1. Introduction

High-resolution optical imaging of deep tissue is fundamentally challenging because of the inhomogeneities of the refractive index of the sample. These inhomogeneities cause aberrations and scattering that prevent light from forming a diffraction-limited focus, thereby reducing the imaging contrast and resolution. Historically, there are two types of approaches to restore the image quality. Adaptive Optics (AO) is used to correct for smooth wavefront aberrations caused by large-scale refractive index inhomogeneities or imperfections in the imaging optics [1]. Wavefront shaping (WFS) techniques [2], on the other hand are designed to compensate for strong multiple scattering caused by wavelength-sized inhomogeneities.

In the framework of AO, many strategies for dealing with aberrations have been proposed [3,4]. In most cases, the phase of the light is decomposed into orthogonal modes, such as Zernike polynomials, to deal with slowly varying aberrations in the pupil plane [5]. Different optimization algorithms, such as hill-climbing, simplex optimization, genetic algorithms, simulated annealing, etc., have been used such that a certain performance metric (e.g., the measured light intensity or the sharpness of the image) is maximized [6–10]. A complication of this approach is that the relation between the coefficients of the modes and the performance metric is usually non-convex. Therefore, such algorithms are not always guaranteed to converge to the global optimum solution. Specially, when the aberration exceeds a certain level AO techniques are ineffective [11].

In contrast, typical WFS algorithms decompose the field of the light into orthogonal modes. Since the propagation problem is linear in the field, the optimum wavefront for focusing light to a single mode can be determined non-iteratively from a predetermined set of measurements [2,12,13]. Finding the optimized wavefronts for multiple targets (output modes) is equivalent to measuring different rows of the scattering medium’s transmission matrix, up to an unknown phase factor for each row of the matrix [13]. It was shown that even for non-linear problems, such as using two-photon fluorescence or speckle variance as a feedback signal [14,15], the resulting optimization problem is often still convex. The performance only deteriorates gradually as the noise increases [16]. The ability of WFS techniques, regarding the optimization speed, has been enhanced by the accompanying development of various wavefront modulators [17,18].

Despite these advantages, WFS is not optimal for compensating forward scattering or low order-amplitude aberrations. These situations require displaying a smooth correction in the pupil, whereas existing methods work in a segment-basis or a Hadamard basis, resulting in patterns with sharp edges [12,13,19,20]. Recently, the cross-over regime between aberrations and strong scattering has received increased interest, both theoretically [21], and experimentally [4,22,23]. However, the approaches of AO and WFS have not been unified so far.

In this work, we combine the advantages of AO and WFS into a single approach that is optimized for forward scattering media. Our new approach, which we call smooth-basis WFS, has the benefits that (1) it is optimized for finding smooth corrections, as is appropriate for forward scattering samples. (2) It is guaranteed to find the global optimum solution in a single iteration, (3) it can be used to simultaneously find the wavefronts to focus light onto multiple targets, (4) it is robust against noise due to the optimum interferometric visibility of the feedback signal.

Our method is a general WFS method that does not have any assumption for the feedback signal. Therefore, different types of feedback such as fluorescence emission, photoacoustic signal, acousto-optic signal, and polarization state of the output mode can be used as a feedback signal [2].

We show that our new method outperforms the best known Hadamard-basis algorithm [20] for forward scattering samples, and performs at least as well as that algorithm for strongly scattering samples.

2. Method

We start by describing the transmission of light through the sample with a transmission matrix with elements $t_{ba}$

Importantly, Eq.  (1) is valid regardless of what basis we use for the modes $a$. Therefore, we are free to choose a basis in which the transmission matrix is sparse. The modes are usually chosen as close to orthogonal as possible to minimize the redundancy in the measurements. In practice, the only constraint is that the elements of the basis can be generated with a phase-only spatial light modulator (SLM). Unlike the usual choice for a basis consisting of square blocks or Hadamard patterns, here we choose a basis consisting of tilted plane waves (a ‘Fourier basis’), which are orthogonal and can also be generated with a phase-only SLM. For forward scattering, all significant contributions in the transmission matrix are for low scattering angles, meaning that the transmission matrix is sparse in this basis.

