Generation of Vector Partially Coherent Optical Sources Using Phase-Only Spatial Light Modulators

A VPCS generator that includes a laser that emits coherent light; an interferometer consisting of PBSs to split the laser light into its vertical and horizontal polarization components; first and second SLMs that respectively control the vertical and horizontal polarization components; and a control system communicatively coupled to the first and second SLMs to adjust beam shape and coherence without physically moving or removing optical elements in the apparatus.

As one might expect, considering their many uses, optical systems to generate vector partially coherent sources have also been developed. The most straightforward and therefore popular design uses a Mach-Zehnder interferometer (MZI) consisting of polarizing beam splitters (PBSs) to split light from a common source into its vertical and horizontal polarization components [5,6,11,[34][35][36][37][38][39]. In each leg of the MZI, the amplitude (beam shape) and coherence of each field component are controlled using absorbing (amplitude) filters and ground-glass diffusers [6,[30][31][32][33]35,36,40,41] or spatial light modulators (SLMs) [5,11,34,37,38,42,43], respectively. While intuitive, the use of separate optical components to control beam shape and coherence complicates the optical setup (alignment, system footprint or size, et cetera) and limits the types of vector partially coherent sources that can be generated (the system's flexibility).
Designs that rely on ground-glass diffusers to affect coherence are generally limited to producing Schell-model sources [2,5,44,45]. The use of SLMs can ameliorate this limitation; however, to the authors' knowledge, they have never been used to produce vector non-Schell-model sources. The most limiting aspect of existing designs is the use of amplitude filters to control beam shape. Amplitude filters adversely affect the flexibility of the system, considering that every new source requires a new set of filters.
In this paper, a simple and flexible design for generating vector partially coherent sources is presented. Like existing designs, the proposed system uses a MZI where, in each interferometer leg, the beam shape and coherence of the field's horizontal and vertical polarization components are controlled separately. In contrast with existing designs, the apparatus presented here uses only SLMs to control coherence and beam shape. This improvement permits many different types of sources to be generated without physically changing the apparatus.
In Sec. II, the system is presented and described in detail. Section III presents the theory necessary for understanding how the apparatus creates vector partially coherent sources. Section IV demonstrates the flexibility of the proposed design by experimentally producing both vector non-Schell-model and Schell-model sources.
The experimental results are compared to theoretical predictions to validate the system. Lastly, key aspects of the proposed approach, which must be considered when designing a system for a specific application, are discussed.

II. APPARATUS DESCRIPTION
A schematic of the vector partially coherent source generator is shown in Fig. 1. Light is emitted from a HeNe laser (632.8 nm) and then expanded. The laser light then traverses a half-wave plate (HWP) before being split along two paths using a PBS. The HWP is used to rotate the angle of polarization (AOP) of the linearly polarized laser light to 45°such that the PBS splits the power equally between the vertical and horizontal legs-so named because of the polarization state that is controlled on that leg.
After passing through the PBS, the light in both legs is incident on reflective, phase-only SLMs. The SLMs used here are Boulder Nonlinear Systems (BNS) model P512-0635 SLMs [46]. Since the BNS SLMs control only vertically polarized light, a HWP is used immediately after the PBS to rotate the vertical leg's AOP 90°. Note that all beam shaping, phasing, and relative amplitude control on the vertical and horizontal legs is performed using the SLMs. This field control is accomplished using the modified phase-screen (MPS) technique [47], which is briefly reviewed in the next section.
The light reflected from the SLMs is scattered into multiple diffraction orders. Here, the desired vertical-and horizontal-leg field instances are produced in the first diffraction orders. Thus, spatial filters (4f system plus an iris) are utilized on each leg to isolate the desired first orders.
The physical layout of the system results in the vertical leg being 100 mm longer than the horizontal leg when measured from the SLMs (the same difference is approximately 45 mm when measured from the laser). To place the output planes of both spatial filters at L3, the horizontal and vertical leg lenses (L1 and L2, respectively) have focal lengths of 350 and 375 mm, respectively.
Note that the coherence length of the source must be considered when designing the system. If the optical path difference (OPD) between the horizontal and vertical legs is greater than the coherence length of the source, only partially coherent beams with diagonal cross-spectral density (CSD) matrices can be generated. The coherence length of the HeNe laser used here is approximately 300 mm; therefore, the 45-mm difference between the legs of the MZI is not a problem.
After passing through the spatial filters, the light in the vertical and horizontal legs is recombined using a PBS. Note that the light in the horizontal leg first transits a HWP to rotate its polarization state from vertical (the SLM control state) to horizontal. The recombined light is then incident on L3-a 1500-mm lens located at the effective source plane of the apparatus. The lens produces the farzone irradiance pattern of the field, which is subsequently recorded by a Lumenera Lw135RM camera [48] placed at the focus of L3.

