1. Introduction

Speckle, a coherent optical phenomenon, has been extensively studied, dating back as far as 1877 [1]. Study intensified following the invention of the laser, thoroughly examining the generation, morphology, and statistics of speckle patterns [2,3]. Much more recently, it was shown that speckle patterns can carry the image of an emissive object, allowing that object to be seen in non-line-of-sight (NLOS) conditions, including through scattering media and around corners [4–6]. Yet, the established theory has a knowledge gap with regard to an important around-the-corner NLOS imaging case: finite-sized, broadband objects with finite propagation distances from the scatterer. Here, we seek to fill the gap through experimentation, theory, and simulation.

In this paper, we specifically examine the theory of statistical correlations between speckle patterns of different wavelengths and positions in space. Speckle correlations play an important role in NLOS imaging because the statistics of the speckle pattern can be leveraged, for example through auto-correlation or bispectrum functions, to recover an image of the hidden target [4–6]. Consequently, a decrease in speckle contrast leads to increased noise in the recovered image. Broadband speckle patterns tend to have reduced speckle contrast when compared to narrowband patterns because the dispersive properties of the scattering media lead to decorrelation of the various spectral components of the pattern, even in reflection. Spectral filtering is a simple way to boost speckle contrast, but leads to a system with decreased information capacity if each component is measured in series, which would require longer image acquisition times, or to increased detector noise if each component is measured in parallel, since each spectral component would require its own detector. It is thus of interest to examine whether the apparent decorrelation can be mitigated by optically and simultaneously combining the most highly correlated regions of each spectral component pattern without resorting to spectral filtering.

Correlation of speckle has, itself, been studied for decades, and has been derived using various physical models. Many publications examine spectral correlation where both the source and detector are in the far-field of the scattering medium [7–14], where the source is effectively omitted and propagation is considered beginning from the scattering medium [15], or where only the source is in the far-field [16,17]. Others consider only the spatial correlations over a quasi-monochromatic speckle field [18–20]. We seek to extend the theory such that neither the source nor the detector need to be in the far field, which would be the case in an around-the-corner NLOS imaging scenario, and to provide a straightforward method for simulating such a situation.

The theory presented here accommodates fields generated by finite-sized, spatially incoherent, broadband sources. Our efforts reveal a set of conditions for imaging a speckle pattern that can be used by an imaging system to maximize speckle correlation and increase contrast under broadband conditions. In turn, this increase in broadband speckle contrast can enable higher-speed acquisition in passive speckle correlation imaging systems as compared to a narrowband filtered case. We also highlight a paraxial requirement for speckle imaging, which is an important consideration for future experiments. To examine the validity of the derived expressions for the scattered field, we present sets of experimental and simulated data and show that the normalized cross-correlation of speckle patterns of different wavelengths, or the spectral correlation, is maximized at positions expected by the derived theory.

2. Scattered field

We begin with the goal of finding an expression for the speckle intensity pattern on the detector in an around-the-corner geometry. This expression will allow us to generate simulated speckle patterns and will provide the insight necessary to finding the conditions for maximizing broadband speckle correlation.

We assume a perfectly conducting reflective scattering surface and a finite-sized, spatially incoherent, broadband source. A schematic of this situation is given in Fig. 1(a). A primary source, such as a broadband LED, is imaged onto an iris by a relay lens. The open aperture of the iris serves as a secondary source whose size can be controlled. Light emitted by the secondary source travels a distance $u$ where it strikes the scattering surface. The size and shape of the illuminated scattering surface is controlled by a second iris. The scattered light travels a distance $v$, controlled by a longitudinal translation stage, where it is absorbed on an array detector. An obstacle blocks the direct line-of-sight between the secondary source and the detector. Our first goal is to find the power spectral density distribution in the detector plane.

 

Fig. 1. A schematic of the experiment and coordinate system. (a) The experimental and simulated system in which light from a spatially incoherent source is scattered and detected. (b) The definition of the coordinate system corresponding to (a). The $z_1=0$ plane corresponds the secondary source plane, the $z_2=0$ plane corresponds to the scattering surface, the $x_2-y_2-z_2$ origin lies a distance $u$ along the the $z_1$ axis from the $z_1=0$ plane, $\alpha$ is both the on-axis incident angle of light striking the scattering surface and the specular reflection angle, the $\xi -\eta$ plane corresponds to the detector plane, and the origin of the $\xi -\eta$ plane is normal to, and lies a distance $v$ along, the specular reflection ray leaving the $x_2-y_2-z_2$ origin.

