## Abstract

We demonstrate digital holographic microscopy that, while being based on phase-shifting interferometry, is capable of single-shot measurements. A two-dimensional (2-D) diffraction grating placed in a Fourier plane of a standard in-line holographic phase microscope generates multiple copies of a sample image on a camera sensor. The identical image copies are spatially separated with different overall phase shifts according to the diffraction orders. The overall phase shifts are adjusted by controlling the lateral position of the grating. These phase shifts are then set to be multiples of π/2. Interferograms composed of four image copies combined with a parallel reference beam are acquired in a single shot. The interferograms are processed through a phase-shifting algorithm to produce a single complex image. By taking advantage of the higher sampling capacity of the in-line holography, we can increase the imaging information density by a factor of 3 without compromising the imaging acquisition speed.

© 2016 Optical Society of America

## 1. Introduction

Measuring the phase information of light is a useful tool for visualizing near-transparent biological specimens without the need for exogenous imaging contrast agents such as fluorescence dyes [1], quantum dots [2], or nano-sized particles [3–6]. Quantification of the phase of the light which interacts with target objects provides more information about the physical status of the samples than that offered by intensity measurements. Over the last few decades, quantitative phase microscopy, which projects the optical phase in either 2-D or three-dimensional (3-D) space in a quantitative way, has become a powerful technique for envisioning structural details of transparent samples [7–13].

To obtain phase information of translucent samples, the most common approach is to record an interferogram generated by superimposing a clean reference beam on a sample beam, using either an in-line (collinear) or an off-axis detection configuration. In the collinear geometry, a reference beam parallel to a sample beam is introduced to produce interference. To acquire a complex field of the light in the sample beam, a phase shifting method has been widely used where the overall phase of the reference beam is precisely modulated in a predetermined fashion [8, 14–18]. The advantage of this scheme is that it utilizes the detection area more efficiently because it does not need to process images over multiple camera pixels to extract single phase information. The size of the diffraction limited spot on the sample plane, which contains the smallest independent object information, can be magnified to the size of two camera pixels. For the collinear configuration, however, multiple images need to be acquired with different phase shifts to retrieve a single complex image. This requirement inevitably compromises the imaging speed and thus makes it unsuitable for fast measurements.

The off-axis configuration has also been frequently used for phase measurements where the reference beam is superimposed on the sample beam at an angle generating a straight fringe pattern modulated in space [7, 10, 19, 20]. A complex field image of an object can then be retrieved by an appropriate transformation [21] with a single image acquisition. This makes the technique appropriate for fast measurements such as the investigation of cell mechanics [7, 9, 22]. However, the spatial modulation of the interference requires multiple pixels to carry single independent object information. Large extra magnification is therefore necessary, such that a diffraction limited spot on the sample plane has to be transferred to at least 7 × 7 pixels at a camera plane when using the spatial modulation of the interference along a diagonal direction. This significantly limits the ability to achieve a higher available sampling density (useful information per pixel) for imaging from the off-axis configuration.

To deal with the trade-off between the acquisition sampling density and an imaging speed, several approaches have been introduced thus far. To maintain the higher sampling density, most approaches employed phase-shifting interferometry, modified with the capability of either a single-shot [23–28] or reduced measurements [29, 30]. These approaches were successful in improving imaging speed by decreasing the number of images taken with phase-shifting interferometry. They were, however, subject to the increased system complexity involved with the need for multiple elements such as multiple cameras [28], phase plates [29, 30], and diffraction gratings in conjunction with many polarizing plates [24, 25, 27], or involved with the necessity for fabrication of polarizer masks with a sub-wavelength scale [23, 26].

In this paper, we demonstrate a simpler method for building a quantitative phase microscope based on four-step phase-shifting interferometry, while maintaining the capability of single-shot measurements. A transmission phase microscope was constructed with a standard type in-line holographic configuration based on a Mach-Zehnder interferometer. A 2-D diffraction grating was placed at a Fourier plane of the detection arm, generating multiple copies of an object on the image plane. By adjusting the position of the grating laterally, the overall phases superimposed on the copies in different diffraction orders can be adjusted. We set the magnification of the microscope in such a way that four different copies simultaneously fall into a camera sensor. By introducing 4 different phases increased by π/2 onto the individual copies of the object image laid on the quadrants of the camera, 4 different interferograms can be acquired at the same time. Then, the complex field image for the object is produced using the standard 4-step phase-shifting algorithm. We call this method single-shot, phase-shifting quantitative phase microscopy (SPQPM).

