Focusing light through complex scattering samples, such as biological tissue, is a long sought-after capability with great importance in many applications, such as deep tissue microscopy and non-line-of-sight imaging. The multiple scattering of light in such complex samples poses a hurdle to high-resolution imaging, since it diffuses any conventionally focused coherent beam to a spatially extended speckle pattern [1]. To overcome scattering, deep-tissue optical imaging techniques either combine light and sound, or computationally tackle the diffuse optical tomography inverse problem [1]. Unfortunately, these approaches suffer from a resolution that is significantly below the optical diffraction limit [1].

Recently, wavefront shaping [2,3] has emerged as a solution for refocusing coherent light through highly scattering samples. In wavefront shaping, a computer-controlled spatial light modulator (SLM) is used to tailor the input optical wavefront such that it interferometrically inverses the effect of scattering, focusing the beam to a diffraction-limited spot on a desired target. However, all wavefront-shaping techniques to date depend on feedback from a “guide-star” at the target [4], requiring either special labeling or a complicated acousto-optical measurement. Here we propose an all-optical wavefront-shaping approach that is based on angular speckle correlations, allowing noninvasive, diffraction-limited focusing. Our approach is based on estimating the spatial optical field distribution inside a complex sample from an image of the scattered light recorded outside the sample. It then utilizes the estimated pattern of the field inside the sample as feedback for wavefront shaping. The approach is applicable when the probed light at the target plane is within the angular range of the speckle correlations.

When a scattering sample is illuminated by coherent light, the interference of the scattered waves gives rise to random speckle patterns [5]. While speckle patterns are usually considered as a hurdle for imaging, their complex structure contains information on the incident wave. In particular, inherent speckle correlations persist even deep beyond the transport mean-free path in scattering samples [6–9]. Such “memory effect” angular correlations were recently exploited for noninvasive imaging through scattering layers, either by scanning a speckled beam on the target [10], or by a single-shot measurement of the scattered light pattern [11]. In the latter, the intensity autocorrelation of spatially incoherent light inside the scattering medium was estimated from the scattered light distribution outside the medium. Here we extend this approach to spatially coherent illumination and utilize the estimated field autocorrelation inside the sample as feedback for wavefront shaping.

To explain the principle of the proposed approach, consider the setup depicted in Fig. 1. Figure 1 presents a simplified scenario where the goal is to focus a coherent monochromatic beam on a target placed between two highly scattering layers. As was recently demonstrated [12,13], in the case of a fluorescently labeled target object, estimating the incoherent spatial intensity autocorrelation at the object plane is straightforward by following the principles of Ref. [11]. However, using such a fluorescence-based approach for iterative focusing is challenging due to the long integration times required to record the low photon flux per speckle grain of scattered fluorescence [12,13] and due to photo-bleaching. Here, by considering coherent illumination, we avoid these limitations, as well as the requirement for fluorescent labeling. Below we show how to estimate the coherent field autocorrelation inside the sample in this scenario and how to use it for wavefront shaping.

 

Fig. 1. Setup for focusing between two scattering layers via speckle correlations: a laser beam is shaped by an SLM. The shaped beam passes through a first diffuser, reaches the target object (a transmission mask), and is scattered again by a second diffuser. The second diffuser’s surface is imaged on a camera to provide feedback for the focusing process ($u=6\mathrm{cm}$, $v=4\mathrm{cm}$, and $f_4=24\mathrm{mm}$). Inset, close-up view of the considered scenario: a target object with a diameter $D_{\mathrm{obj}}$ is illuminated by a speckle field with a speckle grain size of $d_{\text{speckle}}<D_{\mathrm{obj}}$. The target is located at a distance of $L$ from the second diffuser. We aim at focusing the light on the target to the diffraction limit ($d_{\text{speckle}}$).

