Quantitative phase imaging and complex field reconstruction by pupil modulation differential phase contrast

: Di ﬀ erential phase contrast (DPC) is a non-interferometric quantitative phase imaging method achieved by using an asymmetric imaging procedure. We report a pupil modulation di ﬀ erential phase contrast (PMDPC) imaging method by ﬁltering a sample’s Fourier domain with half-circle pupils. A phase gradient image is captured with each half-circle pupil, and a quantitative high resolution phase image is obtained after a deconvolution process with a minimum of two phase gradient images. Here, we introduce PMDPC quantitative phase image reconstruction algorithm and realize it experimentally in a 4f system with an SLM placed at the pupil plane. In our current experimental setup with the numerical aperture of 0.36, we obtain a quantitative phase image with a resolution of 1 . 73 μm after computationally removing system aberrations and refocusing. We also extend the depth of ﬁeld digitally by 20 times to ± 50 μm with a resolution of 1 . 76 μm . phase contrast (PMDPC) imaging method. We use a spatial light modulator (SLM) to modulate a 4f imaging


Introduction
While traditional microscopy only captures the intensity, an object's phase information is highly desired in many situations. For example, most biological tissues are transparent and rich structural information is contained in phase. To obtain the samples' phase, many techniques have been developed. Widely used phase gradient methods such as Zernike phase contrast microscope [1] and Nomarski's differential interference contrast (DIC) [2] are able to get qualitative phase contrast images. However, in both techniques, phase and intensity information are mixed in the images and therefore the phase measurement is not quantitative. Holography [3][4][5][6][7] is another phase imaging method where the object's wavefront interferes with a reference beam and the phase information is obtained from the recorded interference pattern. However, holography suffers from coherent speckle noise and it requires a complicated optical system to generate the path-length matched reference beam. Phase information can also be retrieved by using computational algorithms [8]. For example, Fourier ptychographic microscopy (FPM) takes a series of images captured under different illumination angles [9][10][11][12][13] or different overlapped apertures [14][15][16] to iteratively recover a sample's phase and amplitude; transport of intensity phase imaging [17][18][19][20] uses images captured at different defocused planes to solve the transport of intensity equation (TIE) and retrieve phase. The large number of measurements needed is a disadvantage of these methods. Spiral phase contrast microscopy [21] is another effective quantitative phase imaging method that uses a spiral phase plate for modulation in the Fourier plane. Unlike FPM, it requires only three images at different plate orientations. However, limited spatial coherence negatively influences the absolute phase value accuracy.
Besides the methods mentioned above, differential phase contrast (DPC) is another effective way to get quantitative phase images. In comparison to the majority of the other methods, DPC is experimentally simple to implement. In DPC, spatially asymmetrical imaging methods are applied to generate a phase gradient image [22][23][24][25][26]28], and a sample's quantitative phase information can also be reconstructed by deconvolving the phase gradient images with the phase transfer function [22,25,26,28]. In previous studies, DPC images are achieved in different approaches. D. K. Hamilton et. al first demonstrated DPC in scanning optical microscopy by splitting the detector's area into two halves and detecting the phase gradient by intensity integral difference between the two half sections [22]. Scanning-free asymmetrical illumination DPC (AIDPC) techniques were later demonstrated. Ref. [25] blocks half of the illumination plane for each captured image. Ref. [26] use a programmable LED matrix to generate the phase gradient images. High resolution, speckle-free, quantitative phase images were reconstructed computationally with deconvolution algorithm. Using colored LEDs, dark field, bright field and DPC phase images can also be captured in one shot [27]. However, these methods still have limitations. They only work well for thin samples since oblique illumination components will result in different lateral shifts for different z-planes in thick samples and 2D Fourier shift relation is no longer valid. Although 3D DPC method to overcome this limitation is demonstrated by Lei Tian et. al [28], this approach requires images to be taken separately for each single LED and results in worse reconstructed resolution with increasing defocusing distance.
