## Abstract

This two-part paper demonstrates the use of wave-optics simulations to model the effects of dynamic speckle. In Part I, we formulate closed-form expressions for the analytical irradiance correlation coefficient, specifically in the pupil plane of an optical system. These expressions are for square, circular, and Gaussian scattering spots and four different modes of extended-object motion, including in-plane and out-of-plane translation and rotation. Using a phase-screen approach, we then simulate the equivalent scattering from an optically rough extended object, where we assume that the surface heights are uniformly distributed and delta correlated from grid point to grid point. For comparison to the analytical irradiance correlation coefficient, we also calculate the numerical irradiance correlation coefficient from the dynamic speckle after propagation from the simulated object plane to the simulated pupil plane. Overall, the analytical and numerical results definitely demonstrate that, relative to theory, the dynamic speckle in the simulated pupil plane is properly correlated from one frame to the next. Such validated wave-optics simulations provide the framework needed to model more sophisticated setups and obtain accurate results for system-level studies.

## 1. INTRODUCTION

Speckle plays a pivotal role in directed-energy applications. One cannot actively illuminate a distant object without also introducing speckle. Thus, directed-energy applications such as long-range imaging [1–5], tracking [6,7], wavefront sensing [8,9], phase compensation [10,11], and synthetic-aperture ladar [12,13] depend on the presence of speckle to achieve their desired outcomes. Whenever possible, however, these applications also look to mitigate the effects of speckle to achieve the best possible performance.

Optically rough extended objects (i.e., where the surface roughness is of the order of the wavelength of light) diffusely scatter an incident laser beam to produce a speckled irradiance pattern. The associated speckles, in practice, appear as bright regions of constructive interference. Here, the average size of the speckles is roughly equal to a coherence cell from the scattering spot [14]. These speckles unfortunately act as a noise term that limits performance in the aforementioned directed-energy applications. What is more, different modes of extended-object motion serve to perpetuate this noise term on a frame-by-frame basis, due to the effects of dynamic speckle.

Speckle mitigation, in turn, has been an active area of research since the emergence of the laser itself [15]. For example, researchers often perform speckle averaging to mitigate the effects of dynamic speckle. To quantify the benefits of speckle averaging, one can make use of the signal-to-noise ratio (SNR). In practice, the SNR is inversely proportional to the contrast ratio, $C$, such that

where ${\sigma _I}$ is the standard deviation of the speckled irradiance pattern, and $\bar I$ is the mean [15]. A fully developed speckled irradiance pattern (resulting from fully polarized/coherent light) follows a negative-exponential probability density function (PDF). In turn, $C$ goes to unity [16]. Accumulating $K$ patterns, as a result, decrease $C$ to $1/\sqrt K$ as the PDF becomes more Gaussian like in accordance with the central-limit theorem. This last statement is true only if the individual speckled irradiance patterns are uncorrelated on a frame-by-frame basis [17]. From a systems-engineering perspective, it is therefore of great interest to accurately define when speckle decorrelation occurs, especially in the presence of dynamic speckle.Given a fully developed speckle pattern, the real and imaginary parts of the underlying complex-optical field conform to a complex-circular Gaussian joint PDF [18]. Because of this inherent randomness, there are no deterministic solutions for the size of the speckles and thus for the speckle decorrelation. A general approach to this problem is to derive a correlation function that accounts for the lowest-order statistics of the complex-optical field at two different points in space [19]. Normalizing this function to its peak value yields an analytical irradiance correlation coefficient equal to one for overlapping points and equal to zero for separation by the width of the average size of the speckles. Displacing the speckled irradiance patterns by this distance decorrelates them in time, given some relationship between motion of the extended object and that of the dynamic speckle. Then at a known rate of change in extended-object position, the speckle decorrelation is predictable as a function of time. Through the years, a number of researchers have taken this approach, while many others have studied closely related phenomena that one can easily recast in this manner.

Rigden and Gordon [20], Oliver [21], and Langmuir [22] were among the first scientists to report on dynamic speckle. Allen and Jones [23] offered an explanation of their results based on the diffraction of radio waves. Isenor [24] and Sporton [25] followed up by emphasizing the optical-system geometry and its impact on speckle dynamics in the image plane. Anisimov *et al.* [26] later derived space–time correlation statistics for the first time, and correlation experiments have been underway ever since [27–32].

