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

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 II, we formulate closed-form expressions for the analytical irradiance correlation coefficient in the image plane of an optical system. Part I starts by formulating closed-form expressions for the analytical irradiance correlation coefficient in the pupil plane of an optical system. It is worthwhile to consider the image plane separately from the pupil plane, as the structure of speckle turns out to operate independently in each plane under most conditions of interest. In turn, this paper focuses solely on the theory and simulation of speckle decorrelation in the image plane of an optical system. Because image formation is of concern, the pupil plane (discussed throughout Part I) is equivalent to a plane of observation at some distance from the extended object in a free-space system. Here, 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 II 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 an image plane) for (1) the cases of square, circular, and Gaussian limiting apertures 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 the 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 II so that it complements Part I. 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 I to understand the results in Part II and vice versa), and (2) the reader can pull up Part I alongside Part II 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.

 

Fig. 1. Free-space propagation from an optically rough extended object in the object plane to a limiting aperture in the pupil plane followed by another free-space propagation to an observation screen in the image plane.

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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 Image 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 single-lens system with the $\alpha {\text -} \beta$, $\xi {\text -} \eta$, and $x {\text -} y$ sets of axes placed within the object, pupil, and image planes, respectively. The respective radial coordinates are $\Omega = \sqrt {\alpha + \beta}$, $\rho = \sqrt {\xi + \eta}$, and $r = \sqrt {x + y}$. A distance ${Z_1}$ along the $z$ axis initially separates the object and entrance-pupil planes, whereas ${Z_2}$ is a fixed distance between the exit-pupil and image planes. Note that ${Z_1}$ is simply called $Z$ in Part I. Also note that with the placement of a single lens between the object and image planes, the entrance and exit pupils are coplanar with the lens (which also serves as the aperture stop).

We position an optically rough extended object of width $W$ in the object plane, a limiting aperture of width $D$ in the pupil plane, and an observation screen with infinite field of view (for the time being) in the image 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 $\alpha {\text -} \beta$ plane. As the object moves under fully coherent illumination, the diffusely scattered speckled irradiance pattern changes and eventually decorrelates from its initial state. These changes are generally different in the pupil and image planes as the speckles propagate through the single-lens system. 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 ${\boldsymbol \Omega} = ({\alpha ,\beta})$ within the object plane. In the vicinity of the pupil plane, ${{\boldsymbol p}_1}$ and ${{\boldsymbol p}_2}$ are points located at $({{\xi _1},{\eta _1},{Z_1}})$ and (${\xi _1} + \Delta \xi$, ${\eta _1} + \Delta \eta$, ${Z_1} + \Delta z$), 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 $\ell _1^2 \approx \ell _2^2 \approx Z_1^2$, the result of Eqs. (4), (6), and (12) is

Mathematically speaking, Eq. (13) accounts for speckle decorreclation in the pupil plane of the single-lens system described in Fig. 1. To account for speckle decorrelation in the image plane, we again use Eq. (13), but we first replace the source field $U({\boldsymbol \Omega}) = U({\alpha ,\beta})$, which gives rise to a scattering spot of width $W$, with a pupil function $P(\varrho) = P({\xi ,\eta})$, which gives rise to a limiting aperture of width $D$. We also replace the object distance (${Z_1}$) with the image distance (${Z_2}$). In turn, ${{\boldsymbol p}_1}$ and ${{\boldsymbol p}_2}$ are points located at $({{x_1},{y_1},{Z_2}})$ and (${x_1} + \Delta x$, ${y_1} + \Delta y$, ${Z_2} + \Delta z$), respectively, such that in the vicinity of the image plane,

In effect, speckle decorrelation in the image plane is then independent of the scattering spot from which the pupil-plane speckles originate. Making this approximation requires that the scattering spot spans many coherence areas in the pupil plane and resolution cells in the object plane [14]. It also requires that lens aberrations do not drastically affect the structure of the speckled irradiance pattern.

Much of the foundational work on speckle decorrelation applies a binomial approximation to a power-series expansion of the phasor argument of Eq. (14) prior to integrating. This final paraxial approximation (with respect to phase) ultimately gives rise to a scaled Fresnel diffraction integral, since replacing the impulse response with the well-known Fresnel propagation kernel effectively makes Eq. (14) a normalized Fresnel transform of ${| {P({\boldsymbol \varrho})} |^2}$ in two dimensions.