III. METHODOLOGY
A vector partially coherent source is completely described by the elements of its CSD matrix:  where E i is the ith component of the electric field, W ij is the CSD function of the ith and jth field components, ρ ¼xx þŷy, and ω is the radian frequency [2,5]. Note that, in expressing W as a function of a single frequency ω, it has been assumed that the random optical field is wide-sense stationary. Hereafter, the dependence of the CSD matrix, the CSD functions, and the optical fields on ω is suppressed.
In this work, W ij 's of the form are germane. In Eq. (2), S i is the spectral density of the ith field component and μ ij is the cross-correlation function of the ith and jth components. It is required that μ ij ¼ μ Ã ji [2,5]. If μ ij can be expressed solely as a function of ρ 1 − ρ 2 , then W ij in Eq. (2) takes the Schell-model form [2,5,44,45].
To demonstrate the flexibility of the apparatus in Fig. 1, both a non-Schell-model and a Schell-model source are generated (the results are shown in Sec. IV). For now, the analysis continues with the general CSD function given in Eq. (2).

A. Modified phase-screen technique
An instance of a MPS field takes the form [47] E MPS ðρÞ ¼x ffiffiffiffiffiffiffiffiffiffiffi S x ðρÞ p exp (jfarg½T x ðρÞ þ α x g) þŷ ffiffiffiffiffiffiffiffiffiffiffi S y ðρÞ q exp (jfarg½T y ðρÞ þ α y g); ð3Þ where j ¼ ffiffiffiffiffiffi −1 p , α i is the phase of the ith field component, and T i is the complex transmittance screen for the ith component [49][50][51]. Taking the vector autocorrelation of Eq. (3) yields Comparing Eq. (4) to Eq. (2) reveals the following desired condition: which, in general, is never satisfied. However, it can be shown, by numerically computing χ ij using the joint probability density function of speckle phases [52], that χ ij ≈ μ ij , such that the MPS partially coherent source is practically indistinguishable from the true source. This analysis is presented in Ref. [53] and not reproduced here for the sake of brevity. Methods to synthesize T x and T y to produce the desired μ xx , μ yy , and μ xy 's are presented next.
B. Screen synthesis

Non-Schell-model source
It is now convenient to assume forms for S i and μ ij . An example of a vector non-Schell-model source, which to the authors' knowledge has never been physically generated before, is an electromagnetic nonuniformly correlated (ENUC) beam [5,54]. Its CSD matrix elements take the form where A i and σ i are the amplitude and the rms width of the ith field component, respectively. Also, in Eq. (6), δ ij is the cross-correlation width, B ij is the complex cross-correlation coefficient, and γ i is a real, two-dimensional vector that shifts the maximum of the correlation function away from the origin. Note that In addition to these requirements, the ENUC source must also satisfy the realizability conditions given in Refs. [5,54].
Synthesizing screens that will produce a desired ENUC source can be accomplished using the Choleskyfactorization approach [55,56]. Assuming an N y × N x computational grid, an instance of T i is created by where r i is an ðN y N x Þ × 1 vector of zero-mean, unitvariance, circular complex Gaussian random numbers and T i is the ðN y N x Þ × 1 complex screen. Lastly, R i is an ðN y N x Þ × ðN y N x Þ lower triangular matrix formed from the Cholesky decomposition of μ ii , namely, where μ ii is the ðN y N x Þ × ðN y N x Þ matrix created by evaluating μ ii [the exponential on the second line of GENERATION OF VECTOR PARTIALLY COHERENT … PHYS. REV. APPLIED 6, 064030 (2016) 064030-3 Eq. (6)] at all combinations of points ðx 1 ; y 1 Þ and ðx 2 ; y 2 Þ making up the discrete grid.
Generating T x and T y using Eq. (7) will produce an ENUC source with the desired μ xx and μ yy . The statistical relationship between T x and T y , particularly the crosscorrelation of T x and T y , determines μ xy : where † is the Hermitian transpose. The moment hr x r † y i ¼ 2jB xy j, simplifying Eq. (9) to The right-hand side of Eq. (10) should equal the Cholesky factorization of jμ xy j. In general, it does not; however, there are two cases where it does: (1) If B xy ¼ 0, δ xy is irrelevant, T x and T y are statistically independent, and one is free to choose δ xx and δ yy .
Using the Cholesky-factorization approach outlined above, any vector partially coherent source can be produced. Unfortunately, this flexibility comes at a great cost. Computing Cholesky factors is very computationally intensive-the number of operations is Oðn 3 Þ [56]. In addition, the matrix μ ii is typically very large, requiring a great deal of computer memory to store. For example, synthesizing a screen for a 512 × 512 SLM (the same size as the BNS SLMs used here) requires a staggering 262, 144 × 262, 144 μ ii . To produce T x and T y for the ENUC experiments (the results are presented below), μ xx and μ yy are calculated on 128 × 128 down-sampled versions of the horizontal-and vertical-leg SLMs. From these downsampled μ xx and μ yy , R x and R y are computed. The screens are then synthesized using Eq. (7), reshaped, and interpolated to 512 × 512.

