## 1.

## 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.

## 2.

## Methodology

## 2.1.

### Fresnel Spatial Filtering

The field in the observation plane given the field in the source plane is

## (1)

$${U}^{\mathrm{obs}}(\mathbf{\rho},z)=\frac{\mathrm{exp}(\mathrm{j}kz)\mathrm{exp}\left(\frac{\mathrm{j}k}{2z}{\rho}^{2}\right)}{\mathrm{j}\lambda z}{\iint}_{-\infty}^{\infty}{U}^{\mathrm{src}}({\mathbf{\rho}}^{\prime},0)\mathrm{exp}\left(\frac{\mathrm{j}k}{2z}{\rho}^{\prime 2}\right)\mathrm{exp}(-\frac{\mathrm{j}k}{z}{\mathbf{\rho}}^{\prime}\xb7\mathbf{\rho}){\mathrm{d}}^{2}{\rho}^{\prime},$$^{11}Applying the transform

## (2)

$${\iint}_{-\infty}^{\infty}\{\xb7\}\mathrm{exp}(\frac{\mathrm{j}k}{z}{\mathbf{\rho}}^{\prime \prime}\xb7\mathbf{\rho}){\mathrm{d}}^{2}\rho $$## (3)

$$\mathrm{j}\lambda z\mathrm{exp}(-\mathrm{j}kz){\iint}_{-\infty}^{\infty}{U}^{\mathrm{obs}}(\mathbf{\rho},z)\mathrm{exp}(-\frac{\mathrm{j}k}{2z}{\rho}^{2})\mathrm{exp}(\frac{\mathrm{j}k}{z}{\mathbf{\rho}}^{\prime \prime}\xb7\mathbf{\rho}){\mathrm{d}}^{2}\rho \phantom{\rule{0ex}{0ex}}={\iint}_{-\infty}^{\infty}{U}^{\mathrm{src}}({\mathbf{\rho}}^{\prime},0)\mathrm{exp}\left(\frac{\mathrm{j}k}{2z}{\rho}^{\prime 2}\right){\iint}_{-\infty}^{\infty}\mathrm{exp}[-\frac{\mathrm{j}k}{z}({\mathbf{\rho}}^{\prime}-{\mathbf{\rho}}^{\prime \prime})\xb7\mathbf{\rho}]{\mathrm{d}}^{2}\rho {\mathrm{d}}^{2}{\rho}^{\prime}.$$## (4)

$${\lambda}^{2}{z}^{2}{U}^{\mathrm{src}}({\mathbf{\rho}}^{\prime \prime},0)\mathrm{exp}\left(\frac{\mathrm{j}k}{2z}{\rho}^{\prime \prime 2}\right).$$Solving the resulting expression for the source plane field produces

## (5)

$${U}^{\mathrm{src}}({\mathbf{\rho}}^{\prime \prime},0)=\frac{\mathrm{exp}(-\mathrm{j}kz)\mathrm{exp}(-\frac{\mathrm{j}k}{2z}{\rho}^{\prime \prime 2})}{-\mathrm{j}\lambda z}{\iint}_{-\infty}^{\infty}{U}^{\mathrm{obs}}(\mathbf{\rho},z)\mathrm{exp}(-\frac{\mathrm{j}k}{2z}{\rho}^{2})\mathrm{exp}(\frac{\mathrm{j}k}{z}{\mathbf{\rho}}^{\prime \prime}\xb7\mathbf{\rho}){\mathrm{d}}^{2}\rho .$$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

## (6)

$${U}^{\mathrm{obs},\mathrm{w}}(\mathbf{\rho},z)=W(\mathbf{\rho}){U}^{\mathrm{obs}}(\mathbf{\rho},z),$$## (7)

