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Abstract
Random phase-shifting digital holography based on a self-calibrated system is proposed. In the proposed method, the hologram and the calibration interference fringes can be recorded simultaneously in a single image based on the space-division-multiplexing technique. Three randomly phase-shifted holograms and corresponding interference fringes are recorded, and the phase-shifting amount between each two adjacent holograms is calculated by the sampling Moiré method from the calibration interference fringes. A reflective object is used to demonstrate the effectiveness of the proposed method in the numerical and experiment.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Digital holography [1–3] is a powerful technique for three-dimensional (3-D) measurement because of its ability to acquire both amplitude and phase information of the objects. There are essentially two types of ways to overcome an inherent issue in digital holography that the conjugate image and the zeroth-order diffracted wave are superimposed on the object image, which deteriorates the image quality. One way is called off-axis digital holography [4,5]. In this method, a large angle between the reference wave and object wave is introduced when recording a hologram. The spectra of the spatial-frequency for the zeroth-order diffracted wave, the conjugate image, and the object image will be separated in the direction of the off-axis angle. Then the object image can be easily reconstructed by extracting the spectrum in spatial-frequency of the object image. However, only a small part of bandwidth can be used for image reconstruction, so that the off-axis method’s measurable area is extremely narrow. Another way is to utilize the phase-shifting method to remove the conjugate image and the zeroth-order diffracted wave efficiently.
In the general phase-shifting digital holography [6–9], the phase-shifting amount must be known precisely. A piezoelectric transducer (PZT) is typically used as the phase shifter to record multiple phase-shifted holograms. However, the phase-shifting errors could inevitably be introduced in phase-shifted holograms due to the nonlinearity of the PZT and environmental disturbances. To overcome the problem, numerous phase-retrieval algorithms based on statistics that can estimate the phase-shifting amount from arbitrary phase-shifted interference patterns have been proposed [10–12]. Unfortunately, these methods are formidable to use for an object wave, which is not fully random or zero mean in the diffraction field. A method that sets a reference object beside the measured object to detect the phase-shifting amount was proposed [13]. Unfortunately, this method is arduous for digital holography because of the enormous difference in the reflectivity between the reference object and the measured object. Furthermore, although a closed-loop phase control system [14] and dual camera calibrated system [15,16] were developed to calibrate phase-shifting errors, an additional electronic device such as a single photodiode or camera is required in the systems.
To eliminate this inconvenience, we proposed a calibrated phase-shifting digital holography based on the space-division multiplexing technique, which can detect phase-shifting errors without using any additional electronic devices [17]. In the previous study, four phase-shifted holograms were required for the phase-shifting calculation, and a transparent object was used to demonstrate the effectiveness of the method. Note that it is easy to control the range of the object wave of a transparent object on the image sensor to avoid disturbing the calibration wave.
In this paper, we propose a random phase-shifting digital holography based on a self-calibrated system and records a reflective object for the first challenge. It is well known that the investigation of phase-shifting digital holography has entered to two-step phase-shifting method [8]. However, the power fluctuation of the laser could strongly influence the quality of the reconstructed image in the two-step phase-shifting method. Thus, we use the three-step random phase-shifting method [18] to confirm the effectiveness of the proposed method in the numerical simulation and experiment.
2. Principle
2.1 Random phase-shifting method
Assuming that the complex distribution of an object wave at the image sensor plane is $O({x,y} )= A({x,y} )\textrm{exp}[{i\phi ({x,y} )} ]$, the complex distribution of reference wave is ${R_j}({x,y} )= {A_R}\exp [{i{\delta_j}({x,y} )} ]$, $({j = 1,2,3} )$, the intensity distribution of the recorded holograms can be described as
Then, the amplitude and phase distributions of the object wave at the object plane are obtainable through the diffraction integral calculation. To calculate the complex distribution of the object wave correctly, all the ${\delta _j}$ should be known precisely. Therefore, we proposed a calibrated phase-shifting digital holography based on the space-division multiplexing technique, which can detect the phase-shifting amount precisely.
2.2 Calibrated phase-shifting digital holography
The principle of the calibrated phase-shifting digital holography based on the space-division multiplexing technique is shown in Fig. 1. Make the reference wave and object wave cover the entire area of the image sensor, and the calibration wave impinges a small area of the image sensor. Furthermore, the intensities of the object and calibration