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Abstract
Extreme events (pulses of high intensity) are observed to appear spontaneously in some chaotic regimes of the self-pulsing all-solid-state laser with a saturable absorber (diode-pumped Nd:YVO4 + Cr:YAG). Until now, the mechanism of build-up of these pulses has remained unknown. Besides the practical interest in this device of widespread use, the laser with a saturable absorber is homologous to the Bénard–Rayleigh system with a solute. Experimental observations show the events to appear even if a few different transverse modes oscillate. This leads to a description in terms of a few ordinary differential equations, instead of the more usual description with an equation in partial derivatives. A minimum of five transverse modes and hole burning in the absorber are found necessary elements of the description. On the other hand, phases among the spatial Fourier components do not need to be taken into consideration. The demonstration that, in some systems at least, the formation of extreme events can be explained with a few interacting degrees of freedom has a broad impact.
© 2021 Optical Society of America
1. INTRODUCTION
Solli et al. [1] imported the concept of rogue waves from the field of oceanography to describe large fluctuations in the edge of the spectrum of the light observed in microstructured optical fibers. Since then, the study of optical systems producing what has come to be known as optical rogue waves or extreme events (which can be, broadly speaking, defined by statistical features such as heavy-tailed probability distributions) has expanded rapidly and extended to a variety of systems beyond optical fibers: they have been found, for example, in spatially extended systems [2], fiber lasers [3–7] (where rogue waves appear through the coupling of the radiation of two lasers, and where the observed statistical distributions are qualitatively similar to the ones shown in this work), semiconductor lasers [8–13], plasmas [14], and lasers with modulation of losses [15,16]. Although a strict general definition is still a matter of discussion [17,18], “an optical pulse whose amplitude or intensity is much higher than that of the surrounding pulses” [19] is an intuitive and good definition of an optical extreme event at a qualitative level.
In [20], our group reported, for the first time, the existence of extreme events in a ${\rm Nd}{:}{{\rm YVO}_4} + {\rm Cr}{:}{\rm YAG}$ laser, which is a device of widespread use. The same phenomenon had previously been observed in a similar laser with Nd:YAG as the active medium instead of ${\rm Nd}{:}{{\rm YVO}_4}$ [21], and was also studied later in other lasers with saturable absorbers such as a ${\rm Nd}{:}{{\rm GdVO}_4} + {\rm Cr}{:}{\rm YAG}$ with azimuthal polarization [22] and vertical-cavity surface-emitting lasers with saturable absorbers [23–26] (which are systems similar to the one described here, but with faster dynamics).
In our system, the extreme events were observed in chaotic regimes with a high dimension of embedding, high Fresnel number for the cavity, and complex spatial transverse patterns of the spot. This pointed to the competition of several transverse modes as a necessary feature for the build-up of extreme events. In [27], our group reported direct observation of the pulse-to-pulse evolution of the transverse pattern of this laser in different regimes with extreme events, showing that it alternated chaotically between a limited number of transverse modes (in a typical record, 99% of the spots were described by only five modes), thus confirming our previous assumptions. We also found that in every case, extreme events were linked to specific patterns and that they followed a dynamics more repetitive than average pulses. We observed and described further dynamical features in [28].
Despite the fact that the rogue waves observed in some optical systems do have a proper theoretical explanation through the nonlinear Schrödinger equation (see, e.g., [19,29]), a model that describes the emergence of extreme events in the laser I study in this work is still missing. In [20], our group explored a simple theoretical approach based on rate equations for a single mode [30] and showed that it is able to reproduce many of the dynamical features observed in this system, including chaos, but it does not predict the existence of extreme events. The purpose of this work is thus to find a simple theoretical model that predicts the existence of extreme events in the system under study. By simple, I refer to an approach to the transverse effects problem based on a set of a few ordinary differential equations (as a modal expansion of the Maxwell–Bloch equations) in contraposition to the one that comes from deriving global partial differential equations for slowly varying amplitudes. Even though the latter approach is, a priori, the best suited for systems with a large Fresnel number [31], as is the case for the system under consideration, the experimental observations mentioned above strongly suggest that the first, simpler, approach, which allows more easily a physical interpretation of the underlying mechanisms, is still suitable. Hence, I set out to study models that incorporate the interaction of several transverse modes. All of these stem from the model proposed in [32].
