1. INTRODUCTION

Spatial correlations exist in many different physical systems, and the study of their origin and evolution is one of the primary roles of statistical physics. In optics, the propagation of spatial coherence of partially coherent light sources has attracted much attention ever since the early days of modern optics [1–3]. One of the central theorems in optics is the Van Cittert–Zernike (VCZ) theorem [4,5], which describes the spatial coherence sufficiently far from a spatially incoherent source. This theorem shows a Fourier relation between the intensity distribution on the surface of a spatially incoherent source and the spatial coherence sufficiently far from it. Physically, this implies that the boundaries of the source dictate the coherence properties of the illuminated light sufficiently far from the source. This property of the spatial coherence has been famously exploited in Michelson’s stellar interferometer to measure the size of distant radiation sources [6,7]. Hanbury Brown and Twiss (HBT) later showed that similar stellar information can be extracted by measuring intensity correlations [8–10]. While the classical theorems describe the spatial coherence at large distances from the source, more recent studies consider short propagation distances, in the region near the source (deep Fresnel region) [11,12], and show that therein the spatial coherence is propagation invariant [13–15].

The concepts and theorems derived in linear optics were later extended to interacting photons [16], as well as to atomic and condensed matter systems, demonstrating partial spatial coherence of electrons [17–19] and cold atoms [20–23]. The underlying assumption in all these systems is that they only exhibit ballistic transport, while any diffusive transport is negligible. Although this assumption is well justified in many cases, it does not always hold, and often diffusive transport must be taken into consideration [24–30]. Here we thoroughly investigate coherent diffusion of spatial correlations (spatial coherence) of partially coherent fields, theoretically and experimentally, and compare between diffraction and coherent diffusion of partial spatial coherence. We first present experimental results comparing the two and then present a detailed model explaining the results. Our model is general and is not limited to our specific system.

The comparison between these two distinct physical mechanisms, i.e., diffusion and diffraction, is based on the mathematical similarity between their governing equations:

Clearly the two equations above are identical under the transformation $D\to i\lambda /4\pi$, and accordingly diffraction can be considered as diffusion in imaginary time [27,28]. This mathematical similarity implies exciting physical analogies, where various well-known optical phenomena find their natural analogs in diffusion of complex vector fields [27,33–35]. For example, optical vortices are protected and will not unwind in both diffusion and diffraction [33].

In the past, coherent diffusion of complex fields was demonstrated and studied in various systems, including nuclear magnetic resonance [25], electromagnetically induced transparency (EIT) [27,29], and spintronics [30]. Here, we exploit EIT based on a unique four-wave mixing scheme that was recently presented [36], to study diffusion of partial spatial coherence. Using this scheme, spatially patterned light beams are coherently imprinted onto the spin states of atoms in a hot vapor, such that the spins acquire the spatial amplitude and phase of the incoming light fields. After a temporal duration that is equal to the group delay in the medium, the spatially patterned spin state is coherently imprinted onto the outgoing light field, which is then detected by a camera. Since the atoms diffuse during this temporal duration, the retrieved signal indicates the atoms’ diffusion dynamics.

One of the main advantages of this scheme, as opposed to traditional EIT, is that it can be used to reach relatively long diffusion times with high signal-to-noise ratios. Therefore, it can be used to observe also high spatial frequencies for long diffusion times, which is a key component for this study.

