## Abstract

Without the limitations of fringe number and fringe shape, a dual-channel simultaneous spatial and temporal polarization (DC-SSTP) phase-shifting interferometry system is proposed to achieve rapid and accurate phase retrieval through only one-time phase-shifting procedure with unknown phase shifts. First, an arbitrary phase shifts is simultaneously introduced into two channels of DC-SSTP system by a spatial light modulator (SLM). Second, by performing the subtraction operation between each pair of phase-shifting interferograms captured in the same channel, the background deduction of interferogram can be achieved easily, so the accurate phase can be retrieved rapidly. Especially, it is found even if the fringe number in interferogram is less than one, the proposed DC-SSTP method still reveals high accuracy of phase retrieval. Both the simulation and experimental results demonstrate the outstanding performance of proposed DC-SSTP method in phase measurement.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Phase-shifting interferometry (PSI) [1], showing the advantages of high accuracy, non-intervention, non-damage, full field and rapid speed, has been widely used in optical components detection, fluid dynamics, biological cell detection, refractive index measurement, digital holography and quantitative phase imaging, and many other fields [1–6]. Though the current temporal multi-step PSI method can conveniently achieve phase retrieval with high accuracy, it is needed that a sequence of temporal interferograms are captured in different time points, and the phase shifts is known or uniformly distributed in the integer period [7–10]. So the corresponding accuracy of phase retrieval is easily affected by external disturbance, including the air movement and laser power fluctuation. In simultaneous spatial PSI method, a orthogonal polarization phase-shifting procedure is achieved through using the polarization optical elements, in which three-frame or four-frame phase-shifting interferograms with the phase shifts of π/2 are simultaneously captured by one or more image sensors, then the phase retrieval is performed by three-step or four-step phase-shifting algorithm. Through this single-image sensor based spatial PSI method [11–13] can solve the problem of external disturbance, but it requires the special imaging components, and the corresponding accuracy is limited by the wavelength, incident angle and frequency resolution. Therefore, this single-image sensor based spatial PSI method is not suitable for the phase retrieval with large variation of spatial frequency. Moreover, although the simultaneous multi-image sensors based spatial PSI method, including four-channel method [14] and three-channel method [15], can make full use of spatial resolution of image sensor, but it requires many imaging sensors. Along with the development of PSI, some self-calibration algorithms based on the relationship between fringe patterns are proposed. Though these self-calibration methods can directly achieve the phase retrieval from the interferograms with arbitrary phase shifts [16–22], but they are very time-consuming, and the condition requirement for fringe number and fringe shape are harsh, in which the accuracy of two-step demodulation algorithm is still related with the effect of background deduction [20–22].

Stated thus, a dual-channel simultaneous spatial and temporal polarization (DC-SSTP) phase-shifting interferometry method is proposed, in which only one-time phase-shifting procedure with spatial light modulator (SLM) can simultaneously achieve an arbitrary phase shifts for two channels of DC-SSTP system. And by performing the subtraction operation between each pair of phase-shifting interferograms, the background deduction of interferograms can be removed easily, so the accurate phase can be retrieved rapidly. Following, we will introduce the proposed DC-SSTP method in detail.

## 2. Principle analysis

Figure 1 shows the principle sketch of proposed DC-SSTP method. The linearly polarized light emitted by a He-Ne laser with a wavelength of 632.8nm is attenuated by a variable neutral density attenuator (ND) and the half wave plate (HWP), and then goes through the non-polarized beam splitter (BS1), where the HWP is used to adjust the polarization direction. First, the reflecting beam is vertically illuminated on a liquid crystal SLM, in which the beam reflected from the SLM passes through the BS1 into the polarization beam splitter (PBS), and is separated into P and S polarization lights. The beam reflected from PBS is modulated by the sample and used as the object beam; the transmitted beam used as the reference beam, the microscope objective (MO1) is employed to image the sample on the CCD1 and CCD2. Second, the object beam and reference beam generate the orthogonal common-path superposition in the non-polarization beam splitter (BS2); after transmitting through the quarter wave plate (QWP), the transmission beam is split into two circular polarization beams by another non-polarization beam splitter (BS3) and two polarizers of P1 and P2 with the intersection angle of π/4 relative to the polarization direction, so two-pair of interferograms with the phase shifts of π/2, respectively captured by CCD1 and CCD2, can be simultaneously acquired.

