1. INTRODUCTION

In the roughly 25 years since the development of a unified theory of polarization and coherence, the study of random electromagnetic fields has matured. Building off of the seminal works in Refs. [1–4], today, random electromagnetic beams are designed to have specific coherence and polarization properties. The level of control over shape, polarization, and coherence afforded by these beams makes them particularly useful in a multitude of applications ranging from directed energy, optical communications, and remote sensing, to biology and medicine, and finally to atomic optics and optical tweezing [5–8].

For a planar, wide-sense-stationary, random electromagnetic source, the second-order statistical behavior of the field in the space–frequency domain is described by a $2 \times 2$ dyadic tensor known as the cross-spectral density matrix (CSDM). It is formed by ensemble averaging the outer product of the source-field Jones vector, namely,

To be genuine or physical, a CSDM must be Hermitian and nonnegative definite [6,9,10]. The latter criterion is especially difficult to prove given an arbitrary ${\textbf{W}}$. Motivated by this, Gori et al. [11] employed the vectorial form of Bochner’s theorem of functional analysis to obtain a simple criterion, which (if satisfied) would guarantee that a given ${\textbf{W}}$ was genuine. Their formula, a generalization of their scalar criterion [12], is known as the electromagnetic superposition rule and is given by

The symbol ${\textbf{p}}$ in Eq. (2) stands for the Hermitian matrix

Equation (2) begets two physical interpretations or recipes for producing random electromagnetic sources [11,13,14]. The one that is germane to this work is known as electromagnetic pseudo-modes or elementary fields and is a generalization of scalar pseudo-modal expansions of cross-spectral density (CSD) functions [15]. Originally presented by Martínez-Herrero and Mejías [14], the electromagnetic pseudo-modes interpretation of Eq. (2) considers ${H_x}$ and ${H_y}$ (or linear combinations thereof) as components of a spatially coherent electromagnetic field or mode parametrized by ${\boldsymbol v}$. The matrix ${\textbf{p}}$ weights modes with certain ${\boldsymbol v}$, such that the incoherent sum of all ${\textbf{p}}$-weighted modes (i.e., for all ${\boldsymbol v}$) produces the desired ${\textbf{W}}$. Electromagnetic pseudo-modes can be used to physically generate sources [16]; however, their real benefit is in computational optics, where the savings in computing resources, compared to using the CSDM directly, can be immense [17–23].

In this paper, we present two electromagnetic pseudo-modal expansions of ${\textbf{W}}$ for use in optical simulations with partially coherent, partially polarized light. Both are specializations of the general result discussed in Ref. [14] and are derived from decompositions of the matrix ${\textbf{p}}$. In the next section, we present the derivations of both expansions. We then generate, in simulation, two random electromagnetic beams and compute second-order field moments to validate against theory. Last, we conclude the paper with a brief summary.

2. THEORY

A. Eigendecomposition of ${\textbf{p}}$

Let us return to the matrix ${\textbf{p}}$ in Eq. (3). The conditions on ${\textbf{p}}$ mean that it is Hermitian and nonnegative definite for all ${\boldsymbol v}$. Therefore, it can be diagonalized, such that

Focusing on the integrand in Eq. (6), we find that

Substituting this into Eq. (6), we see at once that

Equation (8) is the main result of this section and mathematically states that a general CSDM can be decomposed into the sum of two CSDMs. The integrands of the “child” CSDMs are in a factorized form (i.e., the outer products of vector fields), and therefore describe spatially coherent, fully polarized (generally non-uniformly) electromagnetic fields ${{\boldsymbol e}_1}$ and ${{\boldsymbol e}_2}$. These fields are the pseudo-modes of the random electromagnetic source. Each child CSDM, via the integrals over ${\boldsymbol v}$, is formed from the weighted, incoherent sum of many ${{\boldsymbol e}_1}$ or ${{\boldsymbol e}_2}$, where ${\boldsymbol v}$, although continuous, can be interpreted as the pseudo-mode’s index. Last, since the “parent” CSDM is formed from the sum of the children, ${{\boldsymbol e}_1}$ and ${{\boldsymbol e}_2}$ are mutually uncorrelated. It is important to note that others have arrived at a similar result by other means [18].

In summary, any genuine stochastic electromagnetic source be generated from the following two electromagnetic fields:

Summing these two fields incoherently over all ${\boldsymbol v}$ produces the desired source. The general forms for ${p_1}$, ${p_2}$, ${{\boldsymbol e}_1}$, and ${{\boldsymbol e}_2}$ can be derived by finding the eigenvalues and eigenvectors of ${\textbf{p}}$. This is straightforward, yet cumbersome. Here, we report the final result:

Here, $\det ({\textbf{A}})$ and ${\rm{tr}}({\textbf{A}})$ stand for the determinant and trace of the matrix ${\textbf{A}}$, respectively. We used the symbol $P$ in Eq. (11) because of its similarity to the degree of polarization [6,9,10,25]; however, we caution that ${\textbf{p}}$ is not the polarization matrix of the source. We discuss this further in the next section.

