Parallel two-step phase-shifting point-diffraction interferometry for microscopy based on a pair of cube beamsplitters is proposed. The first 45°-tilted cube beamsplitter splits object wave into two parallel copies: one copy is filtered by a pinhole in its Fourier plane to behave as reference wave, while the other one remains unchanged as object wave. The second cube beamsplitter combines the object and reference waves, and then split them together into two beams. Along with the two beams, two parallel phase-shifting interferograms are obtained in aid of polarization elements. Based on the proposed configuration, slightly-off-axis interferometry for microscopy is performed, which suppresses dc term by subtracting the two phase-shifting holograms from each other. The setup is highly stable due to its common-path configuration, and has been demonstrated to be suitable for measuring moving objects or dynamic processes.
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In the general type of interferometer, the object and reference waves travel in widely-separated paths and are, therefore, affected by environmental disturbance differently. Thus, if no proper precautions are taken, the fringe pattern in the observation plane is unstable . To solve such problem, one often resorts to common-path interferometers, where the object and reference waves pass the same optical path, thus the measurement is less affected by the environmental disturbance. Phase contrast method is one of the most popular common-path interferometry, and is widely used for investigation of biological features [2, 3]. Off-axis point-diffraction interferometry, another type of common-path interferometry, was proposed by Medecki  in 1996, and latter used to investigate the wave-front aberration . Popescu [6, 7] adapted the point-diffraction interferometry to microscopy and developed the first diffraction phase microscope. Recently, on-axis phase-shifting point-diffraction interferometry for microscopy was implemented in our group by using a pair of gratings .
Parallel phase-shifting interferometry records multiple phase-shifting interferograms with one exposure, thus it can keep real-time measurement ability with on-axis interferometer. Smythe and Moore , and Koliopoulos  employed multiple CCD cameras to record the phase-shifting interferograms simultaneously. However, the employment of multiple imaging sensors brings complexity to the system configuration. Another type of parallel phase-shifting technique is based on pixelated phase-shifting array [11, 12]. Besides, a diffraction grating or Wollaston prism was used to perform parallel phase-shifting interferometry [13–15].
In this paper, we combine the parallel phase-shifting technique with the point-diffraction interferometry to realize real-time phase microscopy, based on a pair of cube beamsplitters. The proposed setup maintains a common-path configuration, and is therefore insensitive to the environmental disturbance. Based on the proposed setup, the slightly-off-axis interferometry for phase microscopy is performed, which suppresses dc term by subtracting the two phase-shifting holograms from each other. Thus, the requirement on CCD resolving power to record the off-axis interferogram is reduced.
2. Experimental setup
The experimental setup of parallel phase-shifting point-diffraction interferometry for microscopy is sketched in Fig. 1 . A He-Ne laser with wavelength 632.8nm is used as light source. The intensity and polarization of the laser beam are controlled respectively by the neutral variable attenuator NF and polarizer P. The laser beam is spatially filtered and expanded by the beam expander BE, and then illuminates the specimen which is located in the front focal plane of the microscopic objective MO. After passing through the specimen, the object wave is magnified by the objective MO, and collimated by lens L 1, thus the magnified image appears in the back focal plane of the lens L 1. A pair of non-polarizing cube beamsplitters and a pinhole filter are placed between two lenses L 2 and L 3, which forms a 4f system. The first 45°-tilted cube beamsplitter BS 1 splits the frequency spectrum of the object wave into two copies. One copy is low-pass filtered by a pinhole to generate the reference wave, whereas the other copy remains unchanged and still behaves as the object wave. The polarizers P 1 and P 2 with horizontal and vertical polarizations are located on the paths of the object and reference waves, respectively.