Here I adopt the same quantitative criterion as in [20,27,28] regarding the extreme events: the regimes of interest are those that yield “heavy-tailed” peak intensity distributions (this means, quantitatively, distributions with kurtosis higher than three, that of the normal distribution) and, within those distributions, I consider as extreme events those pulses with a peak intensity higher than a threshold that is four standard deviations above the mean of the distribution (to which I will refer from now on as “extreme event threshold”). It was shown in [28] that this criterion is appropriate for this system. The pulses fulfilling it are not only of high intensity, but also show specific and interesting dynamical features, namely, (i) a relative (when compared to average pulses) regularity observed in the interspike time intervals between extreme events and surrounding pulses and (ii) comparatively long interspike intervals associated with extreme events (this has also been observed in [33] where, for a model of a ${{\rm CO}_2}$ laser with modulated losses, the threshold criterion was used to relate longer interspike intervals to the onset of extreme events). It must be mentioned that alternative criteria to define extreme events do exist: in some instances, they are “outliers” in a histogram whose statistical properties differ from those of the bulk of the data [34].
This paper is organized as follows: in Section 2, I outline the setup which I aim to model; in Section 3, I report the different models explored in order to find extreme events, including the one that finally yielded them; the main results (corresponding to the latter model) are shown in Section 4; and finally, in Section 5 I summarize the main conclusions of this work.
2. LASER
The scheme of the laser is presented in Fig. 1. The pump is provided by a 2 W (at 808 nm) CW laser diode and collimated and focused (by means of gradient-index lens) to a spot of ${\simeq} 0.8\;{\rm mm}$ of diameter into the active medium: a ${\rm Nd}{:}{{\rm YVO}_4}$ crystal, 1% doped (the operating laser wavelength is 1064 nm, linearly polarized). The layer of the crystal that faces the pump is coated HR at 1064 nm and AR at 808 nm. The cavity is designed in a V-shape, with a folding concave mirror, HR at 1064 nm ($R = 100\;{\rm mm}$), and a plane output coupler (reflectivity 98% at 1064 nm). The geometry of the cavity is intended, first, to allow for a Fresnel number $\# F \approx 10$, and, second, to yield a strongly varying spot size in its second arm (the one that extends from the folding mirror to the output coupler), with the waist on the output mirror. A solid-state saturable absorber (Cr:YAG crystal, 90% transmission when unbleached) is placed in this arm, at a variable distance $X$ from the end mirror. By changing $X$, the mode size at the saturable absorber is modified and hence the condition of saturation. This in turn defines different regimes of $Q$-switch operation: from stable, to several periodic behaviors (periods 2, 4, and 6) and chaotic regimes with and without extreme events. Thus, $X$ is the main experimental control parameter.
Fig. 1. Scheme of the modeled laser. LD, pump laser diode, 2 W CW at 808 nm; GL, GRIN lens; ND, ${\rm Nd}{:}{{\rm YVO}_4}$ slab (active medium, 1 mm in length) coated HR at 1064 nm and AR at 808 nm at the layer that faces the pump; M1, folding mirror ($R = 100\;{\rm mm}$); M2, output mirror (plane); SA, Cr:YAG crystal, transmission (unbleached) 90%, 1 mm in length; ${\alpha = 20^ \circ}$; ${\rm L}1 = 130\;{\rm mm}$; ${\rm L}2 = 70\;{\rm mm}$. The position $X$ of the SA is variable to obtain different dynamical regimes. The operating laser wavelength is 1064 nm.
3. MODELS
I begin by exploring a simple adaptation of the model proposed by Dong et al. [32], which is in turn based in the model by Tang et al. [35], who studied theoretically the simultaneous oscillations of several modes in solid-state lasers, by proposing a suitable rate-equations model. Dong et al. were interested in studying the effects of transverse modes in a laser-diode-end-pumped Cr, Nd:YAG self-$Q$-switched microchip laser, and built a new model upon the previous work of Tang et al., predicting many of the experimental observations reported in the same paper. To do this, they modified the standard multimode rate equation used to describe this system by taking into account the mode coupling dynamics (as a consequence of spatial hole burning) of an arbitrary number of transverse modes and the nonlinear absorption of the saturable absorber. They considered the active medium as a four-level laser system and the saturable absorber as a two-level system with an excited state of absorption, and the laser oscillation as single longitudinal mode. The main virtue of this model is that it allows for the easy incorporation of as many transverse modes as desired. The resulting numerical simulations reproduced some of the features observed in the experiments, such as multi-pulse oscillations (in the form of “satellite” pulses) and pulse trains with different periods. They did not report, either experimentally or numerically, the existence of extreme events.
