Extreme events and single-pulse spatial patterns observed in a self-pulsing all-solid-state laser

The passively $Q$-switched, self-pulsing all-solid-state laser is a device of widespread use in many applications. Depending on the condition of saturation of the absorber, which is easy to adjust, different dynamical regimes are observed: continuous-wave emission, stable oscillations, period doubling bifurcations, chaos, and, within some chaotic regimes, extreme events (EEs) in the form of pulses of extraordinary intensity. These pulses are sometimes called “dissipative optical rogue waves.” The mechanism of their formation in this laser is unknown. Previous observations suggest they are caused by the interaction of a few transverse modes. Here we report a direct observation of the pulse-to-pulse evolution of the transverse pattern. In the periodical regimes, sequences of intensities are correlated with sequences of patterns. In the chaotic ones, a few different patterns alternate, and the EEs are related with even fewer ones. In addition, the series of patterns and the pulse intensities before and after an EE are markedly repetitive. These observations demonstrate that EEs follow a deterministic evolution, and that they can appear even in a system with few interacting modes. This information plays a crucial role for the development of a mathematical description of EEs in this laser. This would allow managing the formation of EE through control of chaos, which is of both academic and practical interest (laser rangefinder).

Figure 1

Sketch of the setup. LD: pump laser diode, 2 W cw at 808 nm; GL: GRIN lens; ND: $\mathrm{Nd}:\mathrm{YV}{\mathrm{O}}_4$ slab $3\times 3\times 1\mathrm{m}{\mathrm{m}}^3$ 1% doped; M1: folding mirror (HR, $R=10\mathrm{cm}$); M2: output mirror (reflectivity 98%, plane); SA: Cr:YAG crystal, transmission (unbleached) 90%; $\alpha ={20}^{\circ }$; distance $\mathrm{ND}-\mathrm{M}1=13\mathrm{cm}$; $\mathrm{M}1-\mathrm{M}2=7\mathrm{cm}$; the position X is adjusted to get different dynamical regimes; FPD: fast photodiode connected to a digital memory oscilloscope; UFC: ultrafast camera recording single pulse spot images; the screen is translucent. By unblocking the laser beam between M2 and the beam splitter, the camera and the oscilloscope, which are both set in the autotrigger mode, start recording at the same time. Yet, the intensities and images series rapidly get out of synchronism, so that a special software code had to be developed to determine what image corresponds to each pulse.

Figure 2

Pulse intensity and corresponding patterns for a period-3 regime. Note the successive labels of the camera shots (from −516 to −510). As an illustration of a typical error, we show here a case where the camera fired between two successive pulses and is an empty image (frame −511, not shown).

Figure 3

Histogram of peak pulse intensities (run 2b9/29/16, 27779 pulses). The horizontal axis is normalized such that the average intensity = 100. The threshold value for EE is 161 (vertical line); there are 64 EE, $d_E=8$, two positive Lyapunov exponents, and $K=5.1$.

Figure 4

Sample zoom of the oscilloscope trace for run 2b9/29/16. Note the EE just before time = 517 ms.

Figure 5

Types of patterns observed in run 2b9/29/16. Their relative frequency is indicated. Probably, $\mathrm{F}=\mathrm{C}+\mathrm{D}$ and $\mathrm{H}=\mathrm{C}+\mathrm{E}$. The spot observed with a standard camera is shown as a reference.

Figure 6

An example of the sequence eabaEabae, which is the most frequent one in the set of EEs. It is, in addition, curiously symmetrical in time. The EE is underlined (frame 12322). See movies of this run and the one in Fig. 2 in Ref. [21].

Figure 7

Superposition of the 14 oscilloscope traces of the sequence eabaEabae. The horizontal line indicates the EE threshold.