Schell-model source
The Cholesky-factorization approach discussed in the previous section can be used to synthesize screens to produce any type of partially coherent source; however, the approach is very computationally intensive. If the partially coherent beam is a Schell-model source, then a simpler and more computationally efficient technique can be used to synthesize T x and T y .
An example of a vector Schell-model source is an electromagnetic multi-Gaussian Schell-model (EMGSM) beam [5,57]. To the authors' knowledge, a general EMGSM beam has never been physically produced before.
The CSD matrix elements of an EMGSM source take the form [5,57] where the normalization constant C 0 is The other symbols were defined previously. The same requirements on δ ij and B ij stipulated above apply to EMGSM sources as well. In addition, EMGSM sources must also satisfy the realizability conditions given in Refs. [5,57].
Since an EMGSM beam is a Schell-model source, T x and T y can be generated using the Monte Carlo spectral method. This technique has been described in the literature many times [5,[49][50][51]58,59]; therefore, only the key relations are included here. An instance of T i can be synthesized by where k, l are the discrete spatial indices of the screen, p, q are the discrete spatial frequency indices, L x and L y are the width and height of the grid in meters, and r i is an N y × N x matrix of zero-mean, unit-variance, circular complex Gaussian random numbers. Lastly, Φ ii is the spatial power spectrum of T i (the Fourier transform of μ ii ), namely, where f ¼xf x þŷf y is the spatial-frequency vector. Equation (13) is in the form of a discrete inverse Fourier transform; thus, T i can be synthesized using the fast-Fourier-transform (FFT) algorithm.
Synthesizing T x and T y using Eq. (13) will produce an EMGSM source with the desired μ xx and μ yy . To determine the form of μ xy , one must take the cross-correlation of Eq. (13), viz., The moment in Eq. (15) evaluates to 2jB xy jδ½p 1 − p 2 × δ½q 1 − q 2 , where δ is the discrete Dirac delta function. Simplifying Eq. (15) further yields The expression jB xy j ffiffiffiffiffiffiffiffiffiffiffiffiffiffi Φ xx Φ yy p should equal the crosspower spectrum, namely, Like the ENUC source discussed above, jB xy j ffiffiffiffiffiffiffiffiffiffiffiffiffiffi Following the above analysis, any vector Schell-model source can be produced. Depending on the forms of μ xx , μ yy , and μ xy , one may have more freedom in choosing B xy , δ xx , δ yy , and δ xy than in the EMGSM case above. Two such sources are the electromagnetic Gaussian Schellmodel (EGSM) [50] and EGSM vortex beams [32].

