## Abstract

Traditional wavefront-sensor-based adaptive optics (AO) techniques face numerous challenges that cause poor performance in scattering samples. Sensorless closed-loop AO techniques overcome these challenges by optimizing an image metric at different states of a deformable mirror (DM). This requires acquisition of a series of images continuously for optimization − an arduous task in dynamic *in vivo* samples. We present a technique where the different states of the DM are instead simulated using computational adaptive optics (CAO). The optimal wavefront is estimated by performing CAO on an initial volume to minimize an image metric, and then the pattern is translated to the DM. In this paper, we have demonstrated this technique on a spectral-domain optical coherence microscope for three applications: real-time depth-wise aberration correction, single-shot volumetric aberration correction, and extension of depth-of-focus. Our technique overcomes the disadvantages of sensor-based AO, reduces the number of image acquisitions compared to traditional sensorless AO, and retains the advantages of both computational and hardware-based AO.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Optical imaging systems have been the driving force behind numerous seminal discoveries in biology and medicine, and their ability to non-invasively probe the structural and functional properties of a sample has driven their introduction into clinical environments at various levels. Conventional microscopes are strictly tethered to physical limits such as those described by Abbe’s diffraction [1], and recent advances have attempted to break-free of this limit using superlenses [2], unique illumination profiles [3], or fluorescence localization [4,5]. While one can engineer the perfect microscope, the other half of this system, i.e. the biological sample, is beyond our control. The inhomogeneous nature of biological samples, at a microscopic level, imparts numerous spatial variations to the refractive index, which reshapes the wavefront and introduces aberrations [6]. This problem is particularly pronounced in ocular imaging and imaging deep within a tissue.

The concept of adaptive optics (AO), borrowed from astronomy, attempts to measure and correct these wavefront distortions or aberrations [7–9]. Sample-introduced optical aberrations are unique and, therefore, warrant a closed-loop solution [10,11]. In traditional hardware-based solutions to AO, the aberrations at the image plane are measured with a wavefront sensor, mapped on to the plane of a compensating element placed in the path of light, and the negative of this distortion is expressed by the compensating element. The most commonly used compensating elements are deformable mirrors (DM) and spatial light modulators (SLMs). The former has high optical efficiency and is wavelength or polarization independent; the latter has a larger pixel count and, therefore, a larger range of response [12].

Alternatively, these aberrations can be corrected computationally in imaging modalities such as optical coherence tomography (OCT) [13], Fourier ptychographic microscopy [14], digital holography [15], and DIC-fluorescence microscopy [16]. Digital wavefront correction is achieved either through deconvolution or with a compensating phase mask. The former is often an illconditioned problem whose performance is dependent on the sample and is extremely intensive computationally [16–18]. In contrast, the latter is a well-conditioned problem [13], but requires complex-valued images [19]. Imaging modalities based on interferometry such as holography and OCT intrinsically generate complex-valued images. For other modalities such as wide-field or confocal microscopy, the phase of the images can be artificially reconstructed with phase retrieval techniques [20,21]. Additionally, real-time algorithms for phase recovery using deep learning and neural networks are driven by the rapid improvement in computational processors and tools [22]. Apart from improving the image quality by correcting sample-introduced wavefront errors, AO has also been used to extend the depth of focus by introducing aberrations such as spherical [23] or astigmatism [24].

One of the major challenges in AO microscopy is wavefront sensing [7]. Traditional sensors, such as
the Shack-Hartmann sensor, exhibit high accuracy with single sources. This
is true in the cases of guide-stars or perfectly reflective surfaces.
However, in biological samples, the plurality of scattering sources
creates a superposition of different wavefronts. The sensed wavefront can
be restricted to a single depth, i.e. the focal plane of the sample by
confocal [25] or
coherence [26]
gating; but the sensitivity of the measurements is still dependent on the
sample structure [8,27]. Moreover, there is an
observed cross-coupling in the measured wavefront between two modes
[28]. Additionally,
the mismatch between the wavefront sensing and the detection optical paths
can lead to inaccuracies, known as non-common path errors [29]. Furthermore, there is a
trade-off between the dynamic range and sensitivity of the wavefront
sensor dictated by the number of lenslet arrays and the camera used.
Despite their disadvantages, the closed-loop AO system with a
Shack-Hartmann wavefront sensor and a DM has been widely used over the
past decade for *in vivo* retinal imaging with AO
ophthalmoscopy [30–32] and AO OCT [33,34].

