Controlling nonlinear rogue wave formation using the coherence length of phase noise

Figure 1(a) shows the schematic of our experimental setup. Our saturable nonlinear medium is a cell containing natural abundance rubidium. We heat the cell to 115 °C, and blue detune our laser source by 600 MHz above the ⁸⁷Rb $D_{2}$ $F=1\rightarrow F^{\prime}=2$ transition in order to have a self-focusing nonlinear response. A horizontally polarized beam from our laser source diffracts from a phase grating impressed on a spatial light modulator (SLM1) and forms a Gaussian beam of diameter 2.5 mm ($D_{0}$) in the first diffractive order. We isolate this diffractive order by letting the light propagate over 2 m, and add a conjugate defocus on SLM1 to compensate for the accumulated defocus on the beam. Both SLM1 and SLM2 are liquid-crystal-on-silicon (LCOS) phase only SLMs from Hamamatsu that have identical resolution ($600\times 800$) and pixel size (20 $\mu$m). SLM2 adds a random phase mask with a spatial coherence length $L_{\rm coh}$ and a maximum amplitude of $\pi$ rad onto the beam. To determine the random phase mask, we generate a $600\times 800$ matrix of uniformly distributed random numbers between 0 and 1, and convolve it with a Gaussian point spread function of width $L_{\rm coh}$ [Eq. (4) in appendix B], which acts as a blur on the salt-and-pepper noise matrix Shirai et al. (2005). We then multiply the matrix by $\pi$ so that the maximum phase amplitude of the added phase noise is $\pi$ rad. Limiting the maximum phase amplitude to $\pi$ rad ensures, as we show later, that rogue waves do not develop after purely linear propagation of the beam through the rubidium cell. The lenses L1 and L2 image the active area of SLM2 onto the entrance facet of the rubidium cell. The waveplates before the cell convert the polarization of the beam to left-handed circular to match the handedness of the $\sigma_{+}$ atomic transition. The lens L3 images the output facet of the cell onto the image plane of the camera, which records the intensity at the cell output.

Figures 1(b)-(e) show the recorded output intensity distributions after linear propagation through the cell for representative phase masks with $L_{\rm coh}/D_{0}$ of 0.135, 0.075, 0.045, and 0.015, respectively. For all linear measurements, we increase the value of detuning from 600 MHz to 65.04 GHz and fix input beam power $P_{\rm in}$ to 4 mW. As shown in Figs. 1(b)-(e), the added phase noise leads to distortion of the beam intensity upon linear propagation, but is weak enough that no sharp caustics are formed. As we decrease the $L_{\rm coh}/D_{0}$ of noise (left to right), more “hotspots” are formed in the beam such that the intensity corresponding to the smallest $L_{\rm coh}/D_{0}$ [Fig. 1(e)] starts to resemble a speckle pattern. Figures 1(f)-(i) show the recorded intensities for the same phase masks as in the top panels (b)-(e), but with the nonlinearity turned on by changing the detuning to 600 MHz, and the beam power $P_{\rm in}$ to 90 mW. The nonlinearity sharpens the hotspots formed during linear propagation while preserving their underlying structure Safari et al. (2017).

To quantify the intensity statistics, we record output intensity patterns for an ensemble of 500 random phase masks with the same $L_{\rm coh}$. We acquire these intensity datasets for nonlinear propagation through the cell at various incident beam powers $P_{\rm in}$ (30 mW, 60 mW, 90 mW, and 115 mW) and various $L_{\rm coh}$ values (varied from 50 $\mu$m to 450 $\mu$m). We also record datasets for linear propagation through the cell. The probability distribution of intensities $p(I/\langle I\rangle_{e})$ in these datasets is well described by the following stretched exponential distribution Pierangeli et al. (2015); Safari et al. (2017); Black et al. (2022)

Here, $\langle I\rangle_{e}$ is the intensity averaged over the entire dataset, $N$ is the normalization coefficient, $A$ describes the width of the distribution, and $B$ is the stretching coefficient that determines its tail. When $B=1$, the distribution is the usual exponential function associated with a fully developed speckle Goodman (1976). As $B$ becomes smaller than 1, caustic formation and rogue wave behavior are more likely to occur, and the intensity statistics become more long tailed. To estimate $B$ for each dataset, we fit Eq. (1) to the tails of the respective intensity histograms using maximum likelihood estimation (MLE) and use Monte-Carlo simulations to obtain the uncertainties of the fit parameters.

