Sampling moiré method: a tool for sensing quadratic phase distortion and its correction for accurate quantitative phase microscopy

The advantages of quantitative phase microscopy (QPM) such as label-free imaging with high spatial sensitivity, live cell compatibility and high-speed imaging makes it viable for various biological applications. The measurement accuracy of QPM strongly relies on the shape of the recorded interferograms, whether straight or curved fringes are recorded during the data acquisition. Moreover, for a single shot phase recovery high fringe density is required. The wavefront curvature for the high-density fringes over the entire field of view is difficult to be discerned with the naked eye. As a consequence, there is a quadratic phase aberration in the recovered phase images due to curvature mismatch. In the present work, we have implemented sampling moiré method for real-time sensing of the wavefront curvature mismatch between the object and the reference wavefronts and further for its correction. By zooming out the interferogram, moiré fringes are generated which helps to easily identify the curvature of the fringes. The wavefront curvature mismatch correction accuracy of the method is tested with the help of low temporal coherent light source such as a white light (temporal coherence ∼ 1.6 μm). The proposed scheme is successfully demonstrated to remove the quadratic phase aberration caused due to wavefront mismatch from an USAF resolution target and the biological tissue samples. The phase recovery accuracy of the current scheme is further compared with and found to better than the standard method called principle component analysis. The proposed method enables recording of the corrected wavefront interferogram without needing any additional optical components or modification and also does not need any post-processing correction algorithms. The proposed method of curvature compensation paves the path for a high-throughput and accurate quantitative phase imaging. © 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement


Introduction
Quantitative phase microscopy (QPM) is a label-free optical technique and within life sciences is primarily used for cell and tissue imaging, such as diagnosis of prostate cancer [1,2], quantifying biophysical parameters like cell membrane fluctuations, dry mass density of cells [1], and other parameters. The main drivers of QPM techniques in biology are quantification of optical thickness with a sub-wavelength accuracy and label-free imaging at a high-speed for the dynamic specimens. To quantify the parameters of interest, a phase map of the object under study has to be generated. For extracting phase map, the light scattered by the specimen and the reference beam, both derived from the same source, are interfered. The interferogram captured by the camera contains both the amplitude and the phase information of the specimen. The phase map of the specimen can then be numerically reconstructed from the interference fringes.
High-throughput, i.e. high-speed imaging over large field-of-view (FoV), with high sensitivity are desirable in QPM to capture fast events happening inside the biological specimens. QPM can be implemented in either in-line or off-axis geometries and the phase recovery can be performed by either a phase shifting [3][4][5][6][7] or a Fourier transform methods [8][9][10], respectively. Off-axis geometry implemented with Fourier transform method enables single-shot QPM, offering high-speed imaging over large field-of-view, which together delivers high-throughput imaging as relevant for live cell imaging [1,9,11,12]. However, for a lossless reconstruction of the object, the zero and the first order peaks in the Fourier domain are to be well separated, which mandates a high density of fringes.
Mostly QPM systems suffer from the aberration caused due to curvature mismatch between the object and the reference wavefronts. Therefore, complete compensation of aberrations in the optical system is mandatory for precise phase recovery of the specimen. In an off-axis arrangement, there are mainly two reasons for phase aberration, firstly, the angle between the reference and the object beam introduces a first-order phase distortion and, secondly, the use of a microscope objective (MO) introduces a quadratic phase distortion [13]. The first order phase distortion can be easily compensated by shifting the first order Fourier peak to the origin [1,9]. However, for quadratic phase distortion removal, both the physical [14] and the numerical [15] methods have been developed in the past.