In order to overcome these challenges, sensorless wavefront sensing techniques estimate
the wavefront computationally using prior knowledge of the pupil function,
the phase of complex-valued images [35, 36], pupil segmentation methods [36, 37], or phase recovery and phase diversity algorithms
[20, 21, 38]. Alternatively, a hill-climbing approach to sensorless
AO involves iteratively optimizing an image metric such as sharpness or
brightness at different states of the DM [39]. This closed-loop approach has been
demonstrated in a variety of imaging modalities including confocal
microscopy [11],
structured illumination microscopy [40], multiphoton imaging [41,42], and OCT [43,44]. These
techniques directly estimate the wavefront correction at the DM plane.
These sensorless closed-loop AO (SCL-AO) techniques are more versatile
than sensor-based AO. However, they require a series of images to be
acquired at the same position for several seconds. This could prove
difficult in dynamically varying samples such as *in vivo*
ophthalmic imaging.

In order to avoid this bottleneck of traditional SCL-AO, our technique,
*AutoAO*, exploits the advantages of all these methods to
enable automated single-shot SCL-AO with a combination of hardware and
computational AO (CAO) [45,46]. We provide
a framework for AO where a single volume is acquired with an initial
setting of the DM, then the wavefront is computationally estimated by
optimizing an image metric, and the output finally mapped onto the DM. The
second volume acquired at the new state of the DM is the desired result,
as any residual aberrations can then be computationally corrected in
post-processing. As only one initial volume is required,
*AutoAO* calibrates the optimal DM state in a single shot.
In contrast to previous SCL-AO techniques, where there is a closed-loop
formed by the DM states, image acquisition, and metric estimation,
*AutoAO* simulates the different states of the DM using
CAO. Figure 1 illustrates the
differences between the three techniques for AO. *AutoAO*
eliminates the need for multiple image acquisitions, reduces laser
exposure on the sample, and dramatically improves the speed of the system.
This implies a 300 × reduction in the number of volumes required
for optimization compared to previous implementations of SCL-AO OCT
[43,47] and SCL-AO laser scanning
ophthalmoscopy [48]. This is demonstrated on an AO optical coherence
microscope (OCM) where the wavefront error is calculated in terms of its
Zernike polynomials [49], based on the method described by Pande et al.
[50] and
implemented on a graphics processing unit (GPU). The wavefront estimation
technique used here exploits the complex-valued images generated
inherently by OCM, and alternate wavefront computation techniques
described above could be used for other imaging modalities. To highlight
the versatility of our technique, we optimized the wavefront for three
different metrics independently corresponding to depth-wise aberration
correction, volumetric aberration correction, and volumetric extension of
the depth-of-focus. The results overcome the challenges of traditional
sensor-based AO, avoid acquisition of multiple images or volumes for
optimization, harness the speed of CAO to achieve real-time correction of
aberrations for a single plane, and increase the energy of the collected
signal that can only be achieved by hardware-based AO (HAO). As the
wavefront is directly inferred from the image, it provides a larger choice
over wavefront sampling and the region-of-interest. Additionally, with its
high-speed and versatility, *AutoAO* can be adapted to any
optical imaging modality in both laboratory and clinical environments.

## 2. Experimental setup

#### 2.1. Description of the OCM system

A custom-built spectral-domain AO OCM system was used to demonstrate our technique, which consisted of a central wavelength of 860 nm sourced by a broadband laser source (SuperLum Inc.) as seen in Fig. 2(a). The sample arm was built in a 4f configuration where the DM (Mirao 52-e, Imagine Optic SA), the pivots of two galvanometer mirrors (Cambridge Technology), and the back focal plane of the objective lens (LUCPLFLN 40X, Olympus Corporation, Numerical Aperture = 0.6) were carefully aligned and conjugated. To compensate for the dispersion in the sample arm, a series of dispersion compensation elements were placed in the reference arm path, with the residual dispersion being corrected computationally [51]. The detection was performed by a spectrometer (Cobra 800, Wasatch Photonics) and camera (Sprint spL4096-140km, Basler AG) with a selected region-of-interest of 2048 pixels corresponding to 177 nm.

Data was acquired with a custom LabVIEW-based software (LabVIEW, National Instruments). A modified version of the resilient backpropagation algorithm, first described by Riedmiller and Braun [52], and real-time CAO display were performed on a GPU (GeForce GTX Titan 1080 Ti, NVIDIA) with a C++-based program using the CUDA libraries. The DM was controlled by a custom C-based program and linked to the LabVIEW software though TCP-IP communication.