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To put this situation into a mathematical context, we use the coordinate system defined in Fig. 1(b). Let the $z_1=0$ plane in Fig. 1(b) correspond to the secondary source plane in Fig. 1(a). Let $U_1(x_1,y_1)$ be the scalar field distribution in the $z_1=0$ plane. The field in another position, $U_2(x_1,y_1,z_1)$, can be found from $U_1$ by applying the Rayleigh-Sommerfeld Diffraction Formula (cf. [21], Eq. (4–9)):

Next, we account for the fact that the scattering surface is not flat, but rather has roughness that causes the scattering. To accomplish this, we modify the phase to include the optical path difference between the mean plane of the surface (the $z_2=0$ plane), which we refer to as the reference plane, and the surface itself. Let $h(x_2,y_2)$ be the zero-mean distribution of height of the surface above the reference plane, where $h$ is positive for a deviation in the $+z_2$ direction. Then the optical path difference (actual - reference) is given by $h(x_2,y_2)\cos \theta _i$, where $\theta _i$ is the incident angle of the light originating from point $(x_a,y_a)$ in the $z_1=0$ plane that strikes point $(x_2,y_2)$, with respect to the reference plane normal vector (see [22,23], especially Fig. 1 of [22]). We find that $\cos \theta _i=\left ( {u\cos \alpha -x_a\sin \alpha } \right )/R_1$. Applying this modification to Eq. (2) gives the adjusted field in the reference plane, $U_3'$:

Similarly, we can find the field $U_4(x_2,y_2,z_2)$ in the space after reflection by applying the Rayleigh-Sommerfeld Diffraction Formula with a correction that accounts for the deviation of the surface from the reference plane and an amplitude function $A(x_2,y_2)$ that accounts for the wall aperture (such as created by iris 2 in Fig. 1(a)):

Ultimately we are interested in the field $U_5(\xi,\eta )$ in the detector plane, which is perpendicular to the specular reflection direction and is a distance $v$ from the origin of the scattering surface (the $\xi -\eta$ plane as shown in Fig. 1(b)). $U_5$ can be found by applying a coordinate transformation to $U_4$:

To make further progress, we apply two paraxial approximations: we assume that both $u$ and $v$ are large compared to other parameters. We find

Several seminal works in non-line-of-sight imaging include an equation that describes the spectral density in terms of a cross-correlation or convolution [4,5,25]. Indeed, Eq. (7) is almost in the form of a cross-correlation between the spectral density distribution of the object, $s$, and an intensity point-spread function, $g$. Although we have already applied the standard paraxial approximation, Eq. (7) still does not contain a true cross-correlation because $g$ depends explicitly on $\xi$ and $x_1$ in its 3rd and 4th parameters. Equation (7) can only be reduced to the form of a cross-correlation if we make the strongly paraxial approximation that

Thus it becomes clear that the recovery of $s$ from a speckle pattern by exploiting the cross-correlation relationship is only strictly applicable under strongly paraxial conditions.

Combining Eqs. (7) and (8) provides a method for rapid simulation of speckle patterns, given a scattering surface distribution $h$. To put Eq. (7) into a more suggestive form, let $S'(\xi,\eta ):=S(v\xi /u,v\eta /u)$ be a coordinate-scaled version of the power spectral density $S$. Then

2.1 Comparison of simulation and experiment

We seek to compare the speckle patterns generated by Eq. (9) with those recorded in a comparable physical experiment. To this end, we used an optical system as shown in Fig. 1(a). A bright, full-visible-spectrum light-emitting diode (LED, Thorlabs SOLIS-2C) was relayed onto ‘iris 1’, which was closed to a diameter of about 0.75 mm so that it approximated a uniform, circular source. We set $u=230$ mm, the diameter of ‘iris 2’ was approximately 1.6 mm, and the scattering surface was set so that the scattering angle $\alpha =45$ degrees, as defined in Fig. 1(b). The scattering surface itself was a rough, silver-coated glass surface with no long-range curvature (1500 grit ground glass diffuser, Thorlabs DG10-1500-P01). An array detector (PixeLINK PL-D755MU-T) was placed at $v=35$ mm and the optical axis passed normal to and through the center of its surface, so that the center of the detector corresponded to $(\xi,\eta )=(0,0)$ as defined in Fig. 1(b).

