Abstract

We introduce a new shearing interferometry module for digital holographic microscopy, in which the off-axis angle, which defines the interference fringe frequency, is not coupled to the shearing distance, as is the case in most shearing interferometers. Thus, it enables the selection of shearing distance based on the spatial density of the sample, without losing spatial frequency content due to overlapping of the complex wave fronts in the spatial frequency domain. Our module is based on a 4f imaging unit and a diffraction grating, in which the hologram is generated from two mutually coherent, partially overlapping sample beams, with adjustable shearing distance, as defined by the position of the grating, but with a constant off-axis angle, as defined by the grating period. The module is simple, easy to align, and presents a nearly common-path geometry. By placing this module as an add-on unit at the exit port of an inverted microscope, quantitative phase imaging can easily be performed. The system is characterized by a 2.5 nm temporal stability and a 3.4 nm spatial stability, without using anti-vibration techniques. We provide quantitative phase imaging experiments of silica beads with different shearing distances, red blood cell fluctuations, and cancer cells flowing in a micro-channel, which demonstrate the capability and versatility of our approach in different imaging scenarios.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Digital holographic microscopy (DHM) is a non-contact, label-free and wide-field imaging technique that can provide quantitative phase imaging of microscopic specimens [116]. In off-axis DHM, the quantitative phase map of the imaged samples can be extracted from a single hologram, acquired in a single camera shot, so that the full frame rate of the camera can be utilized. To achieve this, a reference beam, containing no sample spatial modulation, interferes with the sample beam at a small off-axis angle, creating a spatial separation in the spatial frequency domain, which allows the retrieval of the complex wave front encoded into the hologram from a single camera shot. Due to its fast imaging capability, off-axis DHM has been widely used in measuring dynamic processes, such as rapidly flowing cells [15]. On the other hand, the measurement sensitivity of the acquired phase image is strongly influenced by the instrument design and the surrounding environment [1718]. Traditionally, off-axis DHM is implemented using Mach-Zehnder or Michelson interferometers. In such interferometers, the sample and the reference beams travel separate paths, with the reference beam traveling outside the imaging system, and thus the beams are exposed to different environmental disturbances, such as air fluctuations. This may lead to a lower temporal stability of the setup, inhibiting the ability to measure certain dynamic processes. In common-path interferometers, on the other hand, the reference beam is typically created after the imaging system, so that both interfering beams travel almost the same optical path, and thus the setups are less affected by time-dependent environmental disturbances. Furthermore, as these conventional interferometers are built around the sample, they are bulky, expensive, space-consuming and harder to align, preventing them from being useful for most medical clinics or for low-resource industrial use.

Generally, depending on how the reference beam is generated, off-axis common-path interferometry can be classified into two categories: when the reference beam is created by erasing the sample information, e.g., by spatial filtering [1926], or when the reference beam is created by designating part of the field of view (FOV) to be empty [2736]. In the first category, the reference beam can be generated from a copy of the sample beam, by erasing the sample modulation using pinhole filtering [1926]. In the τ interferometer, for example, the sample beam is Fourier transformed using a lens and split into two copies by a beam splitter. The reference beam is generated by erasing the sample information from one copy of the beam by a pinhole located in front of a mirror [1920]. In diffraction phase microscopy, on the other hand, a diffraction grating rather than a beam splitter is used to generate multiple copies of the sample beam, and one of the copies is spatially filtered with a pinhole to serve as a reference beam [2123]. These schemes allow imaging spatially dense thin samples over the entire FOV. However, pinhole filtering has two limitations: first, it requires precise alignment before an experiment which is usually hard to implement; second, the fringe visibility of holograms is lower due to the fact that a large portion of light intensity is blocked by the pinhole, creating imbalance in the sample and the reference beam intensities. The low visibility of the fringes usually leads to reduced phase sensitivity [17]. In the second class of these interferometers, a portion of the sample beam passes through an empty area in the sample and serves as a reference beam [2736]. This self-referencing technique has been implemented in many different ways. For example, a Lloyd’s mirror configuration can be used to fold part of the sample beam on itself [27]. Although this scheme is simple, the off-axis angle is coupled to the shearing distance. Another way is to create two laterally sheared beams using a trapezoid Sagnac interferometer [28]. A glass plate or a Wollaston prism has also been used to generate two sheared beams in shearing interferometers [2933]. Most of these schemes, however, cannot image dense samples due to overlaps of sample details in the two interfering beams. In addition, the shearing distance in the shearing interferometers is fixed [2935] or can only be adjusted in a very limited range [28,36]. In other words, the shearing distance cannot be adjusted to the sample content and its details in order to avoid unwanted overlaps between the two interfering beams. Flipping interferometry, on the other hand, has a large shearing distance range and can image dense samples but at the cost of half of the optical field of view that must remain empty [3435].

In this paper, we propose the Shearing Interferometry with Constant off-axis Angle (SICA) module, in which the shearing distance and the off-axis angle are completely uncoupled. Furthermore, this approach does not require a half empty optical FOV as does the flipping interferometry module, and the shearing distance can be controlled and set based on the spatial distribution of the empty spaces in the entire sample image. The module is based on a grating and a 4f imaging unit, with a simple and easy alignment. By adding the module to the exit port of a conventional coherently illuminated microscope, the microscope is converted into a digital holographic imaging setup without any internal structural modification, and the microscope camera be positioned at the exit of the module. In the module, two laterally sheared sample beams are generated by the grating, and the part of the sample area containing no sample details serves as the reference beam, as in any shearing interferometer. However, in our module, the shearing distance can be flexibly adjusted by controlling the axial location of the grating, while the off-axis interference angle remains constant. Due to the common-path configuration and the higher efficiency in generating the reference beam (no spatial filtering is used), the setup achieves greater phase sensitivity. To demonstrate the capabilities of our scheme, quantitative phase imaging results of silica beads, red blood cell and SW620 cancer cells flowing in a micro-channel are provided.

2. Experimental setup and characterization

2.1 Optical setup

Figure 1 presents the proposed experimental scheme, in which the proposed SICA module is placed at the exit port of an inverted microscope (Olympus, IX83). The imaging system is illuminated with a HeNe laser (λ = 632.8 nm). The beam passes through the sample, is magnified by microscope objective MO (40×, NA 0.75), and then projected through tube lens TL onto the image plane IP, where the SICA module is positioned. In this module, a 4f imaging system projects the intermediate sample image at IP onto the camera (Thorlabs, DCC1545M). This 4f imaging system consists of lenses L1 and L2 with a focal-length ratio that can provide a secondary magnification (in our case, f2/f1 = 2). A diffraction grating G (100 lines/mm) is placed at a distance z after IP and generates multiple replicas of the sample image. At the Fourier plane of lens L1, mask B is placed, such that only the zero and the first diffraction orders can pass, while other orders are blocked. The two orders that pass the mask are Fourier transformed again by lens L2, hence two laterally sheared sample images are obtained at its back focal plane, where the camera is placed. The shearing distance between the two beams is l. These two beams interfere with an angle α between them to form an off-axis image hologram on the camera.

 figure: Fig. 1.

Fig. 1. An inverted microscope with the proposed SICA module (marked by dashed rectangle), connected to its output. M1, M2, mirrors; S, sample; MO, microscope objective; TL, tube lens; IP, image plane; G, grating; L1, L2, lenses with focal lengths f1=150 mm and f2=300 mm. z, distance of G from IP; B, mask that selects only two diffraction orders; l, shearing distance; α, interference angle.

Download Full Size | PPT Slide | PDF

For better understanding the operating principle of the module, two points need to be clarified. First, the interference angle α is a constant, and independent of the position of grating G. Angle α is unaffected by the position of grating G, since the locations of the diffraction orders at the spectrum appearing in the Fourier plane of L1 remain unchanged. This angle is controlled solely by grating period and the given focal lengths of lenses L1 and L2, according to the following formula:

$$\alpha = {f_1}\lambda /{f_2}d\,\,\,[{rad} ],$$
where d is the grating period. With this angle, the fringe period of the hologram can be calculated as f2d/ f1. This angle can be optimized for off-axis holography by the selection of the grating period according to the numerical aperture (NA) of the optical imaging setup, such that there will not be overlaps between the cross-correlation orders encoding the complex wave front of the sample and the auto-correlation orders in the spatial frequency domain [23]. In our experiments, a grating period of d = 10 µm is chosen, which is smaller than the diffraction limited spot at IP.

Second, as the two sample images after lens L2 coincide at the plane conjugated with grating G, the shearing distance l is the intersection distance at the sensor plane between the two sample beams. To explain this, a ray parallel to the first diffraction beam is plotted, as shown in Fig. 1 by the dashed red line. It originates from point A1 at the IP and intersects the grating plane at point A2. It passes the first diffraction order at the spatial frequency plane, and arrives at the sensor plane at a point conjugated with point A1. As the interception distance (the distance between A2 and A3) at the grating plane is /d, according to geometric relations the shearing distance between the two sample beams is:

$$l = {f_2}\lambda z/{f_1}d. $$
As the shearing distance is dependent on the distance z, it can be adjusted by moving the grating G along the optical axis to match different imaging scenarios based on the sparsity of the sample imaged. When grating G is placed at the image plane IP, i.e., z = 0, the shearing distance is l = 0, which means that the two sample images coincide at the sensor plane without shearing. When z = f1, so that the grating is attached to lens L1, the shearing distance has the maximum value of l = f2λ /d.