In this DC-SSTP system, without placing any mechanical device in the light path, an unknown phase shifts is simultaneously introduced into two channels by SLM, so the stability of DC-SSTP interferometry system is better than that of conventional PSI system. In a word, using this DC-SSTP system, for each channel, a pair phase-shifting interferograms can be captured by this DC-SSTP system, we have that

where*I*

_{11}and

*I*

_{12}denote a pair of phase-shifting interferograms captured by CCD1,

*I*

_{21}and

*I*

_{22}are another pair of phase-shifting interferograms captured by CCD2, $a(x,y)$and $b(x,y)$ represent the background intensity and modulation amplitude, respectively; $\phi (x,y)$ denotes the measured phase; $x=({m}_{x}-{M}_{x}/2)\Delta x$ and $y=({m}_{y}-{M}_{y}/2)\Delta y$represent the pixel coordinates, in which the center of interferogram is thought as the original point; $\Delta x$ and $\Delta y$are the pixel interval along

*x*and

*y*direction, respectively; ${m}_{x}=1,2,3\cdots M{}_{x}$ and ${m}_{y}=1,2,3\cdots M{}_{y}$ denote the pixel order along the

*x*and

*y*direction, respectively; ${\theta}_{1\_1}$ and ${\theta}_{1\_2}$are the phase shifts corresponding to

*I*

_{11}and

*I*

_{21}, ${\theta}_{2\_1}$ and ${\theta}_{2\_2}$ are the phase shifts corresponding to

*I*

_{21}and

*I*

_{22}, so the relationship among these phase shifts can be expressed as:

And the measured phase can be obtained by:

From the above analysis, we can see that the proposed DC-SSTP method can achieve phase retrieval through implementing only one-time phase-shifting procedure with unknown phase shifts. That is to say, though the constant ${\theta}_{t}/2$ in Eq. (8) is unknown, but it will not affect the result and accuracy of phase retrieval. Consequently, the phase shifts does not need to be extracted in our method.

## 3. Numerical simulation and analysis

Numerical simulation is carried out to verify the effectiveness of proposed method. A sequence of simulated trapezoidal fringe patterns with uneven distribution are generated. For comparison, the proposed DC-SSTP method, principal component analysis (PCA) [17] and advanced least square iterative algorithm (AIA) [16] are used for phase retrieval. The size of simulated interferogram is set as 300 × 300 pixels, the pixel interval and pixel range are respectively set as $\Delta x=\Delta y=0.01\text{mm}$ and$x\text{,}y\in [-2.56,2.56]$; the background and modulation amplitude are$a(x,y)=120\mathrm{exp}[-0.1({x}^{2}+{y}^{2})]$and$b(x,y)=100\mathrm{exp}[-0.1({x}^{2}+{y}^{2})]$, respectively; The temporal phase shifts ${\theta}_{t}$ introduced by SLM is 1.2rad. In addition, a Gaussian white noise with mean zero and standard deviation of 1 is added to each fringe pattern. All calculations are performed with the CPU of Intel(R) Core(TM) i3-2120 through using the MATLAB 6.5 software. Figures 2(a) and 2(b) show one-frame simulated fringe pattern and the corresponding theoretical phase distributions. Meanwhile, the phase deviations between the theoretical phase and the retrieved phases with DC-SSTP, PCA and AIA methods are shown in Figs. 2(c)-2(e), respectively. For clarity, Table 1 gives the root mean square error (RMSE) of the difference between the theoretical phase and the retrieved phase, peak to valley error (PVE), as well as the calculation time with the DC-SSTP, AIA and PCA algorithms, respectively. Where the RMSE denotes the square root of the ratio of the square sum of the phase deviation to the pixels number and the PVE is defined as the peak and valley value of the phase deviation. From Fig. 2 and Table 1, it is found that the proposed method reveals obvious advantages of rapid speed, high accuracy and good stability.