#### 2.2. AutoAO algorithm

CAO, described by Adie et al. [13], requires a phase mask,
Φ(*z*, *Q _{x}*,

*Q*), generated as a sum of weighted Zernike polynomials. When multiplied to the image in the two-dimensional (2D) Fourier space, it can alter the aberration state of the image. Given an initial three-dimensional (3D) complex-valued image,

_{y}*S*(

*z*,

*x*,

*y*), the new image,

*S̄*(

*z*,

*x*,

*y*)

_{,}can be described as

*ℱ*

_{[u,v]}denotes the Fourier transform operator along the dimensions

*u*and

*v*, where

*Q*and

_{x}*Q*are the spatial-frequencies of

_{y}*x*and

*y*, respectively, and

*Z*and

_{n}*W*are Zernike polynomials (with the OSA/ANSI index [53]) and their corresponding weights. Ideally, CAO, when used to improve the image quality, corrects all the aberrations for a given volume in post-processing. However, it is known that the presence of aberrations causes a significant decrease in the collected signal energy, which cannot be recovered by CAO [54]. Therefore, if this phase mask were translated to the DM, both the signal energy and image quality are expected to improve.

_{n}An algorithm to automatically estimate the phase mask to improve the image quality was
described by the forward model presented by Pande et al.
[50]. In the
paper cited, the image quality was quantified by a sharpness metric,
described as the sum of the squares of absolute values at every pixel.
In a more generalized form, the metric is chosen based on the
application. The choice and number of Zernike polynomials to optimize
depends on the phase sensitivity of the imaging system and the
response of the DM. Generally, the first three polynomials,
*Z*_{0}, *Z*_{1}, and
*Z*_{2} are ignored because
*Z*_{0} is simply a constant phase offset, and
*Z*_{1} and *Z*_{2} are
tilts introduced by the sample. Additionally, the presence of defocus,
represented by *Z*_{4}, can affect the
convergence of the other polynomials. The corresponding weights are
modeled as a linear function of *z* such that
*W*_{4}(*z*) =
(*z*_{focus} −
*z*)*z*_{df} where
*z*_{focus} is the location of the focal plane
and *z* df is the defocus factor. Therefore, we first
obtain the optimized set of
*W*_{4}(*z*) and the whole
volume is refocused prior to optimization. Figure 2(b) presents a flowchart to elucidate
the *AutoAO* technique.

First, a set of linearly spaced depths *z _{k}* are chosen within
the region of interest

*z*. The optimal value of

*W*

_{4}is estimated as:

*z*

_{focus}and

*z*

_{df}are obtained by linear regression of the estimated

*Ŵ*

_{4}(

*z*). The whole volume is then refocused based on Eq. (3).

_{k}The optimization problem was defined based on the image sharpness metric, 𝒥, as

*Z*

_{3},

*Z*

_{5},

*Z*

_{7},

*Z*

_{8},

*Z*

_{6},

*Z*

_{9},

*Z*

_{12}, and

*Z*

_{24}, which correspond to pairs of astigmatism, coma, trefoil, and spherical aberrations, respectively.

The translation of the phase correction mask, $\sum {W}_{n}{Z}_{n}$, obtained from the weights of the Zernike
polynomial of the DM pattern, *A _{n}*, is
modeled as a linear function:

*α*is the interaction matrix and

*ξ*is the offset. To generate the interaction matrix, each Zernike polynomial among

*Z*

_{3},

*Z*

_{5},

*Z*

_{6},

*Z*

_{7},

*Z*

_{8},

*Z*

_{9}, and

*Z*

_{12}is applied individually at different weights spaced between −0.35 to 0.35 to the deformable mirror (DM). Using the automatic wavefront sensing algorithm, the optimal wavefront was estimated for 8 depths selected from a 60-μm range around the focal plane. Each weight of the optimal wavefront,

*W*, was fit against the DM pattern applied and the goodness of fit was calculated as its coefficient of determination or its

_{n}*r*

^{2}value. If the

*r*

^{2}value of the fit was over 0.5, the slope and the intercept were populated into the interaction matrix,

*α*, and an offset matrix,

*ξ′*, respectively. The offset matrix,

*ξ′*, was later collapsed into a column matrix,

*ξ*, by taking the median value of each row, such that it satisfies Eq. (6). The interaction matrix was later inverted with Tikhonov regularization such that it minimizes the mean square error. This inverted matrix,

*α*

^{−1}, and the offset,

*ξ*, are shown in Fig. 2(c).