The results of a representative experiment with the described setup are given in Figs. 2(a) and 2(b). Figure 2(a) shows an image of the speckle pattern created on the detector with the full source bandwidth. Figure 2(b) shows an image of the same speckle pattern as in Fig. 2(a) except a bandpass filter, with center wavelength 635 nm and passband width 10 nm, was placed in front of the array detector, as diagrammed in Fig. 1(a).

 

Fig. 2. Broadband and spectrally-filtered speckle patterns generated by experiment and simulation. (a) The experimental speckle pattern created by a broadband, circular source scattering from a rough metal surface, in a plane that is centered about and normal to the axial, specular reflection vector. (b) The speckle pattern recorded under the same conditions as in (a) except an optical bandpass filter is used, with center wavelength 635 nm and passband width 10 nm, in front of the detector. (c) The simulated speckle pattern generated by a coordinate transform of Eq. (9) where the source had the same power spectrum as that in (a) and the scattering surface had a translation-invariant Gaussian spatial correlation function. (d) The speckle pattern generated under the same conditions as in (c) except with a source bandwidth of 10 nm about a central wavelength of 635 nm. The yellow arrows in (a) and (b) aid the eye in locating selected shared speckle features, and likewise the green arrows in (c) and (d). The speckle contrast $C$ is listed at the bottom of each panel.

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The results of a representative simulation with parameters approximating the described experimental setup are given in Figs. 2(c) and 2(d). A circle with diameter 0.735 mm was used as the source $s$. A randomly generated array was used as the surface $h$. This surface had a translationally-invariant, Gaussian spatial correlation function with a correlation length of 470 nm and a height standard deviation of 300 nm. These values were chosen to qualitatively produce a pattern similar to that of Figs. 2(a) and 2(b) while approximately matching the average surface roughness (358 nm) of a similar experimental surface reported in the literature [26]. A grid of $8192\times 8192$ points over a window of $120$ mm $\times$ $120$ mm (pixel pitch 14.6 $\mu$m) was used in the simulation, and the random surface $h$ was generated on a grid of 16384 $\times$ 16384 points over a window of 2.3 mm along $x_2$ and 1.6 mm along $y_2$ (pixel pitch 138 nm $\times$ 98 nm). An ellipse, generated by the projection of an iris with diameter 1.6 mm at 45 degrees, as shown in Fig. 1(a), was used as the aperture $A$. We note that Eq. (9) depends implicitly on wavelength. To generate a broadband figure we sum the results of Eq. (9) for an ensemble of wavelengths, keeping $h$ the same for each wavelength but varying $s$ by a scaling factor taken from the manufacturer-published curves of the normalized spectrum of the experimental source and response function of the array detector used for Fig. 2(a). Figure 2(c) contains the sum of 250 spectral components between 400 and 800 nm wavelength and Fig. 2(d) contains the sum of 10 spectral components between 630 and 640 nm. We note that the total intensity starts to fall off in the radial direction faster in experiment than in simulation with these parameters; Fig. 2(a) is dark around the edges where Fig. 2(c) is not. This discrepancy is likely due to the fact that the simulated surface used parameters that may not be optimal and has a height distribution and spatial correlation function that are probably very simple compared to the real surface (cf. [18]).

Patterns like that shown in Fig. 2(a) have been recorded among a significant number of works and are described as “fibrous”, “radial", or “smeared" speckle patterns [1,8,27–32]. The speckles shown in Fig. 2(a) appear to elongate or streak along the radial direction as the radial distance from the center increases. This elongation is due to diffraction from the rough surface that acts as a random phase grating; light diffracts more as wavelength increases, and the elongation increases as speckles scatter further from the optical axis, much like increasing the diffraction order of a grating leads to larger angular separation of spectral components.

