## Abstract

Optical coherence microscopy (OCM) uses interferometric detection to capture the complex optical field with high sensitivity, which enables computational wavefront retrieval using back-scattered light from the sample. Compared to a conventional wavefront sensor, aberration sensing with OCM via computational adaptive optics (CAO) leverages coherence and confocal gating to obtain signals from the focus with less cross-talk from other depths or transverse locations within the field-of-view. Here, we present an investigation of the performance of CAO-based aberration sensing in simulation, bead phantoms, and *ex vivo* mouse brain tissue. We demonstrate that, due to the influence of the double-pass confocal OCM imaging geometry on the shape of computed pupil functions, computational sensing of high-order aberrations can suffer from signal attenuation in certain spatial-frequency bands and shape similarity with lower order counterparts. However, by sensing and correcting only low-order aberrations (astigmatism, coma, and trefoil), we still successfully corrected tissue-induced aberrations, leading to 3× increase in OCM signal intensity at a depth of ∼0.9 mm in a freshly dissected *ex vivo* mouse brain.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Optical imaging in the brain plays a vital role in neuroscience research since it enables visualization of brain structure and function. However, image quality in the brain degrades as the imaging depth increases due to sample-induced aberrations and multiple scattering. Great efforts on hardware adaptive optics (HAO) have been taken to remove aberrations induced by light propagation into the mouse brain [1–4]. HAO typically uses a deformable mirror or spatial light modulator to produce a wavefront that compensates the sample-induced aberration so that a near diffraction-limited focal spot can be formed within the aberrated sample. In general, various AO correction methods can be categorized into wavefront sensor-based and sensorless approaches. Sensor-based approaches [3–6] measure the wavefront from the sample directly using a physical wavefront sensor, and then feed the measured aberration to an active wavefront corrector for a final correction. However, such direct measurement may be corrupted by photons scattered from other depths or transverse locations than from the focus, therefore only works well in weakly scattering samples. Some sensor-based approaches [3,7–10] use fluorescent guide star for wavefront detection, which generally require additional blood vessel labeling and can be impractical when chronic imaging is needed. Sensorless approaches [11–16] use image metrics as feedback to optimize the wavefront, but the optimization process can be susceptible to temporal noise and sample motion, in general adding difficulties for *in vivo* imaging (although there are still successful *in vivo* demonstrations [17–19]).

Optical coherence microscopy (OCM) can provide a good alternative to direct wavefront sensing, especially for deep imaging applications, since it incorporates both coherence and confocal gates to reject multiply scattered light, and offers high sensitivity from heterodyne detection. Previous efforts utilizing long-wavelength OCM have demonstrated its success in deep brain imaging [20–22]. Additionally, due to its access to phase information, wavefront distortion can be sensed and compensated in OCM with computational adaptive optics (CAO) [23–28], reconstructing a better image in the post-processing stage. Given its high sensitivity and excellent background suppression in scattering tissue, CAO-OCM thus has the potential to serve as a suitable wavefront sensor. However, despite these advantages in suppressing multiple scattering, OCM still suffers from signal loss due to tissue-induced aberration, which reduces the intensity of the beam focus within the sample.

Previously, hybrid adaptive optics (hyAO) was introduced as a general approach to integrate HAO and CAO in synergistic ways to achieve various tasks such as an improvement in retinal imaging [29], a throughput enhancement for 4D imaging of cell populations [26], decorrelation of multiply scattered signal with aberration-diverse illumination and coherent averaging [30], and sensorless AO correction based on CAO feedback in resolution phantoms and salmon tissue [31]. In principle, the integration of CAO and HAO can be leveraged to compute a more localized wavefront at the target depth than direct wavefront sensing, due to coherence and confocal gating from OCM. The sensed aberration can then be corrected with HAO in order to enhance the signal deep within the scattering tissue.

Previous work has utilized OCM for the purposes of coherence-gated wavefront sensing [28,32,33]. There are also successful demonstrations of closed-loop aberration sensing and correction using either CAO-OCM [31] or a solely coherence-gated virtual wavefront sensor [34]. Although prior works have applied coherence-gated wavefront sensing to scattering brain tissue [28,32], they were limited to relatively shallow depths (≤ 400* *µm). Here, we perform an analysis of computational aberration sensing performance up to Zernike order 21 for the case of beam-scanned OCM, and by leveraging long-wavelength illumination (1500-1700nm), we utilized CAO-OCM and a deformable mirror to close the loop on hardware-based correction of tissue-induced aberrations to further enhance OCM image quality at a depth of 0.9* *mm in *ex vivo* mouse brain. Our work provides a more in-depth analysis of CAO-based aberration sensing performance for different Zernike aberration orders, gains new insight on the limitations arising from double-pass confocal detection, and showcases the promise of a long-wavelength hyAO approach for deep-brain imaging. Since our method is based on a beam-scanned imaging geometry, it is also compatible with current laser-scanning microscopy methods, such as multiphoton microscopy, that are popular for imaging the mouse brain.