B. Polarization-Matrix-Inspired Decomposition of ${\textbf{p}}$

In addition to the eigendecomposition of ${\textbf{p}}$ presented above, ${\textbf{p}}$ may also be uniquely decomposed into the sum of two matrices, such that

Equation (12) is motivated by the unique, physical decomposition of a polarization matrix into diagonal and singular matrices [6,9,10,25,26]. In the case of the polarization matrix, the diagonal matrix physically models a field with uncorrelated polarization components, and therefore represents the unpolarized portion of the field. The singular matrix, on the other hand, models a field with completely correlated components, and thus describes the polarized portion of the field. From this matrix, one can determine the polarization state of a random light beam [6,10,26].

Just like with $P$ in Eq. (11), one must be cautious applying this physical interpretation to the decomposition of ${\textbf{p}}$ in Eq. (12). The matrix ${\textbf{p}}$ is tenuously related to polarization, and the decomposition in Eq. (12), for the most part, is purely mathematical.

Substituting Eqs. (12) and (13) into Eq. (2) and simplifying produces

The functions $A$, $B$, $C$, and $D$ for decomposition of the polarization matrix were derived in Ref. [26]. They take the same form here, with the appropriate substitutions:

 

Fig. 1. EMGSM Stokes parameter results using the polarization-matrix-inspired decomposition of ${\textbf{p}}$ and the pseudo-modes in Eq. (17): (a), (b) ${S_0}$ theory and simulation; (c), (d) ${S_1}$ theory and simulation; (e), (f) ${S_2}$ theory and simulation; (g), (h) ${S_3}$ theory and simulation.

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Equation (15) states that any genuine CSDM can be uniquely expressed as the sum of three CSDMs, where each is formed from the weighted, incoherent sum of spatially coherent, fully polarized electromagnetic fields. These fields are the pseudo-modes of ${\textbf{W}}$ and are given by

Summing these fields incoherently over all ${\boldsymbol v}$ yields the desired source.

Comparing Eq. (17) to the eigendecomposition pseudo-modes in Eq. (9), we see at once that an additional pseudo-mode is required here. Nevertheless, the computational requirements for both decompositions are the same since a total of four field components (and therefore four computational grids) are required in either case. In fact, the pseudo-modes in Eq. (17) have a potentially significant numerical advantage over those in Eq. (9). In Eq. (10), the quantities $f({\boldsymbol v}) \pm g({\boldsymbol v})$ can equal zero at certain ${\boldsymbol v}$ (they are nonnegative for all ${\boldsymbol v}$). Mathematically, this is handled by the numerators of the $x$ components of ${{\boldsymbol e}_1}$ and ${{\boldsymbol e}_2}$, which are also zero at those same ${\boldsymbol v}$. Computationally, however, this can cause problems. The pseudo-modes in Eq. (17), whose “coefficients” are given in Eqs. (14) and (16), do not have denominators and therefore are numerically stable for all values of ${\boldsymbol v}$.

3. EXAMPLES

In this section, we generate two random electromagnetic sources using the pseudo-modes expansions discussed above. We start with the polarization-matrix-inspired decomposition of ${\textbf{p}}$ and use its corresponding pseudo-modes in Eq. (17) to produce an electromagnetic Schell-model source. We then use the eigendecomposition of ${\textbf{p}}$ and its pseudo-modes in Eq. (9) to generate an electromagnetic non-uniformly correlated (ENUC) beam.

 

Fig. 2. EMGSM ${\textbf{W}}({{x_1},0,{x_2},0})$ results using the polarization-matrix-inspired decomposition of ${\textbf{p}}$ and the pseudo-modes in Eq. (17): (a)–(d) real and imaginary parts of ${W_{\textit{xx}}}$ theory and simulation, (e)–(h) real and imaginary parts of ${W_{\textit{xy}}}$ theory and simulation, (i)–(l) real and imaginary parts of ${W_{\textit{yx}}}$ theory and simulation, and (m)–(p) real and imaginary parts of ${W_{\textit{yy}}}$ theory and simulation.