## 2. Methods and experimental setup

#### 2.1 Phase evaluation with a four-step, phase-shifting method

The evaluation of a phase in the phase-shifting interferometry is performed with a set of interference images. *N* different interference images are taken systemically with *N*-different phase shifts imposed on either a sample beam or a reference beam. The minimum number of images required for the phase extraction in the phase-shifting interferometry is *N* = 3 [18], but *N* = 4 is the most common value used for the phase measurement.

In SPQPM, of which experimental details are described in the next section, a 2-D grating with a grid period Λ along both the horizontal and the vertical directions is introduced in a Fourier plane. The angular separation by the diffraction in the Fourier plane is transferred to position shifts by a lens placed after the grating. Consequently, the grating duplicates and then spatially separates the sample beam on the camera plane. Multiple copies of the sample image, ${E}_{m,n}^{S}$, are then generated with different diffraction orders, where ${E}_{m,n}^{S}$ is the electric field for the image copies by diffraction orders, (*m*, *n*), along the horizontal and the vertical direction. Among all the copies, only four images (*m*, *n* = 0, + 1) fit into the camera sensor and interference images are then produced with a common clean reference beam, ${E}^{R}$, impinging parallel to the sample beam on the camera. The interferogram associated with the diffraction order (*m*, *n*) on the camera is formulated as

*k*is the propagation number,

*z*is the position of the camera along the optical axis, and ${\varphi}^{R}$ is the phase of the reference beam. Similarly, the electric field for the sample beam can be expressed as ${E}^{S}={E}_{m,n}^{S0}\mathrm{exp}\left(-\text{i}kz+\text{i}{\varphi}^{S}+\text{i}{\varphi}_{m,n}^{G}\right)$, where ${E}_{m,n}^{S0}$ is the amplitude, ${\varphi}^{S}$ is the phase of the sample, and ${\varphi}_{m,n}^{G}$ is the phase introduced by the grating. In general, $\left|{E}_{m,n}^{S0}\right|$ strongly depends on the diffraction orders (

*m*,

*n*), but it has almost the same value of $\left|{E}^{S0}\right|$ regardless of (

*m*,

*n*= 0, + 1) in our system (see section 2.2 for further details). Also, the phase is determined by the position of the grating as ${\varphi}_{m,n}^{G}=2\pi \left(mx+ny\right)\ufffd\Lambda $, where

*x*and

*y*are the positions of the grating along the vertical and horizontal directions, respectively. These are adjusted and fixed by $\left(x,y\right)=\left(\Lambda /4,\Lambda /2\right)$, thus the overall phases superimposed on the image copies are given by ${\varphi}_{m,n}^{G}=\pi /2\left(m+2n\right)$.

Each interference image has its own response corresponding to its overall phase. The intensity distribution on the camera is expressed as

Where $a={\left|{E}^{R0}\right|}^{2}+{\left|{E}^{S0}\right|}^{2}$, $b=2\left|{E}^{R0}\right|\left|{E}^{S0}\right|$, and we assume ${\varphi}^{R}=0$ without any loss of generality because it is common for all (*m*,

*n*). Now, the four interferograms can be described as:

*π*/2 to

*π*/2. Thus, an appropriate unwarping algorithm is applied to obtain a continuous phase distribution over the entire imaging area.

#### 2.2 Experimental setup

The schematic of our experimental setup is depicted in Fig. 1(a). The overall shape is similar to that of a typical Mach-Zehnder-type transmission phase microscope, employing an in-line holographic configuration. A He-Ne laser (Thorlabs: HNL150L) with a wavelength of 632.8 nm is used as a light source. The beam from the laser is split by a beam splitter (BS1) and sampled as a sample beam and a reference beam. After transmitting through an object, the sample beam is collected by an objective lens (OL1: Olympus, 40x, 0.5 NA) and a tube lens, and is then further delivered to a camera plane through a 4-*f* telescope with a magnification of 0.4x. This results in a total magnification from the sample plane to the image plane of 16x. A camera (Andor: Neo5.5, sCOMS) is placed at the image plane which has 2560 × 2160 pixels with a 6.5 μm x 6.5 μm pixel pitch. A 2-D grating, which is composed of two 1-D gratings in a configuration in which their grid directions are aligned orthogonally to each other, was placed in a Fourier plane of the 4-*f* telescope, consequently generating different copies of an object in 2-D at the image plane. To minimize the power imbalance among the different diffraction orders, which significantly reduces the interference contrast, transmission grating beamsplitters (Edmund Optics, 80 lines/mm) were used that have optical power equally distributed by 25% of the input power among the 0th and ± 1 orders. The magnification of the 4-*f* telescope is set in such a way that the camera can take four different images, *I _{m}*

