Fresnel spatial filtering of quasihomogeneous sources for wave optics simulations

High-spatial-frequency optical fields or sources are often encountered when simulating directed energy, active imaging, or remote sensing systems and scenarios. These spatially broadband fields are a challenge in wave optics simulations because the sampling required to represent and then propagate these fields without aliasing is often impractical. To address this, two spatial filtering techniques are presented. The first, called Fresnel spatial filtering, finds a spatially band-limited source that, after propagation, produces the exact observation plane field as the broadband source over a user-specified region of interest. The second, called statistical or quasihomogeneous spatial filtering, finds a spatially band-limited source that, after propagation and over a specified region of interest, yields an observation plane field that is statistically representative of that produced by the original broadband source. The pros and cons of both approaches are discussed in detail. A wave optics simulation of light transiting a ground glass diffuser and then propagating to an observation plane in the near-zone is performed to validate the two filtering approaches. © The Authors. Published by SPIE under a Creative Commons Attribution 3.0 Unported License. Distribution or reproduction of this work in whole or in part requires full attribution of the original publication, including its DOI. [DOI: 10.1117/1.OE.56.8.083107]


Introduction
Wave optics simulations are a critical part of optical system design and are used in practically every discipline, including light propagation in random or nonlinear media, rough surface scattering, guided-wave propagation, etc. [1][2][3] The simulation of high-spatial-frequency or spatially broadband optical fields is generally a challenge because the sampling needed to represent and then propagate these fields (typically using fast Fourier transforms) on a discrete grid without aliasing can be prohibitive. The literature is replete with books and papers on this topic. [1][2][3][4][5][6][7] The most intuitive and popular approach for dealing with spatially broadband fields or sources is to spatially filter the source (or the propagation kernel) prior to propagation. 1,3,[7][8][9][10] While the details may differ, the goal of all spatial filtering approaches is the same: to find a spatially band-limited (or near-band-limited) source that produces the exact same observation field as the original, spatially broadband source over a specified region of interest (RoI).
Although spatial filtering is a common optical simulation technique, to the authors' knowledge, it has never been theoretically investigated starting from the Fresnel propagation integral. This analysis is presented in Sec. 2. The result, called Fresnel spatial filtering, is an integral expression that yields the spatially filtered source given the original field. This integral is discussed at length.
In only a few cases, the Fresnel filtering integral can be evaluated in closed form. Thus, a new statistical filtering method is also presented, which is broadly applicable in simulations involving random optical fields, e.g., scattering from rough surfaces, transmission through diffusers, transmission through turbid (highly scattering) media, etc. These situations are commonly encountered in optical phased array systems, high-energy laser weapons, active imaging, remote sensing, and medicine.
Both the field and statistical filtering approaches are validated in Sec. 3. For this purpose, the near-zone propagation of light transmitted by a ground glass diffuser is simulated. The results obtained using both approaches are compared to those of the original, spatially broadband field. Last, this paper is summarized, and a brief discussion of possible applications is provided.

Fresnel Spatial Filtering
The field in the observation plane given the field in the source plane is E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 1 ; 3 2 6 ; 2 2 3 U obs ðρ; zÞ ¼ where k ¼ 2π∕λ, λ is the wavelength, ρ ¼xx þŷy is the observation vector, and ρ 0 ¼xx 0 þŷy 0 is the source vector. 11 Applying the transform to both sides of Eq. (1) and simplifying yields E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 3 ; 6 3 ; 7 4 1 jλz expð−jkzÞ The integral over ρ in Eq. (3) is equal to λ 2 z 2 δðρ 0 − ρ 0 0 Þ. Thus, the second line of Eq. (3) evaluates to E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 4 ; 6 3 ; 6 1 7 λ 2 z 2 U src ðρ 0 0 ; 0Þ exp jk 2z ρ 0 02 : Solving the resulting expression for the source plane field produces E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 5 ; 6 3 ; 5 5 0 U src ðρ 0 0 ; 0Þ ¼ This equation is the inverse Fresnel transform of Eq. (1) and yields the field in the source plane given the field in the observation plane.
The goal here is to find the source plane field that produces the exact observation plane field over a defined RoI. Therefore, let the windowed observation plane field be E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 6 ; 6 3 ; 3 8 3 U obs;w ðρ; zÞ ¼ WðρÞU obs ðρ; zÞ; (6) where W is a window function, e.g., rectangle, circle, etc, which represents the RoI. Substituting Eq. (6) into Eq. (5) produces E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 7 ; 6 3 ; 3 1 9 U src;f ðρ 0 0 ; 0Þ¼ where U src;f is the desired, spatially filtered source. Substituting Eq. (1) into Eq. (7) and simplifying yields The integral on the last line of Eq. (8) is the Fourier transform of the window function W represented hereafter asW.
The final result is E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 0 9 ; 3 2 6 ; 7 4 1 The Fresnel filtered source is the true source multiplied by a quadratic phase factor correlated withW. Evaluating Eq. (9) yields the source plane field that produces the exact observation plane field over the RoI W. Note that the filtered observation plane field will be nonphysical outside the RoI.

