Chaotic Dynamics of Spatial Optical Rogue Waves in SBN Crystals

Abstract

Under strong nonlinearity, the propagation of perturbed Gaussian beams in SBN crystals exhibits two distinct dynamical stages. In the first stage, dominated by the screening nonlinear effect, a rapid modulational instability process occurs, leading to energy redistribution. The Gaussian beam undergoes filamentation, exhibiting statistical properties characteristic of rogue waves. The second stage involves a long-term, slow-varying process of the nonlinear output light field distribution governed by the combined effects of diffusion and screening nonlinearity. It was discovered that the temporal evolution of the degree of correlation between neighboring slow-varying output light spots exhibits chaotic characteristics, which are confirmed by a positive Lyapunov exponent and a chaotic-featured spectrum. Numerical simulation results agree well with experimental observations. These findings reveal a certain intrinsic connection between chaotic dynamics and rogue waves as two distinct nonlinear phenomena.

1. Introduction

Chaos is a pervasive form of motion that manifests across diverse scientific domains, including biology [1], physics [2,3,4], meteorology [5], and finance [6,7]. In optical systems, chaos is the first associated with unstable output of laser light [8,9,10]. The first experimental observations of optical chaos were reported by Laura L. Jones and Ronald Roy in 1984 demonstrated that introducing noise or small disturbances into a laser system could induce irregular intensity fluctuations and chaotic behavior at the laser output [11]. In recent years, chaos has been extensively studied in other nonlinear systems [12,13,14]. In 2021, Xin observed optical chaos after the collision of three solitons in a saturated nonlinear medium [15]. In another study on soliton collisions, it was found that collisions involving multiple solitons with specific energies and quantities often lead to energy exchange, accompanied by the generation of rogue waves (RWs). Chen demonstrated that perturbed wide Gaussian beams (Gb) can exhibit RWs under suitable nonlinear conditions in saturated media [16], while Li showed the probability of such RWs exhibiting chaotic behavior under different nonlinear conditions [17]. Previous research has primarily focused on steady nonlinear states such as solitons. However, our findings indicate that under strong nonlinearity, the evolution of light does not reach a steady state.

In this study, we investigate the temporal evolution of the light-intensity distribution at the exit surface of a saturated nonlinear medium under strong nonlinearity. The long-term dynamics within the medium are characterized by analyzing the Lyapunov exponent and power spectra, revealing the fundamental patterns of its evolution.

2. Experiment

Figure 1 describes our experimental setup. The SBN crystal employed in our experiment featured a cross-sectional area measuring 5 mm × 5 mm and an overall length of 10 mm. The incoming light is a laser at a wavelength of 532 nm, polarized extraordinarily along the direction of the applied voltage. The full width of the beam at half maximum (FWHM) is approximately 100 $\mathsf{\mu }$m, and the incident light has an approximate input power of 20 $\mathsf{\mu }$W. The 1% perturbation applied to it is provided by the amplitude mask.

In previous studies [17], it was found that under 500 V, 800 V, and even 1000 V, the light eventually evolves to a steady state and exhibits statistical properties characteristic of rogue waves. However, at 1500 V, the light cannot stabilize and instead evolves through two distinct stages. The results are shown in Figure 2, where the white line is the normal direction to the initial spot center, and the green line represents the connection line passing through the centers of the various spots. Process 1 (0–40 s, Figure 2) is dominated by screening nonlinearity, which triggers rapid modulational instability. The beam rapidly splits into dispersed, high-intensity localized spots. At this stage, the white and green lines basically overlap. With evolution, the light field gradient becomes larger at the end of process 1, triggering a strong diffusion effect. Process 2 is jointly modulated by the screening nonlinear effect and the slower-reacting diffusion effect, resulting in a downward bending of the spot center by approximately 50 $\mathsf{\mu }$m after 200 s. In process 2, the filaments with varying light energy gradients exhibit different degrees of downward bending. These differences promote interactions among the filaments, leading to a persistently complex and unstable evolution of the optical field.

As can be observed in Figure 3, the probability of RWs exhibits a complex variation characterized by an initial increase, followed by a decrease, and then another increase. Figure 3b clearly illustrates that the probability of RWs undergoes a slow and continuous change, with relatively stable fluctuations in process 2. However, from the actual image in Figure 2, the pattern of spot distribution is obviously changed. Therefore, to investigate the evolution of process 2 in the most intuitive way, the correlation values between neighboring images should be examined [18]. The correlation is defined as follows, where $R_{fg}$ denotes the correlation value of the two neighboring images, and $f(x,y)$ and $g(x,y)$ are the light intensity distributions of the two neighboring images, respectively.

$$R_{fg}=\underset{s}{\int \int }f(x,y)g(x,y)dxdy$$

(1)

In the experimental study, we selected 200 images at 1-second intervals from the 40th second onward. Figure 4 illustrates the evolution of correlation values between neighboring images over time, which can be seen as irregular variations, typically involving two situations: random or chaotic. So, we will discuss whether the system exhibits chaotic properties.

Chaos is a behavior exhibited by deterministic dynamical systems due to their sensitivity to initial conditions, leading to long-term unpredictability. A system is considered chaotic if its largest Lyapunov exponent is greater than zero.

To compute the Lyapunov exponent, we need to reconstruct the experimental data in phase space. The time delay and embedding dimension [19] are determined using autocorrelation and false nearest neighbor methods, respectively. The calculated time delay is denoted by $T$, with $T=2$ and the calculated embedding dimension is denoted by $m$, with $m=8$. Based on the time delay and the embedding dimension, we can then compute the Lyapunov exponent of the system in phase space. In Figure 5, the Lyapunov exponent is indicated by ${\lambda }_i$, with $i$ ranging from 0 to $N$, where $N$ represents the total number of grid points [20]. The spectrum exhibits a continuous range of positive values, indicating that the temporal evolution of the exit surface possesses temporal chaotic characteristics.