## 2. Effect of confocal detection on CAO-OCM

Since OCM detects the interference of the back-scattered optical field from the sample and the reference mirror, the acquired signal is inherently phase-sensitive. Neglecting the wavelength dependency if center wavelength λ >> Δλ (spectral bandwidth), the 2D complex-valued OCM signal $S({x,y} )$ at each depth can be written as

where*η*(

*x*,

*y*) is the scattering potential at the sample plane,

*h*(

*x*,

*y*) is the 2D point spread function (PSF) of the OCM system, and ${\otimes} $ represents a convolution operator. After taking a 2D spatial Fourier transform of the complex OCM signal, we numerically propagate the acquired optical field from the image plane to a computed pupil plane, which provides information about the optical transfer function magnitude |

*H*| with an aberrated phase Φ

*. This can be written as*

_{H}*e*

^{2}width). In this computed pupil, an optimal phase profile $\widehat {{\Phi _H}}$ was searched in order to conjugate the existing aberration ${\Phi _H}$, by adjusting the weights

*c*for each Zernike polynomial

_{n}*Z*,

_{n}Then, the image quality can be optimized after the pupil phase conjugation, based on the principle that a computed pupil with zero phase aberration forms constructive interference, and thus producing the signal with the highest intensity. The Zernike basis convention used in this paper is listed in Fig. 8 in Appendix A.

In a beam-scanned confocal OCM system, the PSF in the space domain is the product between the illumination PSF and the collection PSF [35] (in contrast to a magnitude square operation to compute intensity, the PSF remains complex after this direct multiplication). For an OCM system with the same illumination and collection path, this gives $h(x,y) = g{(x,y)^2}$. Or equivalently, in the spatial frequency domain, the computed pupil is a self-convolution of the physical single-pass pupil function of the optical system [23]

*G*(

*Q*,

_{x}*Q*) and Φ

_{y}*denotes the pupil magnitude and phase in the single-pass pupil. Note that the phase profile for the single-pass pupil and double-pass pupil can both be represented as a linear combination of Zernike basis functions, but the pupil size is defined differently. On the other hand, the pupil self-convolution from the confocal detection smears out the single-pass phase that contains rapid fluctuation in the central area, and causes attenuation in the magnitude of the optical transfer function, because the back-scattered light from certain angles cannot be coupled back into a pinhole efficiently for some aberrations.*

_{SP}Figure 1 shows a simulation of how the pupil self-convolution affects the computed pupil in a confocal OCM. In Fig. 1(a) and Fig. 1(b), aberration described by each Zernike mode is shown in the single-pass and double-pass configurations. After the self-convolution, Φ* _{H}* remains intact for Zernike modes in Fig. 1(a). However, in Fig. 1(b), the computed pupil encounters lower-order phase cross-talk and magnitude attenuation after the pupil self-convolution.

To quantify the cross-talk, Φ* _{H}* is projected onto a standard Zernike basis as shown in Fig. 1(c). For modes shown in Fig. 1(a), the cross-talks are negligible. For modes shown in Fig. 1(b), since the shape of the double-pass aberration is similar to its lower-order counterparts, it is difficult to separate the contribution from the lower order ones cleanly. For example, when a secondary astigmatism (Z

_{11}) is applied physically in HAO, both astigmatism (Z

_{4}) and secondary astigmatism (Z

_{11}) will be detected from the double-pass computed pupil

*H*. For even higher-order modes (e.g., Z

_{21}), the lower-order cross-talks can appear to be more significant than the higher-order aberration itself. Therefore, these lower-order cross-talks reduce the CAO sensitivity to detect these higher-order aberrations, and increase the required search dimensionality (i.e. the optimization process for the detection of higher order aberrations should be combined with and include the detection of the corresponding lower order modes).

The cross-talk effect originates from the shape similarity between the related Zernike modes. As an example, suppose we would like to correct a secondary astigmatism (Z_{11}) that is present in the system. In both HAO and CAO, the AO contour (shown in Fig. 1(d)) for Z_{11} correction is skewed toward Z_{4}, because having a Z_{4} shape can partially match Z_{11}. In this case, certain Zernike modes (Z_{4} and Z_{11}) describing the phase of the pupil are orthogonal only over a unit circle, but this is not necessarily the case for a more general pupil function that includes non-uniform weighting. In particular, cross-talk can be present if the pupil function is Gaussian weighted or self-convoluted, in contrast to a pupil function that has uniform weighting corresponding to a top-flat beam. In the situation without *a priori* knowledge, a typical HAO process searches modes from the lower towards the higher order in a sequential manner [15,16]. If the orthogonality condition is not satisfied, reaching the global maximum could require a large number of iterations to converge if the optimization algorithm starts searching from Z_{4}. This suggests that, in both HAO and CAO, a simultaneous search instead of sequential search can improve convergence to a global maximum when high-order aberrations are present.

