Computational Approaches for Generating Electromagnetic Computational Approaches for Generating Electromagnetic Gaussian Schell-model Sources Gaussian Schell-model Sources

: Two different methodologies for generating an electromagnetic Gaussian-Schell model source are discussed. One approach uses a sequence of random phase screens at the source plane and the other uses a sequence of random complex transmittance screens. The relationships between the screen parameters and the desired electromagnetic Gaussian-Schell model source parameters are derived. The approaches are verified by comparing numerical simulation results with published theory. This work enables one to design an electromagnetic Gaussian-Schell model source with pre-defined characteristics for wave optics simulations or laboratory experiments.


Introduction
The electromagnetic Gaussian Schell-model (EGSM) source/beam was introduced as an extension of the scalar Gaussian Schell-model (GSM) beam [1,2]. Since then, it has attracted special attention due to the interesting polarization evolution that can occur on its propagation and the reduction in scintillation that is possible in free-space optical communications, imaging through turbulence, and remote sensing applications [3][4][5][6][7][8][9][10]. The ability to customize the EGSM attributes can lead to improved performance for particular applications and scenarios.
An EGSM beam can be described by a 2 × 2 cross-spectral density (CSD) matrix that characterizes second-order correlations between two mutually orthogonal components of the fluctuating electric field at a pair of spatial arguments and frequency [10]. Substantial progress has been made on the theoretical understanding of these beams including their propagation aspects, correlation features, and realizability conditions [11][12][13][14][15]. Concurrently, various methods have been proposed to produce EGSM sources numerically and experimentally [16][17][18][19][20][21][22][23]. These efforts include an approach to experimentally synthesize EGSM sources with the same mutually orthogonal electric field components [18], an experimental measurement to verify the validity of the EGSM beam parameters [19,22], and a practical method of producing a general EGSM source [20]. Most recently, a reduction in scintillation for a particular subclass of EGSM beam (completely unpolarized) was successfully demonstrated for propagation in thermally simulated atmospheric turbulence [23]. These studies provide practical techniques to physically realize the EGSM beam and successfully validate the existing theory; however, a practical ability to design and control the EGSM beam characteristics was not the primary emphasis of these efforts.
In this paper, the fundamental relationships between the two orthogonal polarization components of an EGSM beam are examined and a computational approach for creating numerical random screens that are associated with the components is presented. The desired EGSM beam parameters determine the selection of the screen parameters. The concept is that a pair of such screens is applied at the source plane to two orthogonally polarized coherent waves. The two resulting wave components constitute an instantaneous electromagnetic beam realization. Sufficiently large, mutually independent sequences of the screen pairs are then applied and the resulting intensities for each field component are averaged over these ensembles of realizations. The four average intensities, two representing self-correlations and two representing joint correlations, between the two components comprise the EGSM beam. The beams produced in this manner are consistent with the EGSM realizability conditions stemming from the fundamental properties of the CSD matrix [14,15]. The produced ensembles of screens can be used in a numerical wave optics simulation or in the laboratory with spatial light modulators (SLMs).
In Section 2, two screen methodologies, the phase screen (PS) and the complex transmittance screen (CS), are introduced. The relationships between the screen parameters and the desired EGSM beam parameters are explored and the benefits and constraints of the two approaches are discussed. The screen methodologies are validated in Section 3 via numerical modeling of typical EGSM beams and comparison of the results with theoretical predictions. Some final remarks and future research directions are given in Section 4. . The polarization state of the light passing through the system is denoted by two-sided arrows (representing horizontal polarization) and circles (representing vertical polarization). When both are present, the light is in a general polarization state, i.e., polarized, partially polarized, or unpolarized. Figure 1 shows a proposed experimental schematic for generating EGSM sources. Note that this proposed set-up is similar to that presented in Ref [20]. Light leaves a laser and traverses a beam expander (BE) and half-wave plate (HWP) before being split along two paths by a polarizing beam splitter (PBS). The initial HWP is used to control the relative amplitudes of the fields along each path. In paths 1 and 2, the light is polarized vertically (denoted by the circle) and horizontally (denoted by the two-sided arrow), respectively. It is assumed here that the SLMs control only vertically polarized light; thus, a HWP is used in path 2 to transform horizontal linear polarization into vertical polarization. The light in both paths is then incident on the SLMs. Because of their widespread use, it is assumed that the SLMs in Fig. 1 are reflective, phase-only SLMs. The SLMs impart random, correlated phases to the light in paths 1 and 2. After the SLMs, the light enters general lens systems (LS). These LS could be spatial filters, 4-f systems, etc. and are included to remove unwanted diffraction orders, produced by the SLMs, which may corrupt the desired EGSM source output.