While the elongated speckles tend to align with the radial direction, there is some variability in this tendency and there appears to be significant curving and inter-weaving of the speckles. This variability in broadband speckle streaks has been quantified analytically, for Gaussian surfaces and under the far-field approximation, by the partial correlation of monochromatic speckle field components of different wavelengths [27,29]. Under a quasi-dispersionless propagation medium, such as air, the phase depth of the scattering surface depends on wavelength, which leads to reduced correlation between two given spectral components of the speckle field as the difference in their wavelength increases. As surface roughness increases, this decorrelation effect also increases, and leads to shorter streaks and reduced speckle contrast.

With regard to passive, non-line-of-sight speckle correlation imaging, the fibrous nature of a broadband speckle field offers both an opportunity and a problem. From a shot-noise perspective, use of the entire bandwidth of the source with a single detector, such as shown in Fig. 2(a), would greatly decrease the integration time necessary to record an image, simply because of greatly increased power throughput as compared to a narrowband filtered recording of the same object, such as shown in Fig. 2(b). From an image-processing perspective, the fibrous nature of broadband speckle destroys spatial frequency information along the streaking direction and constrains cross-correlation image retrieval methods (eg. [4–6]) to the use of speckle from only a small region around the optical axis where the streaking is reduced. An optical system that mitigates spectral speckle streaking while allowing broadband operation would be highly beneficial in this context. In the next section we describe a hypothetical, dispersive optical system that overlaps the correlated regions of each spectral speckle component and achieves streaking mitigation with broadband speckle.

3. Spectral correlation of speckle

Section 2 shows the streaking effect that broadband sources can have on the scattered speckle pattern. We proceed in this section by examining how the spectral density $S$ can be used to measure the cross-correlation between speckle intensities at different wavelengths and positions in space, with the goal of finding a set of conditions under which correlation can be maximized and the spectral streaking effect shown in Figs. 2(a) and 2(c) can be mitigated while maintaining broadband operation. We accomplish this goal by first examining Eq. (9), and second by explicitly calculating the covariance between spectral densities evaluated at various wavelengths and positions in space.

We begin by examining how the parameters implicit in Eq. (9) might be tuned to generate speckle patterns that are similar between different wavelengths. We reject the tuning of the source $s$, the scattering surface $h$, and source-surface distance $u$ as these would not be controllable during an around-the-corner imaging scenario. The remaining parameters that are available for tuning by a passive, around-the-corner, NLOS optical system are the aperture $A$ (by imaging the scattering surface to an intermediate plane and aperturing the resulting image), the surface-to-detector distance $v(\lambda )$, and a magnification $M(\lambda )$, where we have explicitly reserved the possibility that $v$ and $M$ can depend on the wavelength. Each of the parameters $A$, $v$, and $M$ affect the propagation of speckle through the intensity point-spread function function $g$. Including the magnification parameter $M$, $g$ is evaluated under the strongly paraxial approximation with

Making the substitutions $x_c:=x_b \cos \alpha /(\lambda v M)$ and $y_c := y_b /(\lambda v M)$ gives

Given a reference wavelength $\lambda =\lambda _0$ at which $v=v_0$ and $M=1$, we note from Eq. (10) that $G$, and thereby $g$, is invariant in $\lambda$, up to a scaling factor, if three conditions are met:

Conditions for wavelength invariant speckle

Thus, to meet conditions 1 and 2, a passive optical system would map the fields from different planes $v$, depending on wavelength as given by Eq. (12), with a spectrally dependent magnification factor $M$ given by Eq. (11), onto a single image plane. Such a mapping would thereby overlap the most highly correlated regions of each spectral component of the scattered field. More generally, these conditions describe the positions where maximum spectral correlation is expected. For example, given speckle with wavelength $\lambda _0=500$ nm at position $\left ( {\xi _0,\eta _0,v_0} \right )=\left ( {1\,\text {mm},2\,\text {mm},10\,\text {mm}} \right )$ and with $u=30$ mm, we expect the most highly correlated speckle at $\lambda =600$ nm to be found, on average, at the position $\left ( {\xi,\eta,v} \right )=\left [ {\xi _0/M(\lambda ),\eta _0/M(\lambda ),v(\lambda )} \right ]$ = $\left ( {0.95\,\text {mm},1.9\,\text {mm},7.9\,\text {mm}} \right )$. We note, however, that the magnification factor also affects the speckle size through $s$ as given in Eq. (9) and leads to a spectral dilation blur of each speckle itself; instead of $s$ staying constant and the point-spread function $g$ varying with wavelength, as in Eq. (9), $s$ is variably magnified depending on wavelength while $g$ varies only due to the $h/\lambda$ term in $G$.

