Calibrated Phase-Shifting Digital Holographic Microscope Using a Sampling Moir é Technique

: A calibrated phase-shifting digital holographic microscope system capable of improving the quality of reconstructed images is proposed. Phase-shifting errors are introduced in phase-shifted holograms for numerous reasons, such as the non-linearity of piezoelectric transducers (PZTs), wavelength ﬂuctuations in lasers, and environmental disturbances, leading to poor-quality reconstructions. In our system, in addition to the camera used to record object information, an extra camera is used to record interferograms, which are used to analyze phase-shifting errors using a sampling Moir é technique. The quality of the reconstructed object images can be improved by the phase-shifting error compensation algorithm. Both the numerical simulation and experiment demonstrate the effectiveness of the proposed system.


Introduction
It is arduous to measure three-dimensional (3-D) objects via conventional optical microscopy due to the finite focal depth of an imaging lens. Although confocal laser scanning microscopy (CLSM), which can achieve high-precision 3-D measurement, has been developed, it cannot be used to measure a fast and dynamic object owing to its long scanning time. Digital holographic microscopes [1][2][3][4][5][6] can capture the 3-D information of an object without depth scanning and can focus at arbitrary depths. They can also obtain both intensity and phase information from holograms and thus can measure materials that are mostly transparent, such as biological cells and glass. Thanks to these advantages, the digital holographic microscope has been widely used.
The recording method of digital holographic microscopes is the same as that of digital holography. It is grossly divided into two groups: in-line [7,8] and off-axis [9] recording methods, which depend on the angle between the object wave and the reference wave received by the image sensor. In the off-axis recording method, the spatial-frequency spectra of an object and the 0th-order diffracted wave are not easily separated when large objects are recorded and reconstructed [10]. Therefore, the in-line digital holography using phase-shifting calculation is much more efficacious than the off-axis method. In general, a phase-shifting device such as a mirror mounted on a piezoelectric transducer (PZT) is applied to shift the phase of the reference wave to record multiple phase-shifted holograms for the phase-shifting calculation [11]. However, phase-shifting errors occur due to environmental disturbances, the non-linearity of the PZT, frame loss in the camera, and the wavelength fluctuation of lasers. Numerous methods have been proposed to solve this problem, such as a closed loop phase control system using a single photodiode [12], an algorithm using a random phase-shifting method [13], and self-calibrating algorithms utilizing a statistical method [14]. However, these techniques suffer from certain disadvantages. For example, the output power fluctuation of a laser reduces the detecting precision of phase-shifting errors in the closed loop phase control system. It also dramatically influences the quality of the reconstructed image in the random phase-shifting method. The self-calibrating algorithms are powerless for measuring objects that are mostly transparent because the object must be assumed as sufficiently random in the diffraction field.
Hence, we propose a calibrated phase-shifting digital holography (CPSDH) system that is able to improve the quality of reconstructed images by detecting phase-shifting errors using a sampling Moiré method. The effectiveness of this technique was demonstrated with respect to a reflective object in a preliminary experiment [15]. We are the first to present such a technique. In comparison with conventional phase-shifting digital holographic microscopes, the proposed method markedly improves the quality of the reconstructed images.