Methodology
After traversing the LS, the light in both paths passes through Gaussian amplitude filters (GAFs) which set the desired Gaussian amplitude widths of the EGSM source (discussed in more detail below). The light from path 1 and path 2 is then recombined using a PBS. Note that the HWP, located before the GAF, on path 1 is required to transform the polarization state from vertical to horizontal polarization so that the light from both paths can be recombined. Lastly, a liquid crystal variable retarder (VR) is included to control the relative phasing between the vertical and horizontal polarization states.
It must be stated that the experimental set-up depicted in Fig. 1 is hypothetical. No experimental results are presented in this paper. The approaches presented here for generating EGSM sources are validated via simulation. The above description is included to provide background on how one might generate EGSM sources in practice. An experimental system similar to the one in Fig. 1 is currently in work. Experimental results will be presented in a future paper.
Two methods for generating EGSM sources are presented in this paper-the PS and CS methods. The PS approach involves generating two random phase screens, one for each polarization component. This approach can be implemented in the laboratory with two phaseonly SLMs as shown in Fig. 1. The interested reader is referred to Ref [24]. for the practical aspects of generating a scalar GSM beam with a single nematic phase-only SLM. The PS approach is equivalent to that presented in Ref [20]; however, here, the derivation is presented differently.
While the PS approach is useful for practical implementation purposes, its main disadvantage is that the autocorrelation function of the screen transmittances is typically not of the desired form. This is a significant problem when the desired autocorrelation function is not Gaussian. The CS approach, on the other hand, does not suffer from this shortcoming. This approach involves generating two screens with complex transmittance functions, i.e., both the amplitude and phase of the incident wave are randomized spatially upon transmission through the screen. The CS approach is ideal for numerical simulations, but laboratory implementation is rather difficult because both the amplitude and phase of the source must be controlled.
The elements of the CSD matrix of an EGSM source are [10] ( ) ( where , , x y α β = , S α is the spectral density, αβ μ is the spectral correlation function and In addition, an EGSM source must satisfy the fork inequality to be realizable [15]. It is imperative to show that both proposed approaches produce sources whose parameters obey the above constraints. Hereafter, the dependence on the radian frequency ω is omitted for the sake of brevity.

PS approach
Let the electric field in the source plane, is a complex constant and ( ) α φ ρ is the random phase contribution due to the screen. Performing the autocorrelations necessary to fill the CSD matrix produces The phase screen realizations are sample functions drawn from two correlated Gaussian random processes. Hereafter, for the sake of brevity, functions evaluated at 1 ρ or 2 ρ are denoted with a subscript 1 or 2, respectively. For example, ( ) [ ] ( ) ( ) ; e x p .
The symbol α β φ φ  is the spatial cross-correlation radius of the phase screens α φ and β φ .
Assuming that ( ) can be safely approximated as . Substituting this expression into Eq. (6), then into Eq. (5), and simplifying produces ( ) Note that the relations reported in the left column of Eq. (9) are coupled and cannot be chosen at will. On the other hand, the relations in the right column of Eq. (9) are uncoupled and can be chosen at will. Referring back to Fig. 1, x A and y A are controlled using the initial HWP, x σ and y σ are set by using the appropriate GAFs, and xy B ∠ is set using the VR. The remaining EGSM source parameters are determined by the statistical properties of the phases commanded to the SLMs discussed in detail in Section 2.3.

CS approach
Let the electric field components in the source plane, exp . 4 4 Just like α φ and β φ in the PS approach, T α and T β are sample functions drawn from two correlated Gaussian random processes. This time, however, the random processes are complex.
The expectation in Eq. (11) is recognized as the cross-correlation function of the Gaussian random processes T α and T β : where T α σ and T β σ are the standard deviations of the T α and T β screens, respectively; ; e x p .
The symbol T T α β  is the spatial cross-correlation radius of the complex transmittance screens T α and T β . Substituting Eqs. (12) and (13) into Eq. (11) and simplifying produces By comparing Eq. (14) to Eq. (1), the following relationships are deduced: While not yet evident, the relations reported in the left column of Eq. (15) are coupled and cannot be chosen at will. The relations in the right column are uncoupled and can be chosen at will.