Following, some factors that might affect the performance of proposed DC-SSTP method are analyzed, including the temporal phase shifts introduced by SLM; the spatial phase shifts deviation (SPSD) between two channels, as well as the fringe number and fringe shape in interferogram. Three sequence simulated phase-shifting interferograms with plane wavefront of ${\phi}_{p}(x,y)=2\pi n{}_{f}(x+y)/10.24$, spherical wavefront of ${\phi}_{s}(x,y)=2\pi n{}_{f}({x}^{2}+{y}^{2})/13.1072$and complex wavefront of ${\phi}_{c}(x,y)=2\pi {n}_{f}peaks(x,y)/[max\{peaks(300)\}-min\{peaks(300)\}]$are respectively generated, in which the parameter $n{}_{f}$ denotes the fringe number in interferogram. For each sequence of 4-frame phase-shifting interferograms with different fringe number and fringe shape, the phase retrievals are calculated by DC-SSTP, PCA and AIA, respectively. The “*peaks*” represents the peaks function in Matlab, and $min\{peaks(300,300)\}$ and $min\{peaks(300,300)\}$ denotes the corresponding maximum and minimum of a 300 × 300 matrix, and the pixel interval and pixel range are respectively set as $\Delta x=\Delta y=0.01\text{mm}$ and$x\text{,}y\in [-2.56,2.56]$; the background and modulation amplitude are$a(x,y)=120\mathrm{exp}[-0.1({x}^{2}+{y}^{2})]$and $b(x,y)=100\mathrm{exp}[-0.1({x}^{2}+{y}^{2})]$, respectively.

In order to further present the accuracy variation of proposed DC-SSTP method with the temporal phase shifts, as shown in Fig. 3, three different type interferograms are employed to perform phase retrieval, in which the fringe number in interferogram$n{}_{f}$ is set as 3 and the temporal phase shifts ${\theta}_{t}$ is changed from 0.1 to 3.2 rad. Using the DC-SSTP, AIA and PCA methods, Figs. 4(a)-4(c) give the variation curves of RMSE of phase retrieval with the temporal phase shifts for different wavefront interferograms. We can see that the RMSEs of phase retrieval by the above three algorithms are decreased with the temporal phase shifts. For the plane wavefront interferogram, the DC-SSTP and PCA algorithms reveal good consistency and high accuracy when the temporal phase shifts is changed from 0.5 to 3.2 rad. For the complex or spherical wavefront interferograms, the accuracy of proposed DC-SSTP method is much better than that with AIA or PCA algorithms. Clearly, compared with AIA or PCA algorithms, the proposed DC-SSTP reveals obvious advantages of accuracy and stability regardless of the amount of temporal phase shifts.

Subsequently, we discuss the influence of the SPSD between the two channels on the accuracy of phase retrieval with the proposed method. Assuming that${\theta}_{t}=2rad$and$n{}_{f}=1$, and the SPSD is changed from −0.1rad to 0.1rad. Figures 5(a)-5(c) show the results of phase retrieval with the DC-SSTP, AIA and PCA methods, respectively. Note that for any type pattern fringes, the variation of RMSE with the DC-SSTP method is consistent with the SPSD between two channels, the highest accuracy can be reached if the SPSD is 0, i.e. the spatial phase shifts between two channels is $\pi /2$. In contrast, the accuracy of phase retrieval with PCA or AIA algorithm is changed with the fringe shape in interferogram; for the interferograms with complex wavefront interferogram or spherical wavefront, the accuracy of DC-SSTP method is still higher than that of PCA or AIA algorithm if the SPSD is changed in the range of 0.1rad.