With this rich history in mind, this two-part paper demonstrates the use wave-optics simulations to model the effects of dynamic speckle. In Part I, we formulate closed-form expressions for the analytical irradiance correlation coefficient in the pupil plane of an optical system. Part II then switches gears and formulates closed-form expressions for the analytical irradiance correlation coefficient in the image plane of an optical system. It is worthwhile to consider the pupil plane separately from the image plane, as the structure of speckle turns out to operate independently in each plane under most conditions of interest. As such, this paper focuses solely on the theory and simulation of speckle decorrelation in the pupil plane of an optical system. Because image formation is not yet of concern, the pupil plane (discussed throughout this paper) is equivalent to a plane of observation at some distance from the extended object in a free-space system. In Part II, this distance represents free-space propagation from the object plane to the entrance-pupil plane. A second free-space propagation then focuses the light from the exit-pupil plane to the image plane.

Broadly speaking, Part I aims to fulfill two main goals. The first goal is to establish closed-form expressions for the analytical irradiance correlation coefficient (associated with dynamic speckle in a pupil plane) for (1) the cases of square, circular, and Gaussian scattering spots and (2) four different modes of extended-object motion: in-plane and out-of-plane translation, as well as in-plane and out-of-plane rotation. While meeting this goal does not demand any new theory *per se*, it does fill several gaps in the dynamic-speckle literature that would otherwise require some inference while compiling all of the closed-form expressions in a unified notation. It also frames many of these closed-form expressions for the first time as straightforward functions of extended-object motion. The second main goal is to develop a simulation framework within which to study speckle decorrelation in terms of the numerical irradiance correlation coefficient and thereafter compare the numerical results from simulation to the analytical results from theory.

In service of these goals, the following sections formulate the aforementioned closed-form expressions for the analytical irradiance correlation coefficient (Section 2), the wave-optics simulations used to compute the numerical irradiance correlation coefficient (Section 3), the results that compare analytical and numerical findings (Section 4), and the conclusion to this paper (Section 5). Before moving on to the next section, it is important to note that we wrote Part I so that it complements Part II. In turn, both papers contain overlapping material. So as not to be redundant, this choice enables two things: (1) both papers read independently of each other (i.e., the reader does not have to read Part II to understand the results in Part I and vice versa), and (2) the reader can pull up Part II alongside Part I and compare and contrast the results without too much difficulty. As a result, this two-part paper demonstrates the use of wave-optics simulations to model the effects of dynamic speckle.

## 2. ANALYTICAL IRRADIANCE CORRELATION COEFFICIENT

In this section, we formulate closed-form expressions for the analytical irradiance correlation coefficient, ${\mu _I}({{{\boldsymbol p}_1};{{\boldsymbol p}_2}})$. Strictly speaking, these formulations treat ${\mu _I}({{{\boldsymbol p}_1};{{\boldsymbol p}_2}})$ as a measure of correlation between two points in a static-speckled irradiance pattern. In this way, ${\mu _I}({{{\boldsymbol p}_1};{{\boldsymbol p}_2}})$ offers a sense of the average size of the speckles by solving for the spatial separation between two distinct points in space, ${{\boldsymbol p}_1}$ and ${{\boldsymbol p}_2}$, at which speckle decorrelation occurs. The closed-form expressions formulated in this section are just as effective, however, at defining where speckle decorrelation occurs for dynamic-speckled irradiance patterns [30,33,34]. In practice, we can relate such patterns to the dynamics induced by extended-object motion; thus, ${\mu _I}({{{\boldsymbol p}_1};{{\boldsymbol p}_2}})$ is a useful construct for dynamic speckle, in addition to static speckle, hence the reason we use it in the analysis that follows.

Although speckle is by nature a self-interference effect with respect to the complex-optical field, it manifests as an irradiance measurement (in units of power per unit area) using modern-day optical detectors. Consequently, dynamic speckle involves a correlation function between two speckled irradiance patterns, ${I_1}({\boldsymbol p})$ and ${I_2}({\boldsymbol p})$. The relevant correlation function is

#### A. Propagation from the Object Plane to the Pupil Plane

At this stage in the analysis, it is useful to introduce the rough-surface-scattering geometry proposed in this paper. Figure 1 illustrates this geometry as a free-space system with the $\alpha - \beta$ and $\xi - \eta$ sets of axes placed within the object and pupil planes, respectively. The respective radial coordinates are $\Omega = \sqrt {\alpha + \beta}$ and $\varrho = \sqrt {\xi + \eta}$. A distance $Z$ along the $z$ axis initially separates the object and pupil planes.