A comparison set of broadband images is shown in Fig. 3. The images were created by variably mapping the spectral components of the same broadband speckle field such that they obey one or both of Eqs. (11) and 12. For each part of Figs. 3(a)–3(e), given $\lambda _0=635$ nm, $u=230$ mm, and $v_0=35$ mm, we generated the images using the same method as used in Fig. 2(c). Specifically, we generated 250 images corresponding to spectral components within the bandwidth of the source through a coordinate transformation of Eq. (9), using the same source spectrum and random surface realization of $h$ as used in Fig. 2(c). Upon inspection, all of the images in Fig. 3 share common features and speckles. Many of these shared features are also present in Figs. 2(c) and 2(d), especially near the origin. Each part of Fig. 3 represents a mapping that obeys the unique condition that is listed at the top of the image. The speckle contrast, C, is listed at the bottom of each image and is given by the standard deviation divided by the mean intensity, considering only the approximately 6 mm by 2 mm part of the full speckle pattern that is shown. Figure 3(a) gives the broadband image generated by an optical system that obeys both conditions 1 and 2, and thus both Eqs. (11) and (12). The system of Fig. 3(b) obeys condition 1 for the magnification $M$ while keeping the source plane $v=v_0$ constant. The system of Fig. 3(c) obeys condition 2 for the magnification $M$ while keeping the source plane $v=v_0$ constant. The system of Fig. 3(d) obeys condition 1 for the source plane $v$ while keeping the magnification $M=1$ constant. The system of Fig. 3(e) obeys condition 2 for the source plane $v$ while keeping the magnification $M=1$ constant. The system of Fig. 3(f) gives the monochromatic intensity at $\lambda =635$ nm, $v=v_0$, $M=1$, for comparison.

 

Fig. 3. Simulated broadband speckle patterns in which conditions 1, 2, or both are met. The simulation condition is listed above each pattern and the speckle contrast $C$ is listed at the bottom. The same source spectrum and realization of the surface $h$ is used in each pattern.

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Although Fig. 3 gives examples for only a single realization of the scattering surface $h$, it is useful for making comparisons between the listed speckle mappings and the speckle patterns they create. First, we note that the speckle pattern shown in Fig. 3(f), the monochromatic case, offers the highest speckle contrast because it is not subject to incoherent sums of uncorrelated speckle fields, as compared to the patterns shown in Figs. 3(a)–3(e) or to the bare array detector case of Fig. 2(c). We expect the mapping condition exemplified in Fig. 3(a) to offer relatively high speckle contrast since it overlaps the most highly correlated parts of each spectral component. However, in this case the image shown in Fig. 3(d) offers the highest speckle contrast of all the considered broadband images, implying a discrepancy between maximizing correlation and maximizing speckle contrast. The discrepancy may be due to a trade-off between the intrinsic magnification factor $v/u$ (cf. [5]) of individual speckles and the extrinsically applied magnification factor $M$, leading to various amounts of spectral dilation blur as discussed previously, which affects speckle contrast. We also note that, of the two achievable conditions for satisfying speckle invariance, condition (1) appears to be more important in maximizing speckle contrast than satisfying condition (2). Finally, it is clear that the radial streaking effect seen in Figs. 2(a) and 2(c) is greatly reduced or completely eliminated by meeting conditions (1) and/or (2), under these strongly paraxial simulations.

As discussed in this section, an optical system that satisfies both Eqs. (11) and (12) would have a spectrally dependent focal length and magnification. In terms of design, achromatic Fourier transform (AFT) systems might be a useful reference point. For example, Saastamoinen et al. created an AFT system that spectrally overlapped broadband interference fringes diffracted from a double slit [34]. That system essentially meets condition 1 for wavelength invariant speckle with $v=v_0$ such that $M(\lambda ) = \lambda _0 / \lambda$. With modification we expect that a similar system could produce results like those shown in Fig. 3(b).