In addition to the complication of phase profile after pupil self-convolution, correcting these high-order modes with CAO is difficult due to the signal attenuation in the detected optical transfer function magnitude |*H*|. As shown in Fig. 1(b), using spherical aberration (Z_{8}) as an example, the magnitude transfer function |*H*| misses certain in-band spatial frequencies, which leads to phase instability for that frequency range. In practice, this imposes a higher SNR requirement on the OCM signal (which requires longer camera integration time) in order to bring these attenuated spatial frequencies above the noise floor.

Overall, if high-order aberrations exist in the system, they cause the computed pupil phase to contain lower-order cross-talk, and attenuate certain frequencies in the magnitude transfer function. In practice, sensing and correcting only lower-order aberrations becomes an efficient solution in CAO. This is also applicable to brain imaging in the ballistic photon regime where the low-order aberrations dominate [3].

## 3. Methods

#### 3.1 Optical setup for the HAO-OCM system

We used a beam-scanning spectral-domain OCM system (Fig. 2) to provide coherence and confocal gating, with a deformable mirror (DM, Alpao DM97-15) to shape the wavefront. A Shack-Hartmann wavefront sensor (SH-WFS, Imagine Optic HASO4 First) was used to converge the DM to a desired shape (not used to measure sample aberrations), either to apply known aberrations (for the experiments in Sects. 4.1 and 4.2) or to correct CAO-sensed aberrations (experiments in Sect. 4.3). Since the camera in the wavefront sensor is most sensitive at wavelengths shorter than 1000* *nm, a laser diode at 976* *nm was used to illuminate the DM surface to the wavefront sensor. The imaging system incorporated a super-continuum source (Hamamatsu L15077-C7), which outputs 16* *mW of power before the beam splitting, and the interference signal was collected by a spectrometer (Wasatch Cobra 1600). The spectral range for the OCT system was 1500-1700nm, limited by the spectrometer. The objective lens (Olympus XLPlanN25x, 40% filling ratio) in this OCM setup provided a lateral resolution (FWHM) of 1.5 µm, measured from a resolution phantom (silicone laced with 0.5* *µm TiO_{2} beads). The exposure time was 80 µs per pixel, with sensitivity 105* *dB and 2* *mW power onto the sample. The main source of loss comes from the transmission and reflection loss along the sample arm.

#### 3.2 Procedures for closed-loop CAO sensing and HAO correction

The flowchart for the HAO correction based on CAO-OCM sensing is shown in Fig. 3. After OCM acquisition, a 2D Fourier transform of the complex OCM *en face* signal *S(x,y)* gives access to the computed pupil, in which the magnitude and phase can be modified analogously to an HAO pupil. By adjusting the pupil phase described by standard Zernike modes (including defocus, astigmatism, coma, and trefoil), followed by an inverse Fourier transform that returns to the space domain, a CAO reconstructed plane can be obtained. The aberration profile that maximizes the image metric in the CAO reconstructed plane corresponds to the wavefront distortion sensed from CAO-OCM. The image metric was chosen to be an intensity-based sharpness metric [15,16,36] over the region of interest (ROI) at the focal plane, and can be written as $M = {\sum\nolimits_{x,y} {|{S(x,y)} |} ^4}/N$, where *S*(*x*,*y*) represents the complex OCM image at a specific depth with N total pixels. Note that due to energy conservation in the Fourier transform (i.e. Parseval’s theorem), a nonlinear order of 2 will not change the metric value in CAO. Therefore, a nonlinear order of 4 was adopted, which balances the weighting between energy from isolated bright features against an improvement in overall image contrast. The optimization was automated via a 7-dimensional pattern search (or direct search) [37], with Z_{3}, Z_{4}, Z_{5}, Z_{6}, Z_{7}, Z_{9}, Z_{10} optimized simultaneously using the MATLAB function *patternsearch*. Such a non-gradient based optimization method was adopted because it aims to avoid local variation in the metric values, and can achieve better AO correction efficiency than gradient-based optimization [38]. Briefly, the metric value is iteratively maximized as a function of 7 Zernike coefficients. In each iteration, the centroid of the Zernike coefficients is updated based on the weighted average using the image metric evaluated around the current centroid (e.g. in two dimensional optimization, evaluation points can lie on a circle around the centroid). The Zernike coefficients converge (i.e., the evaluation area shrinks) with the iterations, and the optimization process is terminated when the metric value varies by less than 1% for one iteration. The process required ∼30 seconds for a 512×512 plane (double precision, with dual CPU Intel E5-2650v3 at 30% utilization). Finally, we note that in this study a single-plane metric was used, whereas a prior study [31] utilized a volumetric metric for aberration sensing. The sensitivity of the single-plane metric to physically applied aberrations is given in Fig. 9 (Appendix B). A comparison between using a single-plane vs. volumetric metric is provided in Fig. 10 (Appendix B), which shows that the single-plane metric has a similar performance to a volumetric metric overall, but has the advantage of a faster computation.

