Generating electromagnetic nonuniformly correlated beams

Compared to uniformly correlated or Schell-model partially coherent sources, nonuniformly correlated (NUC) sources are a relatively recent and lightly researched subject. NUC sources have spatially varying correlation properties, giving them fascinating propagation or beam characteristics beyond those of Schell-model sources. Originally introduced by Lajunen and Saastamoinen in 2011 [1] and generalized (vectorized) shortly thereafter [2], NUC sources have since been shown to outperform—in terms of the first and second moments of intensity—both coherent Gaussian and Gaussian Schell-model (GSM) beams when propagated through turbulent media [3].

This has motivated more recent work in synthesizing NUC beams for use in applications such as free-space/underwater optical communications [4–7]. Because of their spatially varying correlation properties, NUC sources are inherently more difficult to generate than uniformly correlated sources. Nearly all of the existing NUC synthesis literature concerns scalar NUC sources and, generally, those techniques cannot be applied to synthesize vector or electromagnetic nonuniformly correlated (ENUC) beams. Only one Letter (Ref. [7]) known to the authors has synthesized ENUC sources, and the results were limited for two main reasons: (1) the ENUC field realizations were synthesized using Cholesky decomposition—a computationally onerous procedure in terms of both memory and processing power, and (2) there was limited control over the cross-polarization correlation functions ($W_{xy}$ and $W_{yx}$) of the resulting ENUC cross-spectral density (CSD) matrix $\underset{\_}{\mathbf{W}}$.

In this Letter, we address both of these limitations and successfully demonstrate how to generate ENUC beams using a spatial-light-modulator (SLM)-based system such as that described in Ref. [7]. We start by presenting how to transform GSM field instances into NUC instances with a simple nonlinear transform [8]. GSM beams, being uniformly correlated, can be synthesized by filtering circular complex Gaussian random numbers (CCGRNs) [9]. The filtering is typically performed in the frequency domain using fast Fourier transforms (FFTs). Being able to take GSM field instances and transform them into NUC realizations, thereby avoiding computing Cholesky factors, is the main contribution of this Letter.

Next, we show how to control $W_{xy}$ and $W_{yx}$ by adapting a recently published technique designed specifically for Schell-model sources [10]. From this analysis, we find the range of physically synthesizable, as opposed to physically realizable [2], ENUC sources. Lastly, we present the simulation results where we generate an ENUC source with desired parameters. We compare the simulated results to the theoretical predictions to validate our method. We conclude with a brief summary of this Letter and a list of potential applications.

We begin with the CSD matrix $\underset{\_}{\mathbf{W}}$ of an ENUC source [2]:

We desire to generate ENUC stochastic field realizations of the form

We note that $T_x$ and $T_y$ are statistically inhomogeneous and, therefore, difficult to synthesize. In Ref. [7], $T_x$ and $T_y$ were generated using a Cholesky factorization, which is computationally prohibitive, i.e., $O(n^3)$ complexity to compute the Cholesky factor and then $O(n^2)$ (matrix-vector products) to synthesize $T_x$ and $T_y$. For example, assume that the desired $T_x$ and $T_y$ are $512\times 512$ grids, which is a common size for commercially available SLMs. The covariance matrix in Eq. (3) would be a staggering $2^{20}\times 2^{20}$ matrix, which would require a supercomputer to store and compute the Cholesky factors. In Ref. [7], the authors generated $T_x$ and $T_y$ on much smaller grids and then interpolated the screens to $512\times 512$.

Considering the above example, it clearly would be beneficial to have another method to generate ENUC screens other than Cholesky decomposition. Here we propose to transform electromagnetic Gaussian Schell-model (EGSM) screens, which are statistically homogeneous and can be generated by filtering CCGRNs [9], directly into ENUC screens. To do so, we begin by expressing the EGSM $\underset{\_}{\mathbf{W}}$ as [11]

Comparing the correlation functions in Eqs. (1) and (4), we see at once that

The proposed ENUC screen generation process is best illustrated by example. Here we wish to generate an ENUC $T_{\alpha }$with ${\delta }_{\alpha \alpha }$, ${\sigma }_{\alpha }$, and ${\mathit{\gamma }}_{\alpha }$—we will generalize this to produce a full ENUC source later in the Letter. We start by determining the size of the grid in $\mathit{\rho }$ space. This grid must be large enough to “fit” the ENUC $W_{\alpha \alpha }$. To keep things general, we assume that $\mathit{\rho }\in [-L/2,L/2]$. Equation (5) maps $\mathit{\rho }$ into $\mathit{r}$ space such that $\mathit{r}\in [0,{(\sqrt{2}L/2+|{\mathit{\gamma }}_{\alpha }|)}^2]$, where the $\sqrt{2}$ comes from considering the square $\mathit{\rho }$ grid at its maximum distance from the origin.