Ideally, *ξ* would be equal to 0 if the microscope has no intrinsic aberrations in the absence of a sample. Additionally, in an ideal scenario, *α* would be a diagonal matrix. However, the number of pixels on the DM is limited, and the effect of discretization of the pattern causes the appearance of non-diagonal elements in *α*. Moreover, *A*_{4} was independently estimated as 0.0085 per pixel shift to the focal plane. The pattern sent to the DM is calculated as $-{\kappa}_{\text{DM}}\mathrm{\Sigma}{A}_{n}{Z}_{n}({q}_{x},{q}_{y})$, where *κ*_{DM} was set at $\frac{0.1}{\pi}$, and *q _{x}* and

*q*are normalized 2D spatial frequency coordinates. Any residual aberrations may be corrected in post-processing through automated CAO that can restore aberration-free focal plane resolution to the entire volume. It should be noted that the calibration for

_{y}*α*and

*ξ*only needs to be performed once during system setup, and that the process could be completely automated.

In the next section, we illustrate three applications of *AutoAO*:
depth-wise aberration correction, volumetric aberration correction,
and extension of depth-of-focus. To demonstrate aberration correction,
both depth-wise and volumetric, a 3D image of a translucent plastic
block is acquired with a transverse dimension of 256 × 256
pixels, corresponding to a transverse area of 100 × 100
μm^{2} at a line-scan rate of 40 kHz and a frame rate
of 128 Hz. Another semi-transparent poly-acrylic sheet was placed at
an angle in the path of light to induce aberrations in the image.
Additionally, to demonstrate our technique for volumetric aberration
correction in a generic biological sample, a block of muscle tissue
was extracted from a defrosted salmon. This sample was imaged with the
focal plane placed 1 mm deep in the tissue with a transverse area of
300 × 300 μm^{2}, corresponding to 1024
× 1024 pixels sampled at a line-scan rate of 60 kHz. To
demonstrate extension of depth-of-focus, the same plastic block was
imaged without any additional elements in the path of light, and with
the same dimensions at a line-scan rate of 20 kHz and a frame rate of
75 Hz. To observe the point spread function (PSF) during extension of
depth-of-focus, a silicone-based phantom, which contained iron-oxide
nanoparticles ranging from 50–100 nm in size and sparsely
distributed within, was imaged with a transverse dimension of 1024
× 1024 pixels, corresponding to 75 × 75
μm^{2}.

## 3. Results

#### 3.1. Depth-wise aberration correction

One important application of AO is deep-tissue imaging where the refractive index of
the sample is constantly varying across depth; consequently, the
aberration state could vary from one depth to another. To demonstrate
the real-time capabilities of our algorithm, a tissue-mimicking
phantom consisting of a translucent plastic sheet, having closely
spaced point-scatterers, was imaged with aberrations introduced by a
semi-transparent poly-acrylic sheet in the path of the imaging beam.
Selected depths from the OCM volume acquired from the plastic sheet,
shown in the top row of Fig.
3(a), are optimized individually. From these images, it is
apparent that the original OCM volume is aberrated. The particularly
elongated shape of the profile suggests that there is a large value of
astigmatism in the images. For this example, it is assumed that
*z*_{focus} (the plane of least confusion) is
known. The image sharpness,
*ℐ*_{DW}[*z _{k}*,

*S*(

*z*,

*x*,

*y*)], is chosen for each depth

*z*independently to find the phase mask for aberration correction. It is evaluated as

_{k}The optimized Zernike weights for the DM pattern, *A _{n}*, is obtained through

*α*

^{−1}and

*ξ*, whereas

*A*

_{4}is calculated as (

*z*−

_{k}*z*

_{focus}) × 0.0085. In order to correct for all aberrations, including defocus, the optimization must be performed depth-wise; each resulting in an independent OCM volume as a result, and whose plane of optimization is aberration-free. As seen in the bottom row of Fig. 3(b), each image appears sharper, aberration-free, and has a focal-plane resolution at each corrected plane. Observing the Zernike weights,

*A*, seen in Fig. 3(b), it is evident that the major aberration corrected is oblique astigmatism. Moreover, there is a constant trefoil subtracted for all planes, along with some depth-variant values of other aberrations. The patterns generated with these coefficients are shown in Fig. 3(c).

_{n}To quantify the improvement to the image numerically, we observe the signal energy collected before and after correction, estimated as |*S*(*z _{k}*,

*x*,

*y*)

*S*(

^{*}*z*,

_{k}*x*,

*y*)| and shown in Fig. 3(d). After correcting these aberrations, it is apparent that there is a minimum of 2× improvement to the collected signal energy. Additionally, in the histogram of the signal levels, i.e. the optical field, we can assume that the peak of the distribution corresponds to the noise (indicated by the black arrow), whereas the higher values correspond to the signal levels (indicate by the red arrow). While the location of the noise peak does not shift before or after correction, the number of occurrences of higher signal values increases. This indicates an improvement to the signal-to-noise ratio of the image.