We position an optically rough extended object of width $W$ in the object plane, while an observation screen with infinite field of view (for the time being) resides in the pupil plane. Each component starts off centered at the origin of its local coordinate system. Distances $\Delta \Omega$ and $\Delta z$ are measures of in-plane and out-of-plane translation, respectively. The $z$ axis and optical axis are collinear with the axis of in-plane rotation ($\Delta \vartheta$), while out-of-plane rotation ($\Delta \varphi$) occurs about some axis in the $\alpha - \beta$ plane. As the object moves under fully coherent illumination, the diffusely scattered speckled irradiance pattern changes and eventually decorrelates from its initial state. Moving forward, we assume that both illumination and observation occur on axis (for ease of modeling). We also assume that deviations from theory (i.e., the closed-form expressions formulated in this section) are appreciable only for large angles of incidence and observation.

With Eqs. (4) and (6) in mind, recall that we can relate the analytical irradiance correlation coefficient, ${\mu _I}({{{\boldsymbol p}_1};{{\boldsymbol p}_2}})$, to the mutual intensity, ${J_U}({{{\boldsymbol p}_1};{{\boldsymbol p}_2}})$. What is more, we can use scalar diffraction theory to propagate ${J_U}({{{\boldsymbol p}_1};{{\boldsymbol p}_2}})$ from plane to plane to determine ${\mu _I}({{{\boldsymbol p}_1};{{\boldsymbol p}_2}})$ in the appropriate plane. For this purpose, if $U({\alpha ,\beta})$ is the source field (i.e., the complex-optical field in the object plane), then the first Rayleigh–Sommerfeld diffraction integral predicts that

To determine the mutual intensity, ${J_U}({{{\boldsymbol p}_1};{{\boldsymbol p}_2}})$, in the pupil plane, we first define a generic point $\Omega = ({\alpha ,\beta})$ within the object plane. In the vicinity of the pupil plane, ${{\boldsymbol p}_1}$ and ${{\boldsymbol p}_2}$ are points located at $({{\xi _1},{\eta _1},Z})$ and (${\xi _1} + \Delta \xi$, ${\eta _1} + \Delta \eta$, $Z + \Delta z$), respectively. Then Eq. (7) yields

According to Goodman [38], the scattered field immediately following an optically rough surface is delta correlated to a first approximation (above the scale of a wavelength). The resulting expression is

Making the paraxial approximation (with respect to amplitude) that $\ell _1^2 \approx \ell _2^2 \approx {Z^2}$, the result of Eqs. (4), (6), and (12) is

#### B. Four Different Modes of Extended-Object Motion

In what follows, we formulate closed-form expressions for the four different modes of extended-object motion proposed in this paper, including in-plane and out-of-plane translation and rotation. For this purpose, we need to first define a set of unit-amplitude source fields, $U(\Omega) = U({\alpha ,\beta})$. Assuming plane-wave illumination, these so-called scattering spots take the following functional forms [39]:

Moving forward, we also need to define the following special functions:

and Here, ${J_1}(\circ)$ is a first-order Bessel function of the first kind (not to be confused with mutual intensity), while $S(\circ)$ and $C(\circ)$ are, respectively, the Fresnel sine and cosine integrals [37]. These special functions readily show up in the closed-form expressions that follow for in-plane and out-of-plane translation and rotation of the object. Furthermore, these special functions [Eqs. (18)–(22)] readily provide cutoff/roll-off conditions. Such conditions define what we mean by speckle decorrelation and offer a sense of the average size of the speckles.### 1. In-Plane Translation

Assuming in-plane translation of the object (Fig. 1), Table 1 provides closed-form expressions for the analytical irradiance correlation coefficient, ${\mu _I}({\Delta \Omega})$, for all three scattering spots (i.e., square, circular, and Gaussian) with corresponding cutoff/roll-off conditions, $\Delta {\Omega _{{\rm{c/r}}}}$. Here, $\Delta \Omega$ is the in-plane translation distance. It is important to note that both the square and circular scattering spots give rise to distinct cutoff conditions (i.e., the special functions go to zero at $\Delta {\Omega _{\rm{c}}}$), whereas the Gaussian scattering spot gives rise to a roll-off condition (i.e., the special function never reaches zero but has a $1/{e^2}$ magnitude at $\Delta {\Omega _{\rm{r}}}$).