The complex wave distribution of the two shifted beams at the sensor plane can also be deduced through the diffraction integral [37], and the intensity of the hologram is (see Appendix for details):

$$I(x, y) = |{K_0}A[(x, y){f_1}/{f_2}] + {K_1}\exp ( - j2\pi {f_1}x/{f_2}d)A[(x - {f_2}\lambda z/{f_1}d, y){f_1}/{f_2}]{|^2}, $$
where (x, y) is the coordinate in the sensor plane, K0 and K1 are diffraction coefficients of the zero and first orders, respectively, and d is the period of the grating. The first term in the bracket of Eq. (3) represents the magnified sample beam traveling along the optical axis, given that A(xy) denotes the sample complex wave at intermediate image plane IP. The second term represents the sample beam that emanates from the first diffraction order, in which the exponential term denotes that its direction of travel is inclined with an off-axis angle α with respect to the optical axis. From Eq. (3), we can also see that the central point of the second term is (f2λz/f1d, 0), which coincides with the result of Eq. (2).

2.2 Demonstration of working principle

To experimentally demonstrate the working principle of the proposed module, a 1951 USAF resolution target was imaged. Figures 2(a) and 2(b) show two off-axis holograms recorded with different shearing distances, and the coinciding insets show magnified regions of the off-axis holograms, demonstrating that although the shearing distances are different, the interference fringe frequency, determined by the off-axis angle, is constant. Indeed, in Figs. 2(a) and 2(b) the off-axis holograms have the same fringe frequency. Figures 2(c) and 2(d) show the corresponding spatial frequency spectra of the holograms appearing in Figs. 2(a) and 2(b), respectively. In these spectra, the diffraction peaks are located at the same positions, which further verifies that both holograms have the same fringe period. This is an obvious advantage as this approach does not require changing the numerical reference wave in the reconstruction procedure each time the shearing distance is changed [38].

 figure: Fig. 2.

Fig. 2. Experimental results of varying shearing distances but with fixed interference angle, when imaging 1951 USAF resolution target. (a,b) Off-axis image holograms with two different shearing distances. White lines on the top right represent 10 µm on the sample. (c,d) Spatial frequency spectra corresponding to (a) and (b), respectively, where the location of the cross-correlation terms remains unchanged.

Download Full Size | PPT Slide | PDF

2.3 Characterization of the system sensitivity

Next, we characterized the phase sensitivity of the setup, which is the minimum distinguishable change in optical path difference (OPD) over one image (spatial) and between images (temporal). It is usually estimated by calculating the spatial and temporal standard deviations of the phase maps when imaging without a sample present [17]. As in the proposed scheme, conventional diffraction phase microscopy (DPM) also adopts a grating, which is placed at image plane IP and followed by a 4f imaging system, to achieve a nearly common-path off-axis interferometry. However, our scheme does not include a pinhole for filtering the zero order, whereas in DPM a pinhole is used in the spatial frequency plane to generate a reference beam. The pinhole filtering leads to a lower efficiency in generating a reference beam, thus, the visibility of holograms in our scheme is enhanced and, in turn, yields an improvement in phase sensitivity.

To evaluate the noise level of our system, we continuously recorded 110 holograms over 10 seconds at a frame rate of 11 fps without the sample present. For comparison, another stack of 110 holograms were also recorded with a DPM under the same conditions. In the DPM setup, in order to generate an ideal reference beam, a 6 mm square mask was placed at the image plane, and a 15 µm pinhole filter was adopted in the spatial frequency plane to filter out the high spatial frequencies in the zeroth diffraction order from the grating, according to analysis in Ref. [18]. The two stacks of holograms were processed to the corresponding OPD maps by subtracting the phase map obtained from the first pre-recorded hologram without the sample present. First, the spatial standard deviation of each OPD map is calculated, resulting in a standard deviation of 3.4 nm in our SICA setup, and 4.4 nm in DPM. Second, the temporal standard deviations of 10000 randomly selected pixels across each stack of images were calculated and the associated distributions are shown in Fig. 3. The average distribution is 2.5 nm in our SICA setup, and it is 3.5 nm in the DPM. These results show that our scheme has better phase sensitivity than DPM, under the same coherent illumination and environmental conditions. As both schemes have the same optical paths, we attribute these results to the fact that the off-axis holograms in our SICA scheme have better fringe contrast, which leads to higher phase sensitivity [17].

 figure: Fig. 3.

Fig. 3. Temporal OPD stability in our SICA setup (orange) and the DPM setup (blue).

Download Full Size | PPT Slide | PDF

3. Experimental results

3.1 Quantitative phase imaging of microbeads with a variable shearing distance

To demonstrate the feasibility of the proposed setup in quantitative phase imaging with adjustable shearing distances, we first imaged 10 µm silica beads with a 60× microscope objective (Plan, Olympus, NA 1.4). The beads (n = 1.46) immersed in an index-matching oil (n = 1.52) were sandwiched between two cover glasses, and were distributed across the FOV with random empty spaces between them. Figure 4 shows two holograms [Figs. 4(a) and (c), respectively] with different vertical shearing distances and the corresponding phase images [Figs. 4(b) and (d)]. In Fig. 4(b), as a smaller shearing is performed, the phase image of the bead in the white rectangle is partially overlapped by the ghost phase of another bead, hence we cannot get the correct phase map. In contrast, Fig. 4(d) shows a quantitative phase image with a larger shearing distance, in which the bead in the rectangle can be measured. A dynamic video of phase maps with shearing distance increasing over time, as well as corresponding holograms with constant fringe period, is provided in Visualization 1. It should be noted that using other schemes that have a fixed shearing distance, as in [2833], the imaging cannot be performed as conveniently as with our scheme. Clearly, the advantage gained by the adjustable shearing distance is that the scheme can be easily transferred to different imaging scenarios with different sample sizes or different empty areas.

 figure: Fig. 4.

Fig. 4. Quantitative phase images of silica beads with (a, b) a small vertical shearing distance and (c, d) a large vertical shearing distance, as obtained under highly coherent illumination. (a, c) Off-axis holograms, with magnified inset showing fringe period and contrast. (b, d) Quantitative phase profiles. White lines represent 10 µm. See gradual changing of the shearing distance over time in Visualization 1.

Download Full Size | PPT Slide | PDF

3.2 Quantitative phase imaging of a red blood cell

Next, we demonstrate the capability of the SICA technique in bio-imaging by imaging human red blood cells (RBC). The sample, provided by Magen David Adom’s Israeli blood bank, was stored for one week before usage. Cells were diluted 1000 times in phosphate buffered saline (PBS) solutions supplemented with 1 mM ethylenediaminetetraacetic acid (EDTA). 10 µl of diluted RBCs were placed between two cover slips and imaged. Figure 5(a) shows the quantitative phase image of an RBC. The temporal fluctuation map of the cell was also measured by recording 100 consecutive holograms in 10 seconds. Figure 5(b) shows the temporal standard deviation of the phase images, which demonstrates the fluctuations of the RBC during this period. From the phase map and its dynamics, one can evaluate the cell functionality and characterize various pathological conditions [3940].

 figure: Fig. 5.

Fig. 5. (a) Quantitative phase image of a single RBC and (b) its fluctuation map, as obtained under highly coherent illumination. White lines represent 5 µm.

Download Full Size | PPT Slide | PDF

3.3 Quantitative phase imaging of cancer cells in flow

Next, we conducted experiments for imaging cells in a microfluidic channel, which demonstrated the value of the ability to fully control the shearing distance to avoid unwanted overlaps between the two sheared images, without undesired effects on the off-axis angle of the hologram. As it requires no labeling of cells, quantitative imaging of cells in flow is useful in biomedical diagnosis for cell sorting and other biological assays. We imaged human colorectal adenocarcinoma cancer cells (SW620) acquired from the American Type Culture Collection (ATCC), grown in a flask. The cells were cultivated in Dulbecco’s modified eagle medium (DMEM) supplemented with 10% heat-inactivated fetal calf serum, antibiotics, and 0.1 mM glutamine (all from Biological Industries, Beit Haemek, Israel), and grown in 5% CO2 at 37° C in a humidified incubator). Cells were grown until ∼80% confluence was achieved, and then harvested using trypsin, washed by PBS supplemented with 1 mM EDTA, and centrifuged at 500 g. The supernatant was then discarded, and the pellet was resuspended in Phosphate-buffered saline (PBS) plus EDTA. In the experiment, a microfluidic channel was used (50 µm width, 50 µm height and 58.5 mm length, ChipShop). A 1 ml syringe containing the sample was connected to the channel and to a peristaltic pump. For this experiment, a 40× microscope objective with NA 0.75 was selected to match sample size to camera FOV. Figure 6 shows the resulting OPD map and the dynamic cell flow is shown in Visualization 2. As the total magnification in this case is 80×, the magnified width of the channel is 4 mm, which matches the height of the camera sensor. Hence, the shearing size in this case needs to be at least 4 mm, which can easily be achieved using SICA. Note that except for flipping interferometry [3435] and the current SICA system, the other previously mentioned shearing schemes [2728,3233,36], in which a replica sample beam with a shearing distance greater than the camera FOV is required, are useless in this case, since this would require too large an off-axis angle which would induce an off-axis fringe frequency that cannot be recorded by the camera due to its pixel size.

 figure: Fig. 6.

Fig. 6. Quantitative phase image of cancer cells (SW620) flowing in a micro-channel, as obtained under highly coherent illumination. White line represents 10 µm. See flow dynamics in Visualization 2.

Download Full Size | PPT Slide | PDF

4. Conclusions

To conclude, we have introduced a new off-axis shearing interferometric imaging module with fully adjustable shearing distance, and demonstrated its application for quantitative phase microscopy. This external SICA module is based on using a diffraction grating, an aperture, and two lenses aligned in a 4f imaging common-path configuration, and is very easy and flexible for alignment. Although the module still requires vacant areas in the sample image, it demonstrates higher phase sensitivity than the pinhole filtering technique under the same coherent illumination and environmental conditions. The main advantage of the module is that in contrast to previous shearing interferometers, the shearing distance between two beams at the sensor plane can be easily adjusted in a large range while the off-axis angle remains constant. As illustrated by the experiments, small shearing distances can be selected for imaging sparsely distributed cells occupying the whole optical FOV, and large shearing distances can be selected for imaging densely distributed cells occupying even the entire sensor FOV, as long as within the optical beam there is a vacant area beside the cell that is of the same size. Hence, compared to other shearing techniques, the proposed module possesses a greater generality and versatility for its ability to enable various shearing distances in diverse imaging scenarios. We therefore believe that when the external SICA module is used in conjunction with a coherently illuminated microscope, it has great potential for real time quantitative phase imaging in biomedicine and other fields.