With this choice of a basis, we are still free to choose a wavefront shaping algorithm. We base our algorithm on a robust approach that was originally developed for use with a Hadamard basis [20]. In this ‘dual reference’ algorithm, the SLM is split into two groups of the same size (group 1 and group 2) with a small overlap. In the first step, basis patterns are displayed on group 1 of the SLM one by one. For each pattern, the phase of the reference (group 2) is varied from 0 to $2\pi$ in a chosen number of steps while the intensity $|E_b|^{2}$ at chosen targets $b$ is measured. In the second step, the basis patterns are displayed on group 2 while group 1 acts as a reference. After these two steps, we have the corrected wavefronts for both groups. Finally, the overlapping segments are used to determine the relative phase between these two corrections so that they can be combined into a single wavefront. The advantage of this algorithm is that it maximizes the interferometric visibility of the feedback signals, thus minimizing the effect of shot noise and readout noise [20].

We now replace the Hadamard basis in the original algorithm by a Fourier basis. Figure 1(a) shows the basis elements displayed on group 1 (left half) of the SLM while group 2 (right half) serves as a reference, and Fig. 1(b) shows the displaying basis elements on group 2 when group 1 is used as a reference. The basis elements correspond to tilted plane waves with transversal wave vectors $\mathbf {k}_a=(k_x, k_y)$. Since we are interested in compensating for low-angle scattering, we only consider basis elements for which $\lVert \mathbf {k}_a\rVert < k_\text {max}$ i.e. we select the elements from a circle in Fourier space, where $k_\text {max}$ can be chosen based on the scattering strength of the sample.

 

Fig. 1. Phase patterns displayed on the SLM for smooth-basis WFS, when group 1 (left half of the segments) (a) and group 2 (right half) (b) are modulated.

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The spacing between the selected wave vectors $(k_x, k_y)$ should be small enough to avoid aliasing in the reconstructed wavefront. Concretely, following the Nyquist-Shannon sampling theorem, the smallest distance between two wave vectors should be such that $\Delta k_y < 2\pi /D$, with $D$ the diameter of the illuminated spot on the SLM, and $\Delta k_x < 2\pi /(D/2)$, where the factor $2$ difference follows from the fact that the SLM is divided into two halves in the $x$-direction.

To find the corrected wavefronts, we perform the same post-processing as is used in [20], except for using a Fourier transform instead of a Hadamard transform in the second step.

3. Experimental results

We used the setup shown in Fig. 2 to measure the required corrections to form a focus through our samples using the Fourier and Hadamard basis algorithms. In the setup, a 632.8 nm HeNe laser (Thorlabs, 2mW) beam is expanded and modulated by a phase-only SLM (Hamamatsu X13138-07). A 4f system images the SLM onto the back focal plane of a microscope objective (Zeiss A-Plan 100x/0.8). The light is collected by an identical objective lens after passing through the sample and recorded by a CMOS camera (Basler acA640-750um). The camera captures the intensity distribution at the sample plane. The wavefront shaping algorithms receive feedback from our chosen targets on the camera.

 

Fig. 2. Schematic of the experimental setup. HWP: half-wave plate, P: polarizer, M: mirror, SLM: spatial light modulator, BS: 50% non-polarizing beam splitter, CMOS: complementary metal oxide semiconductor camera, L1, L2, L3, L4, and L5 lenses with focal length of respectively 200 mm, 75 mm, 75 mm, 150 mm, and 50 mm.

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We used two samples. The first sample is a 120 grit ground glass diffuser and predominantly scatters light in the forward direction. The second is a 11$\pm$3 $\mathrm{\mu}\textrm{m}$ coating of zinc oxide pigment (Sigma Aldrich, average grain size 200 nm) on a microscope coverslip, which is in the strong multiple scattering regime. The transport mean free path of similar zinc-oxide samples was measured to be around 0.6 $\mathrm{\mu}\textrm{m}$ at a wavelength of $\lambda$ = 632.8 nm. [24].

First, we performed wavefront shaping by running smooth-basis WFS for 194 Fourier basis elements and used the glass diffuser as a sample. We displayed 97 Fourier basis elements on group 1 and used group 2 as a reference to perform phase stepping, then displayed the same Fourier basis elements on group 2 while using group 1 as a reference. 5% of the SLM segments are mutual in these two groups.