3.1 Experimentally examining speckle invariance conditions 1 and 2

To experimentally examine the average effects of meeting conditions 1 and 2 we returned to the setup as shown in Fig. 1(a). We recorded the ensemble of speckle patterns created on the detector after varying the optical filter, position of the translation stage, and scattering surface realization, with $\alpha =\pi /4$ radians. Specifically, for each position in the set $v=(35\;\textrm{mm},40.43\;\textrm{mm},42.01\;\textrm{mm},43.04\;\textrm{mm})$ we recorded an image using each bandpass filter, separately, from the set $c/w=$ (635/10, 561/3, 543/3, 532/3), where $c$ indicates the central wavelength in nm and $w$ is the passband width in nm. These $v$ positions were selected by setting $v_0=35$ mm, $\lambda _0=635$ nm, $u=230$ mm in Eq. (12) and evaluating it at each of the listed central wavelengths, $c$. After recording the images, we numerically applied a magnification $M$ given by Eq. (11) using interpolation. These magnified images should contain shared features, when directly overlapped, if Eqs. (11) and (12) are valid. This process was repeated 8 times where the scattering surface was translated, revealing a new surface realization each time, to build up an ensemble of images for statistical purposes.

As an example of speckle intensity correlations that appear under conditions 1 and 2, an arbitrarily chosen set of experimental intensity images is given in Fig. 4. Figure 4(a) gives portions of magnified images taken at the indicated $v$ positions, wavelengths, and magnifications, such that both conditions 1 and 2, and therefore both Eqs. (11) and (12), are satisfied. Figure 4(a) is comparable to Fig. 1(a) in [16], except here the source is not placed at infinity. Indeed, as $u\rightarrow \infty$, $M\rightarrow 1$ by Eq. (11) and conditions 1 and 2 become degenerate, both giving $v=\lambda _0v_0/\lambda$, which is the case demonstrated in [16]. Figure 4(b) gives portions of magnified images taken such that only condition 1 is satisfied, by manipulating the magnification based on the wavelength, where all images are recorded in the same plane with $v=35$ mm: $M=\lambda _0/\lambda$. The white circles in each portion of Fig. 4(a) draw the eye to a selected, shared speckle that is present at the same position in each of the four images. Similarly, black circles draw the eye to the center of a selected feature at the same position in each portion of Fig. 4(b). Many other similarities exist in the image portions of both Figs. 4(a) and 4(b) beyond those selected, and it is clear by eye that speckle correlations persist for light scattered by this particular scattering surface for bandwidths beyond 100 nm even under condition 1 alone.

 

Fig. 4. Examples of correlations in experimental speckle patterns generated by adjusting the longitudinal plane $v$ of the detector and applying a magnification to the recorded image, with $v_0=35$ mm and $u=230$ mm. (a) Spectrally filtered speckle patterns satisfying conditions 1 and 2, where the magnification $M$ and longitudinal plane $v$ each depend on the center wavelength of the filter used to record the pattern. (b) Filtered speckle patterns that satisfy only condition 1, where the magnification depends on wavelength while the longitudinal plane is fixed. The white circles in (a) and black circles in (b) draw attention to selected, persistent features.

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It remains to be examined if Eqs. (11) and (12) give the precise conditions for maximizing spectral correlation. A measurement of the spectral correlation of the recorded ensemble is given in Fig. 5. Figure 5(a) gives the peak value of the two-dimensional normalized cross-correlation, ensemble averaged over independent and identically distributed scattering surface realizations $h$, between $1\times 1$ mm sections of the images recorded at $\lambda _0=561$ nm with $v_0=40.43$ mm, centered at an arbitrarily chosen $\left ( {\xi,\eta } \right )=\left ( {2.4,0} \right )$ mm, and $3\times 3$ mm sections of images recorded at various longitudinal planes, with magnifications given by Eq. (11), centered at the same position $\left ( {\xi,\eta } \right )=\left ( {2.4,0} \right )$ mm (after magnification). As part of the ensemble averaging process, we assumed that the correlation function was rotationally symmetric about the optical axis, which allowed us to rotate the full image frames together about the center by 9 degrees a total of 38 times, thereby increasing our ensemble from 8 to 304 image sections. In each case, the value at the center of the normalized cross-correlation was within 3 $\mu$m of the position containing the peak value. Equations (11) and (12) predict the correlation maximum for $\lambda =635$ nm to occur at $v=35$ mm, for $\lambda =561$ nm to occur trivially at $v=40.43$ mm, for $\lambda =543$ nm to occur at $v=42.01$ mm, and for $\lambda =532$ nm to occur at $v=43.04$ mm. The prediction correctly fits the experimental data in each case.