*AutoAO* for depth-wise aberration correction took an average of 173
± 50 ms for optimization and for transferring the pattern to
the DM. As OCM images are inherently depth-resolved, the computed
wavefront is also restricted to a single depth, overcoming a major
challenge of sensorbased AO techniques. A previous technique for
sensorless OCT [43] iteratively varied the DM pattern directly to
improve the image quality at a plane. This technique needed a series
of ∼300 OCM volumes to be acquired over 65 seconds, where each
*en face* image was only one-fourth the size of those
presented here. Additional studies have reported similar time-scales
on a sensorless AO system for multiphoton microscopy with OCT-guided
wavefront sensing [47]. While these techniques can vary the Zernike modes
at the end of each A-scan, *AutoAO* can only be
performed at the end of a volumetric acquisition. Nonetheless,
compared to these previous methods, our technique offers a
375× improvement in speed and a 300× reduction in the
number of volumes required for optimization. The key differences
between these traditional SCL-AO techniques and
*AutoAO* are illustrated in the schematics shown in
Fig. 1. Compared to a recent
technique for SCL-AO ophthalmoscopy which needed 20 seconds for
optimization [48], *AutoAO* shows a 110×
improvement in speed.

#### 3.2. Volumetric aberration correction

Sample-induced aberrations do not necessarily arise from aberrations introduced within
the depth-of-field. For instance, in the case of *in
vivo* retinal imaging, the aberrations are introduced by the
cornea, intraocular fluid, and lens, which are all before the axial
depth-of-field. These aberrations create a severe decrease in the
collected signal energy and cause distortions to the image. Cases like
this warrant a single correction pattern that can correct the
aberrations for the entire volume, except for defocus. To demonstrate
these capabilities, the same translucent plastic sheet used in the
previous section with introduced aberrations was imaged. Figure 4 shows the initial OCM volume
of the plastic sheet with introduced aberrations, the second OCM
volume after correction with the DM, and after performing CAO on the
latter volume. The 8 *en face* images, which span an
axial range of 54 μm, each show a 50 × 50
μm^{2} transverse section. The whole-volume metric
chosen for optimization is

A total of 8 depths were chosen over a range of 60 μm, *z*_{focus} and *z*_{df} were estimated, and the phase correction mask for the corresponding DM pattern shown in Fig. 5(e) was generated. It is apparent that, even in the presence of non-diagonal elements in the interaction matrix, the computational mask and the DM pattern are almost negatives of each other. No changes were made to the defocus so that the focal plane of the corrected volume coincides with the plane of least confusion of the original image. The estimation of *Ŵ*_{4}(*z*) for the entire volume ∼1.2 seconds; the time taken to estimate the DM pattern is shown in Fig. 5(d). For a fair sample-independent comparison, the number of iterations for the time analysis was set at 50.

First, even before correction with CAO, the second volume has a better focal plane resolution compared to the initial aberrated volume. Second, observing the signal energy collected at each plane in Fig. 5(c), the original volume has two signal peaks, indicating the presence of astigmatism. After correction, the profile appears distinctly Gaussian, with a 2× improvement in the collected signal energy. Third, the histogram of the signal, i.e. the optical field, suggest a dramatic improvement in the signal-to-noise ratio, similar to that of depth-wise correction (Fig. 5(a)). The residual aberrations are minimal and they can be corrected. After correction with CAO, as seen in Fig. 5(b), the maximum value of the OCM signal in a plane increases. Additionally, the volume can be refocused using the methods presented by Adie et al. [13] in post-processing to ensure focal-plane resolution (0.6 μm) for the entire volume [45], as shown in Fig. 4.

While the results shown in Section 3.1 are relevant for real-time processing, one of
the major advantages of OCM is its ability to obtain 3D images through
depth-resolved 2D raster scans of the transverse plane. Therefore, in
order to demonstrate this potential of OCM in biological samples, a
tissue sample from a salmon was imaged with the focal plane placed
∼1 mm below the sample surface, and volumetric aberration
correction was subsequently performed. The correction was performed on
a sub-volume of 256 × 256 transverse pixels. A total of 8
depths chosen over 60 axial pixels were optimized for the metric in
Eq. (8) and the
resulting pattern, shown in Fig.
6(b), was sent to the DM. In Fig. 6(a), appearance of previously suppressed features took
is observed after correction in the cross-sectional images.
Additionally, from the adjacent graph, the regions between 200 and 300
μm have increased signal energy collected per plane. Moreover,
there is an apparent enhancement of features and signal intensity seen
in the *en face* images shown over a region of
∼120 μm in Fig.
6(c). These effects are especially pronounced in the regions
indicated by the yellow arrows. In this case, *AutoAO*
for volumetric aberration correction was sufficient to recover most of
the image features. After correction with CAO, focal-plane resolution
of 0.6 μm was restored to the entire 300 × 300
× 120 μm^{3} volume. By performing the
optimization on a sub-volume of 256×256×8 pixels,
corresponding to 2.2% of the entire volume, the time taken was
similar to that observed in tissue phantoms, while enabling
improvement to the image quality for the entire volume.