It has been reported that a non-telecentric QPI system [16] introduces a parabolic phase perturbation at the image plane. The natural way to compensate for these errors is through the use of a telecentric imaging system. However, realizing this condition in an off-axis geometry is difficult due to the presence of high-density fringes over the entire FoV of the camera. Dobas et al. [17] proposed the use of an electrically controlled liquid lens to produce a spherical wavefront, depending on the microscope objective used, to cancel out the phase perturbation at the image plane. Ferraro et al. [18] described a numerical posteriori procedure -a double exposure method in which the phase map reconstructed from a reference calibration hologram (without the object) is subtracted from the phase map reconstructed from the hologram containing the object. However, these physical methods require the modification of the existing setup or are limited in temporal resolution due to the need for recording of multiple frames and also often depend on post-processing of the image. In [16], the spherical curvature in the object beam is compensated through an in-line geometry setup, by introducing the same physical curvature in the reference beam. However, this method of observing the Fourier spectra of the recorded hologram to ascertain phase errors may fail and leave phase residual error. Post processing numerical methods such as 2D Zernike polynomial fitting [19], phase mask method by Ferraro et al. [18], phase aberration compensation using principal component analysis [20,21], least square fitting technique [22] etc. are also employed for phase error compensation. In [23], a phase mask, the parameters of which are generated via phase gradient fitting and optimization, is subtracted from the acquired data for a phase curvature compensation. Recently, Nguyen et al. employed deep learning by combining convolutional neural network with Zernike polynomial fitting to automatically compensate for phase aberrations in real time [24]. However, the deep-learning approaches requires training of the system which besides being time-consuming process is usually system and application specific.
In this article, we propose a real-time, simple and effective physical technique for off-axis interferometry to eliminate quadrating phase errors arising from wavefront mismatch due to improper optical alignment. The present work does not deal with other optical aberrations, like chromatic and monochromatic aberrations, caused due to the optical system. However, the presence of other optical aberrations in the system can also be discerned using fringes, which is not the scope of this article. In the off-axis setup, the shape of the high fringe density in the interferograms over the entire FoV of the camera cannot be well discerned by a naked human eye which is a low pass filter. To make the shape of the fringes visible to a naked human eye, the concept of sampling moiré method is utilized. When the high-density interferogram is displayed on a pixelated computer monitor at different magnifications -the image will be sampled by different number of the monitor pixels. This enables the high frequency information of the interferogram to be converted into a lower frequency and thereby, allows the naked human eye to discern the shape of the fringes. In other words, any slight deviation in the interferogram becomes visible to human eye in the under-sampled signal due to the generation of moiré patterns. The deviations introduced due to optical path length mismatch can be compensated by adjusting the reference arm before recording the interferograms. It is demonstrated both numerically and experimentally that the proposed method of using moiré fringes can be easily implemented for accurate phase reconstruction. The feasibility of phase reconstruction using the concept of moiré fringes is demonstrated on biological tissue sample as well as on an USAF chart.

Basic theory of QPM
The intensity distribution of the interference pattern due to object wave O (x,y) and a reference plane wave R (x,y) = R 0(x,y) e i2πf R x , where f R = sin θ λ ,at the plane of CCD can be written as [9,25]: where O * (x,y) and R * (x,y) represent the complex conjugates of the object and reference wave respectively. Then a (x,y) = |R (x,y) | 2 + |O (x,y) | 2 , represents a band limited signal centered around the zero frequency, also known as the dc term. If the band width of the object wave along the x-axis of the Fourier plane is taken as 2B, then the bandwidth of the dc term is 4B. Equation (1) may be modified as shown below.
where b (x,y) = 2|R (x,y) ||O (x,y) | and ϕ (x,y) is the argument of the function O (x,y) . The Fourier transform of Eq.
(2) will generate three main lobes -Fourier transform of a (x,y) will be located near the zero frequency and Fourier transform of the cosine function will generate two peaks at (f R , 0) and (−f R , 0). For a faithful reconstruction the central dc term should not overlap with the +1 order or -1 order peaks. Mathematically it means that f R = sin θ λ ≥ 3B. It implies that θ should be chosen to be greater than a certain minimum value. This will result in a high density of fringes as fringe period is inversely proportional to θ.

