Adaptive optics via self-interference digital holography for non-scanning three-dimensional imaging in biological samples

Three-dimensional imaging in biological samples usually suffers from performance degradation caused by optical inhomogeneities. Here we proposed an approach to adaptive optics in fluorescence microscopy where the aberrations are measured by self-interference holographic recording and then corrected by a post-processing optimization procedure. In our approach, only one complex-value hologram is sufficient to measure and then correct the aberrations, which results in fast acquisition speed, lower exposure time, and the ability to image in three-dimensions without the need to scan the sample or any other element in the system. We show proof-of-principle experiments on a tissue phantom containing fluorescence particles. Furthermore, we present three-dimensional reconstructions of actin-labeled MCF7 breast cancer cells, showing improved resolution after the correction of aberrations. Both experiments demonstrate the validity of our method and show the great potential of non-scanning adaptive three-dimensional microscopy in imaging biological samples with improved resolution and signal-to-noise ratio.


Introduction
Quantitative imaging of integral refractive index and thickness of living cells by digital holographic microscopy (DHM) has benefits the study of cellular dynamics [1,2]. While the applications of DHM are restricted to the coherent imaging system, on the other hand, over past several years digital holography of spatial incoherent fluorescent samples has emerged as an attractive scanning-free three-dimensional (3D) imaging technique. The development of Fresnel incoherent correlation holography (FINCH) [3][4][5] and other modalities such as COACH [6,7] have been employed for super-resolution imaging [8,9], multi-spectrum imaging [10], confocal imaging [11] and other imaging modalities [12]. FINCH provides a powerful tool in biological applications where 3D imaging or tracking of the fluorescent labeled particles or structures is required [9,13]. The quality of the images obtained with FINCH, in particular at larger imaging depth, is, however, limited by optical aberrations introduced by the optical system or the sample itself. Adaptive optics (AO) can be utilized to solve this problem by detecting and compensating the aberrations in a direct or indirect way. Traditional direct AO relies on a physical wavefront detection instrument, typically a Shack Hartmann wavefront sensor (SHWS). This approach has, however, its limitations since light originating from or scattered back from different sample depths may obscure the detection of aberrations [14]. Additionally, the precise alignment of the SHWS and the wavefront modulator is often a challenge in practice. time in cases where aberrations are substantial. This can be detrimental for applications to living cells or tissue that suffer from photo-bleaching and photo-toxicity. In special cases, such as holographic adaptive optics, fewer images are sufficient. In this method the convolution of a point-object-hologram and an extended-object-hologram is calculated to compensate the aberrations [20]. The applicability of this method is still limited because not all samples contain a point-like structure that can be used as guide star. Thanks to the high sensitivity of the interferometric imaging modality such as DHM, cell elasticity can be investigated by detail wavefront analysis [21] and further used for adaptive wavefront controlling [22]. In analogy to these interferometric wavefront analysis techniques, a guidestar-free computational adaptive optics (CAO) method has been demonstrated for OCT imaging [23,24]. CAO is, however, hard to implement in conventional fluorescence microscopy, since the phase of the optical aberrations is lost in the detection process.
In FINCH, the light originating from each fluorophore is split into two parts by a spatial light modulator (SLM). The spatially self-coherent two waves can interfere with each other on the camera. For a single fluorophore, the 3D positions as well as the phase aberrations in the optical path are inherently encoded in the recorded hologram. It is thus possible to correct the aberrations computationally without the need for a wavefront sensor and a wavefront modulator. In this article, we propose a FINCH-based 3D fluorescence microscope with guide-star-free anisotropic aberration correction, in which the blurred, complex-value images in FINCH are compensated and improved by a digital optimization procedure. In this procedure, an image quality metric is optimized by changing a 2D phase filter in the Fourier domain. We believe that this guide-star-free method is well fit to practical applications, for the following reasons. First, its operating speed is faster than other methods since only one complex-value hologram is required. Furthermore, the holographic recording and digital AO correction process are independent and therefore the holograms can be captured with a higher speed and then post-processed off-line. Consequently, our method reduces the total amount of light required to excite the sample, compared to other sensor-less AO methods.
Our article consists of four sections. After this introduction, we present in the next section the methodology and simulation results of our self-interference digital holography and computational aberration correction method. In the third section we will describe the optical setup and experimental results obtained with it to demonstrate its validity and resolution performance. We will also show proof-of-principle experiments on a tissue phantom containing sub-resolution fluorescent particles. As a demonstration of the applicability of our approach to biological samples, we show 3D AO images of actin-labeled MCF7 breast cancer cells. In the final section, we will discuss the advantages, limitations and potential applications of our method.