Besides asymmetric illumination, asymmetric Fourier plane modulation can also generate phase contrast images of a sample. A very simple way is Foucault method which uses a knife to block half of the Fourier plane to view the phase gradient. Lowenthal et al. imaged aberrated wavefront and heated air phase changes with a Hilbert filter on the Fourier plane [29]. Hoffman et al. reported modulation contrast microscope which can generate a qualitative phase gradient image with Fourier plane modulation [30]. In this technique, partially coherent light from a slit illuminates the sample and modulated on the Fourier plane. The sample's phase gradient perpendicular to the slit is reflected on the image intensity. However, this is not quantitative phase interpretation and attenuation caused by scattering and absorption is mixed up with phase information. Further improvements on the modulation contrast microscopy were reported later. Pyramid phase microscopy (PPM) [31] applies a pyramid to split the Fourier plane into four subsections, forming four phase gradient images on the detector. Assisted with a calibration of a known sample, phase gradients along two perpendicular directions are reconstructed and quantitative phase can be further obtained. The pyramid can also be replaced with quatrefoil lens [32,33] or a liquid crystal display (LCD) [34], named partitioned detection aperture (PDA). In all of the above methods with asymmetric Fourier plane modulation, the illumination numerical aperture (NA i ) determines the detectable dynamic range of phase gradient and is preferable when NA i is equal to NA/2. Only smooth phase gradients can be correctly recorded, and phase gradient beyond the dynamic range suffers from saturation problem. These phase gradient imaging methods are also partially coherent methods. Although the modified algorithm for out-of-focus samples based on PDA system is introduced in [35], the depth of field can not be extended with full resolution due to partial coherence limitation.
Addressing the above limitations, we develop a pupil modulation differential phase contrast (PMDPC) imaging method. We use a spatial light modulator (SLM) to modulate a 4f imaging system's pupil plane. However, instead of detecting the phase gradient as done in other phase modulation methods [30][31][32][33][34], we generate the phase transfer function to reconstruct the phase directly, similar to [25,26,36]. Unlike aforementioned partially coherent phase imaging methods [24][25][26][27][28][30][31][32][33][34][35][36], we use spatially coherent illumination to obtain a quantitative phase image and demonstrate that we can dramatically extend the depth of field of our system computationally. The sample is not restricted to be smoothly varying and quantitative phase can be reconstructed without calibration.
In PMDPC, a phase gradient image forms at the image plane when half of the pupil plane is blocked by the SLM. Using two such phase gradient images captured separately by applying complementary half-circle pupils with the SLM, we can construct a DPC image that carries the sample's phase information. With DPC images, a quantitative phase image of the sample can be reconstruhted with a deconvolution procedure. Further, we are able to combine this quantitative phase with the sample's intensity image to obtain the complete field of the sample which then allows us to post-process the image. We report experimentally that aberrations arising from the optical elements in the system can be corrected by deconvolving the reconstructed image with a pre-calibrated pupil function. We can also digitally extend the depth of field using angular spectrum propagation algorithm [7]. With our current PMDPC imaging setup with the numerical aperture (NA) of 0.36, a quantitative phase image with a resolution of 1.73μm is obtained. The depth of field for a 20× objective is increased digitally by 20 times as well.
In section 2, we describe the basic principles of PMDPC with simulation results. PMDPC's requirement of the weak object approximation and formation of a full complex field of a sample are also included in this section. We present the experimental setup in section 3. We show the experiment results in section 4, including quantitative phase imaging results of microbeads, resolution measurement with a Siemens star target, depth of field improvement and a biological imaging example of a frog blood sample. At the end of this section, we also compare the reconstructed image qualities given different numbers of input images and compare the result with FPM. Finally, we discuss the advantages and disadvantages of PMDPC compared other techniques and mention potential improvements in section 5.

Weak object transfer function
Consider the following 4f imaging system shown in Fig. 1(a). A complex sample [ Fig. 1(b)] is placed at the front focal plane of lens l 1 . The sample is illuminated by a collimated coherent light. The light field reaching the back focal plane of l 1 would map out the Fourier spectrum of the sample. An SLM placed at this Fourier plane modulates the sample's Fourier spectrum with half-circle pupil patterns [ Fig. 1(c)]. This modulated Fourier spectrum is inverse-Fourier transformed by l 2 to form a phase gradient image [ Fig. 1(d)] on a CCD camera placed at the back focal plane of l 2 .
To simplify our analysis, we set the magnification ratio as 1 : 1 by assuming l 1 and l 2 to have the same focal length. The mathematical relations are as follows: the object field o(x, y) is Fourier transformed by l 1 to become its Fourier spectrum O (u, v). O(u, v) is then modulated by a coherent pupil function P(u, v), resulting in the complex distribution O(u, v)P(u, v) at the pupil plane. O(u, v)P(u, v) is inverse-Fourier transformed by l 2 to form a phase gradient image I (x, y) which is captured by the camera: (1) The Fourier transform of I (x, y) is equal to the autocorrelation of O(u, v)P(u, v): Like other phase retrieval algorithms [22,[25][26][27][28]36], we use the weak object assumption here. In other words, we approximate the complex object as o(x, y) = e −α(x,y)+iφ(x,y) ≈ 1 − α(x, y) + iφ(x, y), where α is the absorption coefficient and φ is the phase induced by the sample. In Fourier domain, it becomes (3), the cross-terms can be assumed to be negligible according to the weak object assumption. Finally, we can separate the phase and amplitude components, and rewrite Eq. (3) as: where A(u, v) and Φ(u, v) are the Fourier spectrum of α(x, y) and φ(x, y), respectively. And H am p and H ph are the amplitude and phase transfer functions defined as:

DPC image formation and reconstruction
DPC reconstruction process is shown in Figs. 1(e)-1(h): DPC image I DPC is constructed using phase gradient images I 1 and I 2 that are captured with a pair of complimentary half-circle pupils P 1 and P 2 : Since the pupil functions satisfy P 1 (u, v) = P 2 (−u, −v), the amplitude transfer functions H 1,am p and H 2,am p cancel each other while the phase transfer functions are doubled, rendering the following relation in the Fourier domain: I DPC is the Fourier transform of I DPC and For a weak object, the absorption can be assumed to be negligible compared to the zero frequency component, and is omitted in the denominator [26].Ĩ DPC can be obtained from captured images, and H DPC is determined by the pupil functions. Therefore, the sample's phase distribution can be reconstructed by a direct deconvolution process according to the following equation [26,27,37]: where is an infinitesimal number added for regularization ( = 10 −6 is applied in both experiment and simulation). The summation in Eq. (10) takes into account multiple pairs of pupil functions in the reconstruction process. It has been previously shown that multi-axis illumination pairs are necessary in AIDPC algorithm since a large portion of the Fourier spectrum information near the asymmetric axis in DPC transfer function is missing [26]. This missing information results in large reconstruction errors in corresponding spatial frequencies (for example, if only top and bottom illuminations are applied, reconstruction errors take the form of vertically distributed strips). However, we show that this multi-axis requirement is not necessary in PMDPC. As shown in Fig. 1(g), PMDPC transfer function covers the whole non-zero frequency region within NA with the same transmission magnitude. The only missing region is the zero frequency center which only affects the offset value of the reconstructed phase image. The zero-frequency component is always transmitted without attenuation in both half-circle pupils. Appendix A discusses in detail about this center frequency treatment. Experimental misalignment and numerical errors may result in inaccuracy along asymmetric axis. Therefore having more pupil pairs improves phase reconstruction accuracy by averaging the reconstruction results to decrease numerical error. However, one pair of asymmetric pupils is sufficient in most cases for a quantitative phase image using PMDPC. In the following experiments, we still apply 2-axis pupil modulation unless otherwise stated. At the end of section 4, the reconstructed phase images of 1-axis versus 2-axis pupil modulation are shown and reconstruction errors are calculated compared against FPM reconstructed phase images. Fig. 2. Red solid line plots the relationship between the reconstruction error, E and the peak-to-trough phase magnitude of the sample, φ peak −to−trough . Blue dashed line indicates where E/φ peak −to−trough equals to 5%. When φ peak −to−trough ≤ 0.74π, E/φ peak −to−trough is below 5%. Inserted figures (a) and (b) show the PMDPC reconstructed phase images when φ peak −to−trough equals to 0.4π and 1.6π, respectively. The ground truth and reconstructed phase profile along the red dashed line are also plotted in both cases on the right side.
residue errors due to this approximation. It is reasonable to expect that when the phase difference becomes larger than a certain value, the reconstruction result may no longer be quantitative. To explore this limit, we simulate a concentric sinusoidal phase object with different peak-totrough phase magnitudes, and then apply PMDPC reconstruction algorithm. Here, eight pairs of pupils are applied to minimize numerical errors. Reconstructing error E is calculated as the standard deviation from true value expressed as [14]: where φ is the ground truth phase. In simulation, we take φ as NA-limited phase information, and in experimental results, it stands for FPM phase reconstruction. φ r is the reconstructed phase image, and N is the number of pixels in the image. Since the offset value of phase can be arbitrary and will strongly affect the calculated E, a parameter, α is applied to compensate for the offset difference and is defined as Fig. 2 with red solid line. At the same time, we show in blue dashed line where E equals to 5% of φ peak −to−trough . When the peak-to-trough phase magnitude φ peak −to−trough is smaller than 0.74π, the reconstruction error E is below 5%. For φ peak −to−trough ≥ 0.74π, E increases more and more rapidly and the reconstruction is no longer quantitatively accurate. However, we notice in Fig.  2 that even when the weak object assumption is invalid, we are still able to obtain qualitative phase images with the PMDPC algorithm.

Forming a complex object field with the reconstructed phase and measured intensity
After having reconstructed the quantitative phase information of a weakly scattering sample, synthesizing the sample's full complex field is readily possible with the addition of its intensity measurement. When an image of the sample is captured with a fully opened pupil in the 4f system of Fig. 1, it contains the amplitude information, |o(x, y)|, of the sample in its intensity measurement, I (x, y), such that I (x, y) = |o(x, y)| 2 . With this information, the sample's full complex field, o(x,y) can be reconstructed because it is simply a multiplication of the sample's amplitude and phase information: In our experiments, we measure our sample's intensity after the PMDPC measurements to reconstruct the sample's complex field and perform post-processing to computationally correct for aberrations and refocus to different z-planes.

Experimental setup
The experimental realization of PMDPC system is shown in Fig. 3. Light from a He-Ne laser source passes through a rotating diffuser and couples into a multimode fiber in order to suppress coherent speckle noise [16,38]. The light from the multimode fiber is collimated and incidents normally on the sample. The randomized light from the multimode fiber can be considered as spatially coherent since NA i is sufficiently small (Appendix B). The sample's optical field is collected by an objective (Olympus 20× 0.4NA, f = 9mm). A reflective SLM is placed at the pupil plane, which is also the back focal plane of the objective. Since the focal plane of the objective lies inside of the objective, a 4f relay system consisting of lens l 3 and l 4 are applied to image the pupil plane onto the SLM's surface. Pupil modulation is achieved by the SLM (LCOS display Model: Holoeye LC-R 1080, refreshing rate: 60Hz), whose fast axis is oriented 45 degrees from the two perpendicularly oriented polarizers LP 1 and LP 2 . The SLM generates phase shift between its two axes for the modulated pixels, thus rotating light polarization by 90 degrees. The modulated components of the SLM reflected light is selected with LP 2 . Finally, a tube lens (Thorlabs ITL200, f = 200mm) focuses the modulated light onto a CCD camera (Allied Vision: Prosilica GX6600, 5.5μm pixel size, 4Hz frame rate) and phase gradient images are captured to be used for our reconstruction algorithm.

Quantitative phase imaging with microbeads
A quantitative phase images of a 10μm polystyrene microbead sample is imaged to show PMDPC's quantitative phase reconstruction capability. 10μm microbeads are immersed in oil (n = 1.580) at room temperature. Eight pairs of phase gradient images are captured to minimize numerical error and misalignment. Figure 4(a) shows vertical and horizontal image pairs and corresponding pupil functions. Reconstructed phase images of PMDPC are shown in Fig.  4(b). Phase reconstruction of the same region is conducted with FPM [14-16] for comparison as well, shown in Fig. 4(c). We compare PMDPC reconstructed phase with FPM retrieved phase information and plot the phase distribution along one of the beads' diameter in Fig. 4(d). We also plot the estimated ideal phase profile of a microbead from the reconstruction result. Our measurements indicate that the refractive index of microbead is 1.587, which is consistent with the refractive index of polystyrene.

Spatial resolution measurement with Siemens star phase target
PMDPC imaging results of phase Siemens star target is shown in Fig. 5. The Siemens star resolution phase target, recommended by [39] is fabricated on a gold-coated glass. Using focused ion beam (FIB), the Siemens star pattern is first etched onto the gold surface and then a 50μm by 50μm area encompassing the whole pattern is further etched with the same exposure time until the glass substrate is exposed within the entire area.
As introduced in section 2.4, we capture four phase gradient images [ Fig. 5(a)] and another full-aperture intensity image [ Fig. 5(c-ii)] in experiment. Quantitative phase [ Fig. 5(c-i)] is reconstructed following PMDPC reconstruction process illustrated in Figs. 1(e)-1(h). The reconstructed phase, after combined with captured intensity, forms the complex optical field distribution of the sample being imaged.
Up to now, the resolution of reconstructed phase is blurred by defocusing and other system aberrations [ Fig. 5(c)]. However, the fact that we have computed the complex field distribution implies that we can computationally post-process the images to remove the aberrations. We characterized the aberration using the method described in [40]. Multiple manually defocused images of 1μm micro-bead are captured, and the optimization algorithm is applied to determine the correct values of different Zernike modes. Figure 5(e) shows the phase image after decon- volving the reconstructed optical field with the characterized aberration function [ Fig. 5(d)]. Finally, we apply angular spectrum propagation method to digitally propagate the images into focus. Figure 5(f) shows an aberration-corrected in-focus phase image. Figure 5(g) plots the phase distribution along the red dashed circle in Fig. 5(f), indicating a periodic phase resolution of 1.73μm, which matches with the theoretical resolution With the object's complex field, we are also able to computationally adjust the focus to different z-planes, thereby digitally extending the depth of field of the system. To demonstrate this, we take the same Siemens star target and manually defocus it in z-direction to ±50μm and ±100μm . We can observe in Fig. 6(a) that captured defocused images are completely blurred at these planes. PMDPC reconstructed optical field distribution at the defocus distance of −48μm is shown in Fig. 6(b) as an example. The complex field distribution at the image plane is a diffraction pattern generated by the Siemens star pattern and also the square edge. After aberration correction and refocusing process described in Fig. 5, sharply focused phase images are formed. Figures 6(c)-6(f) show the refocused images and resolution measurements for each defocus distance. When the sample is defocused by −48μm and +47μm, the periodic resolution of refocused phase images reaches 1.76μm, which closely matches the theoretical resolution. And when the sample is defocused to −96μm and +96μm, the refocused image's resolution drops to 1.84μm and 2.22μm, respectively.
The standard depth of field of 20× microscope objective is 5.6μm defined by λ/N A 2 +e/(M · N A), where e is the camera's pixel size and M is the system's magnification factor. In compari- son, our method increases the depth of field digitally by more than 20 times.
It was pointed out in previous work that AIDPC only applies for thin samples on the focal plane. However, PMDPC is able to reconstruct phase of thick samples and defocused samples with consistent resolution. The difference is explained as follows: AIDPC reconstruction algorithm [25,26] is based on partially coherent illumination, which relies on the 2D Fourier spectrum shifting property under oblique illuminations. This mathematical relation only applies for the sample information on the system's focal plane. Defocused information suffers from problems such as image lateral shift and broadened point spread function, and therefore cannot be correctly reconstructed. Efforts were made in previous AIDPC to reconstruct 3D phase images [28], where images are captured by turning on each single LED in the illumination matrix and then laterally shifted accordingly during reconstruction process. However, this modified method has some problems. First, it requires images to be captured for each single LED, which implies longer acquisition time. Second, it only achieves 6μm resolution under 20× objective when the sample is defocused to 50μm because of broadened PSF at defocused planes. In comparison, PMDPC uses a collimated beam to illuminate the sample. Therefore, the relation of 2D Fourier shifting for different illumination angles is no longer required, and the resolution degradation and lateral shifting are non-issues for the PMDPC method.

Imaging biological samples
We also tested the performance of PMDPC in biological samples. Figure 7 shows the image of a frog blood cell sample. Reconstructed phase and intensity are shown in Figs. 7(a) and 7(b) following the same procedures as above. Under red light illumination, the cells are almost transparent, and only nuclei are clearly seen in the intensity image, while the phase image can better resolve the shape of blood cells and the fine structures around nuclei. We can also generate a phase contrast image by shifting the central spatial frequency of the reconstructed field by to obtain the result shown in Fig. 7(c). At the same time, we capture an image of the same region under a phase contrast microscope, shown in Fig. 7(d). We observe that the phase contrast image generated from PMDPC reconstruction does not suffer from halo and shade-off effects that exist in conventional phase contrast microscopy. Meanwhile, PMDPC image has higher background noise since it is a coherent optical system. We notice that the peak-to-trough phase magnitude of the frog blood cell sample is slightly beyond the quantitative reconstruction region determined in section 2.3. This result shows that PMDPC is still a reasonable qualitative phase imaging technique even when weak object assumption is not accurate in some cases.

Required number of measurements versus image quality
Based on analysis in section 2.2, we stated that PMDPC can still reconstruct good phase image with only one pair of asymmetric pupils. To illustrate this, we reconstruct the phase images of Siemens phase star target, frog blood cells sample and microbeads sample using one pair of phase gradient images and two pairs separately, shown in the first two rows in Fig. 8(a). Reconstruction difference are shown in last row of Fig. 8(a). Comparing the images in both cases, there is no significant missing-frequency artifact when using one pair of phase gradient images for reconstruction. The reconstruction difference can be attributed to experimental misalignment, numerical artifacts, and other imperfections. We also compare the microbead sample reconstructed phase images with FPM reconstructed phase, and calculate the reconstruction error E in Fig. 8(b). In both cases, reconstruction errors are less than 10% of original φ peak −to−trough . Reconstruction with 2 pairs of images is slightly more accurate than with 1 pair. The reconstruction error with 8 pairs is 0.0389 for the reconstructed phase image shown in Fig. 4(b). We expect that more image pairs help improve algorithm robustness and suppress noise. In situations where less images and faster capturing process are needed, only two phase gradient images and one intensity image is sufficient for complex field reconstruction and refocusing process as demonstrated in this paper.

Discussion
We have introduced PMDPC, a phase imaging method via asymmetric pupil modulation. Quantitative high resolution phase images can be reconstructed with a minimum of one pair of phase gradient images captured with complimentary half-circle pupils. After simple post-process procedures of aberration deconvolution and refocusing, we achieve phase imaging with resolution of 1.73μm given the system NA of 0.36. By digitally propagating the reconstructed field, we can also extend the depth of field of a 20×, 0.4NA objective to ±50μm with almost no resolution deterioration, which is more than a 20-fold improvement compared to a conventional microscope.
Unlike other phase gradient imaging methods [30][31][32][33][34], PMDPC doesn't suffer from phase gradient saturation problem. Therefore, it is also suitable for non-smooth samples. Compared with AIDPC described in [25,26], PMDPC has larger depth of field and requires less captured images. We also expect that PMDPC is more robust against Gaussian noise because it has uniform magnitude of phase transfer function across the entire aperture whereas the weak magnitude of AIDPC's transfer function in high-frequency region results in poor signal-to-noise ratio (SNR). The drawback of PMDPC is that it has higher background noise. Although we are able to suppress speckle noise by using a rotating diffuser in our PMDPC system, the coherent illumination means that it is still more sensitive to dust or other imperfections in the optical path compared to partial coherent illumination where extended light sources are employed. The phase detection sensitivity of PMDPC is related with SNR and the sample's attenuation, which is analyzed in detail in Appendix C. In our experimental setup, the sensitivity is measured to be 0.034rad, corresponding to the optical length of λ/200. PMDPC is achieved in a simple 4f system that can be incorporated into a conventional microscope with a setup similar to partitioned detection aperture [34] and Fourier Phase Microscope in [41]. The resolution of PMDPC is limited by the system NA and imaging speed is determined by the camera frame rate, SLM refreshing rate and light intensity. Currently, it takes us 32 seconds to capture an image. The slow frame rate is due to low irradiance on sample, at 1.5 × 10 −4 W/m 2 . This problem is caused by low illumination efficiency rather than PMDPC algorithm since zero frequency information is always totally transmitted. It can be solved by increasing light couple efficiency, using higher power output light source, and using a higher NA collimation lens l 2 . In addition, the reflective SLM can be changed into transmissive SLM to avoid energy loss caused by beam splitter. In addition, this work is focused on the demonstration of PMDPC method and performance, and there is much room for optimization and improvement.
To decrease the cost and improve light efficiency and capturing speed, the SLM can be replaced with alternative modulation methods since only half-circles amplitude modulation is required. For example, we can place a rotating half-circle mask at the pupil plane to achieve the same modulation function as SLM, which can be much faster and cheaper. In summary, PMDPC can be a cost-effective, high-speed, and high-quality quantitative phase imaging technique that can find usage in various fields such as biology, etc..  carry out the reconstruction algorithm. The simulation follows our experimental setup, where NA i = d fiber /f 2 = 0.0033. Simulated FoV is 384μm × 384μm, and the center opening in Fourier modulator is chosen to match NA i . The reconstruction result is shown in Fig. 10(c). As can be seen, there is no significant difference observed in the reconstruction between NA i = 0.0033 and NA i = 0.001. Reconstruction errors E are also calculated, shown in Fig. 10. Therefore, we are able to assume this finite-NA i in our experiment as a coherent illumination.
However, when NA i becomes much larger, we cannot consider it as coherent illumination anymore. For example, when NA i = 0.02, the reconstruction of the same sample will deteriorate as shown in Fig. 10(d). In this case, an algorithm with partially coherent illumination imaging model is needed, which was described in [26,36].

Appendix C: Discussion on phase sensitivity
A system's sensitivity defines the minimum detectable phase, which is closely related with noise level. For interferometric phase imaging methods, a stronger reference beam can improve SNR, while PMDPC is a non-interferometric phase imaging methods that relies on the captured SNR itself.
We first limit our discussion to transparent samples. With no noise and infinite bit depth, ideally PMDPC is able to resolve infinitely small phase change. However, when noise exists, a real imaging sensor is considered, and the phase change is small, the contrast in the captured phase gradient images will be very low. When noise level is larger than signal contrast, the method cannot detect the phase change. When noise exists and the sample has no attenuation, Eq. (4) reduces to:Ĩ In detection domain, it converts into I (x, y) = 1 + H{φ}(x, y) + n(x, y), where H denoting Hilbert transform because the phase transfer function is actually a Hilbert modulator.ñ(u, v) is the Fourier transform of noise, n(x, y). Since any function can be decomposed into sine functions and for simplicity, we discuss on a 1-D sine function, with phase distribution φ(x) = βsin(k x). Since Hilbert transform of sin(k x) is H{sin(k x)} = −cos(k x), detected signal on the image plane will be I ≈ 1 − βcos(k x) + n(x). When the noise level n(x) is comparable with β, the detected signal fluctuation will be obscured by the noise. When β > n, phase signal can be detected. For example, when P signal /P noise = 1/n = 100, (SNR = 20dB), detectable phase signal variation β should be larger than 0.01 ≈ 0.003π, corresponding to 1nm optical length for wavelength of 632.8nm. Therefore, the DPC method is still very sensitive to the phase changes.
For samples with attenuation, the phase sensitivity also depends on the attenuation. When phase changes are very small, while attenuation is much larger than phase changes, the crossterm in expression of O * O cannot be neglected. In this case, reconstructed phase will be mixed with sample's attenuation information. Approximately, when the phase difference is above 1/10 of the attenuation factor, this artifact is almost non-noticeable.
For our setup, we measure phase sensitivity by selecting a region of 40 × 40μm 2 (167 × 167 sensor pixels) where no features are present and calculating the standard deviation of the reconstructed phase over this region. The same method is used in [42,43]. The reconstructed phase fluctuation in this region is shown in Fig. 11. The experimental sensitivity of PMDPC is 0.034 rad, corresponding to the optical length of λ/200, which is comparable to other coherent phase imaging methods such as diffraction phase microscopy (DPM) [43].