I start with a slight modification of the model, the main differences being the following: (i) I take into account the conservation of the population density in the active medium; (ii) I consider the specific design of the cavity described in Section 2, where the active medium and saturable absorber are not contained in the same crystal but separated and at variable distances. I do this by introducing a factor $k$ that stands for the ratio of the area of each mode at the active medium to that of the mode over the saturable absorber [20]. In this way, $k$ is the main control parameter in the simulations presented here: a change in $k$ is equivalent to an experimental modification of the distance $X$ of the saturable absorber to the output mirror (see Section 2), and determines different dynamical regimes. Other minor differences from the model in [32] are as follows: (i) in the equation for the population density of the upper level, the pump term is not constant and takes into account the population of the ground level (via the term $N - {n_u} - {n_l}$); (ii) I consider the saturable absorber as a two-level system, which is the standard approach, thus omitting the term corresponding to the excited state level of the saturable absorber in the photon density equation; (iii) also in this equation, I add a term to account for the spontaneous emission.
I numerically integrate the equations for a wide range of experimentally feasible sets of parameters and up to five transverse modes. For each set, I obtain the bifurcation diagram with $k$ as the parameter of control, i.e., iterating the integration for a grid of $k$ s. The specifics of this integration can be found in Supplement 1. For each value of $k$, I also calculate the kurtosis, the extreme event threshold, and the maximum value event, to detect whether a given set of parameters yields extreme events. I do not find in any case any regime fulfilling the conditions for the existence of extreme events. In Fig. 2, I show an example of this: for an exploration of the model with three modes, I plot, for each $k$, the value of the highest peak intensity (indicated by blue dots) and the extreme event threshold (continuous red line). In every case, the maximum value event is well below the extreme event threshold.
Fig. 2. Extreme event analysis for the model with three transverse modes with $k$ as a control parameter. The red line indicates the extreme event threshold, while the blue dots stand for the maximum value event of the simulated trace for each $k$. The peak intensity value is normalized in such a way that the intensity of the average peak equals 100. It is clearly seen that the system is always far from reaching the extreme event threshold.
Due to the failure of the model to yield extreme events, I seek to increase its complexity by incorporating previously ignored effects, namely, the action of the different transverse modes in the saturable absorber. To take this into account, I add a new variable (and thus a new equation) for each spatial Fourier component of the population difference density of the saturable absorber corresponding to each mode (at this stage, the interaction among the modes is not yet considered: this is done with the aim to try the simplest possible model that yields extreme events). As I do not find extreme events in any of the simulated traces, the next logical step is to improve the model by incorporating the interactions of the different modes inside the saturable absorber. This is done by adding (to the equation corresponding to each spatial Fourier term in the saturable absorber) a term of interaction with the fields of the other modes. These added terms are naturally of the form ${-}\sum_{j = 1}^n {\alpha _{{ij}}}{N_{{s_i}}}{\phi _j}$ (with ${\alpha _{{ij}}}$ constant, ${N_{{s_i}}}$ the Fourier component of mode $i$, and ${\phi _j}$ the photon population density of mode $j$, $i \ne j$). The final, successful model is thus
The following parameters are fixed: $\sigma$ and ${\sigma _g}$ are, respectively, the laser’s stimulated emission and saturable absorber ground state absorption cross sections; ${\gamma _{20}}$, ${\gamma _{21}}$, ${\gamma _{10}}$ are the decay rates between, respectively, the upper laser and ground level, upper and lower laser level, and lower and ground laser level; $l$ and ${l_s}$ are the lengths of the active medium and the saturable absorber, respectively; ${\gamma _s}$ is the decay rate of the saturable absorber; $c$ is the speed of light in vacuum; ${C_i}$ is the coupling coefficient of the $i$th mode to the pump; ${t_r}$ is the time of round trip; $k$ is the ratio of each mode’s area in the active medium to the area in the saturable absorber; ${c_{{ij}}}$ is the coupling coefficient of the $j$th field mode to the spatial distribution of the $i$th mode and is estimated as the fraction of the area of the latter occupied by the former; $R$ is the reflectivity of the output coupler; ${L_i}$ is the linear loss of the $i$th mode; $\Omega$ is the fraction of spontaneous emission that remains in the cavity; ${W_{{th}}}$ is the threshold pump; ${N_s}_0$ is the initial population of the saturable absorber (be aware that this is precisely set as the initial value of every ${N_{{s_i}}}$, i.e., ${N_{{s_i}}}(t = 0) = {N_{{s_0}}}$). The values for each parameter are given in Supplement 1.