Theoretical analysis
A schematic illustrating the principle of recording fluorescence self-interference digital holograms is presented in Fig. 1. It shows a single fluorophore, on-axis, that is excited (not shown) and emits fluorescence (green), which is collected by objective L. Part of this light is phase modulated by the SLM acting as a positive lens and the other part passes through the SLM unaltered. The two spatially self-coherent light beams interfere with each other in the recording plane and the resulting Fresnel-zone-plate-like (FZP-like) hologram is captured by the charge coupled device (CCD). An extended, fluorescently labeled sample can be assumed to be composed of a multitude of fluorophores and in this way the hologram of such a sample can be explained as the incoherent superposition of the FZP-like holograms generated by each fluorophore in the sample. 3D information of the sample can be retrieved by digital reconstruction of the holograms. Direct-current-bias in the hologram and twin image can be suppressed by capturing three, four or more images with different phase shifts introduced by the SLM [3,13]. The focal length of L is f. The distances from fluorophore to L, L to SLM and SLM to CCD are z s , d s and z h respectively. The light field on the front surface of SLM, under the assumption that the emitted fluorescence is quasi-monochromatic, can be expressed as: where Q(1/a) = exp[jπ(x 2 + y 2 )/(λa)], λ is the central wavelength of the fluorescence, x and y are spatial coordinates, and * is the two-dimensional convolution. Optical aberrations will deteriorate the performance of any imaging system, resulting in decreased resolution and signal-to-noise-ratio. System-introduced aberrations are typically caused by misalignment of the optical elements. Most sample-induced aberrations, on the other hand, are caused by inhomogeneities of the refractive index in the sample and mismatch between refractive index of the sample and immersion medium. To simulate such aberrations, we consider here, without loss of generality, an aberration in phase in the sample plane that is conjugate to the front surface of SLM. On the CCD plane the interference pattern has the form of I i , where f SLM is the focal length of the positive lens generated by the SLM, θ i is the i-th phaseshift value and φ a is the phase of the introduced optical aberrations. Zero-order terms (directcurrent-bias) and the twin image in Eq. (2) can be eliminated using the phase-shifting technique. To this end, the complex-value hologram is calculated after recording four holograms with different θ value. Assuming unit magnification, the complex-value hologram of an extended fluorescence object can be expressed as where E is the excitation intensity, n is the density distribution of fluorophores and z r is the reconstruction distance. H p is the aberrant complex-value hologram of a single fluorophore, which can be expressed as the product of the non-aberrant hologram Q(1/z r ) and a phase aberration term Ψ that depends on φ a . In conventional FINCH techniques, a reconstructed image of the object R a can be obtained using the back-propagation algorithm, like: Equation (4) shows that the reconstructed image is blurred by the presence of aberration term Ψ. Different than in conventional fluorescence imaging systems, the phase of the aberrations is preserved in our self-interference holographic system. Therefore, the aberrations in Eq. (4) can be corrected by digitally applying a suitable phase mask φ AO in the Fourier domain: where  F and Most guide-star-based sensor or senor-less AO techniques detect and correct the aberrations in the pupil plane. In contrast, in our approach, the aberrations are compensated by applying a phase mask in the virtual pupil plane of the reconstructed image. Since FINCH provides 3D reconstruction ability, the system provides a series of virtual pupil planes that are arranged axially in the Fourier domain, each one corresponding to a reconstructed image at a certain depth in the sample. This is similar to multi-conjugate adaptive optics [25]. In our AO optimization process, the sharpness metric as defined in [16] can be used to evaluate the quality of the obtained reconstructions. The optimal coefficients for the Zernike modes applied to the phase mask that maximize the sharpness of the reconstructed image are obtained in a similar way as used for sensor-less AO [15,16]. In particular, the optimization is started from the lowest order Zernike polynomials. Orders are optimized one by one by generating several phase masks using random coefficients, measuring the AO corrected images and finding the optimal coefficients by Gaussian fitting of the coefficientmetric curve. Finally the optimal coefficients for all Zernike modes are used together to create the AO correction phase mask. The optimization procedure can be repeated to improve the quality of the corrected image.

