- Chinese Optics Letters
- Vol. 22, Issue 7, 071903 (2024)

Abstract

1. Introduction

In 2007, Solli *et al.* observed optical rogue waves (RWs) during the investigation of supercontinuum in optical fibers^{[1]}. Optical RWs were statistically associated with long-tail distributions, which has been studied in several optical systems, such as fibers^{[2-4]}, optical waveguides^{[5,6]}, and gratings^{[7-9]}. Generally, RWs are generated by a nonlinear process known as modulation instability (MI)^{[10-13]}, which is a complex process and highly sensitive to initial conditions. However, most previous research focused on temporal RWs, with considerably less emphasis on spatial RWs. Research on spatial RWs has been increasing in recent years. In biased strontium barium niobate (SBN) crystals, RWs were induced by a narrow Gaussian beam^{[14]}. It was discovered that the probability of RW occurrence increased with the applied electric field. Chen *et al.* found that a broad Gaussian beam with perturbations also generated RWs in biased SBN crystals^{[15]}. However, the probability distribution of RWs displayed a breathing behavior. Most RWs are generated by MI, which has been also studied in periodic structures. In pyroelectric crystals^{[16]}, the increased rates of MI were influenced by the external biased field. In magnonic crystals^{[17]}, MI was observed only at the frequencies corresponding to the band gaps of the crystal. Optical lattices enable precise control and adjustment of the light field, establishing them as crucial tools for the investigation of optical phenomena. Gomila *et al.* reported that the band gap in an optical lattice may hinder the occurrence of MI^{[18]}. Li *et al.* demonstrated that a plane wave with perturbations could generate PS solitons with RW waveforms in the periodic potential^{[19]}. Rivas *et al.* reported on the first experiments of RWs in one-dimensional (1D) disordered photonic lattices^{[20]}. However, the experiments and statistical results of extreme events in two-dimensional (2D) optical lattice were not investigated yet.

In this paper, we conduct experimental investigations on high-amplitude events within a 2D optical-induced lattice^{[21]} in biased SBN crystal. Our experiment is carried out by injecting a noise-seeded broad Gaussian beam into the lattice. Through the analysis of optical patterns on the output plane of SBN crystal, we examine the transport of the probe beam under both linear and nonlinear conditions. Furthermore, we calculate the probability of RWs and compare it with that in a homogeneous medium.

2. Experiment

The experimental setup is shown in Fig. 1(a). The wavelength of the laser beam is 532 nm. It is split into two beams by BS1. One beam is employed as the 2D lattice-induced light. The power of induced light is 530 µW, and it is ordinarily polarized. A 2D optical diamond lattice [shown in Fig. 1(c)] with a period of 23 µm is established by sending the induced light through a mask. The lattice remains nearly invariant as it traverses the SBN crystal with cross sectional dimensions of $5\text{\hspace{0.17em}}\mathrm{mm}\times 5\text{\hspace{0.17em}}\mathrm{mm}$ and a length of 10 mm. In addition, the spatial spectrum of the lattice is obtained using the spectrum imaging subsystems demonstrated in Fig. 1(a). When the biased voltage $U<800\text{\hspace{0.17em}}\mathrm{V}$, the first photonic band does not open. As shown in Fig. 1(d), the periodic structure’s first photonic band gap is opened with $U=800\text{\hspace{0.17em}}\mathrm{V}$. When $U>800\text{\hspace{0.17em}}\mathrm{V}$, the diffusion effect in SBN crystal is raised, resulting in increased complexity of the nonlinear processes within the system. A broad Gaussian beam, split from the same laser but extraordinarily polarized, is used as the probe beam. The beam width is approximately 450 µm, as shown in Fig. 1(b). Perturbations applied to it are provided by the amplitude mask (AM). The probe beam is injected into the crystal and propagates collinearly with the lattice. The adjustment of nonlinearity is controlled by the biased voltage. Optical patterns at the output plane of SBN crystal are recorded by the CCD camera.

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Figure 1.(a) Experimental setup for observing RWs; the blue square is used to observe the spectrum of the lattice. BS, beam splitter; AM, amplitude mask; HVPS, high-voltage power supply. (b) Probe light at the front plate of SBN crystal. (c) The induced lattice beams in SBN crystal. (d) The Brillouin zone of the lattice.

As the probe beam propagates in the optical lattice, monitoring of the linear and nonlinear transport is obtained by recording its instantaneous and steady-state output patterns from the lattice. Figures 2(a) and 2(b) demonstrate typical output patterns observed in linear and nonlinear situations ($U=800\text{\hspace{0.17em}}\mathrm{V}$), respectively. In Fig. 2(b), several light waves with significantly high amplitude are visible.

Figure 2.The output patterns of the probe beam in the lattice in (a) linear and (b) nonlinear situations in the experiment. U = 800 V.