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In the simplified scenario of Fig. 1, a transparent target object placed between two scattering layers is illuminated through the first layer (Diffuser1) by a monochromatic spatially coherent beam at wavelength $\lambda$. The light intensity at the second layer’s external facet is imaged on a camera. Due to the scattering of the first layer, the object is illuminated by a speckle field with complex amplitude $E_{\text{illum}}(x,y)$. The field after the object is given by $E_{\text{object}}(x,y)=E_{\text{illum}}(x,y)T(x,y)$, where $T(x,y)$ represents the thin object’s transmission. The light passing through the object propagates a distance $L$ before impinging on the second scattering layer. If $L$ is larger than the far-field distance (an assumption that will be later relaxed), the field illuminating the second layer is proportional to the Fourier transform of the object’s field: $E(x^{\prime },y^{\prime },L)\propto \mathcal{F}(E_{\text{object}}(x,y))$. If the second layer is a thin diffuser that can be modeled as a random phase mask (an assumption that is relaxed below), the light intensity on the far side of the second layer, as imaged by the camera is

 

Fig. 2. Retrieval of a hidden object field autocorrelation (e) from a camera image of the second diffuser’s surface (b)–(d), numerical example. Top row, evolution of the speckle intensity distribution diffracted from a hidden target object, as measured by a camera: (a) target object plane; (b)–(d) diffracted intensity distributions at increasing distances from the object. Bottom row, comparison between (e) the object’s field autocorrelation and (f)–(h) the Fourier transform of the camera images (b)–(d). (The autocorrelation coherent peak is removed for visualization.) The Fourier transform of the speckle intensity (f)–(h) provides an estimate for the object field autocorrelation and, thus, a measure of the illuminated area shape. To achieve focusing, a wavefront-shaping algorithm minimizes the width of this autocorrelation, concentrating light on the target.

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The retrieved field autocorrelation, $A(x,y)$, allows us to estimate the size of the illuminated area at the object plane and, thus, can be used as feedback for iterative focusing via wavefront shaping. The field autocorrelation of the coherently illuminated object has two characteristic widths: the first is the speckle grain wide central peak, whose width is dictated by the numerical aperture of the detection or illumination system. The second, broader width, represents the dimensions of the illuminated area on the object, i.e., the focused beam width at the target. The goal of the iterative algorithm is then to “shrink” the estimated width of the focused beam on the target, and, thereby, produce a sharply focused beam at the object plane (Fig. 3).

 

Fig. 3. Experimental results with a round target object (75 μm diameter pinhole). Top row: initial images of (a) the object, (b) the camera image, and the Fourier transform of the camera image, which provides an estimate of the object autocorrelation, $A(x,y)$, after removing the speckle grain wide central coherent peak (c). (d)–(f) same as (a)–(c) after running the iterative optimization algorithm aimed at minimizing the width of (c) (red circle), resulting in sharp focusing on the object (d). Scale bars, 20 μm.

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In the above derivation, the second scattering layer was assumed to be infinitely thin and was modeled as a random phase mask. As such, the intensity pattern measured on its external facet is equal to the light intensity impinging on it from the object side, a phenomenon known as the “shower-curtain effect” [14]. In the case of a realistic scattering layer with a finite thickness, this simple approximation does not hold. In practice, a point illumination on a diffusive sample of thickness $l$ would result in a speckled blob of diameter $\sim l$ on its output facet [9,15]. Thus, for realistic scattering layers, any features of the input intensity pattern that are finer than $\sim l$ would not be recovered on its external facet. However, features that are coarser than $\sim l$ can be recovered [15]. Since the camera image, $I_{\text{camera}}$, and the estimated autocorrelation are Fourier pairs [Eq. (2)], the coarse effective resolution of $I_{\text{camera}}$ for thick layers limits the autocorrelation estimation to an angular field of view no larger than ${\theta }_{\max }\approx \lambda /l$ which, not by coincidence, is the angular range of the memory effect [6,7,15]. This limits the target object transverse dimension to $L\lambda /l$.