Sampling moiré effects
The classical sampling theorem states that the information contained in a continuous signal f(x) can be faithfully reconstructed from its sampled version f(x k ) if the sampling frequency f s satisfies the Nyquist sampling criteria, i.e., f s is atleast twice the highest frequency contained in f(x). When this condition is not satisfied, moiré or aliasing effects start becoming visible. Now, when a digital image is displayed on a monitor, interaction between a periodic signal (pixelated image) and the periodic sampling process (pixelated monitor) takes place and as a result sampled moiré patterns or sub-Nyquist artefacts may arise [26]. A detailed mathematical analysis of the sampling moiré is presented by Issac Amidror [26] and the variety of applications where moiré fringes are employed are presented in [27].
Numerous applications involving moiré fringes have evolved over the years. Moiré fringes based on Talbot effect are employed to generate collimated beams [28,29]. The technique of sampling moiré finds applications in displacement measurements in material science as it provides better resolution at high-speed [30]. Li et al. proposed a nanoscale sensitive 3D scanning electron microscope moiré method for 3D shape measurement [31]. Patorski et al. enhanced the displacement sensitivity of atomic force microscopy by utilizing the higher harmonics of the superposing structures [32].
When a signal with a frequency f is sampled at f s , then the frequency f m of the sampling moiré pattern is given by (when f s ≈ f): It implies that the sampled moiré pattern is generated with a new false lower frequency f m and becomes conspicuous even with the naked eye due to its magnified period. This concept of sampling is illustrated by the simplified one-dimensional case shown in Fig. 1. To understand the concept of moiré, square waves with different spatial frequencies are considered and sampled at a rate ∆x M (monitor pixel size) as illustrated in Fig. 1. The frequency f M = 1 / ∆x M is called the spatial frequency of monitor pixels. In Fig. 1(a), the square wave (signal) has a spatial frequency 'f N ' equal to half of the pixel frequency of the monitor (f M ). This satisfies the Nyquist condition and as a result the bright and dark regions of the square wave get displayed by each monitor's pixels without any loss of information. This is termed as Response in Fig. 1. It has a spatial frequency f N and is the exact replica of the square wave, i.e., no aliasing effects are observed. Figures 1(b) and 1(c) represent the cases where Nyquist criteria is not satisfied. In Figs. 1(b) and 1(c), the square waves have spatial frequencies 3f N /2 and 7f N /3, respectively. The corresponding spatial frequencies of the response signal are found to be equal to f N /2 and f N /3, respectively, which are less than the original signal frequencies. This is due to the under sampling of the signal by the monitor's pixels and leads to the formation of moiré pattern.
The simulation presents the superposition of the interferograms having different fringe shapes and periodic pixel patterns in the computer display as illustrated in Fig. 2. The interference fringes are simulated for different optical path differences (OPDs) between the two interfering beams called object and reference waves. The straight line fringes are formed if the curvatures of both the wavefronts are matched. Any deviation from the wavefront matched condition leads to the formation of curved fringes and resulting in an unwanted spherical phase. Figures 2(a) -2(c) depict the simulations of interference patterns with negative, zero and positive OPDs between the wavefronts, respectively. Figures 2(a 1 )-2(c 1 ) show the zoomed views of the region of interest enclosed within the yellow dotted box. It is difficult to visualize the shape of the fringes from the actual or zoomed images. However, the reconstructed phase maps presented in Figs. 2(d)-2(f) clearly show different phase profiles corresponding to the simulated interferograms shown in Figs. 2(a)-2(c). Therefore, one would need to reconstruct the phase map from the interferogram to know whether complete compensation of the unwanted spherical phase is accomplished or not. On the contrary, when a high fringe density interferogram is demagnified and displayed on a computer screen, the shape of the fringes in the interferogram is easily discerned in the corresponding moiré patterns as shown in Figs. 2(a 2 )-2(c 2 ). Demagnification of the interferogram varies the number of fringes registered on a given pixel of the display and leads to the formation of moiré fringes as explained in Fig. 1. To generate moiré patterns, the interferograms are displayed at 6% magnification using MATLAB. The concept of moiré pattern is further implemented for aberration compensation from the phase images before the data acquisition.