C. Phase-only MPS field
Recall that in the apparatus in Fig. 1, all phasing and amplitude control is accomplished using phase-only SLMs. In the preceding sections, two methods for synthesizing T x and T y to produce partially coherent sources with the desired μ xx , μ yy , and μ xy are discussed. This analysis handles the MPS field component phases that are ultimately commanded to the horizontal-and vertical-leg SLMs. What remains is a discussion of how the MPS field component amplitudes, namely, ffiffiffiffi ffi S x p and ffiffiffiffi ffi S y p , are handled.
Amplitude can be controlled using a phase-only SLM by creating a sawtooth phase grating which produces the desired field in the first diffraction order. By manipulating the height of the grating, the field's amplitude can be precisely controlled. The expression relating sawtooth height to the field in the first diffraction order was derived previously [51,60,61]: where h is the height of sawtooth in waves and sincðxÞ ¼ sinðxÞ=x. Assumed in this relation is that more than four SLM pixels compose each sawtooth.
The vertical-and horizontal-leg phases commanded to the SLMs are where h x and h y are found by solving G is the function that forms the two-dimensional sawtooth phase grating, and F is the function that creates a twodimensional P. The purpose of F is to remove the "phase aberration" caused by G. Note that exp ½jðα x − α y Þ ¼ exp ½j argðB xy Þ. The desired x and y components of the MPS field are produced in the horizontal and vertical legs' first diffraction orders, respectively. It should be noted that, in general, the heights of the sawteeth making up the SLM gratings vary spatially; however, all of the sawteeth contain the same number of SLM pixels. The SLM gratings can be formed in any direction-horizontally, vertically, or diagonally. Figure 2 shows the screen synthesis process. Here, a screen which produces a single instance of E y of an EMGSM source is synthesized.
Step 1 shows the spectral density S y and correlation function μ yy of the source. The sawtooth grating is determined by the spectral density; thus, steps 2, finding the grating heights h y [Eq. (20)], and 3, the functions G and F, are grouped under S y . The phase of the screen is formed from the correlation function-in this case, via the power spectrum. Hence, steps 4, the spatial power spectrum Φ yy [Eq. (14)], and 5, creating an instance of T y [Eq. (13)], are grouped below μ yy . Lastly, the screen θ V is produced in step 6 by using Eq. (19). The same process is used to produce an instance of E x , i.e., to synthesize the horizontal-leg screen θ H .
Note that if jB xy j ≠ 0, then T x and T y must be synthesized from correlated r x and r y [see Eq. (13)]. Since r x and r y are Gaussian, this task is a relatively easy one. Most of the popular mathematical software packages have correlated Gaussian random-number generators. Correlated Gaussian random numbers can also be produced by linearly transforming independent sequences of Gaussian random numbers [62].

IV. VALIDATION
A. Experimental procedure and data processing ENUC and EMGSM sources are produced using the apparatus in Fig. 1 to demonstrate the flexibility and utility of the setup. The measured irradiances, I x and I y , from 5000 ENUC and EMGSM fields are used to compute the normalized spectral densities, and correlations of irradiance,Γ ii ðρ 1 ; 0Þ ¼ Γ ii ðρ 1 ; 0Þ maxfΓ ii ðρ 1 ; 0Þg where Γ ij ðρ 1 ; ρ 2 Þ ¼ hI i ðρ 1 ÞI j ðρ 2 Þi. Each of the 5000 I x 's and I y 's are flat fielded and registered before computing the above moments. The experimentalS i andΓ ij are then compared to the corresponding theoretical expressions (discussed below) to validate the proposed approach. It should be noted that I x and I y are physically separated at the camera or detector (not overlapped as shown in Fig. 1). The intention of this separation is to make calculating the spectral densities and correlations of irradiance more convenient. The experimental ENUC and EMGSM source parameters are reported in Table I. Here, theS x andS y maxima occur off axis, shifted in the x and y directions, respectively (see Table I). EMGSM sources produce far-zoneS i 's with flat-topped profiles. The parameter M determines the flatness ofS i [5,57].
In the experiments, the horizontal-and vertical-leg SLM gratings have a period of eight SLM pixels per sawtooth and are applied in both the x and y directions (i.e., diagonally). The BNS SLMs used in the experiments have a fill factor of 83.4% [46]. The dead space between the SLM pixels produces bright, sinclike, far-zone patterns coinciding with the zeroth diffraction orders in the horizontal and vertical legs of the apparatus. Using diagonal gratings moved the desired first orders away from the zeroth orders' bright sidelobes, thereby minimizing corruption of the resulting fields.