According to the definition of space RWs^{[22]}, light beams exceeding twice the effective intensity (${I}_{e}$) can be classified as extreme events (EEs). ${I}_{e}$ is defined as the average of the first third of the waves with the highest intensity. Events with an abnormality index ($\mathrm{AI}=I/{I}_{e}>2$) are considered as RWs^{[23]}. The experiment was repeated 30 times under different initial conditions, and over 7000 waves were used for statistical analysis. A total of 225 waves were identified as RWs. Additionally, we replicated the aforementioned experiment in a homogeneous medium. The probability density function (PDF) of AI in different media is shown in Fig. 3. A gray dashed line at $\mathrm{AI}=2$ is a criterion to determine RWs. In the nonlinear situation, the probability of RWs in the homogeneous medium is about 13%. The PDF curve (red triangles) with long tails is similar to that of the Nakagami-$m$ distribution ($m<0.25$). In this case, extremely high peak beams are generated. The maximum value of AI is 7.9. Nevertheless, the probability of RWs decreases from 13% in the homogeneous medium to 8% in the optical lattice. There is a significant shortening of the long tails in the lattice (blue circles) compared to a homogeneous medium (red triangles). And the maximum value of AI is 3.9. Under a linear condition, no beams exceeding the gray dashed line were observed. We fail to observe RWs (black squares).

Figure 3.The experimental results of the PDF for AI in an optical lattice (blue circles) and in a homogeneous medium (red triangles) under a nonlinear condition, and in an optical lattice under a linear condition (black squares). U = 800 V.

The phenomenon is primarily attributed to the lattice structure. In a linear situation, the probe beam propagates evenly through an array of waveguides [see Fig. 4(a)]. We failed to observe RWs under these conditions [shown in Fig. 3 (black squares)]. However, under the nonlinear condition, the MI generated from perturbations in the probe beam is hindered by the band gap of the lattice. Consequently, the probe beam is limited to only propagating in the first band [Fig. 4(c)]. Under nonlinear effects, RWS is generated within the lattice [shown in Fig. 3 (blue circles)]. In contrast, a homogeneous medium does not possess such restrictions, as displayed in Fig. 4(d). We can observe RWs in these conditions [shown in Fig. 3 (red triangles)]. In optical lattice, filamentary collisions are not as common as in a homogeneous medium due to lattice waveguide limitations. As a result, the probability of RWs in the lattice decreases significantly.

Figure 4.The spectra of the probe beam in (a), (c) an optical lattice and (b), (d) a homogeneous medium under linear and nonlinear conditions, respectively. U = 800 V.

3. Numerical Simulation

The distributed beam propagation method^{[24]} is used in numerical simulations to verify the results generated experimentally. The dimensionless nonlinear Schrödinger equation used in the simulation is as follows:

In Eq. (1), $z$ represents the propagation distance, $\psi $ denotes the slowly varying light wave envelope, $\beta $ signifies the diffraction coefficient, the Laplacian operator is denoted as ${\nabla}_{\perp}^{2}$, and the nonlinear coefficient $\alpha $ is

The first part of Eq. (2) is used to describe the saturation nonlinearity. The nonlinearity is regulated through the adjustment of $\kappa $. Periodic refractive index modulations are provided by $\mathrm{\Delta}\eta $. A broad Gaussian beam with 20% perturbation is employed as the probe light. Similar to the experiment, the nonlinear propagation of light waves in periodic and homogeneous media is investigated in our simulations. Figure 5 illustrates the cross-section of propagation at $\kappa =2$.

Figure 5.The propagation of probe beam in (a) an optical lattice and (b) a homogeneous medium in numerical simulation. κ = 2.

In comparison to a homogeneous medium, light is limited by an array of waveguides while propagating in a lattice, as illustrated in Fig. 5(a). Moreover, there is a significant decline in wave-to-wave collisions in the homogeneous medium, as depicted in Fig. 5(b). The simulation is repeated 1000 times under different initial conditions, and Fig. 6 presents the PDF of AI in two different media.

Figure 6.The PDF of AI in optical lattice (blue circles), in homogeneous medium (red triangles) under a nonlinear condition, and in optical lattice under a linear condition (black squares) in numerical simulation. κ = 2.

Figure 6 corresponds to Fig. 3. The trend of the curves in both figures is roughly the same. In a homogeneous medium, the probability of RWs is 15.8% (red triangles). The maximum value of AI is 7.7. Furthermore, the probability of the RWs in the optical lattice is 6.5% (blue circles). It is also less than that in the homogeneous medium. A shorter long tail in the lattice is displayed. Under a linear condition (black squares), RWs cannot be observed numerically.

4. Conclusion

In summary, RWs are excited by a broad Gaussian beam with perturbations in a 2D optical-induced lattice. Due to the localization and band structure of the lattice, we cannot observe RWs in a linear situation. Conversely, under a nonlinear condition, we observe less probability of RWs and a shorter long tail in an optical lattice compared to that in a homogeneous medium.

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