I present the main results yielded by this model, including the effective prediction of extreme events, in the next section.
4. RESULTS
For a set of experimentally feasible parameters ${w_p}$ and ${C_i}$ and for a progressively increasing number of transverse modes up to five, I integrate model 1 as mentioned previously, rendering the corresponding bifurcation diagrams with $k$ as the control parameter (see Supplement 1 for details). It is important to note that this change in $k$ is not dynamic, i.e., the followed procedure is run the simulation for a specific value of $k$, extract the values of the peak intensities for such $k$, change $k$, run the simulation again with the same initial conditions as used for the previous value of $k$, and so on. For each bifurcation diagram, I perform an analysis similar to that in Fig. 2 in search of extreme events. For the sets of parameters explored, I do not find these events until I incorporate five transverse modes into the model. Simulations with more than five transverse modes have also yielded extreme events.
As an example, I present a case of a simulation with five modes exhibiting extreme events for a particular range of the control parameter in Fig. 3. The upper graph shows the bifurcation diagram with fixed pump parameter ${w_p}$ and spanning $k$ in an interval from $k = 1$ to $k = 2$, i.e., for each value of $k$, I plot the peak value for every pulse of the corresponding simulated trace. As is typical of many dynamical systems, chaotic regions alternate with periodic windows. I show the analysis in search of extreme events in the lower graph, which is performed in the same manner as that in Fig. 2 (note that in the latter case, the peak intensity value is normalized: this normalization is specific for each $k$, so the relation between the maximum peak values corresponding to different $k$s might differ from the one not normalized); it can be seen that in an interval spanning approximately from $k = 1.4$ to $k = 1.6$, the maximum peak values of the corresponding traces are consistently above the extreme event threshold. (It must be noted that there is another region for which pulses above the extreme event threshold occur, approximately from $k = 1.1$ to $k = 1.2$. However, a close inspection of the traces corresponding to this region reveals that they are not a proper regime of $Q$-switch operation and, hence, are outside the scope of this work.)
Fig. 3. Upper: bifurcation diagram for an integration of model 1 with five transverse modes. Lower: corresponding extreme event analysis showing for each $k$ the normalized intensity of the highest peak (blue dots) and the extreme event threshold (red line).
As a specific example of a simulated trace with extreme events, I show in Fig. 4 one achieved for the value of $k = 1.49$, which is well within the aforementioned region. The plot exhibits a fraction of the trace centered at an extreme event. It evidences the irregular nature of the dynamics involved for this value of the control parameter. In this regard, I performed an analysis of false neighbors, finding that the dimension of embedding is six for this series, and that they have two positive Lyapunov spectrum exponents. These results imply that the dynamics at this point is ruled by deterministic chaos and is consistent with the values found in the experimental series studied in [20].
Fig. 4. Fraction of a simulated time trace for an integration of model 1 with five transverse modes exhibiting extreme events. The plot corresponds to the bifurcation diagram in Fig. 3 for the specific value $k = 1.49$, and thus, the other parameters are the ones reported there. The plot is centered on an extreme event. The horizontal dashed red line shows the extreme event threshold.
I study the statistics of this specific simulated trace in Fig. 5, where I show the histogram of peak intensities for every pulse in that trace (note the logarithmic scale). The kurtosis of the distribution is 5.55 (well above the value of normal distribution). The horizontal dashed line indicates the extreme event threshold: the counts above it correspond to extreme events, of which there are 50 in a total of 10,599 pulses. The frequency of appearance of extreme events is consistent with the one achieved experimentally.
Fig. 5. Histogram for a simulated trace with extreme events (same as in Fig. 4). The vertical dashed red line indicates the extreme event threshold.
Fig. 6. Left: superposition of all the extreme events (total of 50) and their immediate surroundings (400 µs before and after the extreme event) corresponding to the time trace shown in Fig. 4; in the graphic, each extreme event peak is centered (plotted at $t = 0\;\unicode {x00B5}{\rm s}$). Right: superposition of the same quantity of next-to-average pulses and their surroundings (next-to-average peaks are plotted at $t = 0\;\unicode {x00B5}{\rm s}$).