2. EXPERIMENTAL RESULTS

The experimental arrangement used to characterize the diffusion of partially coherent fields is illustrated in Fig. 1(a). We use ${}^{87}\mathrm{Rb}$, which diffuses in 10 Torr of ${\mathrm{N}}_2$ buffer gas, rendering a diffusion coefficient of $D=9.7\pm 0.5{\mathrm{cm}}^2/\mathrm{s}$. The vapor cell is illuminated by two spatially overlapping “control” beams, which are separated by a slight angle, and by a third weak “probe” beam, which is oriented along one of the control beams. Consequently, a fourth beam, denoted as “signal,” is generated in a four-wave mixing process along the orientation of the second control beam. We set the optical frequencies of the probe and control beams such that they couple, respectively, the lower and upper hyperfine states $|1⟩=|5S_{1/2};F=1,2;m=0⟩$ to the excited states $|2⟩=|5P_{1/2};F^{\prime }=1,2;m=1⟩$ and $|3⟩=|5P_{1/2},F^{\prime }=1,2;m=-1⟩$of the $D_1$ transition. The incoming probe beam $E_{\mathrm{in}}(\mathbf{r})$ is shaped using a spatial light modulator. The outgoing signal $E_{\mathrm{out}}(\mathbf{r})$ is imaged onto a CCD camera, and we use digital Fourier filtering to improve the signal-to-noise ratio. Further details regarding the experimental arrangement are given in Supplement 1, and full characterization and analysis of the generation process are described in [36].

 

Fig. 1. Experiment and representative results. (a) Experimental arrangement for diffusion of speckle fields. BS, beam splitter; SLM, spatial light modulator. (b), (d) Detected intensity distribution at the center cross section of the vapor cell for (b) large and (d) small detuning, manifesting short and long diffusion time, respectively. (c), (e) Corresponding autocorrelation of the detected speckle pattern.

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We use the Fourier transformation for the transverse coordinates $\mathbf{r}$, $\tilde{E}(\mathbf{q})=\int \frac{{\mathrm{d}}^2\mathbf{r}}{2\pi }E(\mathbf{r})e^{-i\mathbf{q}·\mathbf{r}}$, and focus on the weak EIT regime with confined spatial frequencies $q^2={|\mathbf{q}|}^2\ll |{\gamma }_{2p}+\mathrm{\Gamma }|/D$. In this regime, the motional broadening of the generation spectrum due to diffusion translates linearity to the generation amplitude, and one finds [36]

The propagator $e^{-D\tau q^2}$ in Fourier space implies diffusion in real space. It follows from Eq. (2) that a structured probe beam in our system can continuously generate a signal which underwent diffusion for an effective temporal duration $\tau$. The easiest way to control $\tau$ in experiment is by changing the two-photon detuning ${\mathrm{\Delta }}_{2\mathrm{p}}$. Figure 1(b) presents a representative retrieved signal for an input Gaussian speckle field, under large detuning ${\mathrm{\Delta }}_{2\mathrm{p}}$ (i.e., short diffusion time, $\tau =4\mathrm{\mu s}$), and Fig. 1(d) shows the retrieved signal for the same speckle pattern under small detuning (i.e., long diffusion time, $\tau =65\mathrm{\mu s}$). Figures 1(c) and 1(e) show the autocorrelation of the retrieved intensity patterns. Based on such measurements and for various values of ${\mathrm{\Delta }}_{2\mathrm{p}}$, we can study the effect of diffusion on the coherence of speckle fields.

Figure 2(a) shows a 1D cross section of a 2D Gaussian speckle pattern as a function of diffusion time, while normalizing the total intensity distribution at every moment, so as to account for the dissipation of the field under diffusion. Figure 2(b) presents the autocorrelation of the speckle pattern as a function of $\tau$, and Fig. 2(c) shows the width $w$ of the autocorrelation versus $\tau$. As evident, the speckles grow in size with diffusion time $\tau$, and the area of the autocorrelation function grows linearly with time, i.e., $w\propto {\tau }^{1/2}$.

 

Fig. 2. Experimental comparison between (a)–(c) diffusion and (d)–(f) diffraction of speckle intensity distributions. (a) Linear cross section of the detected speckle pattern, as a function of diffusion time. (b) Corresponding autocorrelation function of the detected speckle pattern, as a function of diffusion time. Note that, for clarity, this color map presents numerical interpolation of the data circumventing for uneven time steps. (c) Corresponding $1/e$ width squared of the autocorrelation in (b) (with no interpolation) as a function of diffusion time (red circles), as well as the width squared of the autocorrelation for non-diffusive speckles (purple squares). Dashed lines present linear fit to the data. (d) Linear cross section of the detected speckle pattern, as a function of propagation distance. (e) Corresponding autocorrelation function of the detected speckle pattern, as a function of propagation distance. (f) Corresponding $1/e$ width squared of the autocorrelation function in (e) as a function of propagation distance (red solid line), together with a linear fit in the linear regime $z\gg z_{\mathrm{VCZ}}$ (dashed blue line). Dashed vertical line denotes the propagation distance $z=z_{\mathrm{VCZ}}$. The data in (b), (c), (e), and (f) represent the radial mean of the autocorrelation. The errors of these measurements are smaller than the thickness of the data points in (c) and of the solid red line in (f).