$${U}^{\mathrm{src},\mathrm{f}}({\mathbf{\rho}}^{\prime \prime},0)=\frac{\mathrm{exp}(-\mathrm{j}kz)\mathrm{exp}(-\frac{\mathrm{j}k}{2z}{\rho}^{\prime \prime 2})}{-\mathrm{j}\lambda z}{\iint}_{-\infty}^{\infty}W(\mathbf{\rho}){U}^{\mathrm{obs}}(\mathbf{\rho},z)\phantom{\rule{0ex}{0ex}}\times \mathrm{exp}(-\frac{\mathrm{j}k}{2z}{\rho}^{2})\mathrm{exp}(\frac{\mathrm{j}k}{z}{\mathbf{\rho}}^{\prime \prime}\xb7\mathbf{\rho}){\mathrm{d}}^{2}\rho ,$$## (8)

$${U}^{\mathrm{src},\mathrm{f}}({\mathbf{\rho}}^{\prime \prime},0)=\frac{\mathrm{exp}(-\frac{\mathrm{j}k}{2z}{\rho}^{\prime \prime 2})}{{\lambda}^{2}{z}^{2}}{\iint}_{-\infty}^{\infty}{U}^{\mathrm{src}}({\mathbf{\rho}}^{\prime},0)\mathrm{exp}\left(\frac{\mathrm{j}k}{2z}{\rho}^{\prime 2}\right)\phantom{\rule{0ex}{0ex}}\times {\iint}_{-\infty}^{\infty}W(\mathbf{\rho})\mathrm{exp}[-\frac{\mathrm{j}k}{z}({\mathbf{\rho}}^{\prime}-{\mathbf{\rho}}^{\prime \prime})\xb7\mathbf{\rho}]{\mathrm{d}}^{2}\rho {\mathrm{d}}^{2}{\rho}^{\prime}.$$The final result is

## (9)

$${U}^{\mathrm{src},\mathrm{f}}({\mathbf{\rho}}^{\prime \prime},0)=\frac{\mathrm{exp}(-\frac{\mathrm{j}k}{2z}{\rho}^{\prime \prime 2})}{{\lambda}^{2}{z}^{2}}{\iint}_{-\infty}^{\infty}{U}^{\mathrm{src}}({\mathbf{\rho}}^{\prime},0)\mathrm{exp}\left(\frac{\mathrm{j}k}{2z}{\rho}^{\prime 2}\right)\tilde{W}({\mathbf{\rho}}^{\prime}-{\mathbf{\rho}}^{\prime \prime}){\mathrm{d}}^{2}{\rho}^{\prime}.$$## 2.2.

### Fresnel Filtering Discussion

If the observation plane is in the far-zone, Eq. (9) simplifies to

## (10)

$${U}^{\mathrm{src},\mathrm{f}}({\mathbf{\rho}}^{\prime \prime},0)=\frac{\mathrm{exp}(-\frac{\mathrm{j}k}{2z}{\rho}^{\prime \prime 2})}{{\lambda}^{2}{z}^{2}}{\iint}_{-\infty}^{\infty}{U}^{\mathrm{src}}({\mathbf{\rho}}^{\prime},0)\tilde{W}({\mathbf{\rho}}^{\prime}-{\mathbf{\rho}}^{\prime \prime}){\mathrm{d}}^{2}{\rho}^{\prime},$$## (11)

$${U}^{\mathrm{src},\mathrm{f}}({\mathbf{\rho}}^{\prime \prime},0)=\frac{\mathrm{exp}(-\frac{\mathrm{j}k}{2z}{\rho}^{\prime \prime 2})}{{\lambda}^{2}{z}^{2}}{\iint}_{-\infty}^{\infty}{U}^{\mathrm{src}}({\mathbf{\rho}}^{\prime},0)\mathrm{exp}\left(\frac{\mathrm{j}k}{2z}{\rho}^{\prime 2}\right)\phantom{\rule{0ex}{0ex}}\times {W}_{x}{W}_{y}\mathrm{sinc}[\frac{k}{z}\frac{{W}_{x}}{2}({x}^{\prime}-{x}^{\prime \prime})]\mathrm{sinc}[\frac{k}{z}\frac{{W}_{y}}{2}({y}^{\prime}-{y}^{\prime \prime})]\mathrm{d}{x}^{\prime}\mathrm{d}{y}^{\prime},$$A spatially broadband source often encountered in practice is a point source. Letting ${U}^{\mathrm{src}}({\mathbf{\rho}}^{\prime},0)=\delta ({\mathbf{\rho}}^{\prime}-{\mathbf{\rho}}_{c})$ and evaluating the integrals in Eq. (11) yields