Now that we have the grid size in the $\mathit{r}$ plane, we need to determine the grid spacing $\mathrm{\Delta }$. This spacing should be fine enough to capture the variation of $T_{\alpha }$ in $\mathit{r}$ space, which is related to ${\delta }_{\alpha \alpha }$. The required $\mathrm{\Delta }$ has been discussed in many past Letters, and $\mathrm{\Delta }\le {\delta }_{\alpha \alpha }/5$ is sufficient [9]. The number of grid points is $N=\lceil {(\sqrt{2}L/2+|{\mathit{\gamma }}_{\alpha }|)}^2/\mathrm{\Delta }\rceil$.

The next step is to generate an EGSM $T_{\alpha }$. This process has been described in the literature many times [7,9,11]. Here we provide the final result for the reader’s convenience:

 

Fig. 1. ENUC $T_{\alpha }$ generation process. (a) Real part of an EGSM $T_{\alpha }$ on the $\mathit{r}=\widehat{\mathit{x}}\xi +\widehat{\mathit{y}}\eta$ grid, (b) radial slice along $\xi$ axis of the EGSM $T_{\alpha }$ in (a), (c) real part of ENUC $T_{\alpha }$ formed from mapping the $T_{\alpha }$ values in (b) to $\mathit{\rho }$ using Eq. (5), and (d) $y={\gamma }_{\alpha y}$ slice through (c).

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Recall that $\mathit{r}\in [0,{(\sqrt{2}L/2+|{\mathit{\gamma }}_{\alpha }|)}^2]$, and, therefore, we only need a single “radial slice” of an EGSM $T_{\alpha }$ to produce an ENUC $T_{\alpha }$. An example of this is shown in Fig. 1(b), which is the radial slice along the $\xi$ axis of the $T_{\alpha }$ in Fig. 1(a).

Lastly, we map the $T_{\alpha }$ values at $\mathit{r}$ to $\mathit{\rho }$ using Eq. (5). This will likely require interpolation, especially if the points in the $\mathit{\rho }$ grid—such as those in the $\mathit{r}$ grid—are uniformly spaced (as is often the case). Figure 1(c) shows the end result of this process, and the real part of the ENUC $T_{\alpha }$ formed from the EGSM $T_{\alpha }$ in Fig. 1(a).

The ENUC screen [Fig. 1(c)] is rotationally symmetric about the point ${\mathit{\gamma }}_{\alpha }$. Examining Figs. 1(b)–1(d), one can clearly see the effects of the mapping from $\mathit{r}$ to $\mathit{\rho }$: the screen features near $\xi =0$ in (b) map to locations near ${\mathit{\gamma }}_{\alpha }$ in (c). These features are clearly evident in (d), which shows the $y={\gamma }_{\alpha y}$ slice through (c). As a result of the quadratic transformation in Eq. (5), the features near $\xi =0$ in (b) are spatially elongated or stretched [see the region around $x={\gamma }_{\alpha x}$ in (d)]. As we move away from $\xi =0$, the screen features transition from being stretched to compressed, again due to the quadratic mapping in Eq. (5).

The procedure in the preceding paragraphs controls the “self” or diagonal elements of the ENUC $\underset{\_}{\mathbf{W}}$ and, therefore, is sufficient to generate ENUC beams that are linearly and partially polarized. To generate ENUC beams that are circularly or more generally, elliptically polarized, we must precisely control the off-diagonal elements of the ENUC $\underset{\_}{\mathbf{W}}$, namely, $W_{xy}$ and $W_{yx}$.

In Ref. [10], the author controls the full $\underset{\_}{\mathbf{W}}$ of electromagnetic Schell-model sources by using a spatially varying, complex, cross-correlation coefficient $R$ between the random numbers ($r_x$ and $r_y$) that seed $T_x$ and $T_y$. Specialized to the problem here, $R$ takes the form

It is important to check, using the realizability criteria, that indeed $R\in [-1,1]\forall \mathit{f}$. Since we are ultimately interested in producing ENUC sources, we must use the ENUC realizability criteria [2]:

This raises the question: what ENUC ${\delta }_{xy}$ and $|B_{xy}|$ can the technique produce, or, more fundamentally, what are the technique’s synthesizability criteria or limits? Recall that $R\in [-1,1]$ at the lower limit of the fork inequality in Eq. (9). Therefore, this ${\delta }_{xy}$ value sets the lower synthesizability limit. At the upper synthesizability limit, $C\le 1$. Using the expression for $C$ in Eq. (8) and some simple algebra, it is easy to show that