To close the loop between the wavefront sensing and AO correction, we need to map the retrieved wavefront from CAO-OCM to the actual shape on the deformable mirror surface. Defocus, astigmatism, coma, and trefoil were sensed with CAO, and these aberrations except defocus were applied on the DM surface for correction. Defocus was not applied in order to make sure the AO enhancement did not result from merely shifting the focus, but from compensating the intentionally applied or tissue-induced aberration. In the mapping process, a mapping matrix C_{C→H} is desired such that the Zernike coefficients applied in HAO (C_{H}) can be mapped from the detected Zernike coefficients in CAO (C_{C}), i.e. C_{H} = C_{C→H}C_{C.} The mapping matrix C_{C→H} was found by performing a least-square fit between the applied HAO coefficients matrix and the detected CAO coefficients matrix. In this calibration process, 100 random combinations of desired Zernike modes were applied with the DM followed by a CAO wavefront sensing using volumetric OCM imaging of a TiO_{2} beads phantom (TiO_{2}, 500 nm, U.S. Research Nannomaterial Inc.). The coefficient values for the 100 applied aberration profiles on the DM were recorded in C_{H,Calib}, and the detected coefficient values in CAO were recorded in C_{C,Calib}. Both C_{H,Calib} and C_{C,Calib} have size 6 × 100, corresponding to 6 Zernike modes and 100 measurements. The CAO-to-HAO mapping matrix can then be expressed as C_{C→H }= C_{H,Calib}C_{C,Calib}^{+}, where the pseudo-inverse C_{C,Calib}^{+} can be found via inverting its singular value decompositions (SVD). Therefore, after these steps, we were able to sense the wavefront using OCM, to project the designated wavefront onto the deformable mirror with a pre-calibrated mapping matrix, and to apply the HAO correction with a single hardware iteration. In order to have optimal aberration sensing performance, the diameters of the DM and CAO pupils need to be approximately matched. Given that the Zernike polynomials are defined over a unit circle, we can choose a circular pupil aperture (which has a radial coordinate normalized to 1 at the aperture boundary) as the domain for the Zernike polynomials (defined in Fig. 8 in Appendix A). In our experiments, the CAO pupil radius was chosen to be $\sqrt 2 k\textrm{NA}$.

Note that a lateral alignment between HAO pupil, objective pupil, and CAO computed pupil can be crucial. A laterally mismatched pupil generates low-order cross-talks in the Zernike phase in addition to the one from pupil self-convolution. This incurs unnecessary complexity in the mapping matrix and makes it less diagonal. The HAO pupil and objective pupil can be matched by minimizing the beam wobble after applying astigmatism and correcting with CAO [30]. Then, either physically centering the beam onto the galvo mirror or numerically shifting the spatial frequency coordinate system aligns the hardware and the computed pupil. After these operations, a near diagonal mapping matrix can be formed, as shown in Fig. 3.

The mouse brains used in this study were all extracted immediately after the mice (male, 12-32 weeks, C57BL/6J) were sacrificed for each of the imaging experiments. For mounting the fresh adult mouse brain, two layers of 2.5 mm imaging spacer (CoverWell Perfusion Chambers, Electron Microscopy Sciences) with circular cover slides were used. The brain was held tightly within the spacer and was immersed fully within Phosphate Buffered Saline (PBS) solution (1× by volume, pH 7.4). The connection areas between the spacers and cover slides were sealed instantly with glue (Krazy Glue) aided by a super glue accelerator. All our imaging sites were located on the somatosensory cortex region, a region that most neuroscientists are interested in.

## 4. Results

#### 4.1 Performance of CAO-based aberration sensing for different Zernike modes

Given that the computed pupil could be corrupted from the double-passing confocal gating, particularly for high-order aberrations, it can be important to evaluate how well CAO-based aberration sensing works for individual aberration orders, as well as how the overall aberration shape can be mapped between CAO and DM surface.