A Calibrated Phase-Shifting Digital Holographic Microscope System
The sketch of the calibrated phase-shifting digital holographic microscope system for transparent objects is shown in Figure 1. The diameter of the beam used to illuminate the sample is small, such that a beam can be extracted by Beam Splitter (BS) 1 before the beam passes through an expander. The extracted beam passes through the object and arrives at Camera 1 via the microscopic system. The collimated beam is divided into two arms by BS 2. One arm works as the reference beam, and the other one works as the object beam for Camera 2. The beam reflected from the mirror mounted on a PZT is subdivided into two beams by BS 4. One is reflected by BS 5 to arrive at Camera 2 and interferes with the beam reflected from mirror (M) 2. On the other hand, the beam reflected from M 4 and BS 6 arrives at Camera 1 and interferes with the object wave. Hence, the Camera 1 records a hologram including the object information, while Camera 2 records an interferogram with a periodic repetitive fringe pattern generated by two plane waves. The hologram recorded by Camera 1 and the interferogram recorded by Camera 2 will synchronously change if the phase of the reference beam is shifted by the PZT. The sampling Moiré technique is capable of accurately measuring minute displacement from a single repetitive fringe pattern and the accuracy of the technique can theoretically achieve 1/500 of an interference fringe pitch [16][17][18]. Therefore, we introduce the sampling Moiré method to analyze the interferograms recorded by Camera 2 to evaluate the phase-shifting errors. Finally, the 3-D object images are reconstructed by the phase-shifting error compensation algorithm [15].

The Sampling Moiré Technique
The principle of the sampling Moiré technique [17] is represented in Figure 2. If the pitch of the captured grating pattern in the image sensor plane is supposed as P, then the recorded intensity of the grating can be described as In addition, the shifting amount of the PZT is not equal to the theoretical values, because the surface of the mirror mounted on the PZT is not strictly perpendicular to the incoming beam in general. If ϕ s is the set phase-shifting amount of the hologram used for phase-shifting calculation, and the corresponding shifting amount of the PZT is d t , then many phase-shifted interference fringe patterns are recorded by Camera 2, and the average phase difference ∆ϕ m between two neighboring interference fringe patterns can be calculated using the sampling Moiré technique. The real shifting amount of the PZT can be determined using (1)

The Sampling Moiré Technique
The principle of the sampling Moiré technique [17] is represented in Figure 2. If the pitch of the captured grating pattern in the image sensor plane is supposed as P, then the recorded intensity of the grating can be described as  Figure 3 illustrates the principle of the sampling Moiré technique to determine the phase-shifting errors. In the calibrated phase-shifting digital holography system, several interferograms with a periodic repetitive fringe pattern and phase-shifted holograms including the object information are captured by two synchronized cameras. The phase-shifting amount of two neighboring holograms can be calculated from the phase difference of the Moiré fringe patterns. Therefore, if the phaseshifting errors are occurred in the recorded holograms, they will be accurately calculated by Here, ∆ is the phase-shifting amount calculated from the phase difference of two neighboring Here, A g is the amplitude distribution of the grating pattern, B g is the intensity of the background, φ g0 is the initial phase value at position x, and φ g is the phase distribution of the grating pattern. Multiple phase-shifted Moiré fringe patterns can be obtained through down-sampling and intensity interpolation processing. In general, an integer T that is close to P is applied for down-sampling.
T must be larger than or equal to 3 to calculate the phase distribution of the Moiré fringe patterns. The beginning of the down-sampling position is set to the first row of the recorded grating, the pixels at intervals of T-1 rows are extracted, and the vacant pixels are then interpolated using adjacent sampled pixels. T-phase-shifted Moiré fringe patterns can be obtained when the beginning of the down-sampling position increases from the first to the T-th row. The intensity of the phase-shifted Moiré fringe patterns can be represented as where A m , B m and φ m are the amplitude distribution, the intensity of the background, and the phase distribution of the Moiré fringe pattern, respectively. k is the number of the Moiré fringe patterns.
The phase distribution φ m of the phase-shifted Moiré fringe patterns can be obtained by a phase-shifting method [17,18], expressed as Figure 3 illustrates the principle of the sampling Moiré technique to determine the phase-shifting errors. In the calibrated phase-shifting digital holography system, several interferograms with a periodic repetitive fringe pattern and phase-shifted holograms including the object information are captured by two synchronized cameras. The phase-shifting amount of two neighboring holograms can be calculated from the phase difference of the Moiré fringe patterns. Therefore, if the phase-shifting errors are occurred in the recorded holograms, they will be accurately calculated by

Numerical Simulation
In the microscopic field, many specimens are mostly transparent, such as biological cells and glass. Hence, we suppose that the object in the numerical simulation is a transparent object. Images with dimensions of 1024 × 1024 pixel and a 3.45 μm pixel pitch were treated as the amplitude and phase distributions of the object, as shown in Figure 4a,b. The distance between the object and the Here, ∆ϕ n is the phase-shifting amount calculated from the phase difference of two neighboring interference fringe patterns. High-quality object images can be reconstructed using the phase-shifting error compensation algorithm [15].