Generating phase screens (PS approach)
In this section, a method for generating the required discretized x φ and y φ is presented. Of the two approaches discussed above, the PS approach is the most applicable to laboratory research because of the commercial availability of phase-only SLMs.
x y x y x y x y x y x y where r α is a matrix of zero mean circular complex Gaussian random numbers with the real and imaginary parts each having unit variance. In order to generate correlated x φ and y φ , necessary to synthesize the "cross" terms of the Note that the complex exponential terms in the braces are discrete inverse and forward Fourier transform kernels. The discrete function being transformed in Eq. (24), equivalent to the cross-power spectral density, is even in m and n ; therefore, the forward and inverse Fourier transforms yield the same result. Applying these simplifications produces  , e x p , x y x y x y x y x y x y x y one obtains the following relationships: y y x x y y x y x y x y x x y y Using Eq. (9), the general relationships between the EGSM source parameters and the phase screen design parameters are found to be In the above equations, , 0 x y φ φ σ σ π ≥ , and 0 1 < Γ ≤ . Equation (28) expresses the four desired EGSM source parameters in terms of five phase screen design parameters; thus, the system of nonlinear equations is undetermined. Upon closer inspection of Eq. (28), one notes that three of the four desired EGSM parameters can be chosen at will (recall that x A , y A , and While Eq. (28) could be inverted in the manner just outlined, the optimal solution is not guaranteed. Here, the optimal solution is defined as the phase screen design parameters that yield EGSM parameters "nearest to" the desired EGSM parameters. Thus, in this work, the optimal phase screen design parameters are found using constrained nonlinear optimization.

Generating complex screens (CS approach)
In this section, a method for synthesizing discretized x T and y T is shown. Because both amplitude and phase must be controlled, the CS approach is much better suited to research involving simulation. For ease of comparison, the same SLM specifications listed above are used in the simulation results presented in Section 3.
Expanding T α in a Fourier series yields ( ) , , e x p j 2 e x p j 2 , one obtains the following relationships: y y x x y y x y x y x y x x y y x x y y

T T T T T T T T T T T T T T T T T T T T T T
Using Eq. (15), the general relationships between the EGSM source parameters and the complex screen design parameters are It is clear from Eq. (37) that two of the three correlation function widths can be chosen freely (the third is set by the other two). One is generally free to choose the value of xy B subject to the constraint that 1 Γ ≤ . The other EGSM source parameters, x A , y A , and can be chosen at will.

Simulation description
In this section, simulation results are presented to validate the PS and CS approaches described above. As stated previously, 512 points per side and a spacing of 15 m μ were used to discretize the fields along paths 1 and 2 in Fig. 1 with the off-diagonal elements of the CSD matrix equal to zero. Since for this case x y σ σ = , the polarization state was uniform across the source plane [2]. The second was an elliptically partially polarized EGSM source with a fully-populated CSD matrix. Table 1 reports the desired, PS, and CS EGSM source parameters for both cases. The screen parameters for the PS and CS approaches were determined by inverting Eqs. (28) and (37), respectively. For the CS approach, Eq. (37) is easily inverted. When the offdiagonal elements of the desired CSD matrix are zero (Case I), the CS approach can generate an EGSM source with the desired parameters (note that xy δ is irrelevant in these cases). This is not guaranteed when the desired CSD matrix is fully populated (Case II), however.
For the PS approach, Eq. (28) is a coupled system of nonlinear equations and not easily inverted. Here, constrained nonlinear optimization was used to find the phase screen parameters such that where x was a vector of the unknown phase screen parameters. The constraints on x included the conditions given in Eqs. (2) and (3) as well as positivity. In addition, to satisfy the "strongly scattering screen" requirement, i.e., the Gaussian approximation to the joint characteristic function [see Eq. (8)], , x y φ φ σ σ π ≥ . Like in the CS approach, when the offdiagonal elements of the desired CSD matrix are zero (Case I), the PS approach can generate an EGSM source with the desired parameters. Again, this is not guaranteed when the desired CSD matrix is fully populated (Case II).