In a previous paper [Opt. Express 22, 31691 (2014)] two different wave optics methodologies (phase screen and complex screen) were introduced to generate electromagnetic Gaussian Schell-model sources. A numerical optimization approach based on theoretical realizability conditions was used to determine the screen parameters. In this work we describe a practical modeling approach for the two methodologies that employs a common numerical recipe for generating correlated Gaussian random sequences and establish exact relationships between the screen simulation parameters and the source parameters. Both methodologies are demonstrated in a wave-optics simulation framework for an example source. The two methodologies are found to have some differing features, for example, the phase screen method is more flexible than the complex screen in terms of the range of combinations of beam parameter values that can be modeled. This work supports numerical wave optics simulations or laboratory experiments involving electromagnetic Gaussian Schell-model sources.
© 2017 Optical Society of America
The electromagnetic Gaussian Schell-model (EGSM) source is a partially coherent, partially polarized optical beam. It has an intensity profile that is Gaussian, a transverse spatial coherence function that is Gaussian, and partial polarization based on a Gaussian cross-correlation [1,2]. The EGSM exhibits an interesting polarimetric evolution during propagation and can provide performance improvement for free-space optical applications such as communications, imaging, and remote sensing [3–6].
Understanding and exploring the behavior of the EGSM beam is greatly aided by numerical simulation and laboratory experiment [7–9]. Figure 1 illustrates a general approach for constructing the EGSM beam in either a simulation or laboratory. Two Gaussian beams with orthogonal linear polarizations (x and y) are sent through separate random screens and the resulting beams are combined. Correlations between the two polarization channels are embodied in the screens. Averaging the output intensity patterns over many independent realizations of the screens produces the EGSM beam result.
In a recent publication, two types of computational random screen approaches were introduced for use in modeling the EGSM beam: complex screens (CS) and phase-only screens (PS) . In that work, the EGSM source was defined by four transverse spatial parameters: x-correlation length, y-correlation length, xy-cross-correlation length, and xy-correlation coefficient. The last two parameters have some dependence on the first two. It was shown that the four EGSM source parameters map to three deterministic CS parameters. However, the PS is defined by five parameters that represent an overdetermined relationship with the source parameters. A numerical optimization approach based on previously developed realizability conditions [5,6] was used to find the PS parameters; however deterministic relationships between the source and PS parameters were not identified.
In this paper, we introduce a well-known relationship for generating correlated Gaussian random sequences and proceed through an analytic development that precisely defines the relationships between the source and the PS and CS parameters. A computer simulation of an EGSM source and propagation of the beam demonstrates the utility of the approach.
2. Source definition
Consider the polarized electric field vector at the source plane given by :Fig. 1, the components are modeled as
Tx and Ty are transmittance functions or “screens” that embody the random characteristics of the transverse coherence for each component. E0x and E0y describe the deterministic part of the Gaussian beam fields and are generally given by8]Eqs. (2) and (3), Wαβ takes the form
In terms of the beam field, μαβ is known as the complex coherence factor . In our implementation, μαβ physically represents the degree of correlation of Tα and Tβ and is defined as
The degree of correlation is Gaussian in form
The task is to generate random realizations of Tx and Ty given the correlation parameters |Bxy|, δxx, δyy, and δxy. A useful relationship is that Gaussian random sequences X1 and Y1 with correlation coefficient Γ can be computed using Y1 = ΓX1 + X2, where X1 and X2 are independent Gaussian random sequences, and Γ is the correlation coefficient with value 0 ≤ Γ ≤ 1. In the spatial domain, the screen realizations can be synthesized by10] except for introducing a few new parameters.