Numerical Simulation
In the microscopic field, many specimens are mostly transparent, such as biological cells and glass. Hence, we suppose that the object in the numerical simulation is a transparent object. Images with dimensions of 1024 × 1024 pixel and a 3.45 µm pixel pitch were treated as the amplitude and phase distributions of the object, as shown in Figure 4a,b. The distance between the object and the image sensor was set to 0.5 mm. The maximum pixel value of the amplitude image was normalized to 255. The values of the phase distribution were set from −π to π. In the actual experiment, the angle between the object wave and the reference wave was difficult to adjust to zero despite the in-line digital holography. Therefore, we introduced a small angle between the object wave and the reference wave. Figure 4c shows an example of the generated hologram in which the interference fringes appeared, and part of the generated hologram is magnified in Figure 4d. The wavelength of the light source was assumed to be 532 nm. The phase-shifting errors at a maximum of 20% were randomly introduced when holograms and interferograms were generated. The four phase-shifted holograms-I(x, y; 0), I(x, y; π/2 + ∆δ 1 ), I(x, y; π + ∆δ 2 ), and I(x, y; 3π/2 + ∆δ 3 )-and four phase-shifted interferograms were obtained. The reconstructed images by using the conventional phase-shifting method [11] and the phase-shifting error compensation algorithm are illustrated in Figures 5 and 6. We can see that the residual interference fringes appear in Figures 5b and 6b because the phase-shifting calculation of the conventional method is incorrect owing to the phase-shifting errors. On the other hand, both the amplitude and phase images of the object were correctly reconstructed by the phase-shifting error compensation algorithm, as shown in Figures 5d and 6d. Here, the detected phase-shifting errors by the sampling Moiré technique were 0.2846, 0.0399 and 0.2869 rad, respectively. Additionally, the normalized root-mean-square errors (NRMSEs) of the reconstructed amplitude and phase images were calculated. The results are revealed in Table 1. Note that the NRMSE of the proposed method is not equal to zero because a little linear interpolation errors existed in the down-sampling processing [18]. However, the proposed method greatly reduced the errors even though interpolation errors occurred.

Experiment
The experimental conditions and results of the calibrated phase-shifting digital holographic microscope system are presented. The proposed system markedly improves the quality of the reconstructed image compared with the conventional phase-shifting digital holographic microscope.

Experimental Conditions
In the experiment, an Nd:YAG laser working at 532 nm and 30 mW output power was used as the light source. Two complementary metal-oxide semiconductor (CMOS) cameras (VCXU-50, Baumer, Inc., Frauenfeld, Thurgau, Switzerland) with a resolution of 2448 × 2048 pixel and a 3.45 µm pixel pitch were used to record the holograms and interferograms. A transmission-type test target (1951 USAF resolution test chart) was set as the object. The system utilized an infinity-corrected optical microscope that consists of an objective lens with 4 × magnification and a tube lens with a 200 mm focal length. The object image was placed close to the hologram plane via the infinity-corrected optical microscope, and four phase-shifted holograms with π/2 phase-shifting amount were then recorded by driving a PZT (PAZ005, Thorlabs, Inc., Newton, NJ, USA).