Finally, we present the influence of fringe number and fringe shape on the accuracy of phase retrieval. Using the above interferograms shown in Fig. 3, and assuming that ${\theta}_{t}=1.2rad$and $n{}_{f}$ is changed from 0.1 to 10, a Gaussian white noise with mean zero and standard deviation of 1 is added to the interferogram. Using the DC-SSTP, AIA and PCA methods, Figs. 6(a)-6(c) show the variation curves of RMSE of phase retrieval with the fringe number for different wavefront interferograms, respectively.

We can see that the proposed DC-SSTP method still reveals high accuracy and remains almost unchanged in the accuracy of 0.021rad regardless of the fringe shape and fringe number while the accuracy of phase retrieval with PCA or AIA algorithm is low and changed with the fringe shape and fringe number. Especially, when the fringe number in interferogram is less than 1, it is found the PCA and AIA methods have low accuracy and great variation while the proposed DC-SSTP method still remains high accuracy.

## 4. Calculation and analysis of experimental interferogram

Experimental research is employed to verify the flexibility of proposed method, in which 4-frame phase-shifting interferograms of a Jurkat cell are captured for phase retrieval by using the proposed DC-SSTP method. The measured sample is a Jurkat cell, a cellular line of leukemic lymphocyte. The normal Jurkat cell has circular profile with better light transmittance. First, a DC-SSTP system is constructed to capture phase-shifting interferograms with an arbitrary temporal phase shifts. The size of interferogram is equal to 129 × 160 pixels, and the pixel interval of CCD camera is 10μm × 10μm. The reference phase (REF) is achieved by using Schwider-Hariharan five-step algorithm [8] from 5-frame phase-shifting interferograms. Figures 7(a) and 7(b) show one-frame experimental interferogram and the corresponding REF, respectively. The reconstructed phase maps achieved with the DC-SSTP, AIA and PCA methods are shown in Figs. 7(c)-7(e), respectively. Table 2 gives the RMSE of the difference between the REF and the retrieved phase, PVE, as well as the calculation time with the DC-SSTP, AIA and PCA methods, respectively.

As be seen in Fig. 7 and Table 2, the proposed method is much better than the other two methods both in accuracy or calculation time. Note that when the phase variation of interferogram is less than 2π, the proposed method still can achieve accurate phase. Figures 8(a) and 8(b) respectively show one-frame experimental interferogram with less than one fringe and the corresponding REF achieved through using AIA from 136-frame phase-shifting interferograms with size of 58 × 67 pixels. The reconstructed phase maps achieved with the DC-SSTP, AIA and PCA methods are shown in Figs. 8(c)-8(e). Clearly, the proposed method can work well with the RMSE of 0.0664rad while AIA and PCA methods have obvious distortion and deviation.

## 5. Conclusion

In this study, without the limits of fringe number and fringe shape, a novel DC-SSTP method is proposed, in which an unknown phase shifts is first simultaneously introduced into two channels, and for each channel, a pair of phase-shifting interferograms are captured. By performing the subtraction operation between each pair of phase-shifting interferograms, the background deduction of interferogram can be achieved easily, so the accurate phase can be retrieved rapidly. Compared with the current high accuracy phase retrieval methods, the proposed DC-SSTP method reveals outstanding advantages: By implementing only one-time phase-shifting procedure with unknown phase shifts, both the measurement and calculation procedures are simplified; the phase shifts does not need to be extracted; it is found even if the fringe number in interferogram is less than one, the proposed DC-SSTP method still reveals high accuracy of phase retrieval while the current methods have low accuracy and great variation. Therefore, this proposed DC-SSTP method will supply a useful tool for static and slow dynamic phase measurement. For the case of ultrafast lasers, the proposed method requires both the space matching and time synchronization. Besides, the coherent length and monochromaticity of the ultrafast lasers also are the limitations of the DC-SSTP method.

## Funding

National Natural Science Foundation of China (NSFC) (61475048, 61727814, 61575069).

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