In practice, the screens are typically generated by filtering complex random arrays in the spatial frequency domain (f x, f y) :
3. PS approach
The PS approach to creating the EGSM beam is attractive because a common phase-only device, such as a spatial light modulator (SLM), can be used in each polarization leg to implement the screens. In addition, the overdetermined relationship between the screen and beam parameters allows for some screen design flexibility. Either the real or imaginary part can be extracted from Tα for use as a phase screen ϕαPS, therefore, ϕαPS = Re(Tα) or Im(Tα). The associated autocorrelation and cross-correlation functions of the phase-only transmittance functions are 
If the following conditions are trueEqs. (14) and (15) are approximately Gaussian of the form in Eq. (8). Equating Eqs. (14) and (15) with Eq. (8), the following equations can be obtainedEquations (18)‒(21) are essentially identical to Eqs. (28) in  however Γ in this work is well-defined and the equations are provided for reference as they are critical to the following derivations. There are 4 equations that include 5 simulation parameters: , ,,and Γ; and 4 source parameters: δxx, δyy, δxy and |Bxy|. The next task is to explore the limiting conditions on the relationships between the simulation parameters and the source parameters. Substituting Eqs. (18) and (19) into Eqs. (20) and (21) yields
It is apparent from Eqs. (22) and (23) that both Γ and |Bxy| can be defined in terms of the ratios δxx/δxy = Rx and δyy/δxy = Ry. Hence, they can be simplified as
To help with further simplification, we introduce the following inequation where the right side of Eq. (24) is used but the sum is changed to a difference,
A useful approach to study the interrelationship of the parameters is to consider the valid range of |Bxy| as a function of the screen parameter values. Three situations are considered: a) Rx < 1 and Ry < 1, b) Rx < 1 and Ry = 1 and c) Rx < 1 < Ry and Ry < 1/Rx. Since Rx and Ry are interchangeable, the scenarios with x and y reversed are also covered. Figure 2 illustrates representative examples with (a) Rx = 0.75 and Ry = 0.90, (b) Rx = 0.75 and Ry = 1.0, and (c) Rx = 0.75 and Ry = 1.1, when both and are greater than or equal to 6 (black solid), 9 (red dash) and 12 (blue dot). In each plot, the shaded areas underneath the lines indicate the valid |Bxy| values for the stipulated and values. For example, in Fig. 2(b), with the indicated Rx and Ry values and appropriate choice of and , correlation peak values ranging from about 0 ≤ |Bxy| ≤ 0.27 can be modeled with a value of Γ that can range from 0 0.75 ≤ Γ ≤ 1. Figure 2(a) shows that smaller Rx and Ry values (suggestive of a larger δxy value), place a significant upper limit on the value of |Bxy| that can be modeled. On the contrary, when the product RxRy is near 1 (implying δxy2 is similar in value to δxxδyy) then the attainable upper limit of |Bxy| is increased [Fig. 2(c)]. Generally speaking, the available range of parameter values is more limited when modeling larger |Bxy| as compared with modeling smaller |Bxy|.
4. CS approach
In contrast to the PS approach, the relationships for the CS parameters are deterministic. It is also more difficult to implement a complex valued screen in the real world, for example with a SLM, although it is certainly possible . However, the CS approach has no approximation requirement, such as indicated by Eq. (16), to produce the Gaussian correlation functions and the analytic relationships between parameters are relatively simple. The CS is generated directly from the complex transmittance function, or ϕαCS = Tα and the corresponding autocorrelation and cross-correlation functions are
Again, Eqs. (31)‒(34) are substantially the same as Eqs. (37) in  except for the use of the symbols andto clarify the relationships between the PS and CS approaches. Equation (30) restricts the spatial variances in both x and y directions so that 3 simulation parameters, and Γ remain. This implies that the four source parameters δxx, δyy, δxy, and |Bxy| in Eqs. (31)–(34) cannot be chosen independently – one of the three rms widths is determined by the other two. Applying the ratios Rx and Ry again, the following relationships are derived:
Equation (36) indicates that the beam correlation peak value is directly proportional to the correlation coefficient of the random sequences, as well as the rms width ratios.
5. Simulation design example
To illustrate the beam design and simulation approach, we consider the EGSM beam presented as case II in . The parameters for this beam are listed in Table 1. Note that although the deterministic phase values θx and θy of the two component fields are different. This has no effect on the resulting beam intensity patterns. The last 4 columns are the beam parameters that are used in the simulation for both the PS and CS approaches. In the original beam definition of , the cross correlation width was selected as δxy = 0.1714 mm. However, this value is not consistent with Eq. (35) and the values given for δxx and δyy. To satisfy the requirement for the CS approach, the cross correlation width is assigned δxy = = 0.1554 mm.