Simulations
The validity and performance of the proposed method was first tested by simulations. The system shown in Fig. 1 was used and holograms of a 512 × 512 pixels USAF resolution test target were generated and reconstructed using both the conventional back-propagation algorithm and the proposed AO method. A central wavelength of λ = 550 nm was used and z s = 142.5 mm, d s = 95 mm, f = 100 mm, z h = 218 mm, f SLM = 1200 mm, respectively. Gaussian white noise was added to the generated holograms to simulate experimental noise.
To simulate an aberrated system, Zernike polynomials corresponding to astigmatism (z 3 ), coma (z 9 ) and spherical (z 12 ) aberrations were combined with coefficients c 3 = 2 rad, c 9 = −4 rad and c 12 = 2 rad, respectively. The holograms were calculated and the conventional reconstructed image (Fig. 2(a)) was obtained using back-propagation algorithm. The corrected image, obtained using AO optimization with image sharpness as a metric, is presented in Fig. 2(b). Figure 2(b) shows that the image resolution and signal-to-noise ratio can be significantly improved by correcting the aberrations. The optimal retrieved coefficients by improving sharpness are 3 9 1 2 1.8 rad, -1.6 rad, and 1.4 rad.
It should be noted that these values are different than the original aberrations applied to the system, because of the inherent "self-reference" characteristics of self-interference holography: both of the signal and reference beams contain the object as well as the aberrations information. The recorded hologram is actually the difference between these two parts of information.

Experimental setup
Experiments were performed on a custom-built FINCH inverted microscopic imaging system. The scheme of the system is shown in Fig. 3 . The OBJ is placed on an inverted objective lens stage consisting of an objective lens holder (CSA1100, Thorlabs) and a vertical stage (M-MVN80, Newport). The sample is mounted on a sample stage consisting of an insert holder (MP100-MLSH, Thorlabs), a slide holder (MLS203P2, Thorlabs) and wide-field-illuminated by the excitation light. The emission from the specimen is collected by the OBJ and L1, transmitted by the DM, spectrally filtered by a long-pass emission filter F em (HQ500lp, Olympus) and guided to the SLM (1024 × 768 pixels, 19 μm pixel pitch, LC-R2500, Holoeye) via an achromatic lens L 2 (f 2 = 150 mm AC254-150A, Thorlabs). The distance between L 1 and L 2 is 500 mm. A small angle of incidence (about 5°) on the SLM is used to avoid distortions. A cooled CCD camera (2048 × 2048 pixels, 7.4 μm pixel pitch, ML4022, FLI) is used to capture the image. The phase-modulation characteristics of the SLM were measured and user-defined by a look-up-table to ensure a linear phase modulation from 0 to 2π for light with a central wavelength of 550 nm. The wrapped phase map of a lens with focal length of f SLM = 1600 mm is displayed on the SLM. Two polarizer P 1 and P 2 are used before and after the SLM in combination with polarization sensitivity of the SLM to achieve the wavefront division. The orientations of both polarizers are 45° with respect to the sensitive axis of the SLM. The CCD-SLM distance is chosen for maximum overlap of the two interference beams [8,12].

Aberration correction and resolution performance
The fluorescence specimen, consisting of a resolution test target (1951 USAF hi-resolution target, negative, Edmund Optics) and a fluorescent back-plate (365 nm excitation wavelength and 550 nm emission wavelength), was mounted in the focal plane of the objective. Optical aberrations were introduced by a specific Zernike-mode phase map on the SLM. This approach was taken in order to know the magnitude and type of the aberrations in the system, which makes quantitative evaluation of the performance of the AO methods possible. Astigmatism (z 3 ) with coefficient c 3 = −20 rad was first mapped on the SLM. Four phaseshifted holograms I 1 to I 4 (θ 1 = 0, θ 2 = π/2, θ 3 = π and θ 4 = 3π/2) of the aberrant resolution target were recorded sequentially. A complex-value hologram H was calculated using H = I 1 -I 3 -i(I 2 -I 4 ). The blurred sample image ( Fig. 4(b)) was first reconstructed from H using backpropagation algorithm. Then in the AO correction process, starting with five randomly chosen c 3 values, the optimal coefficient was obtained by Gaussian fitting, resulting in 3 6.28 rad c = −  , Fig. 4(a). Finally, a phase mask was generated with this coefficient, and being applied to the reconstructed image ( Fig. 4(c)). The diffraction limited resolving power of the system is 0.61/0.45 × 550 nm = 746 nm. Element 3 of group 9 (with a line width of 775 nm) of the resolution target can be clearly distinguished in the corrected image ( Fig. 4(c)), while this same line pair is severely blurred and hard to resolve without AO correction ( Fig. 4(b)). These sample images demonstrate that correcting the optical aberrations using the proposed method results in diffraction-limited images. To demonstrate the performance of the proposed AO method in a more complex aberrated system, astigmatism (z 3 ), coma (z 9 ) and spherical (z 12 ) aberrations with coefficients of c 3 = −20 rad, c 9 = 20 rad and c 12 = −4 rad were applied to the system. For this experiment, we captured, in addition to the holograms, also regular wide-field images of the sample by using the SLM as an imaging lens. In this case, the unmodulated light was blocked by making the orientations of polarizers P 1 and P 2 collinear with the sensitive axis of the SLM. Wide-field images obtained in this way (Fig. 5(a) and 5(d)) were strongly deteriorated by aberrations, resulting in severe loss of resolution. The complex-value hologram was recorded. The reconstructed image was obtained from the complex-value hologram using the back-propagation algorithm and shown in Fig. 5(b) and 5(e). The image quality of these images is still not optimal. In particular, the resolution appears lower than the diffraction limit. Finally, the optimal coefficients of for aberrations correction were obtained by applying our new AO correction method. A phase mask composed with these values was then applied to the reconstructed image, and a corrected image was calculated, shown in Fig.  5(c) and 5(f). These images obtained after aberration correction show remarkable improvements in sharpness, resolution and signal-to-noise-ratio.