Moiré pattern due to the superposition of interferogram and computer screen pixels pattern
The specifications of the CMOS camera and the workstation monitor given in Table 1 are used for the simulation studies. The length of the simulated interferogram is considered to be equal to the effective sensing area of the CMOS sensor 'L Int '. The effective number of pixels 'N Int ' in the interferogram are considered to be equal to 2048 (H) × 2048 (V). Therefore, the pixel size '∆x Int ' of the interferogram is ∆x Int = L Int / N Int = 6.5 µm. The pixel size of the computer monitor '∆x M ' being used to display the interferogram is ∆x M = 270 µm. There is a difference between the pixel size of the sensor and computer monitor. Therefore, if it is considered that one pixel of the camera is displayed on one pixel of the monitor, then, the additional magnification 'M' due to the pixel size difference is defined as follows: It means that the image captured on the sensor chip is magnified by 41.53 times while being displayed on the workstation monitor. In simulation studies, the fringe period of the interferogram is kept 9 camera pixels (same as the experimentally recorded interferogram), i.e., NP fringe = 9.
The simulated interferogram can be displayed at different magnifications (M disp ) to generate different moiré fringe patterns. The total number of pixels of the monitor used to display the simulated interferogram can be calculated as follows: From Eq. (5), the interferogram of 2048 × 2048 pixels, displayed at a magnification of 1% of its actual size, will use 20 × 20 pixels of the monitor. As the total number of fringes over the length of the simulated interferogram are approximately equal to 227.55 (from Eq. (4)). Therefore, each pixel of the monitor will display the average of approximately 11.3 fringes and leads to the formation of moiré fringes as described in Fig. 3. The moiré pattern, which naturally has a low frequency, is generated as a consequence of under-sampling of the image and helps in deciding the shape of the fringes of the interferogram. Similar moiré effect will also be observed for other magnification levels from 2% -21% as illustrated in Fig. 3. At 21% display magnification, single fringe will be displayed on 2 pixels of the display and satisfy the Nyquist sampling criteria. Thus, the interferogram display magnifications more than 21% will not generate any moiré patterns. The effect of the phase shift in the original interferogram on the moiré pattern is also studied. It is observed that the shift in the fringe pattern does not change the shape of the moiré pattern. It only shifts the moiré pattern as depicted in the bottom row of Fig. 3.

Experimental setup
The schematic of QPM system based on Linnik interferometer is shown in Fig. 4. Light from a continuous laser source is passed through a rotating diffuser RD to reduce speckle noise by destroying its spatial coherence [4,10,[33][34][35]. The output from the RD is coupled into a multi-mode fiber bundle (MMFB) placed at 1 mm away from the diffuser. The size of the spatially incoherent light exiting the MMFB is controlled and kept equal to 1 mm using a variable aperture A. The output of MMFB is nearly collimated by using lens L 1 (FL: 75 mm) and further focused onto the back focal planes of the microscope objectives MO 2 (10×/0.25 NA) and MO 3 (10×/0.25 NA) using lens L 2 (FL: 125 mm) and 50:50 beam splitter BS. MO 2 and MO 3 send nearly collimated light towards the sample and reference mirror, respectively. The light is reflected from the sample and the reference mirror and is collected again by same objective lenses to further recombine at BS. The object and reference wave are collected by lens L 3 (FL: 200 mm) and superimpose at the camera plane to form interference patterns. The interferogram is recorded using CMOS camera (Hamamatsu C11440 ORCA-Flash 4.0 LT) and further displayed on a monitor (Philips 240B4QPYEB/00). The details of the camera and monitor are given in Section 3.1. The magnification and demagnification of the interferograms are done using the camera software before capturing or storing them in the computer.