## (12)

$${U}^{\mathrm{src},\mathrm{f}}(\mathbf{\rho},0)=\frac{{W}_{x}{W}_{y}}{{\lambda}^{2}{z}^{2}}\mathrm{exp}[-\frac{\mathrm{j}k}{2z}({\rho}^{2}-{\rho}_{c}^{2})]\mathrm{sinc}[\frac{k}{z}\frac{{W}_{x}}{2}({x}_{c}-x)]\mathrm{sinc}[\frac{k}{z}\frac{{W}_{y}}{2}({y}_{c}-y)],$$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 ${U}^{\mathrm{src}}$ on a discrete grid are prohibitive, hence wanting to filter ${U}^{\mathrm{src}}$.

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 ${U}^{\mathrm{src}}({\mathbf{\rho}}^{\prime},0)=A({\mathbf{\rho}}^{\prime})\mathrm{exp}[\mathrm{j}k/(2R){\rho}^{\prime 2}]$, where $A$ is the amplitude of ${U}^{\mathrm{src}}$ and $R>0$ is the curvature. Substituting this into Eq. (9) and evaluating the integrals using the MoSP produces

## (13)

$${U}^{\mathrm{src},\mathrm{f}}(\mathbf{\rho},0)\approx \frac{\mathrm{exp}(-\frac{\mathrm{j}k}{2z}{\rho}^{2})}{-\mathrm{j}\lambda z}\frac{A(0)\tilde{W}(-\mathbf{\rho})}{1+z/R}.$$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 ${U}^{\mathrm{src}}({\mathbf{\rho}}^{\prime},0)=A({\mathbf{\rho}}^{\prime})\mathrm{exp}[\mathrm{j}k\varphi ({\mathbf{\rho}}^{\prime})]$, where $\varphi $ is a delta-correlated random phase screen with values uniformly distributed between $-\pi $ and $\pi $. Equation (9) is then evaluated numerically to yield ${U}^{\mathrm{src},\mathrm{f}}$.^{3} This model for the field scattered from a rough surface is accurate if the surface height standard deviation ${\sigma}_{h}>\lambda $ and the surface’s correlation radius ${l}_{h}<\mathrm{\Delta}$, where $\mathrm{\Delta}$ is the grid spacing. Although these conditions are often met in practice, the statistics of the rough surface, i.e., ${\sigma}_{h}$ and ${l}_{h}$, are not included in the model. Note that when using the delta-correlated, $-\pi $-to-$\pi $ phase screen model for simulating rough surface scattering, ${l}_{h}=\mathrm{\Delta}$, where $\mathrm{\Delta}$ is the grid spacing.

Another lesser known approach, which includes ${\sigma}_{h}$ and ${l}_{h}$, is to assume that $A({\mathbf{\rho}}^{\prime})\approx {A}_{x}({x}^{\prime}){A}_{y}({y}^{\prime})$, likewise for $\tilde{W}$, and $h({\mathbf{\rho}}^{\prime})\approx {h}_{x}({x}^{\prime})+{h}_{y}({y}^{\prime})$. Equation (9) then simplifies to

## (14)

$${U}^{\mathrm{src},\mathrm{f}}(\mathbf{\rho},0)\approx {U}_{x}^{\mathrm{src},\mathrm{f}}(x,0){U}_{y}^{\mathrm{src},\mathrm{f}}(y,0)\phantom{\rule{0ex}{0ex}}{U}_{\alpha}^{\mathrm{src},\mathrm{f}}(\alpha ,0)=\frac{\mathrm{exp}(-\frac{\mathrm{j}k}{2z}{\alpha}^{2})}{\lambda z}{\int}_{-\infty}^{\infty}{A}_{\alpha}({\alpha}^{\prime})\mathrm{exp}\left\{\mathrm{j}k[\frac{{\alpha}^{\prime 2}}{2z}+{h}_{\alpha}({\alpha}^{\prime})]\right\}{\tilde{W}}_{\alpha}({\alpha}^{\prime}-\alpha )\mathrm{d}{\alpha}^{\prime},$$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 ${U}^{\mathrm{obs}}$ over the RoI. For simulations involving random fields, the fact that this particular filtered source produces the exact same ${U}^{\mathrm{obs}}$ (over the RoI) as this particular unfiltered, spatially broadband source is generally inconsequential. What is important is that the filtered ${U}^{\mathrm{obs}}$ be representative of the true, unfiltered ${U}^{\mathrm{obs}}$, i.e., the statistics of the filtered and unfiltered ${U}^{\mathrm{obs}}$ match. This new statistical filtering approach is presented in the next section.