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It is well known that speckles can be propagation invariant under diffraction, if they are made by a random superposition of Bessel beams [14,37,38], namely if the field $\varphi$ can be presented in the form $\varphi (r,\theta )=\sum _n{a}_n{J}_n(k_{r}r)\exp (in\theta )$, where $r$ is the radial coordinate and $\theta$ is the azimuthal coordinate, $a_n$ is a complex constants, $J_n$ is a Bessel function of order $n$, and $k_r$ is the radial wave vector. In Fourier space, the last expression transforms to a field with random phases and an intensity distribution in the shape of annular ring. As explained in Supplement 1, Bessel beams are also invariant under diffusion, and therefore a random superposition of Bessel beams with the same radial wave vector would result in a diffusion-invariant speckle field [39]. Consequently, the spatial coherence of such a speckle field would be diffusion invariant as well. Indeed, as demonstrated in Fig. 2(c) (purple squares), the size of these speckles is preserved and does not increase significantly with diffusion time (see Supplement 1 for details). Experimentally, the non-diffracting speckles are generated by encoding axicon phases on to the spatial light modulator (see Fig. 1), with additional phases that are constant in the radial direction and vary randomly with azimuth angle.

Generally, there are great differences between the diffusion and diffraction of speckles. To show this explicitly, we also measure the free-space optical propagation of Gaussian speckles (see Supplement 1 for details). We measure and analyze the propagation of Gaussian speckles in steps of 0.5 mm, over a total distance of 100 mm. The results of these measurements are shown in Figs. 2(d)–2(f). As evident, there are two regimes of propagation distances. Near the source, at the deep Fresnel region, the size of a typical speckle is constant. Far from the source, at the VCZ region, the speckles start pulling apart from one another, and the size of a typical speckle grain gradually grows. In diffusion, on the other hand, Gaussian speckles continuously expand with diffusion time, and the relative intensity of small speckle grains drops while the large speckle grains “take over.” Consequently, the area of the coherence region continuously grows with diffusion time.

To validate and complement our experimental results, we ran numerical simulations comparing between diffraction and diffusion of speckle fields. The calculations were performed by propagating an initial speckle field using the known propagators, shown in the first row of Table 1. Figure 3 compares between diffusion and diffraction and qualitatively agrees with the experimental results of Fig. 2. As evident, the linear relation between the width squared and diffusion time holds even long after the temporal regime measured experimentally (Fig. 2). Furthermore, Figs. 3(c) and 3(f) compare the width of the autocorrelation of a single speckle pattern, to the equivalent width of coherence calculated by averaging many uncorrelated speckle patterns. As evident, the two methods are equivalent, as expected.

Table 1. Diffusion Propagators Derived in the Paper, Compared to Well-Known Diffraction Propagators

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Fig. 3. Numerical calculations comparing between (a)–(c) diffusion and (d)–(f) diffraction of speckle intensity distributions, and of their spatial coherence. (a) Calculated intensity distribution of a speckle pattern as a function of normalized diffusion time $\tau /{\tau }_R$, where ${\tau }_R$ is the diffusion analog of the Rayleigh distance in diffraction ${\tau }_R=w_0^2/4D$. (b) Corresponding autocorrelation function of the simulated speckle pattern, as a function of normalized diffusion time. (c) Corresponding $1/e$ width squared of the autocorrelation in (b) as a function of normalized diffusion time (solid red line). (d) Calculated intensity distribution of a speckle pattern as a function of normalized propagation distance $z/z_R$, where $z_R=\pi w_0^2/\lambda$ is the Rayleigh distance. (e) Corresponding autocorrelation function of the simulated speckle pattern, as a function of normalized propagation distance. (f) Corresponding $1/e$ width squared of the autocorrelation function in (e) as a function of normalized propagation distance (red solid line). The inset in (b) compares between experiment (circles), simulation (solid red line), and analytic expression (dashed black line) for short diffusion times. The calculated intensity correlations of many such speckle realization (ensemble average) are presented in (c) and (f) as well (dashed blue line). The cross sections and widths were obtained by considering the radial mean of the autocorrelation.