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A. Electromagnetic Multi-Gaussian Schell-Model Beam

For our first example, we generate an electromagnetic multi-Gaussian Schell-model (EMGSM) source first described by Mei et al. [27]. The CSDM elements for an EMGSM beam are

Last, $M \gt 0$ is known as the shape parameter. For $M \lt 1$, the function formed by the sum in Eq. (18) (${\mu _{\alpha \beta}}$ or the cross-correlation function) acquires heavy tails as $M \to 0$ [28]; for $M = 1$, Eq. (18) simplifies to the CSDM of an electromagnetic Gaussian Schell-model beam [1,29–31]; and for $M \gt 1$, ${\mu _{\alpha \beta}}$ resembles $\sin (x)/x$, where the central lobe becomes sharper as $M \to \infty$ [32]. The spectral densities radiated to the far zone by EMGSM sources take on cusped, Gaussian, and flat-topped profiles for $M \lt 1$, $M = 1$, and $M \gt 1$, respectively. When $M$ is an integer, the series in Eqs. (18) and (19) are finite with all terms greater than $M$ equal to zero.

For an EMGSM source to be physical or genuine, the parameters in Eq. (18) must satisfy the following [27]:

The ${\textbf{p}}$ and ${\textbf{H}}$ that when substituted into Eq. (2) yield Eq. (18) are

We now have everything we need to produce any genuine EMGSM source using the pseudo-modes in Eq. (17). In the simulation below, we generated an EMGSM beam with the following parameters: ${A_x} = 1.5$, ${\sigma _x} = 0.5\,\,{\rm{mm}}$, ${\delta _{\textit{xx}}} = 0.6\,\,{\rm{mm}}$, ${A_y} = 1$, ${\sigma _y} = 0.8\,\,{\rm{mm}}$, ${\delta _{\textit{yy}}} = 0.4\,\,{\rm{mm}}$, ${B_{\textit{xy}}} = 0.5\exp ({{\rm{j}}\pi /3})$, ${\delta _{\textit{xy}}} = 0.6464\,\,{\rm{mm}}$, and $M = 40$.

To determine the minimum number of pseudo-modes required to accurately generate the above EMGSM source, we first Fourier transformed the product of ${p_{\alpha \beta}}$ and ${H_\alpha}$ given in Eq. (21) with respect to ${\boldsymbol v}$. We then found the maximum width ${f_{v,{\max}}}$ of this function—defined as the location where the normalized magnitude of the function fell below 0.001. The minimum number of pseudo-modes was $P \ge \lceil 2D{f_{v,{\rm{max}}}}\rceil$, where $D$ was the maximum diameter of ${p_{\alpha \beta}}$—also defined as the ${\boldsymbol v}$ location where normalized $| {{p_{\alpha \beta}}} | \le 0.001$. This analysis stipulated that $P \ge 62$ with $D = 23.0175 \times {10^3}\;{{\rm{m}}^{- 1}}$. We note that $P \ge 62$ was the minimum number of pseudo-modes required along one dimension of the ${\boldsymbol v}$ grid. Thus, the total number of modes was ${62^2}$.

We used a similar procedure to determine the minimum number of grid points $N$ to discretize ${{\boldsymbol E}_1}$, ${{\boldsymbol E}_2}$, and ${{\boldsymbol E}_3}$ without aliasing the fields. We first Fourier transformed ${H_\alpha}$ with respect to ${\boldsymbol \rho}$ and found the maximum width ${f_{{\max}}}$ of that function using the same definition as above. The minimum number of grid points per side was $N \ge \lceil 2L{f_{{\rm{max}}}}\rceil \ge 43$, where $L = 10\max ({{\sigma _\alpha}}) = 8\,{\rm{mm}}$ was the side length of the grid. While only 43 grid points per side would be sufficient to represent ${{\boldsymbol E}_1}$, ${{\boldsymbol E}_2}$, and ${{\boldsymbol E}_3}$ without aliasing, we used $N = 128$ to avoid pixelated results.

Figures 1 and 2 report the results. Figure 1 compares the theoretical (column 1) and simulated (column 2) Stokes parameters of the EMGSM source. The rows of the figure correspond to a Stokes parameter, with the theoretical and simulated results plotted using the same color scale defined by the color bar at the end of the row. The rows and the columns of the figure are labeled for the reader’s convenience.

Figure 2 shows the EMGSM ${\textbf{W}}({{x_1},0,{x_2},0})$ results. The layout of the figure was chosen to resemble the $2 \times 2$ ${\textbf{W}}$, with each element of ${\textbf{W}}$ consisting of four images. The four images are themselves arranged in a $2 \times 2$ pattern, where column 1 shows the real (row 1) and imaginary (row 2) parts of the theoretical result. Column 2 shows the same for the simulated result. Both the theoretical and simulated results are plotted on the same color scales defined by the color bars immediately to the right of the simulated images. Like Fig. 1, we labeled Fig. 2 to aid the reader.