Experimental Results and Discussion
One example of the recorded hologram including the object information and one interferogram used for calculating the phase-shifting errors are presented in Figure 7a,b, respectively. We continuously recorded 100 groups of four phase-shifted holograms and found that large phase-shifting errors occurred several times. One example of the reconstructed images is presented. The reconstructed distance was 0.5 mm. The amplitude images reconstructed by the conventional phase-shifting method [11] and the phase-shifting error compensation algorithm [15] are revealed in Figure 8a,c, and the magnified images of the areas indicated in Figure 8a,c are represented in Figure 8b,d, respectively. Furthermore, the phase images were also reconstructed by the conventional method and the proposed method. The results are shown in Figure 9. Figure 9a,c are the original reconstructed phase images, and Figure 9b,d are the magnified images of the areas indicated in Figure 9a,c, respectively. Here, the detected phase-shifting errors were 0.0763, 0.01945 and −0.9727 radian, respectively. As found in the numerical simulation, there are many more residual interference fringes in the images reconstructed by the conventional method than there are in the images reconstructed by the phase-shifting error compensation algorithm because of the large phase-shifting errors. Thus, the effectiveness of the proposed system is experimentally demonstrated. reconstructed distance was 0.5 mm. The amplitude images reconstructed by the conventional phaseshifting method [11] and the phase-shifting error compensation algorithm [15] are revealed in Figure  8a,c, and the magnified images of the areas indicated in Figure 8a,c are represented in Figure 8b,d, respectively. Furthermore, the phase images were also reconstructed by the conventional method and the proposed method. The results are shown in Figure 9. Figure 9a,c are the original reconstructed phase images, and Figure 9b,d are the magnified images of the areas indicated in Figure  9a,c, respectively. Here, the detected phase-shifting errors were 0.0763, 0.01945 and −0.9727 radian, respectively. As found in the numerical simulation, there are many more residual interference fringes in the images reconstructed by the conventional method than there are in the images reconstructed by the phase-shifting error compensation algorithm because of the large phase-shifting errors. Thus, the effectiveness of the proposed system is experimentally demonstrated.      Figure 9b,d. The test target is made of glass whose surface is extremely flat. Obviously, the fluctuation of phase values obtained by the conventional method is much greater than that obtained by the proposed method. In other words, the proposed system is capable of improving the quality of the reconstructed images to achieve a high precision phase measurement.   Figure 9b,d. The test target is made of glass whose surface is extremely flat. Obviously, the fluctuation of phase values obtained by the conventional method is much greater than that obtained by the proposed method. In other words, the proposed system is capable of improving the quality of the reconstructed images to achieve a high precision phase measurement.   Figure 9b,d. The test target is made of glass whose surface is extremely flat. Obviously, the fluctuation of phase values obtained by the conventional method is much greater than that obtained by the proposed method. In other words, the proposed system is capable of improving the quality of the reconstructed images to achieve a high precision phase measurement.

Conclusions
A calibrated phase-shifting digital holographic microscope system has been described. The proposed system used two synchronized CMOS cameras. One was used to record the holograms, which includes the object information, and the other one was used to record the interferograms for evaluating the phase-shifting errors. Both the numerical simulation and experiment demonstrated that the quality of the reconstructed image was greatly improved using the phase-shifting error compensation algorithm. Compared with the conventional phase-shifting digital holographic

Conclusions
A calibrated phase-shifting digital holographic microscope system has been described. The proposed system used two synchronized CMOS cameras. One was used to record the holograms, which includes the object information, and the other one was used to record the interferograms for evaluating the phase-shifting errors. Both the numerical simulation and experiment demonstrated that the quality of the reconstructed image was greatly improved using the phase-shifting error compensation algorithm. Compared with the conventional phase-shifting digital holographic microscope, the proposed system is more stable because of its ability to detect phase-shifting errors. Thus, the proposed system can be applied in various industrial fields, such as product inspection on production lines. Moreover, the cost of the system is low because the low-cost laser and PZT can be used in the proposed system. The digital holographic microscope will become more widespread.