Appendix

In the following, we deduce the image formation in the sensor plane of the SICA module through Fresnel diffraction [37], since it holds in our case even if the grating distance from the image plane is small [41]. For simplicity, we only consider one transverse axis and assume that the coordinates of the five planes in the module are x1, x2, x3, x4 and x5, as shown in Fig. 7. During the process, some constant exponential coefficients are omitted as they do not affect the relative distribution of the complex wave. Assuming that the complex amplitude of the sample image in the IP is denoted as A(x1), after a Fresnel diffraction of distance z the complex wave before grating G is expressed as follows:

$$u({x_2}) = \frac{\textrm{1}}{{j\lambda z}}\exp (j\frac{k}{{2z}}{x_2}^2)\int\limits_{ - \infty }^{ + \infty } {A({x_1})\exp (j\frac{k}{{2z}}{x_1}^2)\exp ( - j\frac{k}{z}{x_1}{x_2})d{x_1}}, $$
where λ is wavelength, and k = 2π/λ is the wave number.

 figure: Fig. 7.

Fig. 7. Schematic of the SICA module. z, z1, distance from planes x1 to x2 and x2 to x3, respectively; α, interference angle.

Download Full Size | PPT Slide | PDF

The diffraction grating G can be expressed in the form of a Fourier series as follows:

$$\begin{array}{l} t({x_2}) = \sum\limits_{n ={-} \infty }^{ + \infty } {{K_n}\exp (jn2\pi {x_2}/d)} \\ {K_n} = \int\limits_0^d {t({x_2})} \exp ( - jn2\pi {x_2}/d)d{x_2} \end{array}. $$
Here, n denotes an integer, d is the grating period, and Kn are the Fourier coefficients. As only the zero (n = 0) and the positive first orders (n = 1) pass through the mask B that is placed in the spatial frequency plane, the transmittance of the grating can be considered to contain only two diffraction orders, as follows:
$$t({x_2}) = {K_0} + {K_1}\exp (j2\pi {x_2}/d). $$
Hence, the complex wave behind grating G is denoted as u(x2)t(x2), and the complex wave before lens L1 is:
$$u({x_3}) = \frac{\textrm{1}}{{j\lambda {z_1}}}\exp (j\frac{k}{{2{z_1}}}{x_3}^2)\int\limits_{ - \infty }^{ + \infty } {u({x_2})} t({x_2})\exp (j\frac{k}{{2{z_1}}}{x_2}^2)\exp ( - j\frac{k}{{{z_1}}}{x_2}{x_3})d{x_2}. $$
Substituting Eqs. (4) and (6) into Eq. (7), we get:
$$\begin{array}{l} u({x_3}) = \frac{\textrm{1}}{{j\lambda {z_1}}} \cdot \frac{1}{{j\lambda z}}\exp (j\frac{k}{{2{z_1}}}{x_3}^2)\\ \{ {K_1}\int\limits_{ - \infty }^{ + \infty } {A({x_1})} \exp (j\frac{k}{{2z}}{x_1}^2)\int\limits_{ - \infty }^{ + \infty } {\exp [j\frac{k}{2}{x_2}^2(\frac{1}{{{z_1}}} + \frac{1}{z})]\exp [ - j2\pi {x_2}(\frac{{{x_1}}}{{\lambda z}} + \frac{{{x_3}}}{{\lambda {z_1}}} - \frac{1}{d})]d{x_2}} d{x_1}\\ + {K_\textrm{0}}\int\limits_{ - \infty }^{ + \infty } {A({x_1})} \exp (j\frac{k}{{2z}}{x_\textrm{1}}^\textrm{2})\int\limits_{ - \infty }^{ + \infty } {\exp [j\frac{k}{2}{x_2}^2(\frac{1}{{{z_1}}} + \frac{1}{z})]\exp [ - j2\pi {x_2}(\frac{{{x_1}}}{{\lambda z}} + \frac{{{x_3}}}{{\lambda {z_1}}})]d{x_2}} d{x_1}\} \end{array}. $$
To calculate the integral over x2, we notice that the Fourier transformation of a Gauss function is still a Gauss function, and thus by analogy having the following relation [37]:
$$\int\limits_{ - \infty }^{ + \infty } {\exp (j\frac{k}{{2m}}{x^2})\exp ( - j2\pi \xi x)} dx = \sqrt {\lambda m} \exp (j\frac{\pi }{4})\exp ( - j\pi \lambda m{\xi ^2}), $$
where m is an auxiliary variable and k is the wave number. Hence, by means of variables substitution 1/m = 1/z1+1/z, ξ1 = (x1/λz + x3/λz1−1/d) and ξ2 = (x1/λz + x3/λz1) and taking the relation z + z1 = f1, Eq. (8) can be simplified as:
$$\begin{array}{l} u({x_3}) = \frac{\textrm{1}}{{i\lambda {f_1}}}\exp (j\frac{k}{{2{f_1}}}{x_3}^2)\\ \{ {K_1}\exp (j2\pi \frac{z}{{d{f_1}}}{x_3})\int\limits_{ - \infty }^{ + \infty } {A({x_1})} \exp (j\frac{k}{{2{f_1}}}{x_1}^2)\exp ( - j2\pi \frac{{{x_3}}}{{\lambda {f_1}}}{x_1})\exp (j2\pi \frac{{{z_1}}}{{d{f_1}}}{x_1})d{x_1}\\ + {K_0}\int\limits_{ - \infty }^{ + \infty } {A({x_1})} \exp (j\frac{k}{{2{f_1}}}{x_1}^2)\exp ( - j2\pi \frac{{{x_3}}}{{\lambda {f_1}}}{x_1})d{x_1}\} \end{array}. $$
In Eq. (10), a constant exponential coefficient is omitted. As the phase transformation of lens L1 is:
$${t_l}({x_3}) = \exp ( - j\frac{k}{{2{f_1}}}{x_3}^2). $$
Hence, the complex wave in the x4 plane can be expressed as:
$$u({x_4}) = \frac{1}{{j\lambda {f_1}}}\exp (j\frac{k}{{2{f_1}}}{x_4}^2)\int\limits_{ - \infty }^{ + \infty } {u({x_3})} {t_l}({x_3})\exp (j\frac{k}{{2{f_1}}}{x_3}^2)\exp ( - j\frac{k}{{{f_1}}}{x_4}{x_3})d{x_3}. $$
Substituting Eqs. (10) and (11) into Eq. (12), and through the same method used in handling Eq. (8), we can get:
$$u({x_4}) = \frac{1}{{j\lambda {f_1}}}[{K_0}\tilde{A}(\frac{{{x_4}}}{{\lambda {f_1}}})\textrm{ + }{K_1}\exp (j2\pi \frac{z}{{{f_1}d}}{x_4})\tilde{A}(\frac{{{x_4}}}{{\lambda {f_1}}} - \frac{1}{d})], $$
where $\tilde{A}$ denotes the Fourier transformation of A(x1). To get from spectrum plane x4 to the back focal plane x5 of lens L2, a Fourier transformation is performed, and thus the complex wave is:
$$u({x_5}) = {K_0}A({x_5}{f_1}/{f_2}) + {K_1}\exp ( - j2\pi {f_1}{x_5}/{f_2}d)A[({x_5} - {f_2}\lambda z/{f_1}d){f_1}/{f_2}]. $$
The coefficient f2/f1 denotes the additional magnification of the image in the IP plane. The first term in the bracket represents the sample image located at the origin, and the second term represents the shifted sample image located at f2λz/f1d with an inclined propagation angle α = f1λ/f2d. The intensity of the hologram is thus the square of the module of u(x5):
$$I({x_5}) = {|{u({x_5})} |^2}. $$

Funding

H2020 European Research Council (678316).

Disclosures

The authors declare no conflicts of interest.

References

1. E. Cuche, F. Bevilacqua, and C. Depeursinge, “Digital holography for quantitative phase-contrast imaging,” Opt. Lett. 24(5), 291–293 (1999). [CrossRef]  

2. P. Ferraro, C. D. Core, L. Miccio, S. Grilli, S. D. Nicola, A. Finizio, and G. Coppola, “Phase map retrieval in digital holography: avoiding the under sampling effect by a lateral shear approach,” Opt. Lett. 32(15), 2233–2235 (2007). [CrossRef]  

3. V. Biaco, B. Mandracchia, V. Marchesano, V. Pagliarulo, F. Olivieri, S. Coppola, M. Paturzo, and P. Ferraro, “Endowing a plain fluidic chip with micro-optics: a holographic microscope slide,” Light Sci. Appl. 6(9), e17055 (2017). [CrossRef]  

4. J. Min, B. Yao, S. Ketelhut, C. Engwer, B. Greve, and B. Kemper, “Simple and fast spectral domain algorithm for quantitative phase imaging of living cells with digital holographic microscopy,” Opt. Lett. 42(2), 227–230 (2017). [CrossRef]  

5. F. Pan, S. Liu, Z. Wang, P. Shang, and W. Xiao, “Digital holographic microscopy long-term and real-time monitoring of cell division and changes under simulated zero gravity,” Opt. Express 20(10), 11496–11505 (2012). [CrossRef]  

6. J. Cho, J. Lim, S. Jeon, G. Choi, H. Moon, N. Park, and Y. Park, “Dual-wavelength off-axis digital holography using a single light-emitting diode,” Opt. Express 26(2), 2123–2131 (2018). [CrossRef]  

7. L. Puyo, J. Huignard, and M. Atlan, “Off-axis digital holography with multiplexed volume Bragg gratings,” Appl. Opt. 57(12), 3281–3287 (2018). [CrossRef]  