 

Fig. 3. ENUC Stokes parameter results using the eigendecomposition of ${\textbf{p}}$ and the pseudo-modes in Eq. (9): (a), (b) ${S_0}$ theory and simulation; (c), (d) ${S_1}$ theory and simulation; (e), (f) ${S_2}$ theory and simulation; (g), (h) ${S_3}$ theory and simulation.

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The agreement among the results in Figs. 1 and 2 is excellent, implying that we have successfully generated the desired EMGSM source.

B. Electromagnetic Non-uniformly Correlated Beam

For our next example, we generated an ENUC source [33,34]. The CSDM elements for an ENUC beam take the form

 

Fig. 4. ENUC ${\textbf{W}}({{x_1},0,{x_2},0})$ results using the eigendecomposition of ${\textbf{p}}$ and the pseudo-modes in Eq. (9): (a)–(d) real and imaginary parts of ${W_{\textit{xx}}}$ theory and simulation, (e)–(h) real and imaginary parts of ${W_{\textit{xy}}}$ theory and simulation, (i)–(l) real and imaginary parts of ${W_{\textit{yx}}}$ theory and simulation, and (m)–(p) real and imaginary parts of ${W_{\textit{yy}}}$ theory and simulation.

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Last, the ${\textbf{p}}$ and ${\textbf{H}}$ that yield an ENUC source via the superposition rule are [33]

For this simulation, we generated an ENUC source using the pseudo-modes in Eq. (9) with the following parameter values: ${A_x} = 1$, ${\sigma _x} = 1.1\,{\rm{mm}}$, ${\delta _{\textit{xx}}} = 0.8\;{\rm{mm}}$, ${{\boldsymbol{\gamma}} _x} = {\boldsymbol {\hat x}}0.35 + {\boldsymbol {\hat y}}0.5\;{\rm{mm}}$, ${A_y} = 1.2$, ${\sigma _y} = 1\,{\rm{mm}}$, ${\delta _{\textit{yy}}} = 0.7\;{\rm{mm}}$, ${{\boldsymbol{\gamma}} _y} = - {\boldsymbol {\hat x}}0.4 - {\boldsymbol {\hat y}}0.25\;{\rm{mm}}$, ${B_{\textit{xy}}} = 0.4\exp ({- {\rm{j}}\pi /6})$, and ${\delta _{\textit{xy}}} = 0.763\;{\rm{mm}}$. We determined the minimum numbers of pseudo-modes $P$ and grid points $N$ in the same manner as above. Because of the Gaussian forms of ${p_{\alpha \beta}}$ and ${H_\alpha}$, we were able to derive closed-form expressions for $P$ and $N$, namely,

Figures 3 and 4 show the Stokes parameter and CSDM ENUC beam results, respectively. The layouts of both are identical to the corresponding EMGSM source figures (Figs. 1 and 2) described in the text above. Again, the agreement between the theoretical and simulated results is excellent, thereby validating the approach.

4. CONCLUSION

In this paper, we presented two electromagnetic pseudo-modal expansions applicable to any genuine CSDM. We derived these expansions from the electromagnetic superposition rule by decomposing the Hermitian, nonnegative-definite matrix ${\textbf{p}}$ in two ways. For the first, we diagonalized ${\textbf{p}}$ using eigendecomposition. This analysis revealed that any CSDM could be written as the sum to two CSDMs, where each was the weighted, incoherent sum of spatially coherent, fully polarized electromagnetic fields. These fields constituted the first set of pseudo-modes.

For the second decomposition, we expressed ${\textbf{p}}$ as the sum of diagonal and singular matrices, mimicking a polarization matrix decomposition, which splits a general polarization matrix into the sum of polarized and unpolarized components. We found, via this decomposition, that any genuine CSDM could be expressed as the sum of three CSDMs. As was the case for the first decomposition, these CSDMs were the weighted, incoherent sums of spatially coherent, fully polarized electromagnetic fields. These fields constituted another set of pseudo-modes, which we briefly compared with the first.

To validate our work, we generated two random electromagnetic sources (in simulation) using one set of pseudo-modes for each. From the simulations, we computed the Stokes parameters and planar cuts through the four-dimensional CSDM elements to compare with theory. We observed excellent agreement among the results, verifying that we had produced the desired sources.

The primary advantage of pseudo-modal expansions of CSD functions (in the scalar case) and CSDMs (in the electromagnetic case) over working with the CSD functions or CSDMs directly is the significant decrease in computing resources required to perform optical simulations. The work presented here will be useful for that purpose. In addition, current procedures for physical generation of exotic CSDMs are also based on Bochner’s theorem [16]. Therefore, our expansions can also benefit the corresponding experimental work.