## 2.3.

### 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 $\mu $, such that its mutual intensity $J$ takes the approximate form^{12}^{,}^{13}

## (15)

$$J({\mathbf{\rho}}_{1},{\mathbf{\rho}}_{2})=\u27e8U({\mathbf{\rho}}_{1}){U}^{*}({\mathbf{\rho}}_{2})\u27e9=A({\mathbf{\rho}}_{1})A({\mathbf{\rho}}_{2})\mu ({\mathbf{\rho}}_{1}-{\mathbf{\rho}}_{2})\phantom{\rule{0ex}{0ex}}\approx I\left(\frac{{\mathbf{\rho}}_{1}+{\mathbf{\rho}}_{2}}{2}\right)\mu ({\mathbf{\rho}}_{1}-{\mathbf{\rho}}_{2}),$$The goal here is to find the filtered mutual intensity ${J}^{\mathrm{src},\mathrm{f}}$, which when propagated will produce the exact mutual intensity ${J}^{\mathrm{obs}}$ over the RoI. From ${J}^{\mathrm{src},\mathrm{f}}$, one can then synthesize instances of ${U}^{\mathrm{src},\mathrm{f}}$ for use in optical simulations.

Taking the autocorrelation of Eq. (9) yields

## (16)

$${J}^{\mathrm{src},\mathrm{f}}({\mathbf{\rho}}_{1},{\mathbf{\rho}}_{2})=\frac{\mathrm{exp}[-\frac{\mathrm{j}k}{2z}({\rho}_{1}^{2}-{\rho}_{2}^{2})]}{{\lambda}^{4}{z}^{4}}{\int \int \int \int}_{-\infty}^{\infty}{J}^{\mathrm{src}}({\mathbf{\rho}}_{1}^{\prime},{\mathbf{\rho}}_{2}^{\prime})\mathrm{exp}[\frac{\mathrm{j}k}{2z}({\rho}_{1}^{\prime 2}-{\rho}_{2}^{\prime 2})]\phantom{\rule{0ex}{0ex}}\times \tilde{W}({\mathbf{\rho}}_{1}^{\prime}-{\mathbf{\rho}}_{1}){\tilde{W}}^{*}({\mathbf{\rho}}_{2}^{\prime}-{\mathbf{\rho}}_{2}){\mathrm{d}}^{2}{\rho}_{1}^{\prime}{\mathrm{d}}^{2}{\rho}_{2}^{\prime}.$$## (17)