8. N. T. Shaked, T. M. Newpher, M. D. Ehlers, and A. Wax, “Parallel on-axis holographic phase microscopy of biological cells and unicellular microorganism dynamics,” Appl. Opt. 49(15), 2872–2878 (2010). [CrossRef]  

9. V. Micó, J. Zheng, J. Garcia, Z. Zalevsky, and P. Gao, “Resolution enhancement in quantitative phase microscopy,” Adv. Opt. Photon. 11(1), 135–214 (2019). [CrossRef]  

10. P. Gao, B. Yao, J. Min, R. Guo, J. Zheng, T. Ye, I. Harder, V. Nercissian, and K. Mantel, “Parallel two-step phase-shifting point-diffraction interferometry for microscopy based on a pair of cube beam splitters,” Opt. Express 19(3), 1930–1935 (2011). [CrossRef]  

11. J. Zheng, P. Gao, X. Shao, and G. Ulrich Nienhaus, “Refractive index measurement of suspended cells using opposed-view digital holographic microscopy,” Appl. Opt. 56(32), 9000–9005 (2017). [CrossRef]  

12. R. Guo and F. Wang, “Compact and stable real-time dual-wavelength digital holographic microscopy with a long-working distance objective,” Opt. Express 25(20), 24512–24520 (2017). [CrossRef]  

13. R. Guo, W. Zhang, R. Liu, C. Duan, and F. Wang, “Phase unwrapping in dual-wavelength digital holographic microscopy with total variation regularization,” Opt. Lett. 43(14), 3449–3452 (2018). [CrossRef]  

14. J. A. Picazo-Bueno, M. Trusiak, J. García, K. Patorski, and V. Micó, “Hilbert-Huang single-shot spatially multiplexed interferometric microscopy,” Opt. Lett. 43(5), 1007–1010 (2018). [CrossRef]  

15. J. A. Picazo-Bueno, Z. Zalevsky, J. García, and V. Micó, “Superresolved spatially multiplexed interferometric microscopy,” Opt. Lett. 42(5), 927–930 (2017). [CrossRef]  

16. M. Trusiak, J. A. Picazo-Bueno, K. Patorski, P. Zdankowski, and V. Micó, “Single-shot two-frame π-shifted spatially multiplexed interference phase microscopy,” J. Biomed. Opt. 24(09), 1 (2019). [CrossRef]  

17. P. Hosseini, R. Zhou, Y. H. Kim, C. Peres, A. Diaspro, C. Kuang, Z. Yaqoob, and P. T. C. So, “Pushing phase and amplitude sensitivity limits in interferometric microscopy,” Opt. Lett. 41(7), 1656–1659 (2016). [CrossRef]  

18. N. A. Turko and N. T. Shaked, “Simultaneous two-wavelength phase unwrapping using an external module for multiplexing off-axis holography,” Opt. Lett. 42, 73 (2017). [CrossRef]  

19. P. Girshovitz and N. T. Shaked, “Compact and portable low-coherence interferometer with off-axis geometry for quantitative phase microscopy and nanoscopy,” Opt. Express 21(5), 5701–5714 (2013). [CrossRef]  

20. N. T. Shaked, “Quantitative phase microscopy of biological samples using a portable interferometer,” Opt. Lett. 37(11), 2016–2018 (2012). [CrossRef]  

21. G. Popescu, T. Ikeda, R. R. Dasari, and M. S. Feld, “Diffraction phase microscopy for quantifying cell structure and dynamics,” Opt. Lett. 31(6), 775–777 (2006). [CrossRef]  

22. I. Frenklach, P. Girshovitz, and N. T. Shaked, “Off-axis interferometric phase microscopy with tripled imaging area,” Opt. Lett. 39, 1525 (2014). [CrossRef]  

23. B. Bhaduri, H. Pham, M. Mir, and G. Popescu, “Diffraction phase microscopy with white light,” Opt. Lett. 37(6), 1094–1096 (2012). [CrossRef]  

24. M. Shan, L. Liu, Z. Zhong, B. Liu, G. Luan, and Y. Zhang, “Single-shot dual-wavelength off-axis quasi-common-path digital holography using polarization-multiplexing,” Opt. Express 25(21), 26253–26261 (2017). [CrossRef]  

25. H. Bai, Z. Zhong, M. Shan, L. Liu, L. Guo, and Y. Zhang, “Interferometric phase microscopy using slightly-off axis reflective point diffraction interferometer,” Opt. Las. Eng. 90, 155–160 (2017). [CrossRef]  

26. S. Mahajan, V. Trivedi, P. Vora, V. Chhaniwal, B. Javidi, and A. Anand, “Highly stable digital holographic microscope using Sagnac interferometer,” Opt. Lett. 40(16), 3743–3746 (2015). [CrossRef]  

27. V. Chhaniwal, A. S. Singh, R. A. Leitgeb, B. Javidi, and A. Anand, “Quantitative phase-contrast imaging with compact digital holographic microscope employing Lloyd’s mirror,” Opt. Lett. 37(24), 5127–5129 (2012). [CrossRef]  

28. C. Ma, Y. Li, J. Zhang, P. Li, T. Xi, J. Di, and J. Zhao, “Lateral shearing common-path digital holographic microscopy based on a slightly trapezoid Sagnac interferometer,” Opt. Express 25(12), 13659–13667 (2017). [CrossRef]  

29. J. Di, Y. Li, M. Xie, J. Zhang, C. Ma, T. Xi, E. Li, and J. Zhao, “Dual-wavelength common-path digital holographic microscopy for quantitative phase imaging based on lateral shearing interferometry,” Appl. Opt. 55(26), 7287–7293 (2016). [CrossRef]  

30. A. S. Singh, A. Anand, R. A. Leitgeb, and B. Javidi, “Lateral shearing digital holographic imaging of small biological specimens,” Opt. Express 20(21), 23617–23622 (2012). [CrossRef]  

31. B. Javidi, A. Markman, S. Rawat, T. O’Connor, A. Anand, and B. Andemariam, “Sickle cell disease diagnosis based on spatio-temporal cell dynamics analysis using 3D printed shearing digital holographic microscopy,” Opt. Express 26(10), 13614–13627 (2018). [CrossRef]  

32. K. Lee and Y. Park, “Quantitative phase imaging unit,” Opt. Lett. 39(12), 3630–3633 (2014). [CrossRef]  

33. Y. Baek, K. Lee, J. Yoon, K. Kim, and Y. Park, “White-light quantitative phase imaging unit,” Opt. Express 24(9), 9308–9315 (2016). [CrossRef]  

34. D. Roitshtain, N. A. Turko, B. Javidi, and N. T. Shaked, “Flipping interferometry and its application for quantitative phase microscopy in a micro-channel,” Opt. Lett. 41(10), 2354–2357 (2016). [CrossRef]  

35. N. Rotman-Nativ, N. A. Turko, and N. T. Shaked, “Flipping interferometry with doubled imaging area,” Opt. Lett. 43(22), 5543–5546 (2018). [CrossRef]  

36. B. Kemper, A. Vollmer, C. E. Rommel, J. Schnekenburger, and G. von Bally, “Simplified approach for quantitative digital holographic phase contrast imaging of living cells,” J. Biomed. Opt. 16(2), 026014 (2011). [CrossRef]  

37. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company Publishers, 2005).

38. R. Guo, F. Wang, X. Hu, and W. Yang, “Off-axis low coherence digital holographic interferometry for quantitative phase imaging with an LED,” J. Opt. 19(11), 115702 (2017). [CrossRef]  

39. B. Bhaduri, M. Kandel, C. Brugnara, K. Tangella, and G. Popescu, “Optical Assay of Erythrocyte Function in Banked Blood,” Sci. Rep. 4(1), 6211 (2015). [CrossRef]  

40. K. Jaferzadeh, I. Moon, M. Bardyn, M. Prudent, J. D. Tissot, B. Rappaz, B. Javidi, G. Turcatti, and P. Marquet, “Quantification of stored red blood cell fluctuations by time-lapse holographic cell imaging,” Biomed. Opt. Express 9(10), 4714–4729 (2018). [CrossRef]  

41. W. H. Southwell, “Validity of the Fresnel approximation in the near field,” J. Opt. Soc. Am. 71(1), 7–14 (1981). [CrossRef]  

