Monte Carlo Simulations of Three-dimensional Electromagnetic Monte Carlo Simulations of Three-dimensional Electromagnetic Gaussian Schell-model Sources Gaussian Schell-model Sources

: This article presents a method to simulate a three-dimensional (3D) electromagnetic Gaussian-Schell model (EGSM) source with desired characteristics. Using the complex screen method, originally developed for the synthesis of two-dimensional stochastic electromagnetic ﬁelds, a set of equations is derived which relate the desired 3D source characteristics to those of the statistics of the random complex screen. From these equations and the 3D EGSM source realizability conditions, a single criterion is derived, which when satisﬁed guarantees both the realizability and simulatability of the desired 3D EGSM source. Lastly, a 3D EGSM source, with speciﬁed properties, is simulated; the Monte Carlo simulation results are compared to the theoretical expressions to validate the method.

For certain scenarios, e.g., near-field beam shaping [13][14][15][16], observing in the near zone of a scatterer, or observing in the vicinity of a caustic, a full, three-dimensional (3D) vector treatment of the field is required. The study of light's behavior under these and other similar conditions motivates research involving 3D stochastic EM fields. Much of the published work on this subject involves defining physically significant statistics of 3D random EM fields, most notably, the degree of coherence [17,18], Stokes parameters [19], and the degree of polarization [20,21]. To date, little work has been performed developing 3D random field models (counterparts to the ubiquitous Schell-model sources in 1D and 2D) or in synthesizing 3D random field instances.
Here, a method to simulate a 3D electromagnetic Gaussian Schell-model (EGSM) source with desired shape and coherence properties is presented. The 3D EGSM source was recently introduced in Ref. [22]. There, the authors derived a set of conditions that all 3D EGSM sources must satisfy in order to be physically realizable sources.
Using the complex screen method (originally developed for the synthesis of 2D random fields) [23][24][25], equations relating the desired 3D EGSM source parameters (source widths, spatial correlation radii, et cetera) to the random complex screen parameters are derived. Then, using these equations and the realizability conditions developed in Ref. [22], a single criterion is found, which when satisfied guarantees that the desired 3D EGSM source is both realizable and simulatable. Lastly, a 3D EGSM source, with specified properties, is simulated using the 3D complex screen method. The simulated results are compared to the analytical expressions to validate the analysis.
It should be noted that no device or method for physically realizing a designer 3D EGSM source currently exists. Nevertheless, the work presented here provides a recipe for generating such a source should the technology become available. This work can be put to immediate use in wave optics simulations of scenarios similar to those mentioned above.

Theory
The cross-spectral density matrix elements of a 3D EGSM source take the form where i, j = x, y, z, A i is the amplitude of the i th field component, B i j is the complex crosscorrelation coefficient between the i th and j th components, σ i is the spectral density width of the i th component, δ i j is the width of the cross-correlation function between the i th and j th components, and r =x x +ŷ y +ẑz [22]. For the 3D EGSM source to be physically realizable, the δ i j and B i j must satisfy [22] In addition, B ii = 1 and B i j ≤ 1 when i j.
Generating 3D EGSM sources can be accomplished by using extensions of the techniques discussed in Refs. [24][25][26]. Since it forms the basis of the modified phase screen technique [27,28], the complex transmittance screen approach (in this context, more accurately termed the complex volume approach) is presented here. The traditional Gaussian phase screen [24][25][26] details can be derived in a similar manner; for completeness, the phase screen expressions are provided in Appendix A.
Let an instance of the i th component of a 3D EGSM source be where C i is the complex amplitude and T i is the random complex transmittance screen. Taking the vector autocorrelation of Eq. (6) produces Comparing Eqs. (7) and (1) reveals the following: One is free to choose the amplitudes A i and the spectral density widths σ i of the 3D EGSM source. The phase angles arg B i j are solutions to where α i = arg (C i ) and θ i j = arg B i j . Solving the above system of equations reveals that θ xy + θ yz = θ xz ; thus, two of the three phase angles can be chosen freely, the last is determined by the other two. This stands in contrast to the 2D case where one is free to choose the phase angle [24]. Lastly, the B i j and δ i j are coupled and even if realizable [i.e., the desired values satisfy Eqs. (2)-(5)], the source they describe may not be simulatable using the complex screen approach. The question of whether a desired 3D EGSM source is simulatable is addressed in the next section.