#### 3.3. Extension of depth-of-focus

Another application of AO is to extend the depth-of-focus by introducing aberrations.
In 2012, Sasaki et al. [23] showed that adding spherical aberration to the
optical system could increase the depth-of-focus with minimal loss to
the image resolution. In 2018, Liu et al. [24] demonstrated that the
depth-of-focus could be extended further by imaging with an astigmatic
beam. In the latter case, the aberration was corrected computationally
in post-processing, and focal plane resolution was restored for the
entire volume. For a perfectly Gaussian beam, both the resolution and
signal energy per plane drop significantly as one moves away from the
focal plane. Introduction of astigmatism was shown to extend the axial
range of OCM by splitting the focal plane of a Gaussian beam into two
line foci. However, this results in a peculiarly shaped PSF that
varies drastically with depth. Introducing spherical aberration has
been shown to preserve the transverse resolution and the image PSF
over a larger set of depths compared to a Gaussian beam. Consequently,
these must exist an aberration pattern that would optimally maintain
an uniform transverse resolution over a large set of depths while
ensuring that the decrease to the signal energy at the focal plane
stays reasonable. From Sections 3.1 and 3.2, it is apparent that the
image sharpness metrics *ℐ*_{DW} and
*ℐ*_{Vol} are related to the image
resolution. Let us define the image sharpness across depth,
*𝒦*[*z*,
*S*(*z*, *x*,
*y*)], as

For a Gaussian beam profile, the variation of *𝒦* along the
*z*-axis is large. In order to maintain uniform
resolution over a larger set of depths, *𝒦*
must vary minimally across depth. Therefore, the image metric for
extending the depth-of-focus,
*ℐ*_{DOF}[*S*(*z*,
*x*, *y*)], modeled the
deviation of *𝒦* with respect to a fixed value,
*μ _{z}*, is defined as:

*N*is the number of depths selected for optimization. Alternatively, if

_{z}*μ*were chosen as the mean image sharpness over these depths,

_{z}*ℐ*

_{DOF}in Eq. (10) would correspond to the standard deviation of image sharpness. An additional stop condition was added to ensure that the drop in the signal-to-noise ratio (SNR) remains within a reasonable limit. Therefore, if any of the depths had an SNR drop of 23 dB, calculated as $\text{SNR}(z)=20{\text{log}}_{10}\frac{Q\left[\left|S(z,x,y)\right|,0.985\right]}{Q\left[\left|S(z,x,y)\right|,0.015\right]}$ where

*Q*is the quantile function, the computational wavefront estimation is terminated and the results are sent to the DM. This pattern is compared to an equivalent amount of spherical aberration and astigmatism applied to the DM. By observing the cross-sectional images of the tissue-mimicking phantom, consisting of a plastic sheet in Fig. 7(a), we see that all the aberrated images have a larger depth-of-focus compared to the flat profile of the DM. The flat profile of the DM is expected to generate a Gaussian beam profile at the sample plane, in the absence of sample-induced aberrations. This is observed in Fig. 7(b) where the flat DM profile appears distinctly Gaussian. We assume that the axial depth-of-field is the range where the signal energy is above 5% of the maximum value for the flat profile. For an astigmatic beam that creates an ∼80% drop from the previous maximum signal energy, the axial range expands from 144 μm to 159 μm. For a spherically aberrated beam, created by adding the Zernike polynomial

*Z*

_{12}, this range extends further to 168 μm. This range for the optimized beam, created by sending the pattern seen in Fig. 7(c) to the DM, is 177 μm, which is a 22% increase to that of the flat DM profile. This pattern is dominated by the weight of

*Z*

_{12}, followed by contribution from

*Z*

_{8}, with reduced contributions from other polynomials.