References

  • View by:
  • |
  • |
  • |

  1. E. Cuche, F. Bevilacqua, and C. Depeursinge, “Digital holography for quantitative phase-contrast imaging,” Opt. Lett. 24(5), 291–293 (1999).
    [Crossref]
  2. P. Ferraro, C. D. Core, L. Miccio, S. Grilli, S. D. Nicola, A. Finizio, and G. Coppola, “Phase map retrieval in digital holography: avoiding the under sampling effect by a lateral shear approach,” Opt. Lett. 32(15), 2233–2235 (2007).
    [Crossref]
  3. V. Biaco, B. Mandracchia, V. Marchesano, V. Pagliarulo, F. Olivieri, S. Coppola, M. Paturzo, and P. Ferraro, “Endowing a plain fluidic chip with micro-optics: a holographic microscope slide,” Light Sci. Appl. 6(9), e17055 (2017).
    [Crossref]
  4. J. Min, B. Yao, S. Ketelhut, C. Engwer, B. Greve, and B. Kemper, “Simple and fast spectral domain algorithm for quantitative phase imaging of living cells with digital holographic microscopy,” Opt. Lett. 42(2), 227–230 (2017).
    [Crossref]
  5. F. Pan, S. Liu, Z. Wang, P. Shang, and W. Xiao, “Digital holographic microscopy long-term and real-time monitoring of cell division and changes under simulated zero gravity,” Opt. Express 20(10), 11496–11505 (2012).
    [Crossref]
  6. J. Cho, J. Lim, S. Jeon, G. Choi, H. Moon, N. Park, and Y. Park, “Dual-wavelength off-axis digital holography using a single light-emitting diode,” Opt. Express 26(2), 2123–2131 (2018).
    [Crossref]
  7. L. Puyo, J. Huignard, and M. Atlan, “Off-axis digital holography with multiplexed volume Bragg gratings,” Appl. Opt. 57(12), 3281–3287 (2018).
    [Crossref]
  8. N. T. Shaked, T. M. Newpher, M. D. Ehlers, and A. Wax, “Parallel on-axis holographic phase microscopy of biological cells and unicellular microorganism dynamics,” Appl. Opt. 49(15), 2872–2878 (2010).
    [Crossref]
  9. V. Micó, J. Zheng, J. Garcia, Z. Zalevsky, and P. Gao, “Resolution enhancement in quantitative phase microscopy,” Adv. Opt. Photon. 11(1), 135–214 (2019).
    [Crossref]
  10. P. Gao, B. Yao, J. Min, R. Guo, J. Zheng, T. Ye, I. Harder, V. Nercissian, and K. Mantel, “Parallel two-step phase-shifting point-diffraction interferometry for microscopy based on a pair of cube beam splitters,” Opt. Express 19(3), 1930–1935 (2011).
    [Crossref]
  11. J. Zheng, P. Gao, X. Shao, and G. Ulrich Nienhaus, “Refractive index measurement of suspended cells using opposed-view digital holographic microscopy,” Appl. Opt. 56(32), 9000–9005 (2017).
    [Crossref]
  12. R. Guo and F. Wang, “Compact and stable real-time dual-wavelength digital holographic microscopy with a long-working distance objective,” Opt. Express 25(20), 24512–24520 (2017).
    [Crossref]
  13. R. Guo, W. Zhang, R. Liu, C. Duan, and F. Wang, “Phase unwrapping in dual-wavelength digital holographic microscopy with total variation regularization,” Opt. Lett. 43(14), 3449–3452 (2018).
    [Crossref]
  14. J. A. Picazo-Bueno, M. Trusiak, J. García, K. Patorski, and V. Micó, “Hilbert-Huang single-shot spatially multiplexed interferometric microscopy,” Opt. Lett. 43(5), 1007–1010 (2018).
    [Crossref]
  15. J. A. Picazo-Bueno, Z. Zalevsky, J. García, and V. Micó, “Superresolved spatially multiplexed interferometric microscopy,” Opt. Lett. 42(5), 927–930 (2017).
    [Crossref]
  16. M. Trusiak, J. A. Picazo-Bueno, K. Patorski, P. Zdankowski, and V. Micó, “Single-shot two-frame π-shifted spatially multiplexed interference phase microscopy,” J. Biomed. Opt. 24(09), 1 (2019).
    [Crossref]
  17. P. Hosseini, R. Zhou, Y. H. Kim, C. Peres, A. Diaspro, C. Kuang, Z. Yaqoob, and P. T. C. So, “Pushing phase and amplitude sensitivity limits in interferometric microscopy,” Opt. Lett. 41(7), 1656–1659 (2016).
    [Crossref]
  18. N. A. Turko and N. T. Shaked, “Simultaneous two-wavelength phase unwrapping using an external module for multiplexing off-axis holography,” Opt. Lett. 42, 73 (2017).
    [Crossref]
  19. P. Girshovitz and N. T. Shaked, “Compact and portable low-coherence interferometer with off-axis geometry for quantitative phase microscopy and nanoscopy,” Opt. Express 21(5), 5701–5714 (2013).
    [Crossref]
  20. N. T. Shaked, “Quantitative phase microscopy of biological samples using a portable interferometer,” Opt. Lett. 37(11), 2016–2018 (2012).
    [Crossref]
  21. G. Popescu, T. Ikeda, R. R. Dasari, and M. S. Feld, “Diffraction phase microscopy for quantifying cell structure and dynamics,” Opt. Lett. 31(6), 775–777 (2006).
    [Crossref]
  22. I. Frenklach, P. Girshovitz, and N. T. Shaked, “Off-axis interferometric phase microscopy with tripled imaging area,” Opt. Lett. 39, 1525 (2014).
    [Crossref]
  23. B. Bhaduri, H. Pham, M. Mir, and G. Popescu, “Diffraction phase microscopy with white light,” Opt. Lett. 37(6), 1094–1096 (2012).
    [Crossref]
  24. M. Shan, L. Liu, Z. Zhong, B. Liu, G. Luan, and Y. Zhang, “Single-shot dual-wavelength off-axis quasi-common-path digital holography using polarization-multiplexing,” Opt. Express 25(21), 26253–26261 (2017).
    [Crossref]
  25. H. Bai, Z. Zhong, M. Shan, L. Liu, L. Guo, and Y. Zhang, “Interferometric phase microscopy using slightly-off axis reflective point diffraction interferometer,” Opt. Las. Eng. 90, 155–160 (2017).
    [Crossref]
  26. S. Mahajan, V. Trivedi, P. Vora, V. Chhaniwal, B. Javidi, and A. Anand, “Highly stable digital holographic microscope using Sagnac interferometer,” Opt. Lett. 40(16), 3743–3746 (2015).
    [Crossref]
  27. V. Chhaniwal, A. S. Singh, R. A. Leitgeb, B. Javidi, and A. Anand, “Quantitative phase-contrast imaging with compact digital holographic microscope employing Lloyd’s mirror,” Opt. Lett. 37(24), 5127–5129 (2012).
    [Crossref]
  28. C. Ma, Y. Li, J. Zhang, P. Li, T. Xi, J. Di, and J. Zhao, “Lateral shearing common-path digital holographic microscopy based on a slightly trapezoid Sagnac interferometer,” Opt. Express 25(12), 13659–13667 (2017).
    [Crossref]
  29. J. Di, Y. Li, M. Xie, J. Zhang, C. Ma, T. Xi, E. Li, and J. Zhao, “Dual-wavelength common-path digital holographic microscopy for quantitative phase imaging based on lateral shearing interferometry,” Appl. Opt. 55(26), 7287–7293 (2016).
    [Crossref]
  30. A. S. Singh, A. Anand, R. A. Leitgeb, and B. Javidi, “Lateral shearing digital holographic imaging of small biological specimens,” Opt. Express 20(21), 23617–23622 (2012).
    [Crossref]
  31. B. Javidi, A. Markman, S. Rawat, T. O’Connor, A. Anand, and B. Andemariam, “Sickle cell disease diagnosis based on spatio-temporal cell dynamics analysis using 3D printed shearing digital holographic microscopy,” Opt. Express 26(10), 13614–13627 (2018).
    [Crossref]
  32. K. Lee and Y. Park, “Quantitative phase imaging unit,” Opt. Lett. 39(12), 3630–3633 (2014).
    [Crossref]
  33. Y. Baek, K. Lee, J. Yoon, K. Kim, and Y. Park, “White-light quantitative phase imaging unit,” Opt. Express 24(9), 9308–9315 (2016).
    [Crossref]
  34. D. Roitshtain, N. A. Turko, B. Javidi, and N. T. Shaked, “Flipping interferometry and its application for quantitative phase microscopy in a micro-channel,” Opt. Lett. 41(10), 2354–2357 (2016).
    [Crossref]
  35. N. Rotman-Nativ, N. A. Turko, and N. T. Shaked, “Flipping interferometry with doubled imaging area,” Opt. Lett. 43(22), 5543–5546 (2018).
    [Crossref]
  36. B. Kemper, A. Vollmer, C. E. Rommel, J. Schnekenburger, and G. von Bally, “Simplified approach for quantitative digital holographic phase contrast imaging of living cells,” J. Biomed. Opt. 16(2), 026014 (2011).
    [Crossref]
  37. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company Publishers, 2005).
  38. R. Guo, F. Wang, X. Hu, and W. Yang, “Off-axis low coherence digital holographic interferometry for quantitative phase imaging with an LED,” J. Opt. 19(11), 115702 (2017).
    [Crossref]
  39. B. Bhaduri, M. Kandel, C. Brugnara, K. Tangella, and G. Popescu, “Optical Assay of Erythrocyte Function in Banked Blood,” Sci. Rep. 4(1), 6211 (2015).
    [Crossref]
  40. K. Jaferzadeh, I. Moon, M. Bardyn, M. Prudent, J. D. Tissot, B. Rappaz, B. Javidi, G. Turcatti, and P. Marquet, “Quantification of stored red blood cell fluctuations by time-lapse holographic cell imaging,” Biomed. Opt. Express 9(10), 4714–4729 (2018).
    [Crossref]
  41. W. H. Southwell, “Validity of the Fresnel approximation in the near field,” J. Opt. Soc. Am. 71(1), 7–14 (1981).
    [Crossref]

2019 (2)

V. Micó, J. Zheng, J. Garcia, Z. Zalevsky, and P. Gao, “Resolution enhancement in quantitative phase microscopy,” Adv. Opt. Photon. 11(1), 135–214 (2019).
[Crossref]

M. Trusiak, J. A. Picazo-Bueno, K. Patorski, P. Zdankowski, and V. Micó, “Single-shot two-frame π-shifted spatially multiplexed interference phase microscopy,” J. Biomed. Opt. 24(09), 1 (2019).
[Crossref]

2018 (7)

2017 (10)

R. Guo, F. Wang, X. Hu, and W. Yang, “Off-axis low coherence digital holographic interferometry for quantitative phase imaging with an LED,” J. Opt. 19(11), 115702 (2017).
[Crossref]

J. A. Picazo-Bueno, Z. Zalevsky, J. García, and V. Micó, “Superresolved spatially multiplexed interferometric microscopy,” Opt. Lett. 42(5), 927–930 (2017).
[Crossref]

J. Zheng, P. Gao, X. Shao, and G. Ulrich Nienhaus, “Refractive index measurement of suspended cells using opposed-view digital holographic microscopy,” Appl. Opt. 56(32), 9000–9005 (2017).
[Crossref]