Results and discussion
4.1. Wavefront curvature mismatch sensing using the concept of moiré pattern In this section, experimental validity of the simulation studies is demonstrated. Figure 5 for optical path length differences of d 0 − 500 µm, d 0 µm, d 0 + 500 µm, respectively, where d 0 represents the distance at which the optical path lengths of both the arms of the interferometer are matched. The OPD change between the object and the reference arms is done by translating the reference objective MO 3 and the mirror simultaneously (Fig. 4). Figure 5(h) represents a flat phase profile, and since no object is involved it may be considered as the system phase. At d 0 the curvatures of wavefronts in both the interferometer arms are matched and straight fringes are generated. The straight line fringes only generate first order phase aberration in the recovered phase images which can be easily compensated by shifting the first order Fourier peak to the origin of frequency domain [9]. The straight line fringe formation requires equal distance between the reference mirror to the detector plane and the sample substrate to the detector plane. Any deviation from this condition results in a spherical phase getting erroneously added into the recovered phase images. It becomes difficult to differentiate between the recorded interferograms corresponding to different OPD positions when interferograms are displayed at their actual size as shown in Figs. 5(d)-5(f). This can be understood by looking at the zoomed view of the fringes enclosed by dotted yellow boxes. However, their corresponding phase maps illustrate an error of almost 3 rad when the reference arm is translated by ±500 µm from zero OPD condition. The insets in red dotted boxes depict the moiré fringe patterns at an initial magnification of 6%. It can be seen that the interferograms corresponding to the three different cases of OPD generate different moiré patterns. Insets in Figs. 5(d) and 5(f) represent the wavefront curvature mismatch condition as curved fringes are observed in the moiré patterns. The recovered phase maps corresponding to the recorded interferograms are depicted in Figs. 5(g)-5(i). Figure 5(h) illustrates an almost flat phase profile which corresponds to the straight line moiré pattern. Thus, the proposed method works well for sensing and correction of a wavefront curvature mismatch.
The moiré fringes can also be generated online easily using LabView framework by overlapping the fringe pattern with digitally generated reference structure. This is called digital moiré principle. This can be another option for the judgment of the shape of interference patterns. Moreover, the proposed sampling moiré approach is not only limited to either vertical or horizontal fringes. It can also be used with any orientation of the interference fringes and consequently be used to control the orientation angle of the interference fringes accurately.

Wavefront curvature mismatch measurement accuracy
It has been demonstrated in the previous section that moiré method for wavefront curvature mismatch compensation works well. A piece of standard silicon wafer (flatness < 1 µm, roughness < 1 nm) is used in both the arms of the interferometer (Fig. 4) to generate good contrast interferograms. Before measuring the accuracy of the method, a white light source having low temporal coherence length (∼ 1.6 µm) is utilized in the interferometer to decide the reference arm step size for the experiments. The reference mirror is kept at a high angle with respect to the vertical to generate the high fringe density interferograms for single shot phase retrieval. Therefore, confinement of the white light fringes would be in a limited region of the camera FoV. For a particular tilt position of the reference mirror during the experiments, first, reference arm of the interferometer is positioned in a way such that the white light interferogram is observed at one extreme end of the camera FoV. Then the reference arm is translated in steps of 5 µm until the white light fringe shifts to the other end of the camera FoV. The shifting of the white light fringe from one end to other end of the camera FoV covers 50 µm translation of the reference arm. Thus, the step size of 50 µm would be sufficient for further experiments and is determined by observing the movement of white light fringes across the field of view (see Visualization 1). In other words, an error of 50 µm in determining the position of reference arm for straight moiré fringes does not lead to the generation of significant quadratic phase aberration.  Fig. 3(a). Curved moiré fringes depict the curvature mismatch between object and reference beams. (c) Zoomed views of the camera software illustrating zoom (i.e., initial magnification: 6%) of displayed interferograms. Zoom 100% represents the original state of the interferograms.