Anisotropic aberrations correction of a tissue phantom
A 3D sample was made by embedding the 500nm fluorescent microspheres (yellow-green 505/515, Invitrogen) in 13.5% polyvinyl alcohol (PVA, n = 1.52) between the microscope slide and cover slip. The thickness of the sample was about 170μm. Holograms of this sample were captured using the setup shown in Fig. 3. After acquisition of four phase-shifted holograms, the complex-value hologram was extracted, the phase of which is shown in Fig. 6. In this phase map, multiple FZP-like holograms can be clearly discerned. Images were reconstructed using the back-propagation algorithm from the complex-value hologram with different propagation distances for refocusing, at depths of about 76μm (Fig. 7(a1)-7(a3)) and 120μm ( Fig. 7(b1)-7(b3)) in the sample. Three microspheres located at different axial and lateral positions are indicated by arrows to demonstrate the digital refocusing ability of the system. Figure 7(a1) and 7(a3) show that microsphere 3 is in-focus at a depth of 76 μm, while microsphere 1 and 2 are out-of-focus. Figure 7(b1) and 7(b3) show the opposite: microsphere 1 and 2 are in-focus at a depth of 120 μm, while microsphere 3 is out-of-focus. It should be noted that, in fluorescence digital holography, in-focus images of microspheres located at different depths in the sample can be obtained from a single complex-value hologram. This is different from scanning microscopy, where multiple images with different focus depth need to be captured. The reconstructed images of the microspheres in Fig. 7 are substantially aberrated (e.g. microsphere 3 looks like an ellipse instead of an Airy disk) and show highly asymmetric outof-focus diffraction patterns. In addition, optical aberrations induced by the sample depend on the spatial location of the microsphere: e.g. with increasing image depth, the reconstructed image of microsphere 2 (Fig. 7(b3)) appears more aberrated (the diffraction pattern looks more arcs-shaped) than that of microsphere 3 (Fig. 7(a1)). Furthermore, the images of microsphere 1 (Fig. 7(b1)) and 2 ( Fig. 7(b3)) show that even at the same depth the aberrations can be different. To solve this problem, we segmented, before AO correction, the reconstructed complex-value image for a given reconstruction depth into multiple sub-regionof-interest (sROIs). Each sROI containing the in-focus image of one microsphere is chosen, thus the aberrations in each sROI are corrected individually. This procedure is repeated for all possible reconstruction depths. All of the 2D-corrected images obtained in this way are finally synthesized to generate a full-field-of-view 3D image with uniform diffraction-limited resolution. In this experiment, we used the summed square of pixel intensities as a metric for AO and used three sROIs at two reconstruction depths containing microsphere 1, 2 and 3 respectively. Four primary aberrations (defocus, astigmatism, coma and spherical aberrations, 3th to 9th and 12th orders of the Zernike polynomials) were corrected as shown in the bottom row of Fig. 8. To more clearly indicate the effects of optical aberrations on the image quality, we show the reconstructed image of each sROI without and with AO correction in Fig. 8. Fig. 8. Effect of aberration correction on reconstructed image of microsphere 1 and microsphere 2 (both located at a depth of 76μm in the sample) and microsphere 3 (located at 120μm). Top row: reconstructed focal-plane images obtained using the back-propagation reconstruction method. Bottom row: reconstructed images with additional computational AO correction. Insets: phase masks used for the AO correction. Images are magnified 20 times using the image interpolation toolbox of the Matlab. The white scale bar indicates 880nm.
The uncorrected images in the top row of Fig. 8 demonstrate the presence of substantial, anisotropic sample-induced aberrations. For optimal correction of the images of microspheres, a different phase mask needs to be applied for each position in the sample (inset in Fig. 8). Using the proposed AO method, the quality of the AO corrected images (bottom row of Fig. 8) is substantially better than those obtained without AO. The aberrations (e.g. coma and spherical aberrations) appear very well compensated. These results demonstrate that with our new AO approach, 3D microscopy can be performed with improved resolution and image quality, uniformly over a large field-of-view. The aberrationcorrected 3D reconstruction method is guide-star-free and can be obtained from a single complex-value hologram of the sample.
As an indication of the computational costs, on our state of the art desktop computer (Xeon E3-1230 3.3 GHz processor and 8 GB RAM), the average time required for correcting seven primary (astigmatism, coma and spherical) aberrations in a 512 × 512 pixels complexvalued hologram was 167 s.