The sample was placed such that scattered light formed a blurred focus at a distance of 400$\pm$10 $\mathrm{\mu}\textrm{m}$ behind the scattering front surface. As feedback signal, we recorded the intensity from the camera pixel with the highest initial intensity. Figure 3(a) shows the captured frame from the initial blurred focus in the image plane. Figure 3(b) shows the focus after displaying the optimized wavefront on the SLM. This optimized focus is 6 times brighter than the original focus. Figure 3(c) shows the intensity profile of the focus before (dot curve) and after (solid curve) displaying the optimized wavefront shown in Fig. 3(d) on the SLM.

 

Fig. 3. (a) Intensity at the image plane with a non-shaped incident beam, and (b) with displaying the optimized wavefront. (c) Intensity profile of the focus at the image plane when displaying a flat wavefront (dotted curve) and the corrected wavefront on the SLM (solid curve). (d) The measured SLM optimized wavefront using 194 modes.

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Since the diffuser predominantly scatters light in the forward direction, we expect the corrected wavefront to be dominated by low-frequency Fourier components. To verify this expectation, we determined the contribution of each basis element to the focus by computing $\langle |t_{ba}|^{2}\rangle$. Here, $\langle \cdot \rangle$ denotes averaging over 30 trials $t_{ba}$, $a$ is the index of the basis element, and $b$ is the index of the camera pixel used for feedback. Figure 4 shows the normalised value of $\langle |t_{ba}|^{2}\rangle$ for group 1 (left half of the SLM) and for group 2 (right half), respectively. As expected from a forward scattering medium, the corrections are concentrated around $(k_x, k_y) = (0, 0)$; effectively causing the transmission matrix to be sparse in Fourier-space.

 

Fig. 4. The normalized averaged amplitude of the measured transmission row in a Fourier basis over 30 trials for group 1: $\langle |t_{ba}^{\textrm {L}}|^{2}\rangle$ and group 2: $\langle |t_{ba}^{\textrm {R}}|^{2}\rangle$. Only basis elements with $\lVert \mathbf {k}_a \rVert < k_\text {max}$ (indicated by the black outline) were used.

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Next, to quantify the effect of the chosen basis, we measured the improvement ratio, which is the ratio between the intensity in the focus before ($\hat {I}_f$) and after ($I_f$) displaying the optimized wavefront on the SLM. This improvement ratio is equivalent to the ratio between the Strehl ratio before and after optimization. Figure 5(a) shows the improvement ratio averaged over 30 trials versus the number of basis elements (modes) used for the Fourier basis (solid curve) and the Hadamard basis (dot curve). The error bars represent statistical errors obtained by repeating the experiments 30 times. Clearly, for almost all number of modes, our new method gives a higher intensity improvement than the algorithm in a Hadamard basis. These results demonstrate the benefit of adapting the set of basis elements to match the sparsity of the transmission matrix.

 

Fig. 5. (a) The measured averaged intensity improvement for the glass diffuser versus mode number using smooth-basis WFS (solid curve), and Hadamard basis (dot curve) for 30 trials. (b) Optimized wavefronts for the glass diffuser using a Hadamard basis algorithm, and (c) smooth-basis WFS .

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For the Hadamard basis, the choice of basis vectors does not correspond directly to scattering angles. Therefore, the sparsity of the transmission matrix is not used optimally. From Fig. 5(a), it can be seen that our new algorithm needs roughly half the number of modes to achieve the same quality of the focus as with the Hadamard algorithm. Therefore, our new algorithm is roughly twice as fast as the Hadamard approach.

The intensity improvement saturates as the number of modes increases. This saturation can be explained from the data in Fig. 4. Increasing the number of modes corresponds to enlarging the circular region in the figure, thus including $k$-vectors corresponding to larger angles. However, the contributions of these vectors decreases rapidly as $\lVert \mathbf {k}\rVert$ increases because the glass diffuser sample predominantly scatters light in the forward direction.

Figure 5(b) shows an example of a measured correction for the glass diffuser using a Hadamard basis with 512 modes, and Fig. 5(c) shows the correction found for the same optimization as Fig. 5(b) using a Fourier basis with 442 modes. The resulting correction is a better match for the forward scattering sample, even though fewer measurements were performed.