Fresnel Filtering Discussion
If the observation plane is in the far-zone, Eq. (9) simplifies to E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 0 ; 3 2 6 ; 5 5 7 which can be considered the Fraunhofer filtered source. If W is rectangular, as it often is, then Eq. (9) becomes E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 1 ; 3 2 6 ; 4 5 8 where W y and W x are the height and width of the RoI and sincðxÞ ¼ sinðxÞ∕x.
A spatially broadband source often encountered in practice is a point source. Letting U src ðρ 0 ; 0Þ ¼ δðρ 0 − ρ c Þ and evaluating the integrals in Eq. (11) yields E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 2 ; 3 2 6 ; 3 0 3 where ρ c ¼xx c þŷy c is the location of the point source in the source plane. Equation (12) is the sinc point source model discussed in Ref. 1. In most cases, Eq. (9) cannot be evaluated in closed form. Thus, the integral must be computed numerically or approximated. The former is generally undesired because the sampling requirements to represent U src on a discrete grid are prohibitive, hence wanting to filter U src .
The latter, approximating Eq. (9), is a very good option for highly divergent fields, like fields transiting negative lenses or reflected from convex mirrors. In these scenarios, Eq. (9) can be evaluated using the method of stationary phase (MoSP). 12 Let U src ðρ 0 ; 0Þ ¼ Aðρ 0 Þ exp½jk∕ð2RÞρ 02 , where A is the amplitude of U src and R > 0 is the curvature.
Substituting this into Eq. (9) and evaluating the integrals using the MoSP produces E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 3 ; 6 3 ; The accuracy of Eq. (13) improves as R → 0 and is generally good for R < z∕10. Unfortunately, Eq. (9) cannot be easily evaluated or approximated when simulating spatially broadband fields with random wavefronts, like fields reflected from rough surfaces. By far, the most popular approach for simulating fields scattered from rough surfaces is to model the scattered field as U src ðρ 0 ; 0Þ ¼ Aðρ 0 Þ exp½jkϕðρ 0 Þ, where ϕ is a delta-correlated random phase screen with values uniformly distributed between −π and π. Equation (9) is then evaluated numerically to yield U src;f . 3 This model for the field scattered from a rough surface is accurate if the surface height standard deviation σ h > λ and the surface's correlation radius l h < Δ, where Δ is the grid spacing. Although these conditions are often met in practice, the statistics of the rough surface, i.e., σ h and l h , are not included in the model. Note that when using the delta-correlated, −π-to-π phase screen model for simulating rough surface scattering, l h ¼ Δ, where Δ is the grid spacing.
Another lesser known approach, which includes σ h and l h , is to assume that Aðρ 0 Þ ≈ A x ðx 0 ÞA y ðy 0 Þ, likewise for W, and hðρ 0 Þ ≈ h x ðx 0 Þ þ h y ðy 0 Þ. Equation (9) then simplifies to E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 4 ; 6 3 ; 4 2 6 U src;f ðρ; 0Þ ≈ U src;f x ðx; 0ÞU src;f y ðy; 0Þ where α ¼ x; y. Because of the reduction in dimensionality, it is now practical to synthesize two independent one-dimensional (1-D) instances of h and to evaluate Eq. (14) numerically. The resulting independent 1-D U src;f x and U src;f y can be downsampled, expanded to two-dimensional (2-D) grids, and then multiplied together to form U src;f . This approach can also be used to filter fields with highly curved random wavefronts, e.g., fields reflected from rough convex surfaces. By letting hðρ 0 Þ ¼ h x ðx 0 Þ þ h y ðy 0 Þ, the simulated l h is not equal to the desired l h and, in general, the statistics of the simulated rough surface are inhomogeneous and anisotropic.
As a consequence of Eq. (9), both rough surface filtering approaches mentioned above filter the random source field directly, so the filtered source, after propagation, is equal to U obs over the RoI. For simulations involving random fields, the fact that this particular filtered source produces the exact same U obs (over the RoI) as this particular unfiltered, spatially broadband source is generally inconsequential. What is important is that the filtered U obs be representative of the true, unfiltered U obs , i.e., the statistics of the filtered and unfiltered U obs match. This new statistical filtering approach is presented in the next section.