Figure 2(a) shows the measured intensity statistics along with their respective MLE fit for $P_{\rm in}$ of 90 mW, and $L_{\rm coh}/D_{0}$ of 0.135 (blue circles and solid line), 0.075 (red diamonds and dot-dashed line), 0.045 (green squares and dashed line), and 0.015 (purple triangles and dotted line). We note that phase noise of smaller $L_{\rm coh}/D_{0}$ has a wider angular spectral bandwidth [see Fig. B1 in appendix B]. This broadband noise seed should cause further broadening of the angular spectrum of the beam through four-wave mixing and lead to sharper caustics and longer-tailed intensity statistics. However, we do not observe a monotonic increase in the ‘tailiness’ of intensity statistics as $L_{\rm coh}/D_{0}$ is reduced in Fig. 2(a), which is also reflected in the associated values of $B$ given in the legend. Instead, $B$ is minimized for $L_{\rm coh}/D_{0}$ of 0.075, and its distribution is the most long tailed.

Figure 2(b) shows the variation of $B$ with $L_{\rm coh}/D_{0}$ for linear measurements (black triangles) and nonlinear measurements with $P_{\rm in}$ of 30 mW (blue circles), 60 mW (red squares), 90 mW (green diamonds), and 115 mW (purple triangles). The shaded gray region represents the range of $B$ for which rogue wave behavior is likely. For linear measurements, we observe that $B$ is larger than 1 for most $L_{\rm coh}/D_{0}$, and we do not see evidence of either caustic or speckle formation. This result is in agreement with the fact that our maximum phase amplitude is $\pi$ rad and hence too low to form either linear caustics Safari et al. (2017) or a fully developed speckle for which the total phase excursion by the scattering induced random walk should be at least $2\pi$ rad Goodman (1976, 2007). The value of $B$ for nonlinear measurements is smaller than $B$ for linear measurements for all $L_{\rm coh}/D_{0}$, which is consistent with the aforementioned increase in sharpness of caustics due to nonlinearity. The noteworthy feature, however, is that for nonlinear measurements, $B$ is significantly more sensitive to the beam power $P_{\rm in}$ when $L_{\rm coh}/D_{0}$ is larger than 0.075 than it is for smaller $L_{\rm coh}/D_{0}$. We further explore this reduced sensitivity to nonlinearity of rogue wave formation for beams with broadband phase noise through numerical simulations.

The propagation of a field $\bm{E}(\bm{r},t)=E(x,y)e^{i(kz-\omega t)}\hat{\bm{e}_{L}}+\text{c.c.}$ through a spatially extended nonlinear medium, such as our rubidium cell, can be described by the (2+1)-D nonlinear Schrödinger equation (NLSE) Boyd (2020) given below

where $E(x,y)$ is the field envelope, $\omega$ is the angular frequency of the laser, $k$ is the wave number, $\nabla^{2}_{\perp}=\partial^{2}/\partial x^{2}+\partial^{2}/\partial y^{2}$ is the transverse Laplacian, $P=\epsilon_{0}\chi(E)E$ is the atomic polarization, and $\chi(E)$ is the total atomic susceptibility that includes the linear as well as all orders of nonlinear response Boyd (2020). In our calculation of total susceptibility, we include the contributions from all the D2 transitions of rubidium. See appendix A for more details. We use the split-step Fourier method Agrawal (2013) to solve Eq. (2), and obtain the field at any location $(x,y,z)$ within the rubidium cell. We use Fresnel propagation for all linear propagation calculations Goodman (2005). For all simulations, we assume a transverse resolution of 2048 $\times$ 2048 pixels, a pixel size of 4.89 $\mu$m, and a longitudinal step size of 0.5 mm.