We repeated the measurement for the ZnO sample. The initial captured frame with a non-shaped incident beam is a fully developed speckle pattern due to the strongly scattering properties of the sample. We chose 100 different camera pixels as our targets. Since multiple scattering is dominant for this sample, we used the enhancement, $\eta$, as a metric to quantify the performance of the algorithms. The enhancement is the optimal intensity at the target point normalized by a reference intensity, which is measured at the same point on the camera, and with the same wavefront displayed on the SLM, but with the sample displaced laterally far enough to destroy the focus [2]. Here, we averaged over 80 positions of the sample.

After measuring the corrected wavefront for each target, we displayed the wavefront and measured the enhancement individually for each target. Figure 6(a) shows the average enhancement $\langle \eta \rangle$ measured for 100 targets and three trials using smooth-basis WFS (solid curve), and using a Hadamard basis (dotted curve). Figure 6(b) shows an example of a measured correction for the ZnO sample using a Hadamard basis with 512 modes, and Fig. 6(c) shows the correction found for the same optimization as Fig. 6(b) using a Fourier basis with 442 modes. The complexity of the measured corrections for the ZnO sample comparing to the glass diffuser (Fig. 5(b) and (c)) is due to the scattering properties of the ZnO sample.

 

Fig. 6. (a) The measured averaged enhancement over 100 targets and 3 trials versus mode number for the ZnO sample using a Fourier basis (solid curve), and Hadamard basis (dot curve). (b) Optimized wavefronts for the ZnO sample using a Hadamard basis algorithm, and (c) smooth-basis WFS .

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Figure 6(a) shows that for the ZnO sample, the enhancement does not saturate even for 1000 modes, because both high and low Fourier modes contribute to the correction. The difference between the two methods is negligible for mode numbers lower than 500.

However, surprisingly, for a high number of modes, smooth-basis WFS provides a higher enhancement than the Hadamard basis even for this strongly scattering sample. This difference is caused by a geometrical effect: since the SLM is conjugated to the back focal plane of the microscope objective, high-frequency components of the shaped wavefront at the SLM are projected onto the sample surface at a large lateral distance from the optical axis. To illustrate this effect, we computed the field distribution at the front surface of the sample by Fourier transforming the field pattern that was displayed on the SLM. For the smooth-basis WFS (Fig. 7(a), measured with a 1006-element Fourier basis), almost all light is concentrated in a circle with the radius equal to the sample thickness (11 $\mathrm{\mu}\textrm{m}$). This circle very roughly indicates the distance over which a light is expected to diffuse laterally and contribute to the target focus at the back surface of the sample.

 

Fig. 7. (a) The normalized amplitude of the computed field at the sample plane corresponding to the measured wavefront using a Fourier basis with 1006 modes and (b) using a Hadamard basis with 1024 modes.

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In contrast, when the Hadamard basis is used (Fig. 7(b), measured with 1024 modes), the high-frequency edges cause side lobes that lie far outside this circle. The contribution of this light will be negligible, causing a lower enhancement of the Hadamard algorithm.

4. Conclusion

To conclude, we demonstrated that it is advantageous to adapt the basis used in WFS to the sparsity of the transmission matrix. Specifically, using our new Fourier basis in which only low spatial frequencies contribute to the wavefront, resulted in a factor of 2 speedup in forward scattering samples, and even caused a modest increase improvement for strongly scattering samples.

In a sense, our approach bridges the gap between WFS and AO. Both methods use smooth patterns to construct a wavefront correction, exploiting the fact that aberrations caused by forward scattering samples are smooth.

However, in contrast to commonly used approaches in AO, our WFS approach reduces the underlying optimization problem to a linear least squares problem, which guarantees convergence of the algorithm [2], robustness to measurement noise [20], and allows simultaneously determining corrections for different target points [13].

In terms of measurement sequences, our new method is similar to the successful pupil segmentation AO technique [4], except for the fact that we use a WFS algorithm to process the data, and use only two ‘pupil segments’ instead of the usual 12-100. The use of the WFS algorithm guarantees finding the global optimum correction in a single optimum, and facilitates theoretical analysis of the convergence behavior even in non-linear cases (see e.g. [14]). The choice for only two pupil segments ensures maximum interferometric visibility during the measurements, resulting in an optimal signal-to-noise ratio.

In summary, we have presented a novel scattering and aberration compensation method, smooth-basis WFS, which allows us to compensate for both scattering and aberrations and acquire smooth corrections. This method combines the best of adaptive optics and wavefront shaping by using a Fourier basis and finding the correction by solving a linear optimization problem.