 

Fig. 5. Experimental results in spectral correlation maximization. (a) Peak values of the normalized cross-correlation between images recorded at $v_0=40.43$ mm, $\lambda _0=561$ nm, $M=1$ and images recorded at other indicated longitudinal positions, and wavelengths, ensemble averaged over scattering surface realizations. The magnification for each data point is given by Eq. (11). (b) Peak values of the normalized cross-correlation between images recorded at $v_0=40.43$ mm, $\lambda _0=561$ nm, $M=1$ and images recorded at the two indicated longitudinal $v$ positions and $\lambda =543$ nm, with M about the optimal expected by Eq. (11). The filled circles indicate the peak in each plot. Error bars in both (a) and (b) indicate plus and minus one standard deviation of peak values in the ensemble. The source-to-scatterer distance $u$ = 230 mm applies in all cases.

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Figure 5(b) gives the peak value of the two-dimensional normalized cross-correlation, ensemble averaged over scattering surface realizations and simultaneous image rotations, between $1\times 1$ mm sections of images recorded at $\lambda _0=561$ nm with $v_0=40.43$ mm, centered at $\left ( {\xi,\eta } \right )=\left ( {2.4,0} \right )$ mm, and $3\times 3$ mm sections of images recorded at 543 nm with $v=40.43$ mm and $v=42.01$ mm with magnification about that given by Eq. (11), centered at $\left ( {\xi,\eta } \right )=\left ( {2.4,0} \right )$ mm (after magnification). In both cases, Eq. (11) correctly predicts the magnification that gives the largest correlation value: $M=1.033$ for $v=40.43$ mm and $M=0.994$ for $v=42.01$ mm. A similar plot can be made for all other recorded images with the same conclusion.

Both Figs. 5(a) and 5(b) provide strong evidence that Eqs. (11) and (12) correctly prescribe the longitudinal planes and magnifications for maximum intensity correlation under given values of $u$, $\lambda _0$, and $v_0$. In the next section, we analytically calculate the equivalent of Fig. 5(a), under a set of simplifying assumptions, to determine if the heuristic argument under which Eqs. (11) and (12) were derived survives more rigorous scrutiny.

3.2 Analytic calculation of spectral correlation

Here, we describe a method and a set of approximations under which the normalized cross-correlation of different spectral components of a speckle intensity pattern, which we call the spectral correlation, can be found by direct calculation. We begin with the simplifying assumption that the secondary source occupies a single point at the $x_1-y_1$ origin. In this case we can use $s(x,y)=\beta s_0\delta (x,y)$, where $\beta$ is a constant with units of distance squared and $s_0$ has units of spectral density. We also assume that the surface $h$ is normally distributed with height standard deviation $\sigma _h$ and has a translation-invariant, Gaussian spatial correlation function with correlation length $\sigma _\rho$. Using the strongly paraxial version of the spectral density, Eq. (9), gives

We are interested specifically in finding the spectral correlation $\gamma$, which is the normalized covariance. We begin with the covariance, $C$, between the spectral densities at two points in space. For convenience, let $C_{mn}:=C(\xi _m,\eta _m,v_m,\lambda _m;\xi _n,\eta _n,v_n,\lambda _n)$, $\gamma _{mn}:=\gamma (\xi _m,\eta _m,v_m,\lambda _m;\xi _n,\eta _n,v_n,\lambda _n)$, and $S_m := S(\xi _m,\eta _m,v_m,\lambda _m)$, where we explicitly show the dependence of the covariance on longitudinal plane $v$ and wavelength $\lambda$ , with $m,n\in \{0,1\}$. The covariance is given by