(d -f) Experimental interferograms due to superposition of object and reference beam for the optical path length separation of d 0 -500 µm, d 0 µm, d 0 + 500 µm, respectively. d 0 is a wavefront curvature matched condition. Insets illustrated in red and yellow color dotted boxes represent corresponding moiré patterns and zoomed interferograms, respectively. Next, in order to check the accuracy of the proposed moiré method, the reference arm (MO3 and M) is kept at an arbitrary position, say dZ, from the zero OPD (wavefront curvature match) condition. The reference arm is then moved in a step of 50 µm sequentially to arrive at the wavefront curvature match condition d 0 . Figure 6(a) shows the high-density interferogram recorded by the camera and displayed on the monitor. Figure 6(b) shows the zoomed view of the interferogram. It can be seen that the curvature of the fringes from the high-density interferogram or the zoomed in region is difficult to determine. However, when the displayed interferogram is demagnified down to 1%, then the shape of fringes can be easily discerned from the moiré pattern as discussed in the earlier section. Figure 6(c) presents the moiré pattern generated due to the superposition of demagnified interferogram at 1% and array of monitor display's pixels. Figure 6(d) illustrates the recovered phase map corresponding to the interferograms shown in Fig. 6(a). The interferogram, zoomed view of the interferogram, corresponding moiré pattern and phase map for each position of the reference arm displaced by 50 µm can be found in Visualization 2. The straight moiré fringes correspond to almost negligible quadratic phase aberration in the recovered phase images. This would allow performing real-time quantitative phase imaging of biological events at high-temporal resolution without any post processing. Note that the focus of the present method is not the removal of other optical aberrations from the system but only the spherical aberration introduced by the MO. However, the effects of other aberrations can also be observed in the moiré pattern.  Fig. 6(a). The color maps show phase in radian. Reference objective MO3 and mirror M both are simultaneously translated in a step of 50 µm in order to obtain the condition of matched object and reference wavefront curvatures (see Visualization 2). The step size of 50 µm is decided by shifting white light interference patterns in a step of 5 µm (see Visualization 1).

Effect of wavefront curvature mismatch in the phase imaging of USAF chart and its compensation
To understand the wavefront curvature mismatch measurement accuracy of the moiré method, experiments are further conducted on a USAF resolution chart (R3L3S1N, Negative 1951 USAF chart, Thorlabs). USAF chart is placed under QPM to record interference pattern for different curvature mismatch positions of reference arm at d 0 -500 µm, d 0 µm, d 0 + 500 µm, respectively. Figures 7(a)-7(c) depict the interference patterns of USAF chart for three different positions of the reference arm. The insets marked with red dotted boxes are the zoomed views of small portion of the interference patterns. Figures 7(d)-7(f) illustrate the recovered phase maps corresponding to reference arm positions at d 0 -500 µm, d 0 µm, d 0 + 500 µm. The reference arm position corresponding to d0 represents the flat phase background condition as shown in Fig. 7(e). This position of almost zero quadratic phase aberration is achieved by adjusting the reference arm to achieve straight line moiré fringes. The other two positions at d 0 -500 µm and d 0 + 500 µm represent the wavefront curvature mismatch conditions and generate quadratic phase aberration in the recovered phase images of the resolution chart. Middle column illustrates the wavefront curvature matched condition between the object and the reference beam. This condition is obtained by using the concept of moiré pattern presented in Fig. 2. The color maps show phase in radian.