The results can also be verified from the perspective of the power spectrum [21]. Chaos is a localized quasirandom system with both periodic and completely random motions. The power spectrum of the system is expected to exhibit a central peak accompanied by continuous low-intensity subsidiary peaks. The power spectrum of the random event shows a flat frequency response with no significant peaks. The power spectrum $S\left(w\right)$ is defined as follows:

$$X\left(w\right)={\sum }_{n=0}^{N-1}x\left(n\right)e^{-iwn}$$

(2)

$$S\left(w\right)={\displaystyle \frac{1}{N}}{\left|X\left(w\right)\right|}^2$$

(3)

Here, $x\left(n\right)$ is the correlation value sequence. The length of the sequence is denoted by $N$.

Figure 6 shows the power spectrum of the sequence of correlation values in the experiment. A high-power primary peak and several low-power secondary peaks can be seen. Based on the characteristics of the power spectrum, it can be determined that the evolution of the system exhibits chaotic characteristics.

3. Simulation Analysis

To corroborate the experimental findings, we conducted a numerical simulation of the second stage of beam propagation under strong nonlinearity in SBN crystals using a distributed beam propagation method(BPM). In the SBN crystal, the Gaussian beam propagation is governed by the nonlinear Schrödinger equation:

$$i{\displaystyle \frac{\partial }{\partial \left(z\right)}}\psi (x,y,z)+{\displaystyle \frac{1}{2k_0}}{\nabla }_{\perp }^2\psi (x,y,z)=-\Delta n{\displaystyle \frac{k_0}{n_e}}\psi (x,y,z)$$

(4)

where ${\nabla }_{\perp }^2$ is the Laplace operator, $k_0=2\pi /\lambda$ is the wave vector, with $\lambda$ being the wavelength of the light. The envelope of the slowly varying light wave is represented by $\psi (x,y,z)$, $z$ denotes the propagation distance. The refractive index for the extraordinary polarized beam is denoted as $n_e$. The beam induces a refractive index change $\Delta n$ in the SBN crystal, which can be expressed as follows:

$$\Delta n={\displaystyle \frac{1}{2}}{n}_e^3{\gamma }_{33}E$$

(5)

where ${\gamma }_{33}$ is the electro-optical coefficient. The expression for E can be formulated as follows:

$$E=E_0{\displaystyle \frac{I_d}{1+I_d}}-{\displaystyle \frac{K_{B}T}{e}}{\displaystyle \frac{\partial I/\partial x}{I+I_d}}$$

(6)

Here,$E_0$ is the external electric field,$I_d$ represents the intensity of dark irradiation, $I$ denotes the intensity of light irradiation, and $e$ stands for the electron charge. The product of the Boltzmann constant and the absolute temperature is denoted by $K_{B}T$. The initial term on the right-hand side of Equation (6) signifies the screening nonlinearity effect, and the second term represents the diffusion effect.

It can be seen from Equation (4) that the propagation equation in the simulation is non-temporal. Therefore, we need to rephrase the propagation equation to simulate temporal propagation. The specific approach is as follows: the beam enters the crystal, and the saturated nonlinearity effect causes changes in the refractive index within the crystal, leaving traces of refractive index perturbations. Subsequent incident beams propagating through the crystal are not only affected by the saturated nonlinearity but also influenced by the preceding refractive index traces. In this way, the simulation achieves temporal propagation.

The full width at half-maximum of the incident Gaussian beam was set to be 100 $\mathsf{\mu }$m, and a noise seed was added to the incident light to simulate perturbations in the incident light, the noise intensity set at 1% of the amplitude of the incident light at the corresponding position. The propagation length is 10 mm, and the remaining parameters are the same as in the experiment. The voltage applied is set at 1500 V.

In the simulation, 200 iterations were performed, as shown in Figure 7a. Similarly, the simulation data were reconstructed in phase space. The time delay and embedding dimension were calculated using the autocorrelation method and the false nearest neighbor method, respectively, with the results of $T=1$, $M=9$. Thereafter, the Lyapunov exponent was calculated in the phase space numerical simulation, as shown in Figure 7b. It was found that the Lyapunov exponents were all greater than zero. The above results were also verified using the power spectrum method, as shown in Figure 7c. The power spectrum is characterized by a central peak and continuous low-intensity subpeaks, which is consistent with a chaotic power spectrum.

In summary, we numerically simulate the evolution of the correlation between exit surfaces over time and find that the fluctuations in a relatively large range are similar to the experimental results. Moreover, by comparing the Lyapunov exponent spectra obtained from both experiments and simulations, we identified a similar downward trend. The power spectra derived from the simulations and experiments also exhibited analogous variation trends.

4. Conclusions

By means of experimental statistics and numerical simulations, we discovered the chaotic temporal evolution of the exit surface intensity distribution of a perturbed Gaussian beam in a saturated nonlinear crystal under strong nonlinearity. The evolutionary process is affected by both the screening nonlinear effect and the slower diffusion effect together. Cross-correlation values are used to describe the correlation between adjacent exit surfaces and to study the changes in these values over time. By observing the Lyapunov exponent greater than zero and the power spectrum exhibiting a central peak and continuous sub-peaks, we demonstrated the chaotic nature of the variation in exit surface intensity distribution. These results reveal the complexity of beam behavior in photorefractive saturated nonlinear systems.