In Fig. 6, I analyze an observed feature specific to the extreme events: the regularity of the dynamics in their vicinity. For this I plot in the left graphic the superposition of each extreme event (and their surroundings) registered in the same simulation as in Figs. 4 and 5, while in the figure on the right, I plot the superposition of an equal number of next-to-average pulses and their surroundings. To be specific, the left plot shows all the extreme events registered in that trace (a total of 50) besides their immediate vicinity (i.e., the intensity registered for 400 µs before and after the extreme event) in a way that every extreme event peak is plotted at $t = 0\;\unicode {x00B5}{\rm s}$. Then, for comparison, in the figure on the right, I perform the same analysis, but instead of taking the extreme events of that run, I choose the same quantity of pulses closest to the average value, and center the graphic on those pulses (thus, at $t = 0\;\unicode {x00B5}{\rm s}$, I plot 50 next-to-average peaks).
In the left graphic, it can be seen that the time interval between an extreme event and the first or second pulses preceding or following it is defined within a fairly narrow interval (and thus, given the time of occurrence of an extreme event, one could determine the time of occurrence of the neighboring pulses). The window of occurrence for later or previous pulses progressively blurs, but at least in the case of the pulses happening before the extreme event, the time of occurrence is clearly defined for up to two pulses. The figure on the right shows instead that the blur for next-to-average pulses happens almost immediately before and after the pulse: i.e., there is not a regular behavior, even in the immediate vicinity of the pulse, and thus one could not predict the time of occurrence of neighboring pulses despite knowing when the next-to-average pulse happens. The behavior shown in the left plot, which is an indication that the trajectories in phase space yielding extreme events are limited to relatively bound manifolds, is a signature of the extreme events observed in both the system described here (see Figs. 7 in [27] and 6 in [28]) as well as in others, such as a model of a loss-modulated ${{\rm CO}_2}$ laser [36], a model of two ${{\rm CO}_2}$ lasers mutually coupled via their saturable absorbers [37], in optically injected semiconductor lasers [10], and in an all-solid-state laser with loss modulation [16].
It is important to mention that for the sets of parameters that yield extreme events, I test the relevance of the number of modes: namely, I perform new integrations of the model for the same interval of $k$ with the same parameters but “switching off” successively each one of the modes involved (thus incorporating four modes in each case). The integrations do not yield extreme events for any of these cases.
5. CONCLUSION
My main goal in this work is to achieve a proper, yet as simple as possible, modelization of extreme events that occur in the all-solid-state self-pulsed laser, specifically a ${\rm Nd}{:}{{\rm YVO}_4} + {\rm Cr}{:}{\rm YAG}$ system. Previous models based on a single mode approach described many of the dynamics observed in the experiments but failed to predict extreme events.
Previous experimental observations have shown that in this system, extreme events appear in chaotic regimes of operation, with dimension of embedding ${\gtrsim} 6$ and a few transverse modes (typically five transverse modes suffice to describe 99% of the pulses happening in a given trace). Based on these results, I set out to sketch a model able to reproduce the existence of extreme events.
To do this, I follow a way of progressively increasing complexity, in each step taking into account additional effects into an initial model [32] and testing whether the resulting one yields extreme events. This is repeated until the final model, which incorporates the effects of the spatial distribution and the interaction of modes in the saturable absorber, exhibits extreme events that appear in chaotic traces with a dimension of embedding ${\simeq} 6$. This is compatible with experimental observations. Extreme events are found only after a minimum number of modes is taken into account. After a thorough scan of the space of parameters around the values corresponding to the actual laser, this number turns out to be five. This too is consistent with experimental observations. It must be warned that this number may be different (but not expected to be too different) for different laser parameters.
One of the main interests in this study is the control of extreme events in the system, which has practical applications. The model presented here, which successfully reproduces extreme events yet keeps relative simplicity compared with the alternative approach of a partial differential equation, lays a useful groundwork to study ways to predict, control, and, hence, optimize the production of these events.
In summary, the model presented in this paper is able to reproduce the following observed features of the dynamics in a solid-state laser with a saturable absorber: the presence of chaos and extreme events, with values of dimension of embedding, positive Lyapunov spectrum exponents, and number of modes and pulse distributions consistent with the observed ones. Also, the dynamic changes as the control parameter is varied, and the regularity in the neighborhood of the extreme events. It is checked that it is the simplest model able to reproduce all these features. The key is the interaction among these modes not only in the active medium, but also in the saturable absorber.
Funding
Agence Nationale De La Recherche (ANR) (Project OCTIROC); Air Force Office of Scientific Research (AFOSR) (FA9550-13-1-0120); Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET) (PIP2011-077).
Acknowledgment
I thank Dr. Alejandro Hnilo, Dr. Jorge Tredicce, and Dr. Marcelo Kovalsky for the fruitful discussions and contributions to this paper.
Disclosures
The author declares no conflicts of interest.
Data Availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the author upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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