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3. DISCUSSION AND THEORETICAL ANALYSIS

We now establish the theoretical framework needed to explain the experimental and numerical results. Due to the strong correspondence between speckle theory and the theory of partial spatial coherence [40–42], we explain the results of diffusion of speckles as diffusion of spatial coherence. We therefore begin with the familiar formalism of diffraction of partially coherent beams and then extend this formalism to diffusion of such beams.

Consider an extended quasi-monochromatic pseudothermal source, which generates a complex field amplitude $E(\mathbf{r},z)$ at axial distance $z$ from the source and at a 2D transverse coordinate $\mathbf{r}$. The field spatial correlations are given by the mutual intensity

In the following we consider a planar quasi-homogeneous source, namely a source whose area $L^2$ is very large compared to the coherence area $l_{\mathrm{c}}^2$ on the source, and any variation in intensity on the source occurs on area scales that are of order $L^2$. If the two scales are well separated, $L\gg l_{\mathrm{c}}$, the mutual intensity $G_0^{(1)}$ at the plain of the source $z=0$ can then be factorized:

Following Gatti [12], far from the source $z\gg z_{\mathrm{VCZ}}$, Eq. (7) can be solved to yield the generalized VCZ theorem [1,42],

Recall that ${\tilde{G}}_0^{(1)}(\overline{\mathbf{q}},\mathrm{\Delta }\mathbf{q})={\tilde{I}}_0(\mathrm{\Delta }\mathbf{q})·{\tilde{\mu }}_0(\overline{\mathbf{q}})$, where ${\tilde{\mu }}_0$ and ${\tilde{I}}_0$ are the Fourier transforms of ${\mu }_0$ and $I_0$, respectively. For $L\gg l_{\mathrm{c}}$ and $z\gg z_{\mathrm{VCZ}}$, the function ${\tilde{\mu }}_0$ varies very slowly as compared to ${\tilde{I}}_0$, and therefore ${\tilde{\mu }}_0$ is approximately constant. In this case the absolute value of the mutual intensity $|G^{(1)}|$ depends only on ${\tilde{I}}_0$, and therefore the coherence region grows as $\lambda z/L$. Indeed, Fig. 2(f) shows a slope of $1.30·{10}^{-3}\pm 0.04·{10}^{-3}$, which agrees with the expected slope of $1.33·{10}^{-3}$.

We now turn to derive equivalent expressions to describe diffusion of partially coherent fields. As before, we begin with Eq. (6), but now we express the fields using the diffusion coherence propagator $H_E^{\mathrm{du}}$, yielding in Fourier space

For $L\gg l_{\mathrm{c}}$, Eq. (9) implies that the width along the relative coordinate $\mathrm{\Delta }\mathbf{r}$ of the mutual intensity $G^{(1)}(\overline{\mathbf{r}},\mathrm{\Delta }\mathbf{r};\tau )$ at any point in time depends mainly on the width of the coherence region on the surface of the source $l_{\mathrm{c}}$, even far from the source. This is very different from the behavior of the mutual intensity under diffraction, where near the source ($z\ll z_{\mathrm{VCZ}}$, even if $z\gg z_R$) the spatial coherence is almost constant; and sufficiently far from the source ($z\gg z_{\mathrm{VCZ}}$), when the boundary of the source begins to play a role, the spatial coherence is dominated by the shape and size of the source. This leads to a significant difference between diffusion and diffraction in a Michelson or Hanbury Brown and Twiss type of interferometer. In diffraction, a HBT interferometer can be used to measure the size of a distant spatially incoherent object, but cannot be used to retrieve information regarding the original size of coherent regions on the source. However, in diffusion the picture is reversed: measuring the spatial coherence indicates the size of coherence regions at a distant source (albeit with accuracy that decays with diffusion time) and will supply little information regarding the size of the source itself.