Filtering Quasihomogeneous Sources
Quasihomogeneous sources are a subclass of the more general and popular Schell-model sources. 12,13 The intensity I of a quasihomogeneous source varies slowly compared to its complex degree of spatial coherence μ, such that its mutual intensity J takes the approximate form 12,13 E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 5 ; 3 2 6 ; 6 8 2 where I ¼ jAj 2 . Equation (15) is a good statistical model for light scattered from rough targets, where the spot on the target' rough surface is much wider than the surface's autocorrelation function.
The goal here is to find the filtered mutual intensity J src;f , which when propagated will produce the exact mutual intensity J obs over the RoI. From J src;f , one can then synthesize instances of U src;f for use in optical simulations.
Taking the autocorrelation of Eq. (9) yields E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 6 ; 3 2 6 ; 5 2 2 Assuming that U src is a quasihomogeneous random field, i.e., the form of J src is given by Eq. (15), and using the integral expression forW given in Eq.
To make further progress, it is necessary to assume that the observation plane is in the far-zone of the quasihomogeneous source. This initially seems like a very prohibitive condition; however, the far-zone criterion for quasihomogeneous sources is typically far less restrictive than that for coherent sources. The far-zone criterion for quasihomogeneous sources is given by Goodman 13 where D, in this context, is the diameter of the illuminated area on the target and d c is the spatial coherence diameter of the field scattered from the target. Since typically d c < 10λ, one obtains the far-zone J only a short distance away from the target's surface. Assuming that Eq. (18) is satisfied and making the common variable substitutions s ¼ ρ 0 1 − ρ 0 2 and t ¼ ðρ 0 1 þ ρ 0 2 Þ∕2 simplifies Eq. (17) to E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 1 9 ; 6 3 ; 6 9 7 J src;f ðρ 1 ; Note that the product of the t and s integrals is the generalized Van Cittert-Zernike theorem applied to J src . 12,13 SubstitutingĨ andμ for the Fourier transforms of I and μ, respectively, yields E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 0 ; 6 3 ; 4 9 3 J src;f ðρ 1 ; The integrand in Eq. (20) is a new J (physically, a new partially coherent source) with the product of W, W Ã , andμ serving as the source's shape andĨ as the source's coherence function. If U src is a circular complex Gaussian, as it is for rough surface scattering (fully developed speckle), then the width ofĨ is the mean speckle size. 15 In most simulations involving speckle, the RoI contains many speckles. Note that the RoI, in general, is not the same as the pupil or detector areas. Thus,Ĩ is much narrower than the product of W, W Ã , andμ, and the integrand in Eq. (20) is another quasihomogeneous source. Making the variable substitutions s ¼ ρ 0 0 1 − ρ 0 0 2 and t ¼ ðρ 0 0 1 þ ρ 0 0 2 Þ∕2 and evaluating the integrals produces E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 1 ; 6 3 ; 2 2 7 J src;f ðρ 1 ; whereĨ ¼ λ 2 z 2 I and E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 2 ; 6 3 ; 1 3 1W The final simplified result is E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 3 ; 3 2 6 ; 7 4 1 From Eq. (23), an instance of U src;f is given by E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 4 ; 3 2 6 ; 6 5 1 where T is a complex amplitude screen whose autocorrelation function isW. The most straightforward and numerically efficient method for synthesizing T is the Monte Carlo spectral method (MCSM). [16][17][18][19][20] The method uses the power spectral density (PSD) to filter or color an array of delta-correlated complex Gaussian random numbers in the frequency domain. The applicable PSD can be derived fromW by applying the Wiener-Khinchin theorem. 12,13 Fourier transforming Eq. (22) and simplifying produces E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 5 ; 3 2 6 ; 4 9 1 where Δρ ¼ ρ 1 − ρ 2 and f ¼xf x þŷf y is the spatial frequency vector. In general, S given in Eq. (25) is not normalized, i.e., the volume under S is not one. 13 This is only a minor mathematical inconvenience; it has no effect on synthesizing T.
Notice that in contrast to Fresnel spatial filtering [see Eq. (9)], quasihomogeneous spatial filtering does not use U src directly. Most importantly, the spatially broadband U src is never discretized, meaning that quasihomogeneous filtering typically requires far fewer grid points than Eq. (9).