3D AO imaging of actin in MCF7 breast cancer cells
To demonstrate that our AO method can be applied to biological specimens, we imaged fluorescently labeled actin in fixed human cells. MCF7 breast cancer cells were grown on a 35mm polystyrene culture dish (35 mm TC-Treated Culture Dish, Corning). Cells were fixed in 4% paraformaldehyde and washed two times with PBS (Phosphate buffered saline). The Factin in the cells was subsequently stained with ActinGreen 488 (ReadyProbes Reagent, Life Technologies). Coverslips (22 × 22 mm, 0.17 mm thickness) were washed twice with PBS and mounted on the culture dish using Prolong Diamond Antifade Mountant with DAPI (Life Technologies).
The optical system as shown in Fig. 3 was used to capture holograms of the cells. In this experiment, a sCMOS camera (2048 × 2048 pixels, 6.5 μm pixel pitch, ORCA-Flash 4.0 V3, Hamamatsu) was used to record the images with high sensitivity. The holograms of the sample were recorded, and the reconstructed images at two depths before and after AO correction are shown in Fig. 9. In Visualization 1 a video is presented of an axial scan through the cell, showing reconstituted images every 0.25 μm in depth over a range of 10 μm. The conventional reconstructed images from the hologram as shown in Fig. 9(a) are blurred by aberrations, for example induced by refractive index mismatch between the immersion medium of our objective (air) and the Antifade Mountant (n = 1.47). Using the proposed AO method with image sharpness as metric, coma and spherical aberrations are detected and corrected in the reconstructed image. The quality of the corrected image is improved as shown in Fig. 9(c). For example, the actin structure indicated by the yellow arrow is narrower in Fig. 9(c) than in Fig. 9(a). Despite the 3D imaging ability, reconstructed, AO corrected images of this sample are very similar to those obtained with the same system working under wide-field imaging mode (Fig. 10). Note that for 3D imaging using wide-field fluorescence microscopy, the sample needs to be scanned in the depth direction. Using our approach, images at different depth can be reconstituted from a single set of holograms obtained without scanning the sample. This is demonstrated in Fig. 9(b) and 9(d) and Visualization 1.

Conclusion
In this contribution, we have proposed and demonstrated a new kind of guide-star-free, sensor-less and modulator-free computational AO method for anisotropic aberration correction in fluorescence self-interference digital holography. We have presented a theoretical discussion of the recording and reconstruction procedure, demonstrating that the phase information of optical aberrations, which are completely lost in conventional fluorescence imaging, can be preserved in a self-interference digital holography system. Using this phase information, an AO optimization procedure that maximizes the sharpness or sum square of pixel intensity of the reconstructed image, can be implemented by computationally applying a suitable phase mask in the Fourier domain. The performance of our approach was first validated using simulations. To further demonstrate the potential of our approach, we performed experiments on a fluorescent resolution test target and a tissue phantom containing sub-resolution fluorescence particles. The results show significant improvements on image quality and signal-to-noise ratio after AO correction using our approach, compared to conventional wide-field imaging and conventional back-propagation algorithm reconstructions. The total exposure time of the sample is greatly reduced, since four exposures are sufficient to record a complex-value hologram that can be used to computationally compensate the optical aberrations. In addition, the anisotropic aberration correction of the proposed method enables 3D fluorescence imaging of thick samples, with uniform diffraction-limited lateral resolution over a large field-of-view. The relatively low sensitivity of the cooled CCD camera we used was a bottleneck for fast hologram acquisition. Furthermore, the calculations for the correction of anisotropic aberrations were relatively time consuming. The acquisition speed can be readily improved using a faster EMCCD or sCMOS camera. At the same time, the correction steps for each sROI are independent of each other and can be parallelized in a GPU. Our method can be readily implemented in any selfinterference digital holographic system. Finally, the experimental results of actin-labeled MCF7 breast cancer cells demonstrate that our approach is compatible with biological samples and can be used to image subcellular structures in cells in 3D while correction for aberrations.