Comparison of the proposed approach with experimental compensation technique
The accuracy of the proposed technique (sampling moiré pattern) is compared with the experimental compensation technique, i.e., recording an additional sample-free interferogram prior to the sample investigation and simply subtracting both the phase maps. Figures 8(a)-8(d) illustrate the results for the displacement of 1 mm from the curvature matched condition. It can be seen from Fig. 8(c), the reconstructed phase map contains a quadratic phase aberration prior to the experimental compensation technique. This aberration is compensated by simply subtracting the phase of the sample free region as shown in Fig. 8(d). Figures 8(e)-8(g) represent the results of the proposed approach based on moiré fringe method. It can be seen from Fig. 8(g) that the proposed approach reconstruct quadratic aberration free phase map of the specimen. The experimental compensation technique has a slight performance edge over the proposed technique. It also removes other optical aberrations, like monochromatic aberrations, caused due to the optical system at the cost of recording two interferograms. However, the proposed approach has an edge like high speed over the experimental compensation technique while dealing with the monochromatic aberration free optical system. The type of the monochromatic aberrations would depend on the quality of the optical components used to develop the phase microscope. If the phase microscope is free from all types of monochromatic aberrations like spherical aberration, astigmatism, coma, field curvature and distortion, then the proposed approach called moiré fringe method will work satisfactorily. In addition, for several applications finding sample-free area might be demanding, such as the target applications, i.e. imaging of pathological tissue sections (see next section) which are usually millimeter to centimeter sized. The standard deviation is found to be equal to 0.0217 and 0.0313 rad corresponding to the experimental compensation technique and proposed approach. Both the results are found to be in a good agreement with each other. The error map obtained from the subtraction of the reconstructed phase maps using physical method and moiré method is shown in the Fig. 8(h). It can be seen that the errors are concentrated on the borders of the bars.

Wavefront curvature mismatch compensation for non-identical objectives in high resolution phase imaging of tissue sample
For all the experimentation shown in the previous sections, two identical objective lenses (10×/0.25NA) are being used in both the object and the reference arm of QPM system. In order to exhibit the potential of the proposed method, the sample objective (MO 2 ) is replaced with an objective lens (60×/1.2NA) for high resolution phase imaging of the biological specimens. Note that for non-identical objective lenses in the object and the reference arm, the zero OPD position of the reference arm does not correspond to wavefront curvature matched position. This is contrary to the experimental settings in case of identical objectives. The reference arm is then translated to achieve zero quadratic phase aberration condition by observing the shape of moiré fringe patterns. In this section, we also compare the accuracy of the proposed moiré method with a standard numerical method for quadratic phase aberration compensation called principle component analysis (PCA) [20]. First, an interferogram of the tissue sample (thickness ∼ 1 µm) is recorded with high NA objective lens with unknown wavefront curvature mismatch between the object and the reference wave. The reference arm of the interferometer is kept at an arbitrary position for interferogram recording of tissue sample as shown in Fig. 9(a). Square shaped fringes present in Figs. 9(a) and 9(d) could be due to imperfection in the camera. When the camera sensor is examined using naked eyes under ceiling light (white light) illumination, these fringes were present in the sensor's active area. This could be due to the imperfect sandwiching of the protecting cover glass on the sensor. This has been further verified experimentally by replacing the camera and observing the disappearance of the square shaped fringes. The reference arm of the interferometer is then translated in a way such that straight moiré fringes are observed in the display of camera acquisition software. The interferogram is displayed at 1% to observe the moiré fringe pattern in Fig. 9. Wavefront curvature mismatch compensation using Moiré and its comparison with PCA based curvature correction in the phase imaging of biological tissues. 60× (1.2 NA) is being used for the interferometric imaging tissue sample. (a, d) Interferometric images of the biological specimen corresponding to the curvature mismatched and matched between the object and the reference arm. This is done by translating the reference arm assembly (objective lens and mirror). The insets depict the zoomed view of the region marked with red dotted box and Moiré fringes generated due to the superposition of zoomed out (Magnification 1%) interferograms and computer display's pixels. (b, e) Corresponding unwrapped phase maps. (c) Reconstructed phase map of tissue sample using PCA based curvature mismatch correction algorithm corresponding to the recorded interferogram depicted in Fig. 9(a). (f) Reconstructed phase map corresponding to the interferogram depicted in Fig. 9(d). The curvature mismatch is sensed and corrected by employing the concept of Moiré pattern presented in Fig. 1. The color maps show phase in rad. the camera software. Figure 9(d) illustrates the interferogram of the same tissue sample when the object and the reference wavefront curvatures are matched by employing the concept of moiré pattern described in the previous section. The insets enclosed in blue color boxes depict the moiré patterns of the interferograms shown in Figs. 9(a) and 9(d).