In Fig. 4(a), a magnitude sweep for each Zernike mode (defined in Appendix A in Fig. 8) was applied with the DM in phantom. CAO was then used to find the corresponding Zernike coefficients for each applied aberration with the same set of Zernike modes from DM. (Note that for the results presented in the main body of the paper, we utilized the standard Zernike functions to describe the computed pupil phase Φ* _{H}*. Figure 12 in Appendix B shows that similar performance can be obtained in phantoms if we utilized the phase of the self-convoluted version of Φ

*.) The magnitude sweep by the DM for each Zernike mode ranged from −0.4 to 0.4 µm rms with 17 spaced points (sampled more densely near the center, shown in Fig. 9). After OCM data collection, a simultaneous search from Z*

_{SP}_{3}to Z

_{21}was performed to optimize the pupil phase, and the correlation coefficients were calculated between the detected CAO mode values and the applied HAO mode values. They are calculated as |cov[C

_{H},C

_{C}]/(σ

_{CH}σ

_{CC})|, where cov[C

_{H},C

_{C}] is the covariance of HAO and CAO coefficient values, with σ

_{CH}and σ

_{CC}the standard deviation of each value array. Zernike orders higher than the applied aberration are excluded from the correlation calculation, since they may incorrectly appear as lower-order aberrations in the CAO sensing.

To understand the trends in Fig. 1(a), let us consider the metric contours in CAO optimization that are plotted in Fig. 4(b-c), for the HAO magnitude of −0.4, 0, and 0.4 µm rms. In this case, only the mode getting swept by the DM was searched in CAO. For modes that can be sensed accurately by tuning only the HAO-applied mode order in CAO, three distinctive peaks are found near the theoretical location, corresponding to the respective HAO applied aberration. The theoretical peak locations are calculated from simulations given the same HAO magnitude, and vary across modes due to different normalization factors in Zernike polynomial after pupil self-convolution (the aperture size of the pupil over which Zernike polynomials are defined has different diameters in HAO and CAO as shown in Fig. 1, and so the rms magnitude is not directly comparable). For modes without good correlations between HAO and CAO (correlation coefficient < 0.5) in Fig. 4(a), peaks are clustered near zero, suggesting an insensitive wavefront detection when performing single-mode CAO sensing. In contrast, the low-order modes (defocus, astigmatism, coma, and trefoil) have good correlations, with CAO metric sensitive to the applied aberration, indicating that single-mode CAO sensing with the HAO-applied mode order works well. When the CAO search range is limited only to the applied HAO aberration order, CAO is still expected to be sensitive to the metric, because the projection of double-pass wavefront onto the single-pass Zernike basis has a non-zero component, as shown in Fig. 1(c). However, the reduced sensitivity of CAO on these high-order aberrations decreases the correlation in Fig. 4(a). This is mainly likely due to the computed pupil degradation from confocal detection (low SNR at certain spatial frequencies) as shown by the |H| images in Fig. 1(b), as well as the shape similarity with low-order aberrations in the double pass pupil phase. (We note that, for the aberration orders where CAO shows degraded performance in Fig. 4(b-c), the metric itself is sensitive to the corresponding HAO-applied aberrations, indicating that the degraded CAO performance is not due to a failure of the image metric.)

Despite the ambiguity arising from shape-similarity, the shape-similarity itself does not limit aberration sensing performance in the noise-free case, and a full-order aberration sensing strategy is able to outperform low-order sensing (noise-free simulation results in Fig. 4(d)). However, when increasing the noise floor in the simulation to the level as in the phantom, high-order aberrations suffer a worse sensing performance than the low-order ones (simulation and experimental results with SNR = 30 dB in Fig. 4(d)). Also note that a low-order sensing strategy may result in inaccurate detection of some higher-order aberration shapes due to incomplete shape description, which can lead to a slight worse correction than not correcting it (e.g. Z_{15} and Z_{18})

Therefore, we limit the closed-loop CAO sensing and HAO correction to use only low-order Zernike modes, since they are the ones that have a good sensitivity. In Fig. 4(d), the performance of CAO sensing and HAO correction using low-order modes and full Zernike orders (Z_{3}-Z_{21}) is compared. Even though the full-order correction could be expected to provide a nearly perfect shape match in theory due to completeness of Zernike polynomials, in practice it does not achieve such correction efficiency, because the signal attenuation in certain spatial-frequency bands reduces the SNR, and less accurate detection of high-order aberration provides an erroneous mapping between CAO and HAO. When the residual wavefront (after mapping the detected double-pass wavefront to the single-pass pupil and subtracting the DM shape) is applied to the single-pass pupil function at the DM plane, a Strehl ratio for the HAO-corrected PSF can be calculated. These results show that low-order aberration can be sensed and corrected to the diffraction limit, whereas the high-order aberrations suffer more inaccuracies. The observation also justifies the strategy of low-order aberration sensing used in the prior CAO sensing [31]. For brain aberration, previous work suggests (and we will also demonstrate in Section 4.3) that using low-order modes can still achieve good correction under the ballistic photon regime [3].

Note that tetrafoil (Z_{16}, Z_{17}) can also be sensed correctly in CAO, since it correlates well with the applied HAO aberration in Fig. 4(a), and 3 distinctive peaks are observed in Fig. 4(c). In general, aberrations from the highest azimuthal degrees (Fig. 8 in Appendix A) can be well-sensed and mapped, due to shape preservation after pupil self-convolution, but we did not include these terms in the CAO optimization in order to achieve a faster speed.