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 pupil functions, $P({\boldsymbol \varrho}) = P({\xi ,\eta})$. These functions take the following functional forms [40]:

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

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 limiting apertures (i.e., square, circular, and Gaussian) with corresponding cutoff/roll-off conditions, $\Delta {\Omega _{{{c}/{r}}}}$. Here, $\Delta \Omega$ is the in-plane translation distance. It is important to note that both the square and circular limiting apertures give rise to distinct cutoff conditions (i.e., the special functions go to zero at $\Delta {\Omega _{c}}$), whereas the Gaussian limiting aperture gives rise to a roll-off condition (i.e., the special function never reaches zero but has a $1/{{e}^2}$ magnitude at $\Delta {\Omega _{r}}$).

Table 1. Closed-Form Expressions for In-Plane Translation

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To formulate the closed-form expressions given in Table 1, we set $\Delta z$ to zero in Eq. (14), such that point ${{\boldsymbol p}_2}$ is at (${x_1} + \Delta x$, ${y_1} + \Delta y$, ${Z_2}$). The radial distance between points of observation in the image plane is then $\Delta r = \sqrt {\Delta {x^2} + \Delta {y^2}}$, which corresponds directly to an in-plane object translation of $\Delta \Omega = \sqrt {\Delta {\alpha ^2} + \Delta {\beta ^2}}$ after accounting for magnification. Thus, by substituting $\Delta r$ with $\Delta \Omega$ and ${Z_2}$ with ${Z_1}$ after integration, the analytical irradiance correlation coefficient, ${\mu _I}({\Delta \Omega})$, becomes a function of the in-plane translation distance, $\Delta \Omega$, and object distance, ${Z_1}$. In so doing, we neglect the effects of boiling as we introduce new speckles into the limiting aperture. This assumption is valid as long as the limiting aperture is larger than the speckles it produces in the image plane.

To make these aforementioned substitutions, one can relate speckle decorrelation in the image plane to a concept known as memory loss, which Cloud describes as the physical movement of a speckled irradiance pattern beyond its original boundaries in any direction [41]. Put another way, the single-lens system described in Fig. 1 becomes anisoplanatic with varying shifts. Such shifts give rise to speckle decorrelation in the image plane.

As shown in Part I, only the wavelength, propagation distance, and scattering-spot width can alter the average size of the pupil-plane speckles, and this size determines what we mean by speckle decorrelation in the pupil plane. Image-plane speckles, on the other hand, are roughly the size of a resolution element (a.k.a. “resel”) in the image plane [42]. The average size of the image-plane speckles, again, determines what we mean by speckle decorrelation in the image plane. However, to relate any extended-object motion in the image plane to that in the object plane, we must project the image of the object into object space. For this reason, a shift by a resel on the object (i.e., the conjugate of a resel in the image) is what causes total decorrelation with in-plane translation [43].

With a square limiting aperture, for example, the width of a resel (and therefore the average size of the speckles in the image plane [42]) is

For a square or circular limiting aperture of width or diameter $D$, the cutoff condition, $\Delta {\Omega _{c}}$, corresponds to the average lateral resel on the object. If dealing with an oblong rectangular aperture, things become separable in the horizontal and vertical directions (using different values for $D$). These findings agree with published results [47,48].

For a Gaussian limiting aperture of $1/{e}$-amplitude diameter $D$, the roll-off condition, $\Delta {\Omega _{r}}$, is consistent with Goodman’s theory [14]. The resulting equation is valid only over small translation distances [49], 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 [50]. 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. Here, we consider soft Gaussian apertures, noting that such apertures use apodization filters to allay the strong diffraction effects associated with hard edges [51].

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 limiting apertures (i.e., square, circular, and Gaussian) with corresponding cutoff/roll-off conditions, $\Delta {z_{{{c}/{r}}}}$. Here, $\Delta z$ is the out-of-plane translation distance. It is important to note that both the square and circular limiting apertures give rise to distinct cutoff conditions (i.e., the special functions go to zero or a minimum at $\Delta {z_{c}}$), whereas the Gaussian limiting aperture gives rise to a roll-off condition (i.e., the special function has a $1/{{e}^2}$ magnitude at $\Delta {z_{r}}$).