Both the object and reference waves are split again by the second cube beamsplitter BS 2, so that two copies of the objective wave overlaps two copies of the reference wave. In each split beam, object wave is superposed with the reference wave, as illustrated in Fig. 2 . When the semi-reflecting layer of BS2 is parallel to the optical axis, the object and reference waves are parallel in both the two split beams . When the semi-reflecting layer of BS2 has an angle θ with respect to the optical axis, the object and reference waves will have also the angle θ with each other in the two split beams. A quarter-wave plate is placed behind the BS 2 to convert the orthogonally linearly polarized object and reference waves into orthogonally circular polarizations. Two polarizers in the polarizer-combination are located respectively on the two split beams, respectively. After transformed by lens L 3, the two split beams become off-axis beams. The object and reference waves interfere with each other in the two split beams, thus the parallel two-step phase-shifting interferograms come into being. If the two polarizers in the polarizer-combination have a polarization angle γ with respect to each other, the two interferograms will have phase-shift of 2γ according to the polarization phase-shifting analysis . The contrast of the interferograms can be controlled by rotating the polarizer P, and the relative intensity of the two interferograms can be adjusted by rotating the polarizer P 3. A CCD camera is located at the back focal plane of lens L 3 to capture the interferograms. The aperture A is used to restrict the imaging area of each split beams, which should be equivalent to or smaller than half of the imaging area of CCD sensor.
3. Reconstruction method
Based on the proposed setup, slightly-off-axis point-diffraction interferometry for microscopy can be performed with the parallel two-step phase-shifting mechanism. For simplicity, the magnified object wave is denoted with O test(x, y). The object wave is split by BS 1 into two copies, one of which is filtered by the pinhole in the Fourier plane and behaves as the reference wave. Considering a tilted angle introduced by BS 2, The reference wave in the two split beams can be expressed as:17] proposed that the dc terms |O test|2 + |R| 2 in the two phase-shifting interferograms can be suppressed by subtracting I 2(x,y) from I 1(x,y):
To separate the real image OR* from the twin image O*R, we perform Fourier transformation on (I 1-I 2)R D. Here, R D=1/A rexp(iΚx) denotes the digital reference wave. The carrier-frequency Κ can be determined from the fringe frequency of the slightly-off-axis interferograms, and A r can be regarded as the unit 1. The spectrum of the real image OR*R D is located at the centre of the frequency domain, and it is separated from that of the twin image. Thus, the complex amplitude of object wave can be calculated with the Angular Spectrum Method:Fig. 3(c) . Within the selected area, (ξ,η)=1; otherwise, (ξ,η)=0. The size of the selected area can be determined by the resolution of the microscopic objective MO. It can be seen from Eq. (4) that the phase-shift δ introduce only a constant phase to the object wave. The constant phase can be offset when subtracting the setup residual phase (in absence of specimen) from the measured phase. Thus the phase-shift δ will not influence the phase distribution of the tested specimen.
Three schemes of point-diffraction interferometry, i.e., the traditional off-axis scheme [4–7], the proposed slightly-off-axis scheme, and the on-axis scheme  are compared here. Assuming the frequency spectrum of tested object lies in [-νmax, νmax], then the twin image, dc term and real image of the interferogram will have spectrum distributions of [-Κ-νmax, -Κ+νmax], [-2νmax, 2νmax], and [Κ-νmax, Κ+νmax], respectively . The traditional off-axis scheme requires the carrier-frequency Κ≥3νmax to separate the spectra of the three terms, thus requires the CCD sampling frequency νCCD≥4νmax to record the off-axis interferogram, considering νCCD≥Κ+νmax. The slightly-off-axis scheme only requires the carrier-frequency Κ≥νmax to separate the real and twin images since the dc term is suppressed by I 1-I 2, resulting in the low requirement of CCD sampling frequency νCCD≥2νmax. The on-axis scheme (Κ = 0) with phase-shifting mechanism does not rely on spectra separation of different terms, thus it only requires the CCD sampling frequency νCCD≥νmax to record the real image. In other words, for a given CCD camera, the traditional off-axis scheme only have maximum 1/4 resolving power of CCD camera; the slightly-off-axis scheme can achieve 1/2 resolving power of CCD camera; the on-axis scheme can make full use of the resolving power of CCD. However, the on-axis scheme  cannot be used for real-time measurement due to the sequential phase-shifting operation. Above all, the slightly-off-axis scheme is the intermediate solution between the off-axis and on-axis schemes, i.e., it has real-time measurement ability and low requirement on CCD resolving power.
4. Experimental results
The experiment of slightly-off-axis point-diffraction interferometry for microscopy is performed based on the experimental setup depicted in Fig. 1. A microscopic objective with magnification M=25X and numerical aperture NA=0.4 is employed in the experiment. A CCD camera with 1024(H)×768(V) pixels and pixel size 4.65μm(H)×4.65μm(V) is used to record the interferograms. The angle θ is set to 0.44°, corresponding to the carrier-frequency of 12 fringes/mm, guaranteeing the separation of the frequency spectra of the real and twin images.