To formulate the closed-form expressions given in Table 1, $\Delta z$ is set to zero in Eq. (13) for in-plane translation, such that point ${{\boldsymbol p}_2}$ is at (${\xi _1} + \Delta \xi$, ${\eta _1} + \Delta \eta$, $Z$). The radial distance between points of observation in the pupil plane is $\Delta \varrho = \sqrt {\Delta {\xi ^2} + \Delta {\eta ^2}}$, which corresponds directly to an in-plane object translation of $\Delta \Omega = \sqrt {\Delta {\alpha ^2} + \Delta {\beta ^2}}$. Thus, by substituting $\Delta \varrho$ with $\Delta \Omega$ after integration, the analytical irradiance correlation coefficient, ${\mu _I}({\Delta \Omega})$, becomes a function of the in-plane translation distance, $\Delta \Omega$. In so doing, we neglect the effects of boiling as we introduce new scatterers into the scattering spot. This assumption is valid as long as the scattering spot is larger than the speckles it produces in the pupil plane.

For a square or circular scattering spot of width or diameter $W$, the cutoff condition, $\Delta {\Omega _{\rm{c}}}$, corresponds to the average lateral size of the speckles. If dealing with an oblong rectangular spot, things become separable in the horizontal and vertical directions (using different values for $W$). These findings agree with published results [40,41].

For a Gaussian scattering spot of $1/e$-amplitude diameter $W$, the roll-off condition, $\Delta {\Omega _{\rm{r}}}$, is consistent with Goodman’s theory [14]. The resulting equation is valid only over small translation distances [42], as are all other Gaussian functions presented in this paper. Such analytical curves decrease monotonically out to infinity, when in practice there are oscillatory outer lobes (as with previous expressions), due to periodic overlap of speckles with large translation distances [43]. Moreover, these analytical curves decay asymptotically, which means there is no zero crossing at which to naturally define the average lateral size of the speckles. Instead, the $1/{e^2}$ point serves as a correlation roll-off condition rather than a cutoff condition. What matters for comparison with discrete irradiance datasets from wave-optics simulations (or experiments) is that there is consistency with theory at least up to this roll-off condition.

### 2. Out-of-Plane Translation

Assuming out-of-plane translation of the object (Fig. 1), Table 2 provides closed-form expressions for the analytical irradiance correlation coefficient, ${\mu _I}({\Delta z})$, for all three scattering spots (i.e., square, circular, and Gaussian) with corresponding cutoff/roll-off conditions, $\Delta {z_{{\rm{c/r}}}}$. Here, $\Delta z$ is the out-of-plane translation distance. It is important to note that both the square and circular scattering spots give rise to distinct cutoff conditions (i.e., the special functions go to zero or a minimum at $\Delta {z_{\rm{c}}}$), whereas the Gaussian scattering spot gives rise to a roll-off condition (i.e., the special function has a $1/{e^2}$ magnitude at $\Delta {z_{\rm{r}}}$).

To formulate the closed-form expressions given in Table 2, $\Delta \Omega$ is set to zero in Eq. (13) for out-of-plane translation, confining point ${{\boldsymbol p}_2}$ to (${\xi _1}$, ${\eta _1}$, $Z + \Delta z$). Unlike with in-plane translation (Section 2.B.1), the results now vary with radial vantage point $\varrho = \sqrt {\xi + \eta}$ in the pupil plane. Thus, the closed-form expressions given in Table 2 are valid only for *on-axis* speckles.

Analogous to the relationship between in-plane translation and the average lateral size of the speckles, the cutoff/roll-off conditions given in Table 2 estimate the average longitudinal size of the on-axis speckles. These speckles all point away from the centroid of illumination, meaning they align with the $z$ axis at $\varrho = 0$ and rotate away from it for $\varrho \gt 0$. They also elongate with increasing distance from the illumination axis, yet they have the same axial projection on average. This behavior implies that they are shortest along the axial dimension with an off-axis length of $\Delta {z_{\rm{c}}} = \sqrt {{\xi ^2} + {\eta ^2} + {Z^2}}$ [40]. A detail worth mentioning is that Li and Chiang numerically derive scaling factors for an exact calculation of the average longitudinal size of the off-axis speckles [44]. Another detail worth mentioning is that Eq. (22) does not cross zero but rather decreases to a minimum value of $6.65 \times {10^{- 3}}$ before increasing again.