$${J}^{\mathrm{src},\mathrm{f}}({\mathbf{\rho}}_{1},{\mathbf{\rho}}_{2})=\frac{\mathrm{exp}[-\frac{\mathrm{j}k}{2z}({\rho}_{1}^{2}-{\rho}_{2}^{2})]}{{\lambda}^{4}{z}^{4}}{\int \int \int \int}_{-\infty}^{\infty}W({\mathbf{\rho}}_{1}^{\prime \prime}){W}^{*}({\mathbf{\rho}}_{2}^{\prime \prime})\mathrm{exp}[\frac{\mathrm{j}k}{z}({\mathbf{\rho}}_{1}\xb7{\mathbf{\rho}}_{1}^{\prime \prime}-{\mathbf{\rho}}_{2}\xb7{\mathbf{\rho}}_{2}^{\prime \prime})]\phantom{\rule{0ex}{0ex}}\times {\int \int \int \int}_{-\infty}^{\infty}I\left(\frac{{\mathbf{\rho}}_{1}^{\prime}+{\mathbf{\rho}}_{2}^{\prime}}{2}\right)\mu ({\mathbf{\rho}}_{1}^{\prime}-{\mathbf{\rho}}_{2}^{\prime})\mathrm{exp}[\frac{\mathrm{j}k}{2z}({\rho}_{1}^{\prime 2}-{\rho}_{2}^{\prime 2})]\phantom{\rule{0ex}{0ex}}\times \mathrm{exp}[-\frac{\mathrm{j}k}{z}({\mathbf{\rho}}_{1}^{\prime \prime}\xb7{\mathbf{\rho}}_{1}^{\prime}-{\mathbf{\rho}}_{2}^{\prime \prime}\xb7{\mathbf{\rho}}_{2}^{\prime})]{\mathrm{d}}^{2}{\rho}_{1}^{\prime}{\mathrm{d}}^{2}{\rho}_{2}^{\prime}{\mathrm{d}}^{2}{\rho}_{1}^{\prime \prime}{\mathrm{d}}^{2}{\rho}_{2}^{\prime \prime}.$$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} and Gori^{14}

Assuming that Eq. (18) is satisfied and making the common variable substitutions $\mathbf{s}={\mathbf{\rho}}_{1}^{\prime}-{\mathbf{\rho}}_{2}^{\prime}$ and $\mathbf{t}=({\mathbf{\rho}}_{1}^{\prime}+{\mathbf{\rho}}_{2}^{\prime})/2$ simplifies Eq. (17) to

## (19)

$${J}^{\mathrm{src},\mathrm{f}}({\mathbf{\rho}}_{1},{\mathbf{\rho}}_{2})\approx \frac{\mathrm{exp}[-\frac{\mathrm{j}k}{2z}({\rho}_{1}^{2}-{\rho}_{2}^{2})]}{{\lambda}^{4}{z}^{4}}{\int \int \int \int}_{-\infty}^{\infty}W({\mathbf{\rho}}_{1}^{\prime \prime}){W}^{*}({\mathbf{\rho}}_{2}^{\prime \prime})\mathrm{exp}[\frac{\mathrm{j}k}{z}({\mathbf{\rho}}_{1}\xb7{\mathbf{\rho}}_{1}^{\prime \prime}-{\mathbf{\rho}}_{2}\xb7{\mathbf{\rho}}_{2}^{\prime \prime})]\phantom{\rule{0ex}{0ex}}\times {\iint}_{-\infty}^{\infty}I(\mathbf{t})\mathrm{exp}[-\frac{\mathrm{j}k}{z}({\mathbf{\rho}}_{1}^{\prime \prime}-{\mathbf{\rho}}_{2}^{\prime \prime})\xb7\mathbf{t}]{\mathrm{d}}^{2}t\phantom{\rule{0ex}{0ex}}\times {\iint}_{-\infty}^{\infty}\mu (\mathbf{s})\mathrm{exp}[-\frac{\mathrm{j}k}{z}(\frac{{\mathbf{\rho}}_{1}^{\prime \prime}+{\mathbf{\rho}}_{2}^{\prime \prime}}{2})\xb7\mathbf{s}]{\mathrm{d}}^{2}s{\mathrm{d}}^{2}{\rho}_{1}^{\prime \prime}{\mathrm{d}}^{2}{\rho}_{2}^{\prime \prime}.$$^{12}

^{,}

^{13}Substituting $\tilde{I}$ and $\tilde{\mu}$ for the Fourier transforms of $I$ and $\mu $, respectively, yields