In the above simplified derivation, $L$ was assumed to be larger than the far-field distance. However, this assumption can be greatly relaxed, since the object is illuminated by a speckle pattern [14] or is diffusive. Marking the transverse coherence size of the field at the object plane by $d_{\text{speckle}}$, where $d_{\text{speckle}}$ is either the illumination speckle grain size or, alternatively, the correlation width of the diffusive object, the diffraction angle of the light propagating from the object is ${\theta }_{\mathrm{diff}}\sim {\sin }^{-1}(\frac{\lambda }{d_{\text{speckle}}})$. The minimal propagation distance where light from the opposite ends of an object mix and interfere to produce the speckles illuminating the second scattering layer is reached at $z_0\approx \frac{D_{\mathrm{obj}}}{2\sin ({\theta }_{\mathrm{diff}})}\approx \frac{D_{\mathrm{obj}}{d}_{\text{speckle}}}{2\lambda }$, where $D_{\mathrm{obj}}$ is the transverse size of the object. For distances larger than $z_0$ ($L>z_0$), each speckle grain in the field impinging on the second scattering layer will result from the interferences of all object points, and will have transverse dimensions of ${\sigma }_x\approx \lambda L/D_{\mathrm{obj}}$. As in the van Cittert–Zernike theorem, the speckle grain size will then reflect the illuminated object dimensions $D_{\mathrm{obj}}$, providing the required feedback for the iterative focusing algorithm. Thus, the autocorrelation of the field at the object plane can be estimated at distances considerably shorter than the far-field distance, as is demonstrated numerically in Fig. 2.

To provide a proof-of-principle experimental verification of our approach, we constructed the setup of Fig. 1. A He–Ne laser beam (Thorlabs HNL008LB) was expanded and shaped by a phase-only SLM (Holoeye PLUTO) to illuminate a target object (a small transmission mask) placed between two scattering layers (Newport light shaping diffuser, 5° and 10°). The light on the second diffuser external facet was imaged by an sCMOS camera (Andor Zyla 4.2 plus). To inspect the focusing results, a beam splitter [Thorlabs 45:55 (R:T) pellicle beam splitter] and a second camera (Thorlabs DCU223M, not shown) were used to record the intensity distribution at the object plane.

Figure 3 displays an experimental result obtained with a circular target object (75 μm diameter pinhole). Using the proposed approach, i.e., iteratively shaping the wavefront to shrink the estimated width of the illuminated object area [Figs. 3(c) and 3(f)], a sharp focus with a size close to a single speckle grain was obtained [Fig. 3(d)] from an initially random speckle pattern [Fig. 3(a)]. The intensity enhancement obtained after 600 iterations was $\eta \approx 65$.

To shrink the width of the illuminated area on the object, we have developed a simple iterative algorithm based on the random partitioning wavefront-shaping algorithm [16]: in each iteration of this algorithm, a phase is added to half of the SLM pixels, selected at random. The added phase is cycled in six steps from 0 to $2\pi$. For each phase step, the broader “width” of the estimated autocorrelation, $A(r)$ [Eq. (2)], is approximated by its effective radius, $⟨r⟩\equiv \sum \frac{A(r)r}{\sum A(r)}$, after removing the central coherent autocorrelation peak and cropping the autocorrelation image to a radius proportional to its width to minimize the influence of the background. For each iteration, the optimization metric, $M=\frac{1}{⟨r⟩}$, was fitted to a cosine as a function of the added phase, and the optimum phase for the maximal $M$ (i.e., minimal $⟨r⟩$) was kept as the SLM phase pattern.

An additional experimental focusing example with a more complex object, consisting of two irregularly shaped apertures of $\sim 30\mathrm{\mu m}$ width each, and a comparison to conventional optimization of the total intensity is shown in Fig. 4. As expected, conventional optimization of the total transmitted intensity [Fig. 4(d)] enhances the intensity on the entire object area, without forming a localized sharp focus [2]. In contrast, our speckle correlation-based approach [Fig. 4(g)] forms a sharp focus.

 

Fig. 4. Experimental results: comparison of the proposed approach and conventional optimization of the total intensity on a complex object. Top row: initial images of the object (a), camera image of the diffuser surface, $I_{\mathrm{cam}}$ (b), and the log of the Fourier transform of $I_{\mathrm{cam}}$, providing the estimate for the object’s autocorrelation (c). (d)–(f) same as (a)–(c) after a conventional iterative optimization of the total intensity, (g)–(i) the same as (d)–(f) with the proposed approach, showing sharp focusing. Scale bars, 20 μm.