T i generation
A complex screen T i can be generated using the Monte Carlo spectral method (MCSM) [23,24,29,30]. Using the MCSM (not reviewed here), a realization of T i takes the form where i, j, k are discrete spatial indices, l, m, n are discrete spatial frequency indices, r i is a 3D matrix of zero-mean, unit-variance circular complex Gaussian random numbers, L x , L y , L z are the physical sizes of the grid in the x, y, and z directions, respectively, and Φ ii is the spatial power spectrum of the screen: Note that Eq. (10) is in the form of a discrete inverse Fourier transform; thus, T i can be generated quickly using the fast Fourier transform algorithm. Generating T i using Eq. (10) will produce a screen with the desired δ ii and, via Eq. (6), a 3D EGSM source with the desired W ii . To produce a 3D EGSM source with the desired W i j requires examination of the cross-correlation of the T i and T j screens, namely, .
The moment in Eq.
where δ is the discrete Dirac delta function and 0 ≤ Γ i j ≤ 1 is the correlation coefficient between the 3D matrices of Gaussian random numbers r i and r j . Substituting in the expressions for the spatial power spectra [recall Eq. (11)] and simplifying yields .
(13) The expression in the braces must equal the spatial cross-power spectrum, to produce the desired 3D EGSM source. Comparing the braced expression in Eq. (13) with Eq. (14), it is clear that Thus, the relationships between the 3D EGSM source and complex screen parameters are These expressions are very similar to the 2D EGSM source complex screen relations given in Ref. [24]-the only difference being the square root term in Eq. (19). Note that the values of δ i j are set by the values of the corresponding "self" correlation radii exactly the same as in the 2D case [see Eq. (18)]. Equation (18) always satisfies the δ i j realizability condition in Eq. (4).
Examining Eq. (19), it initially appears, like in the 2D case, that one is generally free to choose the values of B i j granted 0 ≤ Γ i j ≤ 1. This, however, is not the case. Returning to the B i j realizability condition given in Eq. (5) and substituting in Eqs. (18) and (19) yields A 3D plot of this simulatability condition is shown in Fig. 1. Figure 1(a) shows a view of the volume's surface with the interior removed and (b) shows a view of the surface from a different angle. The volume intersects each primary plane in a quarter circle. Any Γ i j combination that results in a point within the volume (underneath the surface in Fig. 1) is both a realizable and simulatable 3D EGSM source. If Γ xz = Γ yz = 0, implying that |B xz | = B yz = 0 (physically reducing a 3D EGSM source to a 2D one, assuming that W zz = 0), then Eq. (20) reduces to Γ xy ≤ 1 which is always true. Thus, satisfying Eq. (20) guarantees realizability and simulatability of that particular 3D EGSM source as well as all 2D EGSM sources which can be derived from it.

Summary of 3D EGSM source simulation process
Before proceeding to the simulation results, a step-by-step description of the complex screen 3D EGSM source simulation process is warranted: 1. Choose values for the free source parameters σ i and A i , which are related to the screen parameters |C i | via Eq. (16).
2. Choose values for the phase angles arg B i j = θ i j , which are related to arg C i C * j via Eq. (17). Recall that θ xy + θ yz − θ xz = 0. The pseudoinverse of the matrix in Eq. (9) can be used to find simulatable phase angles which are closest (minimum Euclidean norm) to a set of three desired phase angles. can be combined into a single step where the desired coupled parameters are inputs into a constrained nonlinear optimization routine that returns the screen parameters δ ii , δ j j , and Γ i j which yield realizable and simulatable source parameters closest (by minimizing a user specified norm) to the desired parameters.
5. Generate 3 3D matrices of correlated, zero-mean, unit-variance circular complex Gaussian random numbers r i . The covariance matrix is formed from the values of Γ i j found in step 4.
7. Form 3D EGSM source instance E i using Eq. (6). To validate the above analysis, a 3D EGSM source was simulated using the complex screen technique. The EGSM source parameters are listed in Table 1. The computational volume was discretized using 512 points per side with a sample spacing equal to 1.37 mm, making the total physical size of the grid 0.7 m × 0.7 m × 0.7 m. The component spectral densities S i and spectral degrees of correlation µ i j ,