The interferograms are further post processed using Fourier transform based phase recovery algorithm to obtain their wrapped phase maps depicted in Figs. 9(b) and 9(e), respectively. The wrapped clearly illustrate the presence of quadratic phase aberration in the interferogram shown in Fig. 9(a). In contrast, the quadratic phase aberration is not present in the interferogram depicted in Fig. 9(d) as can be confirmed from its wrapped phase map as well. Figures 9(c) and 9(f) illustrate the recovered phase maps of the tissue sample by employing PCA and Fourier transform based phase retrieval algorithms [9,20]. It can be seen from Fig. 9(c) that PCA based recovery removes the quadratic phase aberration successfully from the phase images; however, it generates some fringing artifacts at the edges of the image. Further, numerical compensation of the quadratic phase aberration would increase some computational loads. On the contrary, Fourier transform based phase recovery, from the interferogram recorded after implementing the proposed method, does not generate such artifacts. In addition, the reconstructed phase map does not suffer from the quadratic phase aberration. To measure the quadratic phase aberration compensation capability of both the approaches, a small tissue free regions of the interferograms are selected. The standard deviations corresponding to PCA and proposed approach are found to be equal to 0.104 rad and 0.035 rad, respectively. It can be seen that the proposed method generates less phase error in the measurement comparative to PCA. Thus, the present method can be employed for accurate and high-throughput QPI of the cells and thin tissues sections.

Conclusion
High-speed and high phase sensitivity of QPM is useful when capturing dynamics of cell and sub-cellular organelles; and when combining QPM with other functionalities such as optical trapping [9,11], Raman spectroscopy [12] or fluorescence microscopy [36]. Single shot phase recovery algorithm, i.e., Fourier transform method allows for high-imaging. However, for a lossless reconstruction of the object, the zero and first order peaks in the Fourier domain are to be well separated, which mandates a high density of fringes. High fringe density creates problem in discerning the shape of the fringes over the whole FoV of the camera due to the resolution limitation of the human eyes. Moreover, for a faithful and a fast phase reconstruction, the shape and the density of the fringes in the interferograms should be straight and satisfy the Nyquist sampling criteria, respectively. To overcome these limitations, we proposed a new method called sampling moiré method to visualize the shape of the fringe pattern over the entire FoV of the camera. Moiré fringe pattern, which is a low frequency signal compared to the original interferometric images, helps to clearly visualize the shape of the fringe pattern prior to the data acquisition. The interferogram is displayed in the camera acquisition software at lower magnification to identify and correct the shape of the fringes. The proposed method is a simple and real-time physical technique for the quadratic phase aberration compensation. The applicability of the method is demonstrated by performing QPI of USAF resolution chat and 1 µm thick tissue sample.
The proposed sampling moiré method enables high-throughput QPM. In microscopy, highthroughput refers delivering imaging of large number of objects (cells or tissues) over large FoV (areas) in a minimal time. In the proposed approach, quadratic phase distortion is corrected from the entire FoV in a single step, without acquiring multiple images or a need of numerical postprocessing, thereby reducing both the acquisition time and the post-processing time. Contrary to other methods based on background removal, the proposed approach is also suitable for applications such as high cell-density and tissue imaging where sample can be present over the entire FoV. Although, present implementation of the method is based on the visual judgement criteria, the entire process can be automated i.e. procedure of under-sampling the image can be programmed for the removal of quadratic phase distortion.
The method is not only limited to the current experimental setup. It can be easily implemented with any QPM system for quadratic phase aberration compensation. The proposed method is a physical quadratic phase compensation technique. Therefore, it can be used with those QPM configurations, like, Michelson and Mach Zehnder type etc., where the optical path length of either an object or a reference arm can be adjusted. The optical path length can be changed either by mechanically translating the optical component or by using a tunable lens and electrically controlling the wavefront curvature of the object or the reference fields. The proposed method will also work for multiple wavelengths used simultaneously in QPM system, provided the optical microscopy system used does not have any chromatic aberration and it can be used to generate straight fringes for multiple wavelengths for curvature matched conditions, i.e. when the zero optical path length condition is satisfied.