. 6 shows the pulse-to-pulse distance and its Fourier transform with the DC component removed. The oscillations for these two gain values supports the experimental findings by showing a dominant periodic signature (*g*
_{0}=2.68) and a chaotic signature (*g*
_{0}=2.72). Thus the multi-pulsing transition behavior is qualitatively consistent in both theory and experiment. A quantitative comparison is not possible due to aliasing in the experimental video footage as well as the fact that the theoretically based WGA model is different from the SWCNT mode-locking of the experiment. Specifically, in the experimental movies in which each frame was on the order of a fraction of a second, several million round trips of the cavity will have elapsed. Thus the visualization misses many of the faster timescale events that are captured in the simulations. Thus when comparing, it should be noted that the separation between the pulses of simulation are not expected to match the experimental video. At best, we can investigate the Fourier spectra in order to ascertain the periodic or chaotic signatures. Regardless, both models exhibit the key features predicted, suggesting the broader nature of the phenomena [18].

## 5. Conclusion

The phenomenon of multi-pulsing in mode-locked lasers has been well-known for almost two decades, with energy quantization and pulse-to-pulse interactions being studied and quantified by numerous researchers [3–16]. Indeed, there are still many open problems and competing theories concerning the fundamental physical mechanisms that drive the multi-pulse interaction. Despite the various studies, this manuscript provides for the first time, to our knowledge, a detailed study of the transition dynamics when the cavity bifurcates form *N* to *N*+1 pulses per round trip. As is clearly evident in the theory and experiment, periodic and chaotic behavior occurs very close to the transition point of the instability. The initial instability is shown to be driven by a Hopf bifurcation as a band of unstable eigenvalues cross into the right half plane, i.e. the real part of the eigenvalues are positive, leading to growth and instability. It is conjectured that as the gain is increased further the large number of oscillatory modes that cross into the right-half plane generate a complicated oscillatory (chaotic) structure, thus leading to the observation of seemingly random oscillations in the pulse-to-pulse interactions. A further increase in gain leads to stable *N*+1 mode-locking once again.

Although the theory and experiments are in qualitative agreement, there remain open questions about the pulse-to-pulse interactions. Specifically, the role of the translationally invariant zero mode in forcing the translation of the pulse as observed in the panels of Fig. 5. Additionally, what are the phase dynamics and how do they drive the pulse-to-pulse separation. And finally, what determines the length scale of separation for the mode-locked pulses once stabilized in the *N*+1 configuration. Future work will investigate these issues.

## Acknowledgements

The authors are indebted to A. Chong and W. Renninger for valuable discussions concerning the multi-pulsing behavior and its dynamic onset.

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