R. Guo and F. Wang, “Compact and stable real-time dual-wavelength digital holographic microscopy with a long-working distance objective,” Opt. Express 25(20), 24512–24520 (2017).
[Crossref]

N. A. Turko and N. T. Shaked, “Simultaneous two-wavelength phase unwrapping using an external module for multiplexing off-axis holography,” Opt. Lett. 42, 73 (2017).
[Crossref]

V. Biaco, B. Mandracchia, V. Marchesano, V. Pagliarulo, F. Olivieri, S. Coppola, M. Paturzo, and P. Ferraro, “Endowing a plain fluidic chip with micro-optics: a holographic microscope slide,” Light Sci. Appl. 6(9), e17055 (2017).
[Crossref]

J. Min, B. Yao, S. Ketelhut, C. Engwer, B. Greve, and B. Kemper, “Simple and fast spectral domain algorithm for quantitative phase imaging of living cells with digital holographic microscopy,” Opt. Lett. 42(2), 227–230 (2017).
[Crossref]

C. Ma, Y. Li, J. Zhang, P. Li, T. Xi, J. Di, and J. Zhao, “Lateral shearing common-path digital holographic microscopy based on a slightly trapezoid Sagnac interferometer,” Opt. Express 25(12), 13659–13667 (2017).
[Crossref]

M. Shan, L. Liu, Z. Zhong, B. Liu, G. Luan, and Y. Zhang, “Single-shot dual-wavelength off-axis quasi-common-path digital holography using polarization-multiplexing,” Opt. Express 25(21), 26253–26261 (2017).
[Crossref]

H. Bai, Z. Zhong, M. Shan, L. Liu, L. Guo, and Y. Zhang, “Interferometric phase microscopy using slightly-off axis reflective point diffraction interferometer,” Opt. Las. Eng. 90, 155–160 (2017).
[Crossref]

2016 (4)

2015 (2)

S. Mahajan, V. Trivedi, P. Vora, V. Chhaniwal, B. Javidi, and A. Anand, “Highly stable digital holographic microscope using Sagnac interferometer,” Opt. Lett. 40(16), 3743–3746 (2015).
[Crossref]

B. Bhaduri, M. Kandel, C. Brugnara, K. Tangella, and G. Popescu, “Optical Assay of Erythrocyte Function in Banked Blood,” Sci. Rep. 4(1), 6211 (2015).
[Crossref]

2014 (2)

2013 (1)

2012 (5)

2011 (2)

B. Kemper, A. Vollmer, C. E. Rommel, J. Schnekenburger, and G. von Bally, “Simplified approach for quantitative digital holographic phase contrast imaging of living cells,” J. Biomed. Opt. 16(2), 026014 (2011).
[Crossref]

P. Gao, B. Yao, J. Min, R. Guo, J. Zheng, T. Ye, I. Harder, V. Nercissian, and K. Mantel, “Parallel two-step phase-shifting point-diffraction interferometry for microscopy based on a pair of cube beam splitters,” Opt. Express 19(3), 1930–1935 (2011).
[Crossref]

2010 (1)

2007 (1)

2006 (1)

1999 (1)

1981 (1)

Anand, A.

Andemariam, B.

Atlan, M.

Baek, Y.

Bai, H.

H. Bai, Z. Zhong, M. Shan, L. Liu, L. Guo, and Y. Zhang, “Interferometric phase microscopy using slightly-off axis reflective point diffraction interferometer,” Opt. Las. Eng. 90, 155–160 (2017).
[Crossref]

Bardyn, M.

Bevilacqua, F.

Bhaduri, B.

B. Bhaduri, M. Kandel, C. Brugnara, K. Tangella, and G. Popescu, “Optical Assay of Erythrocyte Function in Banked Blood,” Sci. Rep. 4(1), 6211 (2015).
[Crossref]

B. Bhaduri, H. Pham, M. Mir, and G. Popescu, “Diffraction phase microscopy with white light,” Opt. Lett. 37(6), 1094–1096 (2012).
[Crossref]

Biaco, V.

V. Biaco, B. Mandracchia, V. Marchesano, V. Pagliarulo, F. Olivieri, S. Coppola, M. Paturzo, and P. Ferraro, “Endowing a plain fluidic chip with micro-optics: a holographic microscope slide,” Light Sci. Appl. 6(9), e17055 (2017).
[Crossref]

Brugnara, C.

B. Bhaduri, M. Kandel, C. Brugnara, K. Tangella, and G. Popescu, “Optical Assay of Erythrocyte Function in Banked Blood,” Sci. Rep. 4(1), 6211 (2015).
[Crossref]

Chhaniwal, V.

Cho, J.

Choi, G.

Coppola, G.

Coppola, S.

V. Biaco, B. Mandracchia, V. Marchesano, V. Pagliarulo, F. Olivieri, S. Coppola, M. Paturzo, and P. Ferraro, “Endowing a plain fluidic chip with micro-optics: a holographic microscope slide,” Light Sci. Appl. 6(9), e17055 (2017).
[Crossref]

Core, C. D.

Cuche, E.

Dasari, R. R.

Depeursinge, C.

Di, J.

Diaspro, A.

Duan, C.

Ehlers, M. D.

Engwer, C.

Feld, M. S.

Ferraro, P.

V. Biaco, B. Mandracchia, V. Marchesano, V. Pagliarulo, F. Olivieri, S. Coppola, M. Paturzo, and P. Ferraro, “Endowing a plain fluidic chip with micro-optics: a holographic microscope slide,” Light Sci. Appl. 6(9), e17055 (2017).
[Crossref]

P. Ferraro, C. D. Core, L. Miccio, S. Grilli, S. D. Nicola, A. Finizio, and G. Coppola, “Phase map retrieval in digital holography: avoiding the under sampling effect by a lateral shear approach,” Opt. Lett. 32(15), 2233–2235 (2007).
[Crossref]

Finizio, A.

Frenklach, I.

Gao, P.

Garcia, J.

García, J.

Girshovitz, P.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company Publishers, 2005).

Greve, B.

Grilli, S.

Guo, L.

H. Bai, Z. Zhong, M. Shan, L. Liu, L. Guo, and Y. Zhang, “Interferometric phase microscopy using slightly-off axis reflective point diffraction interferometer,” Opt. Las. Eng. 90, 155–160 (2017).
[Crossref]

Guo, R.

Harder, I.

Hosseini, P.

Hu, X.

R. Guo, F. Wang, X. Hu, and W. Yang, “Off-axis low coherence digital holographic interferometry for quantitative phase imaging with an LED,” J. Opt. 19(11), 115702 (2017).
[Crossref]

Huignard, J.

Ikeda, T.

Jaferzadeh, K.

Javidi, B.

Jeon, S.

Kandel, M.

B. Bhaduri, M. Kandel, C. Brugnara, K. Tangella, and G. Popescu, “Optical Assay of Erythrocyte Function in Banked Blood,” Sci. Rep. 4(1), 6211 (2015).
[Crossref]

Kemper, B.

J. Min, B. Yao, S. Ketelhut, C. Engwer, B. Greve, and B. Kemper, “Simple and fast spectral domain algorithm for quantitative phase imaging of living cells with digital holographic microscopy,” Opt. Lett. 42(2), 227–230 (2017).
[Crossref]

B. Kemper, A. Vollmer, C. E. Rommel, J. Schnekenburger, and G. von Bally, “Simplified approach for quantitative digital holographic phase contrast imaging of living cells,” J. Biomed. Opt. 16(2), 026014 (2011).
[Crossref]

Ketelhut, S.

Kim, K.

Kim, Y. H.

Kuang, C.

Lee, K.

Leitgeb, R. A.

Li, E.

Li, P.

Li, Y.

Lim, J.

Liu, B.

Liu, L.

M. Shan, L. Liu, Z. Zhong, B. Liu, G. Luan, and Y. Zhang, “Single-shot dual-wavelength off-axis quasi-common-path digital holography using polarization-multiplexing,” Opt. Express 25(21), 26253–26261 (2017).
[Crossref]

H. Bai, Z. Zhong, M. Shan, L. Liu, L. Guo, and Y. Zhang, “Interferometric phase microscopy using slightly-off axis reflective point diffraction interferometer,” Opt. Las. Eng. 90, 155–160 (2017).
[Crossref]

Liu, R.

Liu, S.

Luan, G.

Ma, C.

Mahajan, S.

Mandracchia, B.

V. Biaco, B. Mandracchia, V. Marchesano, V. Pagliarulo, F. Olivieri, S. Coppola, M. Paturzo, and P. Ferraro, “Endowing a plain fluidic chip with micro-optics: a holographic microscope slide,” Light Sci. Appl. 6(9), e17055 (2017).
[Crossref]

Mantel, K.

Marchesano, V.

V. Biaco, B. Mandracchia, V. Marchesano, V. Pagliarulo, F. Olivieri, S. Coppola, M. Paturzo, and P. Ferraro, “Endowing a plain fluidic chip with micro-optics: a holographic microscope slide,” Light Sci. Appl. 6(9), e17055 (2017).
[Crossref]

Markman, A.

Marquet, P.

Miccio, L.

Micó, V.

Min, J.

Mir, M.

Moon, H.

Moon, I.

Nercissian, V.

Newpher, T. M.

Nicola, S. D.

O’Connor, T.

Olivieri, F.

V. Biaco, B. Mandracchia, V. Marchesano, V. Pagliarulo, F. Olivieri, S. Coppola, M. Paturzo, and P. Ferraro, “Endowing a plain fluidic chip with micro-optics: a holographic microscope slide,” Light Sci. Appl. 6(9), e17055 (2017).
[Crossref]

Pagliarulo, V.

V. Biaco, B. Mandracchia, V. Marchesano, V. Pagliarulo, F. Olivieri, S. Coppola, M. Paturzo, and P. Ferraro, “Endowing a plain fluidic chip with micro-optics: a holographic microscope slide,” Light Sci. Appl. 6(9), e17055 (2017).
[Crossref]

Pan, F.