## (20)

$${J}^{\mathrm{src},\mathrm{f}}({\mathbf{\rho}}_{1},{\mathbf{\rho}}_{2})=\frac{\mathrm{exp}[-\frac{\mathrm{j}k}{2z}({\rho}_{1}^{2}-{\rho}_{2}^{2})]}{{\lambda}^{4}{z}^{4}}{\int \int \int \int}_{-\infty}^{\infty}W({\mathbf{\rho}}_{1}^{\prime \prime}){W}^{*}({\mathbf{\rho}}_{2}^{\prime \prime})\tilde{\mu}\left(\frac{{\mathbf{\rho}}_{1}^{\prime \prime}+{\mathbf{\rho}}_{2}^{\prime \prime}}{2}\right)\phantom{\rule{0ex}{0ex}}\times \tilde{I}({\mathbf{\rho}}_{1}^{\prime \prime}-{\mathbf{\rho}}_{2}^{\prime \prime})\mathrm{exp}[\frac{\mathrm{j}k}{z}({\mathbf{\rho}}_{1}\xb7{\mathbf{\rho}}_{1}^{\prime \prime}-{\mathbf{\rho}}_{2}\xb7{\mathbf{\rho}}_{2}^{\prime \prime})]{\mathrm{d}}^{2}{\rho}_{1}^{\prime \prime}{\mathrm{d}}^{2}{\rho}_{2}^{\prime \prime}.$$^{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, $\tilde{I}$ is much narrower than the product of $W$, ${W}^{*}$, and $\tilde{\mu}$, and the integrand in Eq. (20) is another quasihomogeneous source.

Making the variable substitutions $\mathbf{s}={\mathbf{\rho}}_{1}^{\prime \prime}-{\mathbf{\rho}}_{2}^{\prime \prime}$ and $\mathbf{t}=({\mathbf{\rho}}_{1}^{\prime \prime}+{\mathbf{\rho}}_{2}^{\prime \prime})/2$ and evaluating the integrals produces

## (21)

$${J}^{\mathrm{src},\mathrm{f}}({\mathbf{\rho}}_{1},{\mathbf{\rho}}_{2})\approx \frac{\mathrm{exp}[-\frac{\mathrm{j}k}{2z}({\rho}_{1}^{2}-{\rho}_{2}^{2})]}{{\lambda}^{4}{z}^{4}}\tilde{\tilde{I}}\left(\frac{{\mathbf{\rho}}_{1}+{\mathbf{\rho}}_{2}}{2}\right)\tilde{\mathcal{W}}({\mathbf{\rho}}_{1}-{\mathbf{\rho}}_{2}),$$## (22)

$$\tilde{\mathcal{W}}({\mathbf{\rho}}_{1}-{\mathbf{\rho}}_{2})\approx {\iint}_{-\infty}^{\infty}{|W(\mathbf{t})|}^{2}\tilde{\mu}(\mathbf{t})\mathrm{exp}[\frac{\mathrm{j}k}{z}({\mathbf{\rho}}_{1}-{\mathbf{\rho}}_{2})\xb7\mathbf{t}]{\mathrm{d}}^{2}t.$$The final simplified result is

## (23)

$${J}^{\mathrm{src},\mathrm{f}}({\mathbf{\rho}}_{1},{\mathbf{\rho}}_{2})=\frac{\mathrm{exp}[-\frac{\mathrm{j}k}{2z}({\rho}_{1}^{2}-{\rho}_{2}^{2})]}{{\lambda}^{2}{z}^{2}}I\left(\frac{{\mathbf{\rho}}_{1}+{\mathbf{\rho}}_{2}}{2}\right)\tilde{\mathcal{W}}({\mathbf{\rho}}_{1}-{\mathbf{\rho}}_{2}).$$## (24)

$${U}^{\mathrm{src},\mathrm{f}}(\mathbf{\rho})=\frac{\mathrm{exp}(-\frac{\mathrm{j}k}{2z}{\rho}^{2})}{\lambda z}A(\mathbf{\rho})T(\mathbf{\rho}),$$^{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 from $\tilde{W}$ by applying the Wiener–Khinchin theorem.^{12}^{,}^{13} Fourier transforming Eq. (22) and simplifying produces

## (25)

$$\mathcal{S}(\mathbf{f})={\iint}_{-\infty}^{\infty}\tilde{\mathcal{W}}(\mathbf{\Delta \rho})\mathrm{exp}(-\mathrm{j}2\pi \mathbf{f}\xb7\mathbf{\Delta \rho}){\mathrm{d}}^{2}\mathrm{\Delta}\rho \phantom{\rule{0ex}{0ex}}={\lambda}^{2}{z}^{2}{|W(\lambda z\mathbf{f})|}^{2}\tilde{\mu}(\lambda z\mathbf{f}),$$^{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 ${U}^{\mathrm{src}}$ directly. Most importantly, the spatially broadband ${U}^{\mathrm{src}}$ is never discretized, meaning that quasihomogeneous filtering typically requires far fewer grid points than Eq. (9).