Table 2. Closed-Form Expressions for Out-of-Plane Translation

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To formulate the closed-form expressions given in Table 2, $\Delta \rho$ is set to zero in Eq. (14) for out-of-plane translation, confining point ${{\boldsymbol p}_2}$ to (${x_1}$, ${y_1}$, ${Z_2} + \Delta z$). Thus, for a single-lens system and out-of-plane translation, one can substitute ${Z_2}$ with ${Z_1}$ after integration, and the analytical irradiance correlation coefficient, ${\mu _I}({\Delta z})$, becomes a function of the object distance, ${Z_1}$.

Analogous to the relationship between in-plane translation and the average lateral resel on the object, the cutoff/roll-off conditions given in Table 2 estimate the average longitudinal resel on the object. In turn, the average size of the longitudinal image-plane speckles is proportional to $\lambda {({{Z_2}/{D_{{\rm XP}}}})^2}$, or more generally $\lambda F_{w}^2$. A detail worth mentioning is that although off-axis observation can change the behavior of the speckle decorrelation in the pupil plane, as shown in Part I, it has very little influence on the behavior of the speckle decorrelation in the image plane [52]. Another detail worth mentioning is that Eq. (24) 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 limiting apertures (i.e., square, circular, and Gaussian) with corresponding cutoff/roll-off conditions, $\Delta {\vartheta _{{{c}/{r}}}}$. Here, $\Delta \vartheta$ is the in-plane rotation angle. It is important to note that both the square and circular limiting apertures give rise to distinct cutoff conditions (i.e., the special functions go to zero at $\Delta {\vartheta _{c}}$), whereas the Gaussian limiting aperture gives rise to a roll-off condition (i.e., the special function has a $1/{{e}^2}$ amplitude at $\Delta {\vartheta _{r}}$).

Table 3. Closed-Form Expressions for In-Plane Rotation

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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 $\Delta \vartheta r$ for linear distance, $\Delta \Omega$. Doing so produces the relationships given in Table 3.

Similar to the pupil-plane expressions (Part I), the closed-form expressions in this case vary with radial vantage point $r = \sqrt {x + y}$. Churnside’s work confirms these expressions after appropriate simplifications [53], as does further analysis by Yura et al. [54]. 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 $r = 0$. This result is physically accurate, since the speckle at the very center of rotation remains stationary, independent of in-plane rotation $\Delta \vartheta$.

Table 4. Closed-Form Expressions for Out-of-Plane Rotation

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Fig. 2. Analytical exploration of the trade space in terms of the four different modes of extended-object motion.

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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 limiting apertures (i.e., square, circular, and Gaussian) with corresponding cutoff/roll-off conditions, $\Delta {\varphi _{{{c}/{r}}}}$. Here, $\Delta \varphi$ is the out-of-plane rotation angle. It is important to note that both the square and circular limiting apertures give rise to distinct cutoff conditions (i.e., the special functions go to zero at $\Delta {\varphi _{c}}$), whereas the Gaussian limiting aperture gives rise to a roll-off condition (i.e., the special function has a $1/{{e}^2}$ magnitude at $\Delta {\varphi _{r}}$).

In essence, out-of-plane rotation is a unique case of extended-object motion. Here, the same set of scatterers occupies the scattering spot within the object plane over a small range of angles. For this reason, the object never moves by a resel in any direction, and therefore no speckle decorrelation occurs by memory loss. Cloud’s other criterion for speckle decorrelation [41], which says that the phase difference across a resel attains a value of $2\pi$, comes into play now. For example, in terms of object-plane tilt, one can apply a phase change of $\phi = k\Delta \varphi \Omega$ across the optically rough surface in a transmission geometry. The reflection geometry instead requires $\phi = 2k\Delta \varphi \Omega$. As such, one can set the radial difference $\Delta \Omega$ equal to the resel width $\lambda {Z_1}/D$ and solve for $\Delta \phi = 2\pi$. The out-of-plane rotation angle $\Delta \varphi$ then prompts a cutoff of $D/({2{Z_1}})$, which proves to be consistent with theory [55,56].