B. ENUC and EMGSM theoretical CSD matrices
The analytical expressions for the CSD matrix elements of an ENUC source corresponding to the experimental setup are where f ¼ 1.5 m, λ ¼ 632.8 nm, and W ij in the integrand is the ENUC CSD function given in Eq. (6). The above integrals cannot be evaluated in closed form; however, they can be computed using the FFT algorithm. As stated above, an interesting characteristic of ENUC sources is that they exhibit self-focusing in the near field [5,54]. To demonstrate this effect in the experiments, a R ¼ −3 m phase curvature is applied to the horizontaland vertical-leg SLMs, in addition to the ENUC phases given in Eq. (19).
Unlike ENUC sources, the theoretical EMGSM CSD matrix elements observed at the focus of a lens are easily derived in closed form: The spectral densities are easy to compute using Eqs. (23) and (24), i.e., S i ðρÞ ¼ W ii ðρ; ρÞ. Since the fields are Gaussian distributed, the theoretical correlations of irradiance can be expressed in terms of Eqs. (23) and (24) by using the Gaussian-moment theorem [6,30,41,52]: C. Results 5 show the near-field normalized spectral density and correlation of irradiance results for the ENUC source, respectively. Figures 6 and 7 show the corresponding far-field results for the EMGSM source. The first row in  Except for a few minor discrepancies, e.g., Figs. 5(f), 5(l), and 6(c), the agreement between the experimental and theoretical results is excellent. These results validate the proposed approach. It should be noted that EGSM vortex [32] and electromagnetic Bessel-Gaussian Schell-model beams [11] are also produced as part of this study. The results are similar to those for ENUC and EMGSM sources; they are omitted here for the sake of brevity.

D. argðB xy Þ
One parameter that has been difficult to control is argðB xy Þ, which is required if one desires to generate elliptically polarized sources. Precise control of this term requires that the OPD between the horizontal and vertical legs of the apparatus be known and, most importantly, stable. The OPD between the legs can vary significantly due to slight variations in temperature or small vibrations. Unfortunately, we are not able to stabilize the OPD of our system. In our most recent attempt, we try to actively correct the system OPD using a three-bin phase-shifting interferometer and a variable retarder operating in a closed loop. The OPD drift is too fast to correct. Subsequent measurement of our system OPD finds that it varies randomly over 0.5 waves (roughly 300 nm) and faster than 120 Hz. Our inability to control argðB xy Þ is a limitation of our equipment, not the proposed approach. The control of argðB xy Þ is left to future work.

V. DESIGN CONSIDERATIONS
Before concluding, it is worth discussing some key aspects of the proposed approach which must be considered when designing or building a system like that in Fig. 1 for a specific application. Two of these aspects have already been discussed-controlling the OPD between the horizontal and  vertical legs of the MZI and the coherence length of the light source. In addition to these factors, the size D, pixel pitch Δ, and type of SLM also play an important role in the types of sources that can be generated.

A. SLM size D
Clearly, sources with correlation or beam widths (δ and σ, respectively) greater than D cannot be accurately generated. Even sources with δ, σ ≈ D will have correlation functions μ or spectral densities S that are driven by the shape of the SLM's active area. Therefore, D sets the effective "coherent" and "maximum" source sizes. In practice, both S and μ should "fit" comfortably on the SLM face. An empirically derived criterion which ensures this fit is that both μ and ffiffiffi S p should be no greater than 1% of their maximum values at the edges of the SLM.

B. SLM pitch Δ
To adequately sample or represent μ requires Δ < δ=5, where δ, in this context, is the 1=e point or first zero crossing of μ (whichever is appropriate). Sources with correlation radii δ < 5Δ are not accurately produced, making 5Δ the effective "incoherent" limit of the SLM. Similarly, adequate sampling of S generally requires that Δ < σ=ð5LÞ, where σ is the 1=e point or first zero location of ffiffiffi S p (whichever is appropriate) and L is the sawtoothgrating period. This requirement is more stringent than the correlation condition because SLM pixels are binned to form the sawtooth grating (eight pixels are used in this work). Thus, the amplitude of the desired source is generally harder to produce than is the phase [47,51]. The requirement that σ > 5LΔ sets the effective "minimum" source size of the SLM.
The grating period L affects system performance in two competing ways. A large L better approximates a continuous sawtooth and results in more precise amplitude control. Recall that the expression relating sawtooth-grating height to the field in the first diffraction order [Eq. (18)] requires that L > 4. L also affects the angular separation of the diffraction orders: where m is the order number [63]. Clearly, L and θ m are inversely related. L must therefore be chosen so that the field's amplitude can be accurately produced, while also providing a large enough angular separation of the orders such that the desired first order can be passed by a spatial filter with little corruption from the m ≠ 1 orders.