Apart from signal energy, the profiles of the PSF must also be
considered in the choice of aberrations. The PSF of the different beam
profiles was observed on a silicone-based phantom containing
sparsely-distributed iron-oxide nano-particles with mean diameter of
100 nm. As seen in Fig. 7(d),
the flat DM profile has a PSF that varies dramatically with depth. In
fact, the profiles far below focus are severely aberrated due to
sample-induced aberrations. In the case of an astigmatic beam, the
PSFs are distinctly elongated and need computational correction to
recover any discernible features. This could restrict the application
of an astigmatic beam in real-time clinical applications. The
corresponding increase to the axial range is less than that of the
spherically aberrated or optimized beam. While the spherically
aberrated beam has a more uniform resolution for a larger range of
depths compared to the flat DM profile, the PSFs away from the focus
appear irregular and may distort image features. The optimized beam
has a uniform PSF for the depth range shown, with a 25%
increase to the focal plane resolution compared to the flat DM
profile. This effect is reflected in the *en face*
images shown in Fig. 7(e). The
features in the flat DM profile appear distorted for the depths
**z**_{1}, **z**_{6}, and
**z**_{7}. All the images from the astigmatic beam
are distorted and need computational correction in post-processing.
While the spherically aberrated beam performs well above the focal
plane, the profiles at depth **z**_{6} have
side-lobes and the features are distorted at depth
**z**_{7}. These effects are minimized for the
optimized beam, which was expected to have a uniform beam profile for
the entire range of depths shown here. Even in the absence of
computational correction, the extension to the depth-of-focus could be
considered as an acceptable trade off to the minimal loss of
resolution at the focal plane.

Figure 8 shows the images of the tissue-mimicking phantom, generated with the flat and optimized DM profiles after appropriate correction with CAO, where focal-plane resolution is restored to the entire volume. We define depth-of-focus as depth range over which the image remains sufficiently in focus. In OCM, one of the limiting factors to depth-of-focus is the signal energy collected from the depths away from the focal plane and introducing aberrations. Introducing aberration spreads the signal energy over a larger set of depths and is therefore expected to extend the depth-of-focus. Initially, at depth **z′**_{1}, the signals have nearly equal values for both the flat and optimized profiles. The *en face* images at this depth corroborate these observations. As expected, at the depths **z′**_{2}, **z′**_{3}, and **z′**_{4}, the images obtained with a flat profile have a higher SNR compared to the aberrated beam. These correspond to the peak of the black curve in Fig. 7(b). The corresponding OCM images reflect these observations, where the images from the flat DM profile have higher SNR. However, it must be noted that the image features are recovered for an aberrated beam profile with an acceptable loss to the SNR at these depths. Beyond these depths, at **z′**_{6}, and **z′**_{7}, the SNR of the images obtained with an optimized DM profile is higher than that of the flat DM profile. At these depths, it is apparent that the images in the second row of Fig. 8 have higher SNR than their counterparts in the first row. This demonstrates the advantage of optimizing the beam profile to extend the depth-of-field to image deeper into the sample without any significant loss to the image quality at the focal plane. Additionally, instead of choosing a fixed aberration to extend the depth-of-field, *AutoAO* provides a method to selectively target regions of interest in the sample dynamically. This could prove especially useful in multimodal multiphoton imaging systems [57] where control over the depth-of-focus could help in choosing and co-aligning the optimal axial section.

## 4. Discussion

In this paper, we have presented a technique for SCL-AO optics that combines the
advantages of HAO, CAO, and previous sensorless techniques. While
traditional SCL-AO techniques optimize an image metric at different states
of a DM, CAO can simulate these states instead, thereby requiring only one
initial volume for optimization. Additionally, *AutoAO*
overcomes the challenges of sensor-based AO and the performance is,
therefore, independent of sample properties. Moreover, algorithms for CAO
and wavefront sensing can show accelerated performance when implemented on
a GPU, enabling real-time operation. Furthermore, *AutoAO*
can be used for a variety of applications such as depth-wise aberration
correction, volumetric aberration correction, and extension of
depth-of-focus. *AutoAO* is demonstrated here on an AO-OCM
system where aberration correction was performed for a single depth in 173
ms, and for a whole volume within a few seconds. Its performance was also
evaluated on a generic biological sample for deep-tissue imaging. In this
case, we have also shown that *AutoAO* can be performed on
a small fraction of the entire final volume to achieve high speeds, when
the aberrations are assumed to be transversely invariant within the
field-of-view. Finally, *AutoAO* has also been shown to
extend the depth-of-field for OCM by 22% with acceptable reduction
to the SNR.