Simulation details
were computed from 500 complex screen field instances. The simulated S i and µ i j were then compared to the corresponding theoretical quantities to verify that the desired 3D EGSM source had been realized. Figures 2 and 3 show spectral density S results. To give the reader an understanding of what a complex screen 3D EGSM source looks like, Fig. 2 shows two views of the spectral density volume computed from a single realization of the 3D EGSM source whose parameters are given in Table 1. Figure 2(a) shows a view of the volume's surface, while Fig. 2(b) shows the x-y, x-z, and y-z planar slices through the volume. Figure 3 shows the theoretical (solid traces) and simulated (circles) (a) x, (b) y, and (c) z spectral densities, respectively. The agreement between the simulated and theoretical results is excellent. These results verify that the complex screens, on average, are producing a 3D EGSM source with the desired shape. To verify that the resulting source has the desired coherence properties, the spectral degrees of correlation µ i j must be examined.  Figure 4 shows the theoretical (solid traces) and simulated (circles) (a) µ xx , (b) µ yy , and (c) µ zz results, respectively; Fig. 5 shows the Fig. 2. Single realization spectral density volume of the complex screen 3D EGSM source whose parameters are given in Table 1-(a) surface view and (b) x-y, x-z, and y-z planar slices.  theoretical (solid traces) and simulated (circles and squares) (a) µ xy and µ * yx , (b) µ xz and µ * zx , and (c) µ yz and µ * zy , respectively. In Figs. 4 and 5, both the real and imaginary parts of µ i j are plotted.

Spectral degrees of correlation µ i j
Again, the agreement between the simulated and theoretical results is excellent. These results combined with the spectral density results in Fig. 3 validate the complex screen simulation procedure developed in Section 2.

Conclusion
In this paper, the simulation of a 3D EGSM source using the complex screen technique was investigated. Relationships between the source parameters (A i , B i j , σ i , and δ i j ) and the complex screen parameters were derived and discussed. Using these expressions and the recently published 3D EGSM source realizability conditions [22] yielded a new, single simulatability criterion. When satisfied, this condition guaranteed both realizability and simulatability of the desired 3D EGSM source using the complex screen method.
To validate the analysis, a 3D EGSM source was simulated using the complex screen technique. The simulated directional spectral densities and spectral degrees of correlation were compared to those of the true 3D EGSM source. The agreement between the simulated and theoretical results was excellent.
Although no device or method for physically realizing a desired 3D EGSM source yet exists, this paper provides the basic recipe for generating such a source if or when the technology becomes available. In the meantime, the technique and procedure for simulating 3D EGSM sources developed here will find use in optical simulations involving non-paraxial, partially coherent EM fields, e.g., in simulations where a random field is observed in the near zone of a scatterer or in the vicinity of a tight focus, or caustic.

A. Phase screen 3D EGSM Sources
An instance of the i th component of a phase screen 3D EGSM source is where φ i is the random 3D phase screen. The relationships between the 3D EGSM source and phase screen parameters are where 0 ≤ Γ φ i φ j ≤ 1 is the correlation coefficient between the i th and j th component phase screens and σ 2 φ i and φ i φ i are the variance and spatial correlation radius of the i th component phase screen, respectively.
Via the MCSM, a phase screen instance can be generated by using where Φ φ i φ i is the spatial power spectrum of the i th component screen, namely, and r i is a 3D matrix of zero-mean, unit-variance circular complex Gaussian random numbers. The steps for generating a phase screen 3D EGSM source instance are generally the same as the complex screen steps listed above. Because of the complicated nonlinear equations relating the source parameters to the phase screen parameters [see Eqs. (25)-(28)], steps 3 and 4 of the source simulation process are much more difficult than in the complex screen case.

Funding
Air Force Office of Scientific Research (AFOSR) (FA9550-121-0449); Russian Federation Ministry of Education and Science.