## 3.

## Validation

## 3.1.

### 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 ${U}^{\mathrm{src}}$ and Eq. (9)—the sampling requirements to propagate ${U}^{\mathrm{src}}$ were impractical using 2-D grids.

The source field for this simulation was

## (26)

$${U}^{\mathrm{src}}(x)=\mathrm{exp}(-\frac{{x}^{2}}{{\sigma}^{2}})\mathrm{rect}\left(\frac{x}{D}\right)\mathrm{exp}[\mathrm{j}kh(x)],$$^{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 ${\sigma}_{h}=3\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$ and a correlation length ${l}_{h}=60\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$. Instances of $h$ were produced using the MCSM.

Equation (26) and the two filtered sources discussed below were propagated $z=15\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{m}$ (corresponding to a 1-D Fresnel number ${N}_{F}\approx 3.3$) 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 $11\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$ sample spacings, respectively. The RoI in the observation plane was ${W}_{x}=10\text{\hspace{0.17em}\hspace{0.17em}}\mathrm{cm}$ wide.

The Fresnel-filtered source was computed directly from Eq. (26) using a 1-D form of Eq. (9), namely,

## (27)

$${U}^{\mathrm{src},\mathrm{f}}(x)=\frac{\mathrm{exp}(-\frac{\mathrm{j}k}{2z}{x}^{2})}{\lambda z}{\int}_{-\infty}^{\infty}\mathrm{exp}(-\frac{{x}^{\prime 2}}{{\sigma}^{2}})\mathrm{rect}\left(\frac{{x}^{\prime}}{D}\right)\mathrm{exp}[\mathrm{j}kh({x}^{\prime})]\phantom{\rule{0ex}{0ex}}\times \mathrm{exp}\left(\frac{\mathrm{j}k}{2z}{x}^{\prime 2}\right){W}_{x}\mathrm{sinc}[\frac{k}{z}\frac{{W}_{x}}{2}({x}^{\prime}-x)]\mathrm{d}{x}^{\prime}.$$An instance of the quasihomogeneous filtered source was

## (28)

$${U}^{\mathrm{src},\mathrm{f}}(x)=\frac{\mathrm{exp}(-\frac{\mathrm{j}k}{2z}{x}^{2})}{\sqrt{\lambda z}}\mathrm{exp}(-\frac{{x}^{2}}{{\sigma}^{2}})\mathrm{rect}\left(\frac{x}{D}\right)T(x),$$## (29)

$$\mathcal{S}(f)=\lambda z\mathrm{rect}\left(\frac{\lambda zf}{{W}_{x}}\right)\sqrt{\pi}\frac{{l}_{h}}{k{\sigma}_{h}}\mathrm{exp}[-{\left(\frac{{l}_{h}\lambda f}{2{\sigma}_{h}}\right)}^{2}]$$## (30)

$$\u27e8\mathrm{exp}[\mathrm{j}kh({x}_{1})]\mathrm{exp}[-\mathrm{j}kh({x}_{2})]\u27e9\approx \mathrm{exp}[-{k}^{2}{\sigma}_{h}^{2}\frac{{({x}_{1}-{x}_{2})}^{2}}{{l}_{h}^{2}}]$$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. (26) and (27). Synthesizing and propagating Eq. (28) requires significantly less resources and, if desired, can easily be done in 2-D. For example, the same simulation described above using Eq. (28) requires only 2048 points and $80\text{\hspace{0.17em}\hspace{0.17em}}\mu \mathrm{m}$ spacings in both the source and observation planes. In addition, the 15-m propagation can be performed in three steps instead of eight.