Figures 3(a)-(d) show the simulated output intensities for the same set of phase masks used in the experiment that were used for the measured output intensities shown in Figs. 1(f)-(i). We also include an amplitude mask on the Gaussian beam to match the intensity of the Gaussian beam in our experiment [see Fig. C1(a)]. The simulated intensities in Figs. 3(a)-(d), and the measured intensities in Figs. 1(f)-(i) have very similar underlying intensity structures and sharpness of caustic features. Figure 3(e) shows the simulated intensity statistics for $P_{\rm in}$ of 90 mW, and $L_{\rm coh}/D_{0}$ of 0.135 (blue circles), 0.075 (red diamonds), 0.045 (green squares), and 0.015 (purple triangles). We use 200 realizations of random phase masks of a particular $L_{\rm coh}/D_{0}$ to calculate these intensity statistics. The simulated statistics show a good qualitative agreement with the measured statistics shown in Fig. 2(a) for the same set of parameters, and in both scenarios, the histogram corresponding to $L_{\rm coh}/D_{0}$ of 0.075 is the most long tailed. There are several contributing factors in the experiment that could lead to the observed differences between simulations and measurements, such as nonlocality in the nonlinear response of rubidium vapor Suter and Blasberg (1993), aberrations in the imaging optics and the windows of the cell, and the pixel size of SLMs. However, our simplified numerical model agrees reasonably well with the measurements, and can be used to study the propagation dynamics of caustic and rogue wave formation within the cell.

We use the scintillation index $\beta^{2}$ as a metric for the sharpness of caustics, which is defined as follows Gbur (2014); Kravtsov and Orlov (2012)

Here, $\langle\cdots\rangle$ denotes the transverse spatial average over the entire field. For fully developed speckle patterns, $\beta^{2}$ is unity. In contrast, caustic patterns with a very sharp concentration of light have $\beta^{2}$ larger than unity. To understand the interaction between nonlinearity and the grain size of phase noise present on the beam, we monitor the variation of $\beta^{2}$ with propagation distance $z$ over a 15 cm long nonlinear medium for various $P_{\rm in}$ and $L_{\rm coh}$. Figure 4(a) shows the evolution of $\beta^{2}$ for a noisy Gaussian beam with $L_{\rm coh}/D_{0}$ of $0.135$ (blue, solid), 0.075 (red, dot-dashed), 0.045 (green, dashed), and 0.015 (purple, dotted) during linear propagation. The black horizontal dashed line indicates the threshold value of $\beta^{2}$ above which rogue wave behavior is likely. For a specific set of input parameters ($P_{\rm in}$ and $L_{\rm coh}$), we average $\beta^{2}$ at each $z$ over 100 different phase masks. This averaged $\beta^{2}$ is represented by the lines, and the shaded regions around the lines represent its standard deviation. Figures 4(b) and (c) show the evolution of $\beta^{2}$ with $z$ for nonlinear propagation with $P_{\rm in}$ of 90 mW and 180 mW, respectively, and the same set of values of $L_{\rm coh}/D_{0}$ as in Fig. 4(a).