Because OCM generates complex-valued images intrinsically, it simplifies the wavefront
reconstruction problem. The image wavefront for OCM can be computationally
altered simply by multiplying a phase-mask in the Fourier domain. For
intensity-based microscopy techniques, *AutoAO* could be
implemented with an additional step that involves phase retrieval or phase
diversity techniques. Phase retrieval of confocal images has been in
development since the development of modern microscopy [58–60]. Additionally,
instead of sensing the image wavefront with phase masks, alternative
techniques such as pupil segmentation or phase diversity could be used
[36–38].
While the applications used in Sections 3.2 and 3.3 are specific to 3D
volumes, section 3.1 provides a generic framework to achieve correction
for other imaging modalities. However, such imaging modalities might need
additional hardware modifications to support phase reconstruction.
*AutoAO* could also be readily implemented in multimodal
OCM and multiphoton AO microscopy systems, such as the one described by
Cua et al. [47].
Additionally, in such a multimodal imaging system, a volumetric aberration
correction pattern estimated for the OCM images based on Section 3.2 could
also be used to image and correct the aberrations for the multiphoton
imaging system at different depths just by varying the value of
*A*_{4}.

The concept of using digital wavefront sensing with correction of aberrations in the
system has been previously demonstrated on guide-star samples in
fluorescence microscopy with phase retrieval [61]. However, we have shown that the
performance of *AutoAO* is sample-independent. Furthermore,
rather than using AO to improve the PSF of a single plane, our technique
can be expanded to a multitude of applications by choosing an appropriate
metric, *ℐ*. Similar to our metrics for aberration
correction based on Eqs.
(6) and (8), there
have been numerous techniques that have used an image-based sharpness
metric taken at different states of a DM to optimize the correction
pattern [43,44,47,62–65]. Figure
1 illustrates the key difference between these traditional
techniques and *AutoAO*. These techniques are susceptible
to motion artifacts and require the acquisition of several stable images
over tens of seconds, which is particularly challenging in *in
vivo* studies. Alternatively, these techniques could employ axial
tracking algorithms which would limit the sensitivity and accuracy of the
correction pattern [44,47]. In
contrast, *AutoAO* finds the optimal pattern by acquiring a
single stable 3D volume. The phase stability that *AutoAO*
demands for optimal performance is the same as the phase stability needed
for implementing CAO on *in vivo* samples. CAO has been
successfully used for wavefront sensing or aberration correction for
*ex vivo*, *in vitro*, and *in
vivo* OCM volumes of various biological samples [45, 46, 66, 67]. Also, the optimal pattern can be
estimated for an initial volume of a smaller size and field-of-view,
thereby reducing the effect of motion artifacts. Additionally, a
300× reduction in the number of volumes acquired, compared to a
previous study [43], implies a 300× reduction in laser exposure on
the sample. Our technique also shows an improvement in speed and laser
exposure compared to a later study for lens based AO-OCT for *in
vivo* retinal imaging that required an acquisition of 55 OCT
volumes over 3 seconds at very high A-scan rates [44]. In the demonstration of
*AutoAO* presented in this paper, the most time-consuming
step was to acquire the initial volume for acquisition, which took 2
seconds, compared to the calculation of the DM pattern, which took a small
fraction of a second. Therefore, at these high image acquisition speeds
described by Jian et al. [44], the whole process of *AutoAO* could be
completed within a fraction of a second.

Finally, in this paper, we used modal estimation of the wavefront in terms of its Zernike polynomials. As the response of the DM is restricted by the limited number of actuators, the number of Zernike polynomials on the DM used in this paper was restricted to 8. To avoid having an ill-defined or over-fitted interaction matrix, the number of polynomials for the computational wavefront estimation was restricted to 9. If the DM were to have a better response for higher-order Zernike polynomials, the wavefront estimation could include higher order Zernike modes. Additionally, some algorithms determine the wavefront zonally in terms of its spatial distribution [36,68,69], and this is the preferred method in astronomical and ophthalmic AO [7,70]. The resolution and sampling of the wavefront sensor is usually much higher than that of the compensating element, and additional discretization and regularization must be done to ensure that the translation from the estimated wavefront to the DM pattern remains well-defined.

In conclusion, we have presented a technique, *AutoAO*, which finds a phase function optimized for a chosen metric using CAO, and translates this pattern to the DM. This eliminates the need for a wavefront sensor, harnesses the high speed of CAO to enable real-time correction, avoids acquisition of multiple volumes for closed-loop operation, and recovers both the signal energy and SNR post correction. While there are numerous optimized microscopes that have been designed, we believe that our technique brings us a step closer to building an optimized optical imaging system.

## Funding

Air Force Office of Scientific Research (AFOSR) (FA9550-17-1-0387); National Institutes of Health (NIH) (R01 CA213149).

## Disclosures

Stephen A. Boppart is co-founder and Chief Medical Officer of Diagnostic Photonics, which is licensing intellectual property from the University of Illinois at Urbana-Champaign related to computational optical imaging. The other authors declare no competing interests or disclosures.

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