## 3.2.

### Simulation Results

Figures 1Fig. 2–3 show the results. Figure 1 shows instances of ${U}^{\mathrm{src}}$ [Eq. (26), Figs. 1(a) and 1(b)], ${U}^{\mathrm{src},\mathrm{f}}$ [Eq. (27), Figs. 1(c) and 1(d)], and ${U}^{\mathrm{src},\mathrm{f}}$ [Eq. (28), Figs. 1(e) and 1(f)]. The effects of filtering are quite clear in Figs. 1(d) and 1(f), where the filtered phases (or wavefronts) are much smoother than the [Eq. (26)] result [Fig. 1(b)].

Figure 2 shows the corresponding ${U}^{\mathrm{obs}}$ instances. As expected, the ${U}^{\mathrm{obs}}$ corresponding to Eq. (26) [(a) and (b)] and Eq. (27) [(c) and (d)] are identical over the RoI; Eq. (27) is nonphysical outside the RoI. The ${U}^{\mathrm{obs}}$ corresponding to Eq. (28) [(e) and (f)] is different than the other two. Recall that the goal here is to ensure that the ${U}^{\mathrm{obs}}$ obtained from propagating Eq. (28) is representative of Eqs. (26) and (27), not necessary equal.

To show this, Fig. 3 reports the observation plane mutual intensities ${J}^{\mathrm{obs}}$ computed over the RoI from 2500 instances of ${U}^{\mathrm{obs}}$ corresponding to source fields Eq. (26) [(a) and (b)], Eq. (27) [(c) and (d)], and Eq. (28) [(e) and (f)]. Except for very minor disagreements in the locations of phase wrapping cuts [Fig. 3(f)], caused by small numerical differences in the positions of ${J}^{\mathrm{obs}}$ zero crossings [notice that the wrapping cut discrepancies occur at the “nulls” in the ${\mathrm{log}}_{10}|{J}^{\mathrm{obs}}|$ plot [Fig. 3(e)], the curves are in excellent agreement. Figure 3 validates the quasihomogeneous filtering approach discussed in Sec. 2.

## 4.

## 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 ${U}^{\mathrm{src},\mathrm{f}}$ that produced the exact same ${U}^{\mathrm{obs}}$ over a specified RoI as the original, spatially broadband ${U}^{\mathrm{src}}$. 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 ${U}^{\mathrm{src}}$ were onerous.

This motivated development of the second filtering technique—statistical or quasihomogeneous spatial filtering. Instead of requiring that ${U}^{\mathrm{src},\mathrm{f}}$ produces the exact same ${U}^{\mathrm{obs}}$ over the RoI as ${U}^{\mathrm{src}}$, this approach only required that the filtered source produces a ${U}^{\mathrm{obs}}$ that was representative of that produced by ${U}^{\mathrm{src}}$ (i.e., the statistics of the filtered and unfiltered ${U}^{\mathrm{obs}}$ matched), making this approach very applicable in simulations involving scattering from rough surfaces or transmission through diffusers. Because ${U}^{\mathrm{src}}$ 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., ${\sigma}_{h}$ and ${l}_{h}$. This stood in contrast to the more common method of using a delta-correlated, $-\pi $-to-$\pi $ 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 ${U}^{\mathrm{src}}$. 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.

## Acknowledgments

The views expressed in this paper are those of the authors and do not reflect the official policy or position of the US Air Force, the Department of Defense, or the US government.

## References

## Biography

**Milo W. Hyde IV** received his BS degree in computer engineering from Georgia Tech in 2001 and his MS degree and PhD in electrical engineering from AFIT in 2006 and 2010, respectively. Currently, he is an associate professor in the Department of Electrical and Computer Engineering at AFIT. He is a senior member of IEEE and SPIE. He is also a member of OSA, AGU, DEPS, and ACES.

**Santasri R. Bose-Pillai** received her PhD in electrical engineering with a focus in optics in 2008 and her MSEE degree in 2005, both from New Mexico State University. She received her BSEE degree with honors from Jadavpur University, India, in 2000. Currently, she is a research assistant professor at AFITs Center for Directed Energy within the Engineering Physics Department. She is a senior member of SPIE. She is also a member of OSA and DEPS.