The above results suggest that high-order aberration shape (including the commonly used spherical aberration Z_{8}) can be difficult for experimental detection in CAO, making it hard to establish a one-to-one mapping between CAO and HAO pupil. The reason is that the self-convolution of high-order aberration degrades the pupil magnitude and makes the pupil phase more similar to the low-order ones. However, we may still use low-order modes to partially sense and correct higher-order aberrations. The performance of this approach in a tissue environment will be evaluated in Sections 4.2 and 4.3.

#### 4.2 Sensing intentionally applied aberrations in an ex vivo brain

So far we have investigated the aberration sensing performance for different aberration orders, and now we test the CAO-based aberration sensing inside an *ex vivo* adult mouse brain. Based on the results in Section 4.1, we excluded the higher-order modes that in general are not well-sensed by CAO. We used the DM to apply 100 random combinations of astigmatism, coma, and trefoil, to phantom and different depths across the brain. Each mode magnitude was uniformly distributed within the range of ±0.5 µm rms, the resulting aberration applied to the DM, and CAO was used to sense the applied aberration.

One of the applied aberrations is shown in Fig. 5(a), as an example for the CAO sensing. The DM applied wavefront was directly measured by a SH-WFS, as well as by CAO at different depths in an *ex vivo* adult mouse brain. In Fig. 5(b), Pearson correlation coefficients for each Zernike mode are calculated between CAO and HAO for low-order aberrations (see Fig. 11 and associated text in Appendix B for correlation measurements up to Zernike mode *Z*_{21}, which shows that for Zernike terms (Z_{8}, Z_{11}-Z_{21}) the correlations are lower than the phantom results in Fig. 4(a)). Figure 5(b) shows that strong correlations exist above depth 800 µm in brain (the white matter layer). By maximizing the image metric, CAO senses the total aberration that includes the stationary sample-induced aberration in addition to the time-varying aberration imposed by the DM. The effect of internal tissue-induced aberration can be mostly excluded in the correlation as it appears as a constant offset that does not contribute to the correlation coefficient. To quantitatively compare the sensed aberration to the applied aberration, we mapped the resulting CAO wavefront to the DM surface, and subtracted it from what has been applied to the DM (the stationary sample-induced aberration was estimated from the average of the residual wavefront across the 100 measurements, and was excluded in the comparison). The histograms of wavefront rms value before and after the subtraction are plotted in Fig. 5(c). A good CAO sensing produces a well-matched wavefront to the DM surface shape, and thus shifting the histogram of residual wavefront rms closer to zero. As we go deeper, the magnitude of the residual wavefront becomes closer to the applied wavefront, indicating an unmatched CAO sensing with the DM shape. Since we observe both a decrease in correlation, and an increased deviation between applied and CAO sensed wavefront, we conclude the CAO sensing performance degrades along the depth inside the mouse brain, which may be caused by the corruption of computed pupil from multiple scattering signal and high-order aberration. It also indicates the white matter layer induces relatively stronger scattering than the cortex layer in the adult mouse brain. This is reasonable as the overall refractive index in the white matter layer is greater than the neocortex layer in the brain due to its densely packed myelinated axons, which are lipid-rich [39,40].

#### 4.3 Closed-loop wavefront sensing and correction in an ex vivo brain

Finally, we applied CAO to detect low-order (astigmatism, coma, and trefoil) tissue-induced aberrations from a fresh *ex vivo* mouse brain and used DM to compensate the sensed aberration in real time. As shown in Fig. 6, at 850 µm depth in the brain, CAO is used to sense a small region that contains myelinated fibers. By mapping the sensed wavefront to the DM, an HAO correction was performed physically. The improvement is quantified by drawing a line profile across a thin strand of fiber before and after HAO, which shows a ∼1.7× improvement in the optical field magnitude, giving ∼3× for the intensity improvement. Note that HAO correction has a larger improvement compared to CAO, because HAO physically improves photon collection at the time of imaging, whereas CAO can only re-distribute the signal from collected photons. Therefore, when there is a non-negligible background from either multiple scattering or system noise, correcting image with HAO instead of reconstructing image with CAO leads to superior imaging performance.

At 950 um depth in Fig. 7, abundant white matter features (the myelinated fiber bundles) start to appear. After the CAO correction, we see multiple speckle points become brighter, acting like the intrinsic guide stars in the white matter layer. After correcting the aberration with DM using the CAO sensed wavefront, the overall band structure becomes cleaner. Note that outside the area of interest, there is a degradation in signal, indicating the applied wavefront does not correct the aberration in these area. The area of interest inside the box then represents approximately the isoplanatic patch (∼ 190×190 µm^{2}) for the AO correction in the brain. Since different region in the brain requires different aberration correction, region dependent sensing and correction can be a good option, as discussed in [28].