### 3. In-Plane Rotation

Assuming in-plane rotation of the object (Fig. 1), Table 3 provides closed-form expressions for the analytical irradiance correlation coefficient, ${\mu _I}({\Delta \vartheta})$, for all three scattering spots (i.e., square, circular, and Gaussian) with corresponding cutoff/roll-off conditions, $\Delta {\vartheta _{{\rm{c/r}}}}$. Here, $\Delta \vartheta$ is the in-plane rotation angle. It is important to note that both the square and circular scattering spots give rise to distinct cutoff conditions (i.e., the special functions go to zero at $\Delta {\vartheta _{\rm{c}}}$), whereas the Gaussian scattering spot gives rise to a roll-off condition (i.e., the special function has a $1/{e^2}$ amplitude at $\Delta {\vartheta _{\rm{r}}}$).

In essence, in-plane rotation is an extension of in-plane translation (Section 2.B.1), given a circular path around the rotational axis. Accordingly, we can substitute arc length $\Delta \vartheta \varrho$ for linear distance $\Delta \Omega$. Doing so produces the relationships given in Table 3.

Similar to out-of-plane translation (Section 2.B.2), the expressions in this case vary with radial vantage point $\varrho = \sqrt {\xi + \eta}$. Unlike for out-of-plane translation, however, the closed-form expressions given in Table 3 readily account for off-axis speckles with the inclusion of variable $\varrho$. Churnside’s work confirms these results after appropriate simplifications [45], as does further analysis by Yura *et al.* [46]. Saleh makes the point that in-plane rotation at sufficiently large angles warrants the inclusion of a sinusoidal argument factor to account for periodic replication of the signal in time [27]. A detail worth mentioning is that the on-axis correlation is unity with a cutoff/roll-off condition of infinity, since $\varrho = 0$. This result is physically accurate, since the speckle at the very center of rotation remains stationary, independent of in-plane rotation $\Delta \vartheta$.

### 4. Out-of-Plane Rotation

Assuming out-of-plane rotation of the object (Fig. 1), Table 4 provides closed-form expressions for the analytical irradiance correlation coefficient, ${\mu _I}({\Delta \varphi})$, for all three scattering spots (i.e., square, circular, and Gaussian) with corresponding cutoff/roll-off conditions, $\Delta {\varphi _{{\rm{c/r}}}}$. Here, $\Delta \varphi$ is the out-of-plane rotation angle. It is important to note that both the square and circular scattering spots give rise to distinct cutoff conditions (i.e., the special functions go to zero at $\Delta {\varphi _{\rm{c}}}$), whereas the Gaussian scattering spot gives rise to a roll-off condition (i.e., the special function has a $1/{e^2}$ magnitude at $\Delta {\varphi _{\rm{r}}}$).

In essence, out-of-plane rotation also mimics the behavior of in-plane translation (Section 2.B.1), given small-angle rotations. Accordingly, we can substitute linear distance $2Z\Delta \varphi$ for linear distance $\Delta \Omega$. Doing so produces the relationships given in Table 4.

For near-normal angles of incidence and observation, the speckled irradiance patterns arise from the same set of rough-surface scatters [47]. As such, a linear tilt across the object’s surface imposes a linear shift in the far field. This last point relates to the shift theorem of the Fourier transform [39]. Thus, under the small-angle approximation, the appropriate substitution comes about through geometric considerations as the object surface normal subtends an angle of $\Delta \varphi \approx \Delta \varrho /Z$.

Recalling that $\Delta \varrho$ and $\Delta \Omega$ are functionally equivalent for in-plane translation and free-space propagation, the result is that $Z\Delta \varphi$ replaces $\Delta \Omega$ in a transmission geometry. The reflection geometry then requires that $2Z\Delta \varphi$ replaces $\Delta \Omega$, as an angle doubling occurs due to the double pass through the depth of the tilted object. With this last point in mind, the rough-surface scattering geometry under consideration leads to the closed-form expressions given in Table 4. The same results follow from Goodman’s use of scattering vectors to characterize the speckle decorrelation at normal incidence and observation [14].