C. SLM type
Many different types of SLMs exist [63]. The two most popular are liquid-crystal (LC) SLMs (the type used here) and MEMS SLMs (also known as segmented deformable mirrors). Both have pros and cons which are briefly discussed here. LC SLMs generally possess a large number of small pixels-the current state of the art is a 10 megapixel (4094 × 2464) SLM with a 3.74-μm pitch [64]. The sheer number and size of the pixels in modern LC SLMs permits practically any partially coherent source to be generated. The major drawback of LC SLMs is speed, with the fastest operating at about 1 kHz. Note that most LC SLMs operate well below 1 kHz; the 10-megapixel SLM mentioned previously operates at 24 Hz [64].
MEMS SLMs, on the other hand, generally possess a much smaller number of much larger pixels (or actuators). The current state of the art in MEMS SLMs is a 4092actuator (64 × 64) device with a 400-μm pitch [65]. When one considers the importance of SLM size and pitch in producing partially coherent sources, it is clear that MEMS SLMs cannot match the range of LC SLMs. MEMS SLMs, however, are orders of magnitude faster than their LC counterparts, e.g., the 4092-actuator MEMS device mentioned previously operates at 10 kHz [65].
Choosing which type of SLM to employ in the apparatus in Fig. 1 comes down to the importance of speed. If speed is of lesser importance than flexibility in the application, LC SLMs are the best choice. If, on the other hand, speed is paramount, MEMS SLMs are the better option.

D. Design optimization
What makes the apparatus in Fig. 1 very powerful is the ability to generate many different types of vector partially coherent sources using phase-only SLMs and a simple optical setup. The range of sources that can be produced is much greater than existing systems which use separate optical components to manipulate beam shape and coherence. The most significant drawback of the proposed approach over existing partially coherent source generators, particularly those which use rotating ground-glass diffusers and amplitude filters, is speed. As stated above, if speed is paramount, MEMS SLMs can be used instead of LC SLMs at the expense of flexibility.
It should be noted that the apparatus in Fig. 1 is built for proof-of-concept purposes only, with no particular application in mind. Thus, the total size of the system (its footprint) has not been optimized. The largest components (in terms of physical size) in Fig. 1 are the spatial filters. Long-focal-length plano-convex lenses are used here solely for ease of setup, alignment (i.e., to minimize aberrations), and cost. By changing the layout of the system and employing standard optical design techniques [66,67], the entire apparatus could be made to fit within a 1-m 2 area.

VI. CONCLUSION
An alternative design for generating vector partially coherent sources is presented. In contrast to existing designs which use separate optical components to manipulate beam shape and spatial coherence (commonly, amplitude filters and ground-glass diffusers or SLMs), the system described here uses only SLMs to control these aspects of the field. In addition to having a simpler optical setup, this improvement significantly increases the flexibility of vector partially coherent source generators by permitting many different types of sources to be produced without physically changing the apparatus.
System design is thoroughly discussed in Sec. II. Section III presents the theory underpinning the design, namely, the MPS technique, screen synthesis for both Schell-model and non-Schell-model sources, and phaseonly field control. Section IV demonstrates the effectiveness and flexibility of the proposed system by producing ENUC and EMGSM sources, neither of which, to the authors' knowledge, have been physically generated before. The apparatus is validated by comparing experimental ENUC and EMGSM results to theoretical predictions. The agreement between theory and experiment is excellent. Lastly, key aspects of system design-the effects of SLM size, pixel pitch, and SLM type on partially coherent source generation-are discussed.
The system design and subsequent approach for generating vector partially coherent sources presented in this paper will be useful in any application where control over beam shape, spatial coherence, and polarization is required. These applications include-but are not limited tofree-space optical communications, directed energy, remote sensing, manufacturing, and medicine.

ACKNOWLEDGMENTS
This research was supported in part by an appointment to the Postgraduate Research Participation Program at the Air Force Institute of Technology (AFIT) administered by the Oak Ridge Institute for Science and Education through an interagency agreement between the U.S. Department of Energy and AFIT. The views expressed in this paper are those of the authors and do not reflect the official policy or position of the U.S. Air Force, the Department of Defense, or the U.S. Government.