We note that in all of the scenarios shown in Figs. 4(a)-(c), $\beta^{2}$ at first increases with $z$, and then peaks as the phase noise on the beam morphs into intensity distortion. This rate of increase in $\beta^{2}$ depends strongly on the grain size of the phase noise, as well as the nonlinearity. In the absence of nonlinearity, as observed in Fig. 4(a), $\beta^{2}$ peaks when the beam comes to an initial focus along the minima of phase gradients of the added phase mask. When the grain size of the noise is much smaller than the beam diameter (such as when $L_{\rm coh}/D_{0}=0.015$), the phase variations occur over a smaller area within the beam and so the phase gradients are larger and more densely packed [see Fig. C1(b)]. These grains with large phase gradients within the beam come to a sharp focus after some propagation at which point $\beta^{2}$ reaches a maximum. For purely linear propagation, these hotspots then diverge, thereby causing $\beta^{2}$ to decrease with $z$. As the grain size of phase noise becomes larger, the phase gradients decrease in magnitude and become less densely packed [see Fig. C1(c)], which leads to fewer grains within the beam that focus into hotspots at larger $z$.

In the presence of nonlinearity, the hotspots formed after the initial reorganization of the beam continue to self-focus. Hence, $\beta^{2}$ increases beyond unity and maximizes when at least one of the hotspots reaches a full width at half maximum (FWHM) size $\Delta r$ of $25\pm 2.5\mu$m. A Gaussian beam of this FWHM size and an average power of 1.4 mW (say, $P_{\rm cr}$) forms a self-trapped filament that propagates for at least 1.3 cm in the rubidium vapor without any change in its width before diverging due to absorption and diffraction. Filaments of the same width but smaller power than $P_{\rm cr}$ diverge more quickly, while those with power larger than $P_{\rm cr}$ undergo multiple self-focusing and defocusing cycles depending on their power Feit and Fleck (1988). For $P_{\rm in}$ of 90 mW and $L_{\rm coh}/D_{0}$ of 0.015, more than two filaments of size $\Delta r$ are formed when $\beta^{2}$ is maximized such that the power in each filament is smaller than 0.9 mW [see Fig. C1(f)]. In contrast, for $P_{\rm in}$ of 90 mW, and $L_{\rm coh}/D_{0}\geq 0.045$, a single filament of size $\Delta r$ with average power larger than 1 mW is formed when $\beta^{2}$ is maximized [see Fig. C1(k)]. As shown in Fig. 4(b), this sharper intensity contrast between the “rogue” filaments and the background intensity in the beam for $L_{\rm coh}/D_{0}\geq 0.045$ results in a higher peak of $\beta^{2}$ for these cases than when $L_{\rm coh}/D_{0}\leq 0.045$. When $P_{\rm in}$ is increased to 180 mW, the caustics become even sharper, and more filaments of size $\Delta r$ are formed when $\beta^{2}$ is maximized, which as shown in Fig. 4(c) occurs at even smaller $z$ for all cases. For $L_{\rm coh}/D_{0}$ of 0.015 ($\geq 0.045$), the average power in each filament is smaller (larger) than 1.4 mW [see Fig. C2]. Hence, for noisy beams with $L_{\rm coh}/D_{0}\geq 0.045$, the propagation after the initial peak of $\beta^{2}$ is followed by another cycle of self-focusing of filaments and subsequently, by diffraction. Nevertheless, even at such large beam powers, the small-grained phase noise seeds the formation of several filaments each containing less than $P_{\rm cr}$ power. This phenomenon limits the maximum intensity in a rogue feature and the tailiness of the intensity statistics.

In summary, we have shown that the grain size of phase noise on a laser beam can be used to control rogue wave formation in media with a self-focusing nonlinearity. The likelihood of rogue wave formation is minimally affected by nonlinearity when the coherence length of phase noise is much smaller than the beam diameter. Our numerical simulations show that small-grained phase noise causes the beam power to be redistributed into multiple filaments rather than a single filament, which is formed when the phase noise has a longer correlation length. This redistribution of beam power into several filaments of smaller intensity limits the maximum intensity in rogue features relative to the background. Understanding the role of nonlinearity in amplifying the phase noise-induced intensity fluctuations on a field could be helpful in devising efficient mechanisms to mitigate these fluctuations for intense structured light propagating through a turbulent medium Cox et al. (2020); Watkins et al. (2020), and developing efficient radiance limiters using saturable nonlinear media Schweinsberg et al. (2011).