Marron and Morris studied this problem in the case of rotating objects, deriving a spatiotemporal correlation function with an envelope that follows the analytical irradiance correlation coefficient as a function of rotation angle [55]. The derivation involves propagating mutual intensity through to the image plane, making similar arguments to those preceding Eq. (14). Rather than convolve with the free-space impulse response, however, the convolution is between object-plane mutual intensity and the coherent point spread function (PSF) of the imaging system. A rectangular aperture generates the normalized PSF

 

Fig. 3. Example irradiance and phase datasets from the wave-optics simulations.

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The closed-form expressions in Table 4 are in line with familiar forms of the modulation transfer function (MTF) for equivalent incoherent systems [58]. Taking the squared magnitude of the coherent PSF is an operation equivalent to autocorrelating the pupil function, which determines the MTF of an incoherent system. The second column of Table 4 represents an angular cutoff beyond which speckle fully decorrelates, though this is not the same cutoff that describes the coherent spatial-frequency modulation limit [59]. Total decorrelation takes place once the speckles in the pupil plane translate by the aperture width and a new, independent realization takes its place. As a side note, the cutoff in a coherent transmission geometry is twice that of a reflection geometry due to the single pass in optical path length.

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 limiting apertures and the roll-off conditions for Gaussian limiting apertures.

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

A. Simulating Propagation from the Object Plane to the Image Plane

Analogous to Fig. 1, Fig. 4 depicts the imaging 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 [62]. The wave-optics simulations also made use of a free-space wavelength of ${\lambda _0} = 1\;{\unicode{x00B5}{\rm m}}$ and a limiting-aperture width/diameter of $D = 30\;{\rm cm} $, which are typical values for long-range propagation studies.

 

Fig. 4. Illustration of the imaging system simulated in the wave-optics simulations. Here, we use an optically rough three-bar object (for illustrative purposes).

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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 [63]. As such, the grid spacing, $\delta$, was 1.5 mm, and the grid side length, $S$, was 76.8 cm. Critical sampling [64] (a.k.a. Fresnel scaling [61]) then stipulated that

Recalling that the scattering-spot (square-only) width $W$ 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]. In the pupil plane, we used two thin-lens transmittance functions to collimate the light after propagation from the object plane to the entrance pupil and focus the light upon propagation from the exit pupil to the image plane (Fig. 4). Unity scaling, in turn, dictated that ${Z_1} = {Z_2} = Z$, which gave rise to unit-magnification imaging in the wave-optics simulations. Table 5 summarizes all of the parameters of interest in the wave-optics simulations.

Table 5. Parameters of Interest in Wave-Optics Simulations

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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.

 

Fig. 5. Numerical exploration in terms of the average RMSE versus the number of Monte Carlo trials.

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Fig. 6. Analytical and numerical results for in-plane translation, given (a) square, (b) circular, and (c) Gaussian limiting apertures.

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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. (33)] as we moved the simulated object plane closer to the simulated pupil plane. Varying this propagation distance also changed the image size and added some defocus, meaning we had to recrop and downsample each speckled image (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 $\Delta z$.

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.

The simulated in-plane rotation 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. (32)]. 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 3.

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. (35) to find that the average RMSE becomes asymptotically stable in the neighborhood of 40 Monte Carlo trials. Choosing this number keeps the error below $\sim1\%$. 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.

 

Fig. 7. Analytical and numerical results for out-of-plane translation, given (a) square, (b) circular, and (c) Gaussian limiting apertures.

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4. 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.

 

Fig. 8. Analytical and numerical results for in-plane rotation, given (a) square, (b) circular, and (c) Gaussian limiting apertures.

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Fig. 9. Analytical and numerical results for out-of-plane rotation, given (a) square, (b) circular, and (c) Gaussian limiting apertures.

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The data points in Figs. 6–9 also indicate $\pm1$ 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 $\sim3\%$; 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 I and vice versa.

5. CONCLUSION

This paper demonstrated the use of wave-optics simulations to model the effects of dynamic speckle. It serves as Part II of a two-part paper. In this paper, we formulated closed-form expressions for the analytical irradiance correlation coefficient, specifically in the image plane of an optical system. These expressions were for square, circular, and Gaussian limiting apertures 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 an image plane. Overall, the analytical and numerical results definitely demonstrated that, relative to theory, the dynamic speckle in the simulated image 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.

 

Fig. 10. Example annular mask for radial isolation of the irradiance datasets (a) without speckle and (b) with speckle.

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