To see this more formally, consider a Gaussian speckle field with Gaussian envelope, $\mu (\mathrm{\Delta }\mathbf{r})=e^{-\mathrm{\Delta }{\mathbf{r}}^2/l_{\mathrm{c}}^2}$, $I(\overline{\mathbf{r}})=I_0{e}^{-{\overline{\mathbf{r}}}^2/L^2}$. In this case, Eq. (9) yields

Notice that Fig. 2(c) indeed shows a slope of $40.5\pm 0.4{\mathrm{cm}}^2/\mathrm{s}$, which agrees with an independent measurement of the diffusion coefficient, $4D=38.8\pm 2.2{\mathrm{cm}}^2/\mathrm{s}$, as described in Supplement 1. Also, comparing Figs. 1(c) and 3 indicates that the small discrepancies between theory and experiment are similar to those between theory and simulation, and are dominated by the limited number of speckles analyzed.

The number of speckles $N$ at the plane of the source is estimated by $N\sim {(L/l_{\mathrm{c}})}^2$. While $N$ is conserved under diffraction, it is not conserved under diffusion. In diffusion $N$ decreases with diffusion time, $N\sim (L^2+4D\tau )/(l_{\mathrm{c}}^2+4D\tau )$.

4. CONCLUDING REMARKS

We analyzed diffusion of partially coherent complex fields and compared between diffusion and diffraction of the spatial coherence. For this purpose, we presented a new theoretical model, which we derived in analogy and comparison to the familiar theorems in diffraction. Our theoretical analysis is general and applies to any diffusive physical system. We showed, both in theory and experiment, that the complex field and the spatial coherence of partially coherent beams both undergo diffusion in a similar manner, and that in the case of a Gaussian input their widths grow with the square root of diffusion time. This is in contrast to the well-known linear dependence under diffraction. While diffraction of partially coherent beams behaves differently in the deep Fresnel region and the VCZ region, diffusion of the partial coherence behaves the same in all temporal regimes.

As we showed, diffusion of partial coherence leads to a diffusion analog of the classical diffraction Michelson or HBT interferometers. In this diffusion analog, the boundary of the source has little effect on the spatial coherence, and measuring the spatial coherence far from the source can be considered as a measurement of the original region of coherence at the source. This behavior is, of course, very different than the classical Michelson of HBT interferometers, where the boundary of the source dominates the spatial coherence far from the source, while any information regarding the coherence region on the surface of the source is lost.

Finally, we showed that while the number of speckle grains in a field is conserved under diffraction, in diffusion it decreases with diffusion time. The exception of diffusion-free speckle fields, where the width of the spatial coherence and the number of speckles remains constant, was explored as well.

The work presented here extended concepts and theorems from statistical optics to the field of coherent diffusion. While we focused here on polariton diffusion, our analysis is general and provides a first step in applying the VCZ theory and HBT interferometry to various diffusive systems, such as astronomical stellar atmospheres [43] and imaging through turbulent or complex scattering media [44–46]. Furthermore, our model can be highly relevant for multi-pixel quantum memories [47–49], where realistic systems are expected to suffer from partial spatial coherence. In such systems it is therefore extremely important to understand the diffusion dynamics of spatial coherence, to appropriately estimate the temporal duration over which information can be stored. Recently, dissipation has been exploited for computational resources in advanced photonic systems [50–52]. It has been shown that dissipation in highly non-linear systems can rapidly anneal a system to a global ground state and serve as a physical simulator for hard computational problems [53–55]. Such systems are often governed by diffusive transport, and they are often only partially spatially coherent. The model we established and verified in this work could serve as an important step in fully understanding the physical mechanisms behind such applications and will help develop next-generation computational resources.