In the first experiment, a rectangular phase step (70μm×20μm) was used as specimen. The phase step was etched in silica glass plate, and the thickness corresponds to an optical path difference (OPD) of λ/4 (λ=632.8 nm). Based on the proposed setup, two interferograms with phase-shift of π/2 for the specimen were obtained through one exposure, and given in Fig. 3(a). It is seen that the object in the two interferograms are reverse with each other due to the beam-splitting scheme of the cube beamsplitter . The phase step is modulated by low-frequency fringes because of the slightly-off-axis scheme. Figure 3(b) shows a part of the frequency spectrum of (I 1-I 2)R D, where the dc terms of the interferograms is eliminated, and the spectra of real and twin images are separated with each other. By using Eq. (4), the complex amplitude of the object wave was retrieved. The reconstructed OPD of the specimen is given in Fig. 3(c), after the OPD of setup is subtracted. The reconstructed OPD is consistent with the real value, and the noise is low and approximately constant.
In the second experiment, we demonstrate the application of the proposed setup for real-time measurement of moving objects or dynamic processes. PMMA beads (refractive index n=1.49) suspended in fluid water were used for experiment. It was measured continuously for 40s at internal of 1s. The movie (Media 1) of Fig. 4(a) contains the recorded parallel phase-shifting interferograms of the dynamic specimen. The exposure time for each interferogram is set to 5ms. Using the reconstruction method described in section 3 and the least-square phase unwrapping algorithm, the OPD of the specimen is retrieved and shown in the movie (Media 2) of Fig. 4(b), which shows the movement of the PMMA beads (a large one and a small one) together with the fluctuation of the fluid water.
To assess the stability of the proposed setup against the environmental disturbance thus quantify its sensitivity to optical path variation, we perform continuously phase measurement in absence of any specimen for a period of 35mins at intervals of 1min. The temporal optical path difference (OPD) fluctuation associated with a randomly-selected point over the 35mins is given with the black rectangles in Fig. 5 . The phase of the point was averaged over an area of 1.5μm×1.5μm, which corresponds to the transverse resolution of the microscope. It is known from Fig. 5 that the standard deviation of the OPD fluctuations for the point during 35mins is 4.2nm, which means the setup has long-term OPD stability. Besides, the repeatability of the setup, which is defined as the standard deviation of the difference between two successive measurements OPD i-OPD i-1, was also tested and the results were shown with triangles in Fig. 5. The average standard deviation on the OPD change between two successive measurements is 3nm (equivalent to λ/210), which means the repeatability of the setup is high. The high stability and repeatability attribute to the common-path configuration of the setup.
It should be noted that the proposed setup can be also used for on-axis parallel phase-shifting point-diffraction interferometry. When the semi-reflecting layer of BS2 is placed parallel to the optical axis, two on-axis parallel two-step phase-shifting interferograms can be obtained with one exposure. Compared with the off-axis scheme, the on-axis scheme can make full use of CCD camera spatial resolving power and thus can capture finer sample spatial details. However, it requires that the intensity of reference wave is pre-measured and no less than two times maximum intensity of the object wave to reconstruct the complex amplitude of the object wave from the two phase-shifted interferograms [18, 19], thus the visibility of the interferograms is reduced.
We have proposed a common-path point-diffraction interferometry for microscopy, where two parallel two-step phase-shifting slightly-off-axis interferograms are obtained through one exposure, thus the phase of the specimen can be measured simultaneously. In the setup, a tilted cube beamsplitter splits the object wave into two parallel copies: one of them is pinhole filtered in its Fourier plane to behave as reference wave, while the other one remains unchanged as object wave. The second tilted cube beamsplitter combines the object and reference waves, and split them into two beams at the same time. Together with the polarization phase-shifting unit, two phase-shifting interferograms are obtained along with the two beams. The setup has demonstrated to measure moving objects or dynamic processes with high stability and low requirement on CCD resolving power. The proposed method can be used for investigation of dynamic biological processes.
This research is supported by the Natural Science Foundation of China (NSFC) (61077005, 10874240), and the Chinese Academy of Sciences (CAS)/State Administration of Foreign Experts Affairs of China (SAFEA) International Partnership Program for Creative Research Teams.
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