To show how Eq. (10) differs from Eq. (9), we set ${\delta }_{xx}={\delta }_{yy}=\delta$ and plot both inequalities versus $|B_{xy}|$ in Fig. 2. The solid black curve demarks the maximum physically allowed values of ${\delta }_{xy}/\delta -1$ and corresponds to the upper limit in Eq. (9). The values of ${\delta }_{xy}/\delta -1$ above this curve (the solid red region in the figure) are not physically realizable. The dashed black curve denotes the maximum synthesizable values of ${\delta }_{xy}/\delta -1$ and corresponds to the upper limit in Eq. (10). The area below this curve (shaded in green) is the synthesizable region. Between the curves, shaded in yellow, is the region that is physically realizable, but not synthesizable. Although physically permitted, the values of ${\delta }_{xy}/\delta -1$ in this region cannot be generated using the technique presented here. For general ${\delta }_{xx}$ and ${\delta }_{yy}$, the synthesizable region is always inside the realizability volume and, therefore, for a given ${\delta }_{xx}$, ${\delta }_{yy}$, and $|B_{xy}|$, $\max ({\delta }_{xy}^{\text{real}})\ge \max ({\delta }_{xy}^{\mathrm{synth}})$.

 

Fig. 2. ENUC physical realizability [Eq. (9)] versus synthesizability [Eq. (10)] for ${\delta }_{xx}={\delta }_{yy}=\delta$.

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Having derived the synthesizability criteria and having discussed how to generate ENUC $T_x$ and $T_y$ from the corresponding EGSM screens, we now present the simulation results where we synthesize an ENUC source with the parameters listed in Table 1. Note that we chose these values for proof-of-concept purposes only. When designing an ENUC beam for use in a specific application, it is imperative to understand how these parameters affect the beam shape, coherence, and polarization of the propagating ENUC beam. This analysis is presented in Refs. [2,11] and is not included here for brevity. Before presenting the simulation results, we briefly discuss the setup.

Table 1. ENUC Parameters

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As we show in Code 1, Ref. [12], we used grids with $N=1024$ points per side and side lengths $L=10\mathrm{mm}$. The grid spacing in $\mathit{\rho }$ space was $L/N=9.77\mathrm{\mu m}$. These numbers, combined with the ${\mathit{\gamma }}_x$ and ${\mathit{\gamma }}_y$ in Table 1, corresponded to a “side length” in $\mathit{r}$ space of $118.0077{\mathrm{mm}}^2$ and a spacing of $\mathrm{\Delta }=0.1152{\mathrm{mm}}^2$.

We generated correlated EGSM $T_x$ and $T_y$ using Table 1 and Eqs. (6)–(8). We then transformed those EGSM screens to ENUC screens following the procedure described above and shown pictorially in Fig. 1. We lastly produced an ENUC field instance using Eq. (2).

We used 50,000 ENUC field realizations to compute the spectral density, polarization ellipses, and $\underset{\_}{\mathbf{W}}(x_1,0,x_2,0)$. These results are shown in Fig. 3. Overall, the agreement between the simulation and theory is excellent. Note that the visually conspicuous differences between the theoretical and simulated $\mathrm{Im}(W_{xx})$ [Figs. 3(g) and 3(h)] and $\mathrm{Im}(W_{yy})$ [Figs. 3(q) and 3(r)], in fact, are quantitatively small. The color scales of those results are $[-0.005,0.005]$ and $[-0.0098,0.0098]$, which are approximately 200 and 150 times smaller than the corresponding real part results.

 

Fig. 3. ENUC simulation results: subfigures (a) and (b) show the theoretical and simulated spectral densities overlaid with the polarization ellipses. Subfigures (c)–(r) show the elements of $\underset{\_}{\mathbf{W}}(x_1,0,x_2,0)$. Each element (labeled for the reader’s convenience) is a $2\times 2$ group of images, where the theoretical and simulated $W_{\alpha \beta }$ are in columns 1 and 2 and the real and imaginary parts of $W_{\alpha \beta }$ are in rows 1 and 2, respectively.

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In summary, we presented a method to generate ENUC beams from EGSM sources. Previous work used Cholesky decomposition—a computationally expensive process—to synthesize ENUC beams [7]. Here we transformed EGSM field instances (generated by filtering CCGRNs) directly into ENUC realizations (without computing Cholesky factors), saving significant computing time and resources. We simulated the generation of an ENUC source with specified parameters. The simulated results were in excellent agreement with the theory, thus validating our approach.

The method for generating ENUC beams presented in this Letter can be directly implemented on existing SLM-based vector beam generators and can be used in any application that utilizes ENUC beams such as free-space or underwater optical communications.