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Figures 4(c), 4(f), and 4(i) display the autocorrelations estimated from the camera images, together with their calculated radii $⟨r⟩$ (thin red circles). Note the two peripheral peaks in Fig. 4(f), which reflect the autocorrelations of the double-aperture object, which are greatly reduced in Fig. 4(c). In the case of a too small memory effect range (i.e., too large object), these autocorrelation side-lobes would be impossible to estimate.

To study the dependence of the proposed focusing approach on the experimental parameters, we performed a numerical study of the optimization performance for different object dimensions and the number of controlled SLM pixels. The simulated objects are assumed to be within the memory effect range. If no aperture is present, the autocorrelation would only represent the speckle grain size dictated by the illumination area on the first layer. The results of the dependence on the object dimensions are shown in Fig. 5: each point represents the average enhancement obtained after 3000 iterations, averaged over 10 realizations of the scattering layers. Interestingly, for very small objects containing only a few speckle grains, the simple proposed focusing algorithm is less effective. We attribute this to be the result of the imperfect removal of the central coherent autocorrelation peak in the simple processing we have used: the estimated autocorrelation width of small objects (having dimensions close to the speckle grain size) will be more affected by such a simplified peak-removal procedure. However, more robust autocorrelation-width estimation algorithms can be developed for this task. The obtained intensity enhancement and focus size for larger objects are largely independent of the object size (Fig. 5). This is in agreement with the theoretical expected enhancement of $\eta \approx N_{\mathrm{DOF}}/N_{\text{speckle}}$ [2,3], where $N_{\mathrm{DOF}}$ is the number of controlled degrees of freedom, limited by the number of SLM pixels or the number of iterations, and $N_{\text{speckle}}$ is the number of speckle grains contained in the intensity enhanced focus (i.e., the focused spot area), calculated to be $\sim 4.7$ in our simulations results. Additional simulations with a varying number of controlled SLM pixels, $N_{\mathrm{SLM}}$, showed a linear dependence of the enhancement on $N_{\mathrm{SLM}}$, as expected (not shown).

 

Fig. 5. Numerical comparison of the focus intensity enhancement ($\eta$) for different (circular) object sizes, averaged over 10 realizations of scatterers. The error bars are one standard deviation. The object size (horizontal axis) is given as the ratio between its diameter to the speckle grain diameter. Insets: final intensity distributions at the object plane.

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To summarize, we presented an all-optical approach for noninvasive focusing through scattering layers. Unlike previous works [17–19], our approach does not require fluorescence labeling or optical nonlinearities. We demonstrated the approach using thin diffusers. However, other scattering samples can be used, as long as the detected light originates from an area on the target plane that is within the memory effect range. Specifically, the memory effect range was recently shown to be non-negligible also in relatively thick anisotropically scattering samples, such as soft tissues [20]. In our experiments, apertures were used to limit the detected light on the target to be within the memory effect range. In practical imaging scenarios, this could potentially be realized by acousto-optic tagging, [4], allowing us to focus light to dimensions below the ultrasound focus size. Alternative applications may be found in focusing on reflective targets through a diffusive layer.

As in any iterative optimization approach, the main limitation of the presented approach is the relatively long timescales required for the hundreds of iterations: a typical experiment with the slow refresh rate (10 Hz) liquid crystal SLM used in our experiments required tens of minutes. However, SLMs that are orders of magnitude faster [21–23] could significantly reduce the optimization time. More advanced optimization algorithms, such as genetic algorithms [24], may also reduce the number of required iterations. While we obtained nearly diffraction-limited foci, we were not able to experimentally achieve a focus having strictly single speckle grain dimensions. We attribute this to the relatively slow convergence of the chosen optimization metric and algorithm, and to an imperfect removal of the autocorrelation coherent peak. This is not a fundamental limitation, and more advanced optimization metrics are expected to yield improved performance.