#### C. Analytical Exploration

Figure 2 plots the closed-form expressions formulated in Tables 1–4. In particular, Fig. 2(a) plots the case of in-plane translation (Table 1), Fig. 2(b) plots the case of out-of-plane translation (Table 2), Fig. 2(c) plots the case of in-plane rotation (Table 3), and Fig. 2(d) plots the case of out-of-plane rotation (Table 4). All plots include the respective cutoff conditions for square and circular scattering spots and the roll-off conditions for Gaussian scattering spots.

## 3. NUMERICAL IRRADIANCE CORRELATION COEFFICIENT

All of the closed-form expressions formulated in Section 2 make use of continuous speckled irradiance patterns, ${I_1}({\boldsymbol p})$ and ${I_2}({\boldsymbol p})$. In this section, we make use of discrete irradiance datasets, ${I_1}$ and ${I_2}$, from wave-optics simulations (or experiments). With this last point in mind, the numerical irradiance correlation coefficient, ${\hat\mu_I}$, takes the following form:

With Eq. (23) in mind, the wave-optics simulations setup in this section makes use of the following procedure.

- 1. Create an optically rough extended object using a phase-screen approach.
- 2. Propagate from the object plane to the pupil plane.
- 3. Crop the irradiance dataset ${I_1}$ and save for reference.
- 4. Modify the optically rough extended object with the appropriate mode of extended-object motion.
- 5. Repeat as necessary, saving the frame-to-frame irradiance dataset ${I_2}$.
- 6. Calculate the numerical irradiance correlation coefficients as a function of extended-object motion.

To illustrate steps 1–3, Fig. 3 displays example irradiance and phase datasets. These wave-optics simulations make use of the WaveProp Toolbox in MATLAB [49].

#### A. Simulating Propagation from the Object Plane to the Pupil Plane

Analogous to Fig. 1, Fig. 4 depicts the free-space system simulated in the wave-optics simulations. These wave-optics simulations used an $N \times N$ grid resolution with $N = 512$. This choice provided an acceptable balance between physical accuracy and computational efficiency [50]. The wave-optics simulations also made use of a free-space wavelength of ${\lambda _0} = 1\;{{\unicode{x00B5}{\rm m}}}$ and a limiting-aperture (circular-only) diameter of $D = 30\,\,{\rm{cm}}$, which are typical values for long-range propagation studies.

For simplicity, the wave-optics simulations used unity scaling between the simulated object and pupil planes. They also used 200 grid points across the aperture diameter, while padding the circular pupil with zeros to exceed the recommended factor of 2.4 [51]. As such, the grid spacing, $\delta$, was 1.5 mm, and the grid side length, $S$, was 76.8 cm. Critical sampling [52] (a.k.a. Fresnel scaling [49]) then stipulated that

Satisfying critical sampling typically gives wave-optics results that are free of aliasing. However, the high spatial frequencies contained in diffuse speckle made the wave-optics simulations especially prone to aliasing even with Eq. (24) satisfied. Tailored methods such as pupil-plane filtering [53] aim to combat this problem by eliminating the high spatial frequencies that would cause aliasing. Nonetheless, empirical evidence suggests that first doubling the grid resolution, then propagating the field (via the impulse-response method [52]) and cropping back down has greater resistance to aliasing [49]. Taking this approach, we set $Z = 2.30\,\,{\rm{km}}$.Recalling that the scattering-spot width/diameter $W$ varies inversely with speckle size, it cannot be so large as to cause insufficient sampling of the speckle in the simulated pupil plane. As a result, we set $W = 30.7\;{\rm{cm}}$, so that the object Fresnel number, ${N_{{\rm{obj}}}} = DW/({\lambda Z})$, was 40. This choice populated the pupil plane with roughly 40 speckles across $D$ (Fig. 3), yielding five grid points per speckle for an average pupil-plane error of ${\lt}1\%$ [7,8]. Table 5 summarizes all of the parameters of interest in the wave-optics simulations.