Simulation Details and Methodology
It is best to present the utility of the above analysis through example. Here, the transmission and subsequent propagation of a collimated Gaussian beam through a ground glass diffuser was simulated. The simulation was performed in 1-D, so the results using Eq. (24) could be directly compared to those using the unfiltered, spatially broadband U src and Eq. (9)-the sampling requirements to propagate U src were impractical using 2-D grids.
The source field for this simulation was E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 6 ; 3 2 6 ; 1 7 8 where λ ¼ 1 μm, σ ¼ 5 mm, D ¼ 25.4 mm, rect is the rectangle function defined by Goodman, 11 and h is the diffuser's optical path length function. The path lengths h were assumed to be Gaussian distributed and Gaussian correlated with a standard deviation σ h ¼ 3 μm and a correlation length l h ¼ 60 μm. Instances of h were produced using the MCSM. Equation (26) and the two filtered sources discussed below were propagated z ¼ 15 m (corresponding to a 1-D Fresnel number N F ≈ 3.3) in eight steps to the observation plane using the angular spectrum method. 1,2 The source and observation planes were discretized using 239,884 points with 3 and 11 μm sample spacings, respectively. The RoI in the observation plane was W x ¼ 10 cm wide.
The Fresnel-filtered source was computed directly from Eq. (26) using a 1-D form of Eq. (9), namely, E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 7 ; 6 3 ; 6 5 3 An instance of the quasihomogeneous filtered source was E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 8 ; 6 3 ; 5 2 8 U src;f ðxÞ ¼ where T was synthesized from the following 1-D PSD E Q -T A R G E T ; t e m p : i n t r a l i n k -; e 0 2 9 ; 3 2 6 ; 7 5 2 using the MCSM. The PSD of T requires computing the moment hexp½jkhðx 1 Þ exp½−jkhðx 2 Þi. For Gaussian distributed and Gaussian correlated h, this moment is approximately E Q -T A R G E T ; t e m p : i n t r a l i n k -; s e c 3 . 1 ; 3 2 6 ; 6 7 3 assuming that σ h > λ, which is generally the case. Note that l h ∕ðkσ h Þ is the approximate spatial coherence radius of the scattered field. It is important to note that the large number of points and small spacings mentioned above were required to perform the propagation simulations with the source plane fields given in Eqs.     Fig. 1(b)]. Figure 2 shows the corresponding U obs instances. As expected, the U obs corresponding to Eq.  Fig. 3(f)], caused by small numerical differences in the positions of J obs zero crossings [notice that the wrapping cut discrepancies occur at the "nulls" in the log 10 jJ obs j plot [ Fig. 3(e)], the curves are in excellent agreement. Figure 3 validates the quasihomogeneous filtering approach discussed in Sec. 2.

Conclusion
In this a paper, two-source plane spatial filtering techniques for use in wave optics simulations were presented and discussed. The first, called Fresnel spatial filtering, provided the spatially band-limited source U src;f that produced the exact same U obs over a specified RoI as the original, spatially broadband U src . In general, the Fresnel spatial filtering integral could not be evaluated analytically. Evaluating the integral numerically was undesirable because the number of points and sample spacing required to accurately represent U src were onerous.
This motivated development of the second filtering technique-statistical or quasihomogeneous spatial filtering. Instead of requiring that U src;f produces the exact same U obs over the RoI as U src , this approach only required that the filtered source produces a U obs that was representative of that produced by U src (i.e., the statistics of the filtered and unfiltered U obs matched), making this approach very applicable in simulations involving scattering from rough surfaces or transmission through diffusers. Because U src never needed to be discretized, this statistical approach required far fewer points than Fresnel spatial filtering. In addition, the quasihomogeneous spatial filtering approach included the underlying statistical properties of the random scatterer, i.e., σ h and l h . This stood in contrast to the more common method of using a delta-correlated, −π-to-π phase screen for modeling rough surface scattering or transmission through diffusers.
Both Fresnel and quasihomogeneous spatial filtering were validated in Sec. 3. For this purpose, the near-zone propagation of light transmitted by a ground glass diffuser was simulated. The results obtained using both filtering approaches were compared to those of the original U src . All were in excellent agreement.
The spatial filtering techniques discussed in this paper will be useful for mitigating aliasing in wave optics simulations involving spatially broadband fields, e.g., simulating rough surface scattering, speckle, etc. These scenarios or optical phenomena are often encountered in the deployment of optical phased arrays, high-energy lasers, active imaging and remote sensing systems, and medicine. Thus, the spatial filtering techniques presented herein will be applicable in the design and subsequent simulation of those systems. In addition, these techniques can also be used in laboratory experiments where spatial light modulators serve as filtered rough targets. This approach has many advantages over using actual rough, diffuse objects-the main one being light conservation.