Overall, the closed-loop OCT signal enhancement for sharp features like myelin fibers is demonstrated to reach ∼1.7× improvement in the optical field magnitude and hence ∼3× for the intensity, which is comparable to the enhancement factor achieved inside salmon tissue in a previous CAO-OCM sensing work [31], and also inside brain with a confocal fluorescence microscope [9].

## 5. Discussion

In this paper, we have shown that CAO is able to sense and correct tissue-induced aberration at 0.9 mm depth in freshly dissected *ex vivo* adult mouse brains, but there are also practical limitations. Since CAO performs optimization based on the collected OCM signal, a good tissue structure with >10 dB signal-to-background ratio [26] is beneficial for CAO sensing based on feedback from image metrics (When the contrast of tissue structure is low and the OCT image is dominated by speckle, in future work it may be worth investigating CAO sensing based on the sub-aperture correlation method [24,41], which in principle can utilize spatial shifts in speckle patterns rather than requiring tissue structure for CAO optimization). For deep wavefront sensing with CAO, attenuation of the illumination power and increase of the multiple scattering background both degrade the aberration sensing accuracy. In addition, since CAO utilizes a 2D spatial Fourier transform to obtain the computed pupil, which averages the phase variation across the ROI, a uniform aberration has the best sensing and correction efficiency for CAO. Therefore the choice of proper region for CAO wavefront sensing can be important [28], which is analogous to the traditional iterative HAO correction.

CAO-OCM has limited capability in correctly mapping the high-order aberrations in the experimental settings (SNR < 30 dB) as shown in Fig. 4, due to a confocal type detection, leading to degraded SNR in certain spatial-frequency bands after self-convolution of the single-pass pupil function. An approach to improve the sensing performance in the future for high-order aberrations is to eliminate the confocal gate (such as detecting the optical field before the collection pinhole [34], or using plane-waves to illuminate the sample [25]). These hardware modifications could lead to better sensitivity in high-order aberration, but at the cost of increased multiple scattering background. If the confocal gate needs to be maintained, the detected transfer function will be attenuated in spatial frequencies due to the self-convolution of the single-pass pupil. However, if the sample motion permits longer acquisition time, ‘missing’ spatial frequencies with low SNR could be potentially acquired by coherently accumulating selected spatial frequencies from different acquisitions, such as scanning the focus across depth or laterally moving the collection pinhole. Note that it is also possible to gather sample responses from plane waves at different angles based on a wide-field detection scheme and then synthesize a confocal gate [42]. In this case, aberrations can be individually corrected at each angle without computed pupil degradation due to the separate data acquisition at each plane wave illumination angle.

For large aberration that significantly reduces the beam intensity, or if higher-order aberrations are prominent, a single loop of correction may be insufficient, because the aberration sensing performance of CAO is limited by low SNR. In the future, one approach to address this problem is to employ iterative CAO sensing and HAO correction, since the CAO detection becomes more accurate as SNR increases with each iteration. Besides, even though the majority of aberrations can be compensated within a single iteration using the CAO sensor in this paper, having an extra sensorless HAO optimization loop can compensate high-order aberrations, which fine tunes the image quality. For example, after completing a low-order correction (which improves SNR and fine tissue structure), a full-order correction can be attempted to expand the hardware correction to higher orders. Such high order correction after the low order compensation may be especially beneficial when small features need to be extracted. Since high-order aberration can be averaged out in a relatively large FOV, having the low-order aberration being removed first can reduce the dimensionality of search after concentrating on small features. In principle, closed-loop CAO and HAO should have the advantage of reducing the number of hardware iterations when compared to sensorless HAO alone, which translates to a reduction in sample exposure to the illumination light.

Since CAO-OCM has both coherence and confocal gate for rejection of multiply scattered light, it may also serve as a suitable wavefront sensor for other types of microscopes. One of the most promising future applications can be the integration of CAO-OCM sensor with a multiphoton microscope. Since CAO-OCM is label-free and requires low illumination power, it is more favorable in practical biological imaging settings compared to the AO multiphoton system with direct wavefront sensing, which normally requires extra blood vessel labeling as fluorescence guide star [3,4] or using sensorless iterative approaches. Since the fluorescence signal after AO has an exponential increase with respect to the nonlinear order [43], if aberration sensing with CAO-OCM can be demonstrated in brain tissue *in vivo* (see paragraph below), we could expect a ∼3x intensity improvement in two-photon and a ∼5x intensity improvement in three-photon imaging for small features, based on our current 3x intensity enhancement achieved in the confocal-OCM. Given that the nonlinear microscopes have more significant signal enhancement from AO than linear scattering in AO-OCM, the integration of OCM and multi-photon microscopy can have both multi-modal information and wavefront sensing capability that can potentially guide the HAO.