#### B. Simulating Four Different Modes of Extended-Object Motion

To simulate an optically rough extended object, we used a phase-screen approach [7,8,14]. In so doing, we assumed that the surface heights were uniformly distributed and delta correlated from grid point to grid point. At each grid point within the scattering spot, we then took a random draw from a uniform phase distribution on the interval $[{- \pi ,\pi})$ and examined four different modes of extended-object motion.

### 1. *Simulating In-Plane Translation*

Simulating in-plane translation required that we move the phase screen laterally across the scattering spot. Since the phase-screen approach used in this paper assumed that the surface heights were uniformly distributed and delta correlated from grid point to grid point, we set the minimum in-plane translation distance to a single grid point of motion between each captured frame. Implementing in-plane translation in this way involved a circular shift of the phase screen in one direction. Since the object width, $W$, was considerably smaller than the grid side length $S$, the resultant scattering spot had zero magnitude (or near-zero magnitude in the case of a Gaussian scattering spot) near the edges of the grid. Thus, the phase wraparound resulting from a small circular shift did not affect the phase screen’s frame-to-frame randomness.

### 2. *Simulating Out-of-Plane Translation*

Out-of-plane translation was perhaps the most laborious mode of extended-object motion to simulate properly, as it required a different propagation distance between the simulated object and pupil planes for each successive value of $\Delta z$. This outcome meant that we inevitably violated critical sampling [Eq. (24)] as we moved the simulated object plane closer to the simulated pupil plane. Varying this propagation distance also changed the lateral speckle size, meaning we had to recrop and upsample each speckle pattern (for comparison with the original) as the object moved closer to the pupil plane. Nonetheless, we empirically determined that the wave-optics simulations were robust against the effects of aliasing and resampling for all values of $\Delta z$.

As discussed in Appendix A, the simulated out-of-plane translation exhibited a radial dependence. Thus, masking the irradiance datasets restricted the viewing region to a certain radius to calculate the numerical irradiance correlation coefficient [Eq. (23)]. These masks were of the same thickness as the size of the speckles, where speckle size was defined by the cutoff/roll-off conditions given in Table 2.

### 3. *Simulating In-Plane Rotation*

To simulate in-plane rotation, we applied a rotation matrix at the specified angle $\Delta \vartheta$. We also applied nearest-neighbor interpolation. In turn, we observed reasonable rotation in the resulting dynamic speckle (as expected) without a noticeable loss of fidelity.

Similar to out-of-plane translation, the simulated in-plane rotation also exhibited a radial dependence, as discussed in Appendix A. In turn, masking the irradiance datasets restricted the viewing region to a certain radius to calculate the numerical irradiance correlation coefficient [Eq. (23)]. In accordance with the cutoff/roll-off conditions given in Table 3, these masks were of the same thickness as the average size of the speckles.

### 4. *Simulating Out-of-Plane Rotation*

Simulation of out-of-plane rotation involved multiplying the simulated object plane by the following complex reflectance function:

#### C. Numerical Exploration

In the next section, we compare the results obtained for the numerical irradiance correlation coefficient to those obtained for the analytical irradiance correlation coefficient. To do so, we need to perform Monte Carlo averaging on the numerical results. To explore this numerical trade space, we use root mean square error (RMSE), such that

Figure 5 plots Eq. (26) to find that the average RMSE becomes asymptotically stable in the neighborhood of 40 Monte Carlo trials. Choosing this number keeps the error below ${\sim}1\%$. Note that the average RMSE results displayed in Fig. 5 are fairly representative for all four modes of extended-object motion. Also note that we averaged over 100 realizations at each datapoint for curve-smoothing purposes.

#### D. ANALYTICAL AND NUMERICAL RESULTS

Figures 6–9 provide the analytical and numerical results for this paper. Overall, the analytical results from theory are in agreement with the numerical results from simulation. With this agreement in mind, we discuss the four different modes of extended-object motion in the following list.