Park, N.

Park, Y.

Patorski, K.

M. Trusiak, J. A. Picazo-Bueno, K. Patorski, P. Zdankowski, and V. Micó, “Single-shot two-frame π-shifted spatially multiplexed interference phase microscopy,” J. Biomed. Opt. 24(09), 1 (2019).
[Crossref]

J. A. Picazo-Bueno, M. Trusiak, J. García, K. Patorski, and V. Micó, “Hilbert-Huang single-shot spatially multiplexed interferometric microscopy,” Opt. Lett. 43(5), 1007–1010 (2018).
[Crossref]

Paturzo, M.

V. Biaco, B. Mandracchia, V. Marchesano, V. Pagliarulo, F. Olivieri, S. Coppola, M. Paturzo, and P. Ferraro, “Endowing a plain fluidic chip with micro-optics: a holographic microscope slide,” Light Sci. Appl. 6(9), e17055 (2017).
[Crossref]

Peres, C.

Pham, H.

Picazo-Bueno, J. A.

Popescu, G.

Prudent, M.

Puyo, L.

Rappaz, B.

Rawat, S.

Roitshtain, D.

Rommel, C. E.

B. Kemper, A. Vollmer, C. E. Rommel, J. Schnekenburger, and G. von Bally, “Simplified approach for quantitative digital holographic phase contrast imaging of living cells,” J. Biomed. Opt. 16(2), 026014 (2011).
[Crossref]

Rotman-Nativ, N.

Schnekenburger, J.

B. Kemper, A. Vollmer, C. E. Rommel, J. Schnekenburger, and G. von Bally, “Simplified approach for quantitative digital holographic phase contrast imaging of living cells,” J. Biomed. Opt. 16(2), 026014 (2011).
[Crossref]

Shaked, N. T.

Shan, M.

M. Shan, L. Liu, Z. Zhong, B. Liu, G. Luan, and Y. Zhang, “Single-shot dual-wavelength off-axis quasi-common-path digital holography using polarization-multiplexing,” Opt. Express 25(21), 26253–26261 (2017).
[Crossref]

H. Bai, Z. Zhong, M. Shan, L. Liu, L. Guo, and Y. Zhang, “Interferometric phase microscopy using slightly-off axis reflective point diffraction interferometer,” Opt. Las. Eng. 90, 155–160 (2017).
[Crossref]

Shang, P.

Shao, X.

Singh, A. S.

So, P. T. C.

Southwell, W. H.

Tangella, K.

B. Bhaduri, M. Kandel, C. Brugnara, K. Tangella, and G. Popescu, “Optical Assay of Erythrocyte Function in Banked Blood,” Sci. Rep. 4(1), 6211 (2015).
[Crossref]

Tissot, J. D.

Trivedi, V.

Trusiak, M.

M. Trusiak, J. A. Picazo-Bueno, K. Patorski, P. Zdankowski, and V. Micó, “Single-shot two-frame π-shifted spatially multiplexed interference phase microscopy,” J. Biomed. Opt. 24(09), 1 (2019).
[Crossref]

J. A. Picazo-Bueno, M. Trusiak, J. García, K. Patorski, and V. Micó, “Hilbert-Huang single-shot spatially multiplexed interferometric microscopy,” Opt. Lett. 43(5), 1007–1010 (2018).
[Crossref]

Turcatti, G.

Turko, N. A.

Ulrich Nienhaus, G.

Vollmer, A.

B. Kemper, A. Vollmer, C. E. Rommel, J. Schnekenburger, and G. von Bally, “Simplified approach for quantitative digital holographic phase contrast imaging of living cells,” J. Biomed. Opt. 16(2), 026014 (2011).
[Crossref]

von Bally, G.

B. Kemper, A. Vollmer, C. E. Rommel, J. Schnekenburger, and G. von Bally, “Simplified approach for quantitative digital holographic phase contrast imaging of living cells,” J. Biomed. Opt. 16(2), 026014 (2011).
[Crossref]

Vora, P.

Wang, F.

Wang, Z.

Wax, A.

Xi, T.

Xiao, W.

Xie, M.

Yang, W.

R. Guo, F. Wang, X. Hu, and W. Yang, “Off-axis low coherence digital holographic interferometry for quantitative phase imaging with an LED,” J. Opt. 19(11), 115702 (2017).
[Crossref]

Yao, B.

Yaqoob, Z.

Ye, T.

Yoon, J.

Zalevsky, Z.

Zdankowski, P.

M. Trusiak, J. A. Picazo-Bueno, K. Patorski, P. Zdankowski, and V. Micó, “Single-shot two-frame π-shifted spatially multiplexed interference phase microscopy,” J. Biomed. Opt. 24(09), 1 (2019).
[Crossref]

Zhang, J.

Zhang, W.

Zhang, Y.

H. Bai, Z. Zhong, M. Shan, L. Liu, L. Guo, and Y. Zhang, “Interferometric phase microscopy using slightly-off axis reflective point diffraction interferometer,” Opt. Las. Eng. 90, 155–160 (2017).
[Crossref]

M. Shan, L. Liu, Z. Zhong, B. Liu, G. Luan, and Y. Zhang, “Single-shot dual-wavelength off-axis quasi-common-path digital holography using polarization-multiplexing,” Opt. Express 25(21), 26253–26261 (2017).
[Crossref]

Zhao, J.

Zheng, J.

Zhong, Z.

M. Shan, L. Liu, Z. Zhong, B. Liu, G. Luan, and Y. Zhang, “Single-shot dual-wavelength off-axis quasi-common-path digital holography using polarization-multiplexing,” Opt. Express 25(21), 26253–26261 (2017).
[Crossref]

H. Bai, Z. Zhong, M. Shan, L. Liu, L. Guo, and Y. Zhang, “Interferometric phase microscopy using slightly-off axis reflective point diffraction interferometer,” Opt. Las. Eng. 90, 155–160 (2017).
[Crossref]

Zhou, R.

Adv. Opt. Photon. (1)

Appl. Opt. (4)

Biomed. Opt. Express (1)

J. Biomed. Opt. (2)

B. Kemper, A. Vollmer, C. E. Rommel, J. Schnekenburger, and G. von Bally, “Simplified approach for quantitative digital holographic phase contrast imaging of living cells,” J. Biomed. Opt. 16(2), 026014 (2011).
[Crossref]

M. Trusiak, J. A. Picazo-Bueno, K. Patorski, P. Zdankowski, and V. Micó, “Single-shot two-frame π-shifted spatially multiplexed interference phase microscopy,” J. Biomed. Opt. 24(09), 1 (2019).
[Crossref]

J. Opt. (1)

R. Guo, F. Wang, X. Hu, and W. Yang, “Off-axis low coherence digital holographic interferometry for quantitative phase imaging with an LED,” J. Opt. 19(11), 115702 (2017).
[Crossref]

J. Opt. Soc. Am. (1)

Light Sci. Appl. (1)

V. Biaco, B. Mandracchia, V. Marchesano, V. Pagliarulo, F. Olivieri, S. Coppola, M. Paturzo, and P. Ferraro, “Endowing a plain fluidic chip with micro-optics: a holographic microscope slide,” Light Sci. Appl. 6(9), e17055 (2017).
[Crossref]

Opt. Express (10)

F. Pan, S. Liu, Z. Wang, P. Shang, and W. Xiao, “Digital holographic microscopy long-term and real-time monitoring of cell division and changes under simulated zero gravity,” Opt. Express 20(10), 11496–11505 (2012).
[Crossref]

J. Cho, J. Lim, S. Jeon, G. Choi, H. Moon, N. Park, and Y. Park, “Dual-wavelength off-axis digital holography using a single light-emitting diode,” Opt. Express 26(2), 2123–2131 (2018).
[Crossref]

P. Gao, B. Yao, J. Min, R. Guo, J. Zheng, T. Ye, I. Harder, V. Nercissian, and K. Mantel, “Parallel two-step phase-shifting point-diffraction interferometry for microscopy based on a pair of cube beam splitters,” Opt. Express 19(3), 1930–1935 (2011).
[Crossref]

R. Guo and F. Wang, “Compact and stable real-time dual-wavelength digital holographic microscopy with a long-working distance objective,” Opt. Express 25(20), 24512–24520 (2017).
[Crossref]

Y. Baek, K. Lee, J. Yoon, K. Kim, and Y. Park, “White-light quantitative phase imaging unit,” Opt. Express 24(9), 9308–9315 (2016).
[Crossref]

A. S. Singh, A. Anand, R. A. Leitgeb, and B. Javidi, “Lateral shearing digital holographic imaging of small biological specimens,” Opt. Express 20(21), 23617–23622 (2012).
[Crossref]

B. Javidi, A. Markman, S. Rawat, T. O’Connor, A. Anand, and B. Andemariam, “Sickle cell disease diagnosis based on spatio-temporal cell dynamics analysis using 3D printed shearing digital holographic microscopy,” Opt. Express 26(10), 13614–13627 (2018).
[Crossref]

P. Girshovitz and N. T. Shaked, “Compact and portable low-coherence interferometer with off-axis geometry for quantitative phase microscopy and nanoscopy,” Opt. Express 21(5), 5701–5714 (2013).
[Crossref]

M. Shan, L. Liu, Z. Zhong, B. Liu, G. Luan, and Y. Zhang, “Single-shot dual-wavelength off-axis quasi-common-path digital holography using polarization-multiplexing,” Opt. Express 25(21), 26253–26261 (2017).
[Crossref]

C. Ma, Y. Li, J. Zhang, P. Li, T. Xi, J. Di, and J. Zhao, “Lateral shearing common-path digital holographic microscopy based on a slightly trapezoid Sagnac interferometer,” Opt. Express 25(12), 13659–13667 (2017).
[Crossref]

Opt. Las. Eng. (1)