For convenience, let

Upon evaluating the ensemble averaged terms in Eq. (17) (details are given in Appx. 1), we find

To examine the validity of Eqs. (11) and (12) from the analytical perspective, a set of curves corresponding to Eq. (15) are given in Fig. 6. The same parameters $u$, $\lambda _0$, and $v_0$ are applied as in Fig. 5 to find the values of $v_1$ that are expected to maximize speckle correlation under a given $\lambda _1$. Each curve in the figure represents $\gamma _{01}$ vs. $v_1$ for the listed values of wavelength $\lambda _1$ with $u=230$ mm, $\xi _0=0$ mm, $\eta _0=1$ mm, $v_0=40.43$ mm, $\lambda _0=561$ nm, $\xi _1=0$ mm, $\eta _1=\eta _0/M$, where $M=\left ( {\lambda _0v_0} \right )/\left ( {\lambda _1v_1} \right )$ as prescribed by condition (1), $\sigma _h=425$ nm, $\sigma _\rho =3$ $\mu$m, and $A(x,y)$ is a circular pinhole of radius 0.8 mm. We note that while these values of $\sigma _h$ and $\sigma _\rho$ are somewhat different from those used in Sec. 2.1 (chosen here for numerical stability), the $v$ position of the correlation peaks appear to be completely insensitive to these values. The three insets correspond to two-dimensional plots, normalized for contrast, of $\mu _{01}$ vs. $\xi _1$ and $\eta _1$ over an area of $4\times 4$ $\mu$m, centered on $\xi _1=0$ mm, $\eta _1=\eta _0/M$, with $\lambda _1=561$ nm, at the indicated values of $v_1=v$, and show the structure of $\mu _{01}$ about the maximum correlation point identified by condition (1). By inspection, the structure resembles a point-spread function induced by defocus of an optical system. As $v_1$ decreases from 40.43 mm the central peak reduces and lobes begin to form, eventually forming a null at the center near 37.7 mm and peaking again near 36.7 mm. We note that the curves given in Fig. 6 are significantly narrower and peak at a value higher than those given in Fig. 5, due to the simplifying assumptions that are inherent in Eqs. (18–20) and to noise, which serves to reduce correlation, in the experimentally recorded images. However, the main finding of this section is that each of the four curves in Fig. 6 reach their maximum at exactly the positions expected by Eqs. (11) and (12), matching those of Fig. 5(a).

 

Fig. 6. Spectral correlation curves calculated analytically using Eq. (15). Each curve plots the central value of the cross correlation, under magnification given by condition 1, between speckles at the listed wavelengths with longitudinal positions $v$ and speckles at $\lambda =561$ nm with $v=40.43$ mm, $M=1$. The insets show examples of the structure of the full 2-dimensional spatial cross-correlations, the central values from which the plotted curves are extracted (here, the insets correspond to the red $\lambda =561$ nm curve and are normalized for contrast, with colorbar on the right). The dotted lines aid the eye in identifying the $v$ distance for each curve peak, and the dashed-dotted lines indicate the $v$ distance corresponding to the associated points on the red curve for each inset.

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4. Conclusion

Here, we have examined the speckle field generated from radiation emitted by a finite-sized, broadband, spatially incoherent source at intermediate distances from a scattering surface. We have shown that, under strongly paraxial conditions, a realization of a speckle pattern can be quickly generated using the Fourier transform relation given by Eq. (9). From this relation we heuristically derived a set of three conditions under which the cross-correlation between speckle fields of different wavelengths can be maximized. The first of these conditions is a known consequence of diffraction gratings, the second is previously unknown, and the third is impossible to meet under normal conditions. By combining the first two conditions, we derived Eqs. (11) and (12) that determine the average three-dimensional position to find a maximally correlated speckle of one wavelength given the three-dimensional position of a speckle of another wavelength. These two equations were examined experimentally in Sec. 3.1 and analytically in Sec. 3.2 and found to hold. We expect our findings to enable the design of optical systems for fast, high-contrast, around-the-corner speckle correlation imaging with broadband sources.