- 1. Figure 6 shows the analytical and numerical results for
*in-plane translation*(also see Visualization 1). Here, the sampling is relatively coarse due to the fact we set the minimum in-plane translation distance to a single grid point of motion between each captured frame. Future efforts could look at using interpolation to increase this sampling. However, doing so could violate the assumptions used throughout this paper, in particular, that the optically rough surface is delta correlated to a first approximation. - 2. Figure 7 shows the analytical and numerical results for
*out-of-plane translation*(also see Visualization 2). Here, we show results for several values of $\varrho$ but relative to some scattering-spot width $W$. To calculate numerical results for off-axis observation, where $\varrho \ne 0$, we made use of an annular mask, which we illustrate in Fig. 10 in Appendix A. By normalizing the numerical results using the approach presented in Ref. [41], we derived nonlinear scale factors [54] to modify the closed-form expressions presented in Table 2. In particular, we normalized the off-axis radial position to the scattering-spot width $W$, since the scale factors themselves are a normalization of the longitudinal correlation lengths. These scale factors allowed for comparison with the analytical results for off-axis observation, where $\varrho \ne 0$. - 3. Figure 8 shows the analytical and numerical results for
*in-plane rotation*(also see Visualization 3). Here, we show results for several values of $\varrho$ relative to some position $P$. Similar to out-of-plane translation, we made use of the annular mask, which we illustrate in Fig. 10 in Appendix A, to calculate numerical results for off-axis observation, where $\varrho \ne 0$. Recalling that the closed-form expressions in Table 3 are already set up to handle off-axis observation, one does not need scale factors in this particular case. Moreover, we observe the off-axis speckle at relative rather than absolute radial positions because the correlation falloff is linear with radial position. - 4. Figure 9 shows the analytical and numerical results for
*out-of-plane rotation*(also see Visualization 4). These results mirror those contained in Fig. 6 for in-plane translation, but the use of Eq. (25) at arbitrary out-of-plane rotation angles allowed for better sampling. This similarity comes about because translation of speckle dominates the pupil-plane decorrelation in both cases.

The data points in Figs. 6–9 also indicate ${\pm}1$ standard deviation about the Monte Carlo average (i.e., the average with respect to 40 Monte Carlo trials). A general observation is that these standard deviations seem to grow with increasing extended-object motion, which is not surprising. Even so, the error bars maintain an upper bound of ${\sim}{{3}}\%$; thus, the Monte Carlo averaging did not dramatically affect the mean result for any one trial. Before moving on to the next section, it is important to note that Visualizations 1–4 help in comprehending the results presented in this section. These visualizations show results for a square scattering spot and circular limiting aperture. This particular setup is common between Parts I and II of this two-part paper. Thus, we include both pupil and image planes in these visualizations, so that the results presented here complement those contained in Part II and vice versa.

## 4. CONCLUSION

This paper demonstrated the use of wave-optics simulations to model the effects of dynamic speckle. It serves as Part I of a two-part paper. In this paper, we formulated closed-form expressions for the analytical irradiance correlation coefficient, specifically in the pupil plane of an optical system. These expressions were for square, circular, and Gaussian scattering spots and four different modes of extended-object motion, including in-plane and out-of-plane translation and rotation. Using a phase-screen approach, we then simulated the equivalent scattering from an optically rough extended object, where we assumed that the surface heights were uniformly distributed and delta correlated from grid point to grid point. For comparison to the analytical irradiance correlation coefficient, we also calculated the numerical irradiance correlation coefficient from the dynamic speckle after simulated propagation from the object plane to a pupil plane. Overall, the analytical and numerical results definitely demonstrated that, relative to theory, the dynamic speckle in the simulated pupil plane is properly correlated from one frame to the next. Such validated wave-optics simulations provide the framework needed to model more sophisticated setups and obtain accurate results for system-level studies.

## APPENDIX A

Both the simulated out-of-plane translation and the simulated in-plane rotation exhibited a radial dependence. Thus, masking the irradiance datasets restricted the viewing region to a certain radius to calculate the numerical irradiance correlation coefficient [Eq. (23)]. On-axis observation simply required a circular mask, but off-axis observation required an annular mask, as shown in Fig. 10(a). In this work, the mask had the same thickness as the average size of the speckles, as portrayed in Fig. 10(b). In general, the annular mask had inner and outer radii ${\varrho _1}$ and ${\varrho _2}$, respectively, with an average radius

## Acknowledgment

The authors of this paper thank the Joint Directed Energy Transition Office for sponsoring this research, as well as T. J. Brennan for many insightful discussions regarding the results presented within. Approved for public release; distribution is unlimited. Public Affairs release approval AFRL-2021-0982.

## Disclosures

The authors declare no conflicts of interest.

## Data Availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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