H. Bai, Z. Zhong, M. Shan, L. Liu, L. Guo, and Y. Zhang, “Interferometric phase microscopy using slightly-off axis reflective point diffraction interferometer,” Opt. Las. Eng. 90, 155–160 (2017).
[Crossref]

Opt. Lett. (17)

S. Mahajan, V. Trivedi, P. Vora, V. Chhaniwal, B. Javidi, and A. Anand, “Highly stable digital holographic microscope using Sagnac interferometer,” Opt. Lett. 40(16), 3743–3746 (2015).
[Crossref]

V. Chhaniwal, A. S. Singh, R. A. Leitgeb, B. Javidi, and A. Anand, “Quantitative phase-contrast imaging with compact digital holographic microscope employing Lloyd’s mirror,” Opt. Lett. 37(24), 5127–5129 (2012).
[Crossref]

N. T. Shaked, “Quantitative phase microscopy of biological samples using a portable interferometer,” Opt. Lett. 37(11), 2016–2018 (2012).
[Crossref]

G. Popescu, T. Ikeda, R. R. Dasari, and M. S. Feld, “Diffraction phase microscopy for quantifying cell structure and dynamics,” Opt. Lett. 31(6), 775–777 (2006).
[Crossref]

I. Frenklach, P. Girshovitz, and N. T. Shaked, “Off-axis interferometric phase microscopy with tripled imaging area,” Opt. Lett. 39, 1525 (2014).
[Crossref]

B. Bhaduri, H. Pham, M. Mir, and G. Popescu, “Diffraction phase microscopy with white light,” Opt. Lett. 37(6), 1094–1096 (2012).
[Crossref]

K. Lee and Y. Park, “Quantitative phase imaging unit,” Opt. Lett. 39(12), 3630–3633 (2014).
[Crossref]

D. Roitshtain, N. A. Turko, B. Javidi, and N. T. Shaked, “Flipping interferometry and its application for quantitative phase microscopy in a micro-channel,” Opt. Lett. 41(10), 2354–2357 (2016).
[Crossref]

N. Rotman-Nativ, N. A. Turko, and N. T. Shaked, “Flipping interferometry with doubled imaging area,” Opt. Lett. 43(22), 5543–5546 (2018).
[Crossref]

R. Guo, W. Zhang, R. Liu, C. Duan, and F. Wang, “Phase unwrapping in dual-wavelength digital holographic microscopy with total variation regularization,” Opt. Lett. 43(14), 3449–3452 (2018).
[Crossref]

J. A. Picazo-Bueno, M. Trusiak, J. García, K. Patorski, and V. Micó, “Hilbert-Huang single-shot spatially multiplexed interferometric microscopy,” Opt. Lett. 43(5), 1007–1010 (2018).
[Crossref]

J. A. Picazo-Bueno, Z. Zalevsky, J. García, and V. Micó, “Superresolved spatially multiplexed interferometric microscopy,” Opt. Lett. 42(5), 927–930 (2017).
[Crossref]

P. Hosseini, R. Zhou, Y. H. Kim, C. Peres, A. Diaspro, C. Kuang, Z. Yaqoob, and P. T. C. So, “Pushing phase and amplitude sensitivity limits in interferometric microscopy,” Opt. Lett. 41(7), 1656–1659 (2016).
[Crossref]

N. A. Turko and N. T. Shaked, “Simultaneous two-wavelength phase unwrapping using an external module for multiplexing off-axis holography,” Opt. Lett. 42, 73 (2017).
[Crossref]

J. Min, B. Yao, S. Ketelhut, C. Engwer, B. Greve, and B. Kemper, “Simple and fast spectral domain algorithm for quantitative phase imaging of living cells with digital holographic microscopy,” Opt. Lett. 42(2), 227–230 (2017).
[Crossref]

E. Cuche, F. Bevilacqua, and C. Depeursinge, “Digital holography for quantitative phase-contrast imaging,” Opt. Lett. 24(5), 291–293 (1999).
[Crossref]

P. Ferraro, C. D. Core, L. Miccio, S. Grilli, S. D. Nicola, A. Finizio, and G. Coppola, “Phase map retrieval in digital holography: avoiding the under sampling effect by a lateral shear approach,” Opt. Lett. 32(15), 2233–2235 (2007).
[Crossref]

Sci. Rep. (1)

B. Bhaduri, M. Kandel, C. Brugnara, K. Tangella, and G. Popescu, “Optical Assay of Erythrocyte Function in Banked Blood,” Sci. Rep. 4(1), 6211 (2015).
[Crossref]

Other (1)

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company Publishers, 2005).

Supplementary Material (2)

NameDescription
» Visualization 1       Gradual changing of the shearing distance over time
» Visualization 2       Flow dynamics of cells

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1.
Fig. 1. An inverted microscope with the proposed SICA module (marked by dashed rectangle), connected to its output. M1, M2, mirrors; S, sample; MO, microscope objective; TL, tube lens; IP, image plane; G, grating; L1, L2, lenses with focal lengths f1=150 mm and f2=300 mm. z, distance of G from IP; B, mask that selects only two diffraction orders; l, shearing distance; α, interference angle.
Fig. 2.
Fig. 2. Experimental results of varying shearing distances but with fixed interference angle, when imaging 1951 USAF resolution target. (a,b) Off-axis image holograms with two different shearing distances. White lines on the top right represent 10 µm on the sample. (c,d) Spatial frequency spectra corresponding to (a) and (b), respectively, where the location of the cross-correlation terms remains unchanged.
Fig. 3.
Fig. 3. Temporal OPD stability in our SICA setup (orange) and the DPM setup (blue).
Fig. 4.
Fig. 4. Quantitative phase images of silica beads with (a, b) a small vertical shearing distance and (c, d) a large vertical shearing distance, as obtained under highly coherent illumination. (a, c) Off-axis holograms, with magnified inset showing fringe period and contrast. (b, d) Quantitative phase profiles. White lines represent 10 µm. See gradual changing of the shearing distance over time in Visualization 1.
Fig. 5.
Fig. 5. (a) Quantitative phase image of a single RBC and (b) its fluctuation map, as obtained under highly coherent illumination. White lines represent 5 µm.
Fig. 6.
Fig. 6. Quantitative phase image of cancer cells (SW620) flowing in a micro-channel, as obtained under highly coherent illumination. White line represents 10 µm. See flow dynamics in Visualization 2.
Fig. 7.
Fig. 7. Schematic of the SICA module. z, z1, distance from planes x1 to x2 and x2 to x3, respectively; α, interference angle.

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

α = f 1 λ / f 2 d [ r a d ] ,
l = f 2 λ z / f 1 d .
I ( x , y ) = | K 0 A [ ( x , y ) f 1 / f 2 ] + K 1 exp ( j 2 π f 1 x / f 2 d ) A [ ( x f 2 λ z / f 1 d , y ) f 1 / f 2 ] | 2 ,
u ( x 2 ) = 1 j λ z exp ( j k 2 z x 2 2 ) + A ( x 1 ) exp ( j k 2 z x 1 2 ) exp ( j k z x 1 x 2 ) d x 1 ,
t ( x 2 ) = n = + K n exp ( j n 2 π x 2 / d ) K n = 0 d t ( x 2 ) exp ( j n 2 π x 2 / d ) d x 2 .
t ( x 2 ) = K 0 + K 1 exp ( j 2 π x 2 / d ) .
u ( x 3 ) = 1 j λ z 1 exp ( j k 2 z 1 x 3 2 ) + u ( x 2 ) t ( x 2 ) exp ( j k 2 z 1 x 2 2 ) exp ( j k z 1 x 2 x 3 ) d x 2 .
u ( x 3 ) = 1 j λ z 1 1 j λ z exp ( j k 2 z 1 x 3 2 ) { K 1 + A ( x 1 ) exp ( j k 2 z x 1 2 ) + exp [ j k 2 x 2 2 ( 1 z 1 + 1 z ) ] exp [ j 2 π x 2 ( x 1 λ z + x 3 λ z 1 1 d ) ] d x 2 d x 1 + K 0 + A ( x 1 ) exp ( j k 2 z x 1 2 ) + exp [ j k 2 x 2 2 ( 1 z 1 + 1 z ) ] exp [ j 2 π x 2 ( x 1 λ z + x 3 λ z 1 ) ] d x 2 d x 1 } .
+ exp ( j k 2 m x 2 ) exp ( j 2 π ξ x ) d x = λ m exp ( j π 4 ) exp ( j π λ m ξ 2 ) ,
u ( x 3 ) = 1 i λ f 1 exp ( j k 2 f 1 x 3 2 ) { K 1 exp ( j 2 π z d f 1 x 3 ) + A ( x 1 ) exp ( j k 2 f 1 x 1 2 ) exp ( j 2 π x 3 λ f 1 x 1 ) exp ( j 2 π z 1 d f 1 x 1 ) d x 1 + K 0 + A ( x 1 ) exp ( j k 2 f 1 x 1 2 ) exp ( j 2 π x 3 λ f 1 x 1 ) d x 1 } .
t l ( x 3 ) = exp ( j k 2 f 1 x 3 2 ) .
u ( x 4 ) = 1 j λ f 1 exp ( j k 2 f 1 x 4 2 ) + u ( x 3 ) t l ( x 3 ) exp ( j k 2 f 1 x 3 2 ) exp ( j k f 1 x 4 x 3 ) d x 3 .
u ( x 4 ) = 1 j λ f 1 [ K 0 A ~ ( x 4 λ f 1 )  +  K 1 exp ( j 2 π z f 1 d x 4 ) A ~ ( x 4 λ f 1 1 d ) ] ,
u ( x 5 ) = K 0 A ( x 5 f 1 / f 2 ) + K 1 exp ( j 2 π f 1 x 5 / f 2 d ) A [ ( x 5 f 2 λ z / f 1 d ) f 1 / f 2 ] .
I ( x 5 ) = | u ( x 5 ) | 2 .

Metrics