For the PS approach, Fig. 3(a) is generated for the parameters given in Table 1 along with the assumption that and ≥ π2. The green solid line in Fig. 3(a) shows the valid range is Γ [0.9982, 1] for |Bxy| = 0.15. If we choose Γ = 1 and solve for the spatial standard deviation values of the random phase screens (,) we find two solutions pairs: (17.21, 15.39) or (39.38, 38.36). On the other hand, if we choose Γ = 0.9985, which is at the other end of the valid range, then the solution pairs are (19.48, 17.85) or (26.00, 24.71). In fact, there is a continuum of choices available for the PS screen parameters, but we generally find that seeking to make the screen width parameters and values closer to the initial source beam width parameters and provides better simulation design flexibility in terms of pixel number, grid size, and computation time trade-offs. In this case we choose to use Γ = 1 and (,) = (17.21, 15.39).
Figure 3(b) illustrates the relationship between |Bxy| and Γ for the CS approach and the specific choice of Γ = 0.1503 necessary for |Bxy| = 0.15. Table 2 shows our choice of PS screen parameters and the required CS screen parameters for the EGSM simulation results that are presented in section 6.
Prior to presenting the simulation results, it is instructive to illustrate the wider applicability of the PS approach. Consider the value of δxy = 0.1714 mm that was originally proposed in ref . As noted above, this value does not provide a valid definition for the EGSM beam in Table 1 for simulation with the CS approach. However, when applying the PS approach, Fig. 4 shows the validity relationship between |Bxy| and Γ. This beam belongs to the scenario where both Rx and Ry are less than 1 and the attainable |Bxy| has a relatively small upper limit of 0.1732. But the desired |Bxy| = 0.15 is under this value and the available range for the screen correlation is Γ [0.82, 0.85], which is marked as the green solid line. Thus, it is possible to use the PS approach to simulate this beam.
6. Simulation results
In this section we show the results of modeling the EGSM beam of Table 1 in the source plane using the PS and CS screens. The screens were created using a grid of 1024 × 1024 pixels corresponding to a physical area of 15mm × 15mm. The pixel number and physical size were chosen to ensure that the array physical side length was at least 5 times the maximum value of any of the width parameters (σx, σy,,or) and, meanwhile, the pixel sample interval is small enough to provide 10 samples for the minimum width parameter value . To display the intensity pattern of resulting beam, we use the Stokes parameters. Analytically, the Stokes parameters are obtained by [11,16]
Figure 5 presents the PS, CS, and analytical beam results for each component of the Stokes vector. A relative intensity scaling is used where the same color scale is applied to all plots. Both the PS and CS Stokes parameters were obtained by averaging 20,000 realizations. The agreement with the analytic beam definition (theory) is excellent. We note that the CS approach generally seems to take more realizations to converge to the smooth analytical predictions. The beam intensities are shown in the S0 frames (top row) and a preference for horizontal polarization can be observed in S1 (second row). In the third row, the yellow in the center of S2 indicates a slight preference for 45° linear polarization, which is a result of the small correlation between the components |Bxy| = 0.15. The red in S3 (representing a negative value) in the last row indicates a slight left-hand circularly polarization preference. Although the results shown in Fig. 5 represent the beam in the source plane, propagation of the beam can be simulated by applying the fields created for each screen realization to a numerical propagation algorithm and then applying the averaging .
We have introduced a practical approach for generating random phase-only screens and complex screens for modeling EGSM beam sources. These results can be applied in numerical wave optics simulations or in laboratory experiments where the screens can be implemented on a device such as spatial light modulators. Our work provides a more direct, deterministic approach to EGSM beam modeling design than a previous method that relies on numerical optimization based on analytic realizability conditions. Our results can also be easily applied as a test of whether a set of selected EGSM parameters is physically realizable. The approach incorporates a common numerical recipe for the generation of correlated random Gaussian sequences. Average autocorrelation and cross-correlation functions for the screens were derived and the exact relationships were established between the screen parameters and the defined EGSM source parameters. Both the PS and the CS methodologies were demonstrated in a wave optics simulation framework for an example EGSM source. The Stokes image results show excellent consistency with the analytic beam definitions. There are several differences between the implementation and application of the PS and CS methods. For example, in the CS method the beam correlation peak value |Bxy| is directly proportional to the correlation coefficient Γ of the random sequences whereas this correlation relationship is more flexible in the PS method and is a function of the phase variance values and . In general, the PS method is more flexible than the CS in terms of the range of combinations of beam parameter values that can be modeled.
Air Force Office of Scientific Research (AFOSR) through the Multidisciplinary Research Program of the University Research Initiative (MURI), (FA9550-12-1-0449).
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