The experiments in this paper were performed in an *ex vivo* brain. The main challenge to perform aberration sensing using CAO for *in vivo* brain imaging is the phase stability in the presence of the animal motion. Since phase correlation within the PSF acquisition interval is required for CAO sensing, the sample needs to be stable across several x-z cross-sectional frames along the slow axis. Therefore, faster OCM acquisition, such as only collecting the focal plane like *en face* OCM [44] can be helpful to preserve the phase correlation. Alternatively, phase registrations based on adjacent A-scans [45] can be used to overcome moderate live motion.

## 6. Conclusions

In this work, we investigated the advantages and limitations of CAO wavefront sensing in a beam-scanned OCM system that incorporates a double-pass confocal gate. Overall, CAO sensing based on confocal-OCM works well for low-order aberrations such as defocus, astigmatism, coma, and trefoil, but has degraded performance for higher-order aberrations. The degradation resulted from a self-convolution of the single-pass system pupil, which causes signal magnitude attenuation (and lower SNR) in certain frequency bands of the double-pass computed pupil, and generates cross-talk in the pupil phase due to shape similarity with lower-order aberrations. Simulations identified SNR as a key factor influencing the aberration sensing performance, particularly for higher-order aberrations. However, by limiting the correction to only low-order aberrations, we still managed to close the loop between the computational CAO-OCM wavefront sensing and hardware wavefront correction, achieving ∼3× intensity enhancement at ∼0.9 mm depth within a freshly dissected *ex vivo* mouse brain. This pushes the depth limit of the hybrid AO approach and demonstrates its potential applications for deep tissue microscopy.

## Appendix A. Definition of Zernike polynomials used in this paper.

## Appendix B. Further details related to CAO sensing.

As discussed in Section 3, the image metric used in the CAO optimization is given by $M = {\sum\nolimits_{x,y} {|{S(x,y)} |} ^4}/N$. This type of nonlinear intensity metric can be a good choice for CAO [26,46], but the performance was not evaluated in high-order aberration for OCM images,

In Fig. 9, we swept each Zernike mode using DM from −0.4 to 0.4 µm rms. The image metric shows a sensitivity for high-order aberration in HAO, with its peak near zero, when a magnitude sweep was applied on DM for each mode from Z_{3} to Z_{21}. This validates our assumption that the metric can be used for detecting high-order aberration. Note that the metric curve from certain aberration orders does not reach its peak at precisely zero. This is caused by sweeping a single Zernike mode at a slightly defocused region. This artifact can be addressed by a simultaneous search of Zernike modes that include the defocus (but does not need be applied), as we did in the experiments.

Also, in the previous CAO sensing work [31], a sharpness metric assessed from the volume around focal plane is adopted rather than from the single focal plane. Here, we showed in Fig. 10, a CAO sensing with a single focal plane and volume achieved similar performance with our fourth order intensity metric. Therefore, in the experiments, we computed the metric only from the focal plane for a faster speed.

In Fig. 11, we compared the CAO sensing of different Zernike orders of aberrations across depth in an *ex vivo* mouse brain in response to a linear sweep in HAO magnitude from −0.4 to 0.4 µm rms for each Zernike mode, similar in Fig. 4. There is a decay in the correlation between HAO and CAO coefficient across depth, as agreed with the low-order aberration sensing in Fig. 5.

Previous work proposed a forward model to produce a CAO wavefront that better matches the wavefront after pupil self-convolution [47]. It generates a wavefront in the single-pass pupil and obtains the equivalent double-pass wavefront from numerical pupil self-convolution. Similar methods as in Fig. 11 were used to compute the correlation coefficients. We compared the performance in Fig. 12 between the forward model that uses a double pass wavefront, and the sensing approach adopted in this work that uses the single-pass wavefront, in a bead phantom. The single-pass wavefront achieves a slightly higher accuracy than the double-pass wavefront for high-order aberration, even though it is still not considered good based on the correlation coefficient >0.5 threshold. The double-pass wavefront in principle matches perfectly to a confocal gated pupil phase, but in practice does not show superior performance in the beads phantom, because the standard Zernike basis can approximate the double pass shape good enough (as shown in Fig. 4(a)), and the high-order aberration under noise can be much less distinguishable with lower-order ones in the experiments.

## Funding

National Science Foundation (Career: CBET-1752405, DBI-1707312); National Institutes of Health (NIBIB-R21EB022927, NINDS-R01NS120819); Cornell Neurotech Mong Fellowship.

## Acknowledgments

We thank Hamamatsu for loaning us the super-continuum laser source. We also wish to thank Dr. Yuechuan Lin for his analysis and measurement of OCM system sensitivity.

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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