_{,}

*with*

_{n}*m*= 0, + 1 and

*n*= 0, + 1 simultaneously. These four images are exactly the same, except for the overall phase superimposed by the grating. The phase for each diffraction order relative to that for the 0th-order is a function of the lateral position of the grating. We precisely adjusted the position of the grating using two computer-controlled translation stages (Thorlabs: PT3-Z8, 25mm). We systematically measured the interference characteristic curves between adjacent diffraction orders while scanning the position of the 2-D grating. From a harmonic fitting of the modulated interference, the appropriate positions of the grating were determined precisely. Thus the error caused by the uncertainty of the grating position was minimized. The grating is placed at (Λ/4, Λ/2) from the origin (the position where phase difference does not exist among the diffraction orders). This formation introduces the

*π*/2 and

*π*phase difference to the 1st diffraction orders along the horizontal and vertical directions, respectively. Consequently, the phase associated with each copy is given by ${\varphi}_{m,n}^{}=\pi /2\left(m+2n\right)$. Four of the image copies with different phases are combined with the reference beam at another beam splitter, BS2, at a zero angle and then produce 4-step phase shifted interference images. By taking the resulting interferogram, a complex field image of an object can be obtained using Eq. (7) without the need for multiple image acquisition.

## 3. Experimental results

#### 3.1 Quantitative phase image of polystyrene beads

In order to demonstrate the capability of SPQPM to measure a quantitative phase image of an object with a single shot, we first carried out a test experiment for imaging polystyrene beads; the beads have a diameter of 16 μm with a refractive index of 1.59, immersed in a refractive index matching oil (reflective index 1.56). Four copies of the interference images of the polystyrene beads were simultaneously acquired with SPQPM. The images were then clipped from the corresponding quadrants such that the object structure in each image coincided as shown in Fig. 2(a). Due to the in-line configuration, the images have a uniform level of interference in the backgrounds, which is associated with the superimposed phase difference by the grating; however, they have a certain interference variation involved with the object structure in the area where the beads are located. By using Eq. (7), a phase image for the bead cluster is processed and the wrapped phase map is presented in Fig. 2(b). The unwrapped phase distribution is also obtained using a standard unwrapping method based on the Goldstein algorithm and is converted into a thickness map for the bead cluster. The diameter and thickness concur with the values provided by the manufacturer, as shown in Fig. 2(c).

#### 3.2 Preservation of detection bandwidth

Usually, the off-axis configuration requires a large magnification due to the need for multiple camera pixels to acquire object information using the spatial modulation of interference. An interference fringe spacing is also carefully set to avoid the overlap between bandwidths of the spatial frequencies associated with the 0th and ± 1st orders in the Fourier space during the image processing, and simultaneously, to achieve the maximum imaging field of view [31]. These requirements set a strict imaging condition on the off-axis configuration, where a diffraction limited spot in the sample plane should be sufficiently magnified such that it occupies the minimum area of 7 × 7 camera pixels, which is the minimum integer number due to the discrete nature of the camera pixels while the actual number is ${\left(2+3\sqrt{2}\right)}^{2}$from the analytic derivation, in the image plane when tilting the reference beam along the diagonal direction at 45°. Otherwise, a significant overlap among the different spatial frequency components interrupts the object information and consequently ruins both the imaging quality and the resolution.

In contrast, in SPQPM, the magnification which maps a diffraction limited spot to 2x2 camera pixels is sufficient to preserve the system resolution limit. However, due to the use of four image copies, the actual number of camera pixels needed for imaging a diffraction limited spot should be 4x4. To compare these two schemes more systematically, we performed imaging of the same sample using the two methods. First, we operated the setup as a SPQPM configuration. The diffraction limit of the objective lens is about 1.6 μm on the sample plane, which is transferred to 13 μm on the camera plane by the overall magnification of 8x. Thus, the diffraction limited spot occupies about 2 × 2 pixels on the camera. The cartoon in Fig. 3(a) shows the use of the camera pixels in this imaging mode. This formation is certainly sufficient for SPQPM to obtain the correct information for an object without loss of system resolution. A United States Air Force (USAF) resolution target was used as a test object for this experiment. Figure 3(c) shows the acquired intensity image of the target. The finest features of element 3 in group 9, which have a center-to-center distance of about 1.6 μm, can be barely resolved because SPQPM can utilize the full bandwidth of the spatial frequency of the objective lens and thus maintain the resolution provided by the imaging system.

In order to verify the extent of the spatial frequency bandwidth supported by SPQPM, a diffuser was used as a sample. The speckle pattern generated by the diffuser fully covered the numerical aperture (NA) of the objective lens. After imaging the speckle pattern, the corresponding complex field image was obtained and Fourier transformed. Figure 3(e) shows the *k*-space of the complex speckle pattern, visualizing the range of the detection NA. The circular boundary of the spatial frequency fits into the entire *k*-space, which means that SPQPM preserves all the information captured by the objective lens.

Next, the microscope was converted into a traditional off-axis configuration by tiling the reference beam with respect to the sample beam along the diagonal direction. The interference fringe pattern was aligned at 45° on the camera and the fringe spacing was set so that the highest spatial frequency component captured by the objective lens was included. For a fair comparison with SPQPM, the overall magnification was doubled so that the same number of camera pixels participated in imaging the same area of the sample. The object diagram in Fig. 3(b) shows the use of camera pixels for this imaging configuration. The same part of the resolution target was imaged and the processed image is presented in Fig. 3(d). All the features in group 9 were then washed out due to the crosstalk between the 0-th and ± 1-st order spatial frequency distributions.

For a more comprehensive understanding of this behavior, we imaged the same diffuser using the off-axis configuration. The Fourier transform of the interferogram for the speckle pattern is shown in Fig. 3(f). Significant overlaps are observed among the components of different orders, the boundaries of which are indicated with yellow circles. Due to the insufficient magnification, the extents of the distributions are too large to be fully separated from each other in *k*-space. In this case, we simply applied a standard image processing algorithm for an off-axis configuration, assuming that all the distributions are entirely separate and the diameter of the 0-th order distribution is assumed to be two times larger than that for the ± 1-st order distributions. As a result, the algorithm estimated the extents for the distributions as presented by red circles in Fig. 3(f). Only the region within one of the smaller red circles in Fig. 3(f) is selected for the inverse Fourier transformation through image processing; consequently, the imaging resolution is significantly limited.

Considering that with the off-axis configuration, the minimum number of camera pixels for imaging one diffraction limited spot is 7x7, the additional magnification of 1.75x is needed to preserve the system resolution. This means that the amount of information acquired by SPQPM with the same camera sensor is 3 times more than that taken by the off-axis measurement. From the investigation of spatial frequency bandwidths, the advantage of SPQPM, more effective utilization of camera pixels with a single-shot measurement, is clearly verified.

#### 3.3 Measurement of living cells

One of the crucial requirements for a microscopy system capable of quantifying a phase associated with an object is its appropriateness for biological samples. To demonstrate the capability of SPQPM for imaging biological samples, we performed an experiment with living cells. Live cancer cells (FRO: squamous cancer cell-line derived from a human pharyngeal cancer) were prepared in a Roswell Park Memorial Institute (RPMI) 1640 medium supplemented with 10% heat-activated fetal bovine serum (FBS) and 1% penicillin, and sandwiched between a slide glass and a cover slip. The slide was kept in an incubator for 30 minutes for cell stabilization. After the incubation, the cells were imaged with SPQPM and the taken interferogram was processed using Eq. (6). The results are presented in Fig. 4(a) and 4(b) for different cells. The morphological profiles are clearly visualized by phase measurements similar to conventional quantitative phase microscopy.

## 4. Conclusion

We demonstrated a wide-field and single-shot quantitative phase microscope based on four-step, phase-shifting interferometry using the spatial separation of an object image with different phase shifts. A 2-D diffraction grating located in the Fourier plane of the detection port of the microscope generated multiple copies of the object image and separated the images laterally in the image plane. In addition, the overall phase shift of each image copy was precisely set by the lateral positions of the grating. By superimposing a plane reference beam introduced parallel to the sample beam, four interferograms modulated with a phase step of π/2 can be acquired within a single field of view. Through the standard imaging process algorithm for phase-shifting interferometry, a quantitative phase image of an object can be retrieved.

Since the only optical component that is required for the construction of SPQPM is a simple 2-D grating positioned in a Fourier plane of a standard-type transmission phase microscope, the complexity for the implementation was minimized. Furthermore, SPQPM takes advantage of phase-shifting interferometry, carrying three times more information than the usual off-axis detection scheme. However, unlike the conventional phase-shifting method, SPQPM is free from the multiple recording of object images, and thus is able to acquire an object image with a single-shot measurement with a speed equivalent to the camera frame rate. Therefore, SPQPM will facilitate the precision quantification for the fast dynamics of transparent specimens, utilizing the imaging sensor more effectively.

## Acknowledgments

This research was supported by a grant of the Korea Health Technology R&D Project through the Korea Health Industry Development Institute (KHIDI), funded by the Ministry of Health & Welfare, Republic of Korea (grant number: HI14C3477 & HI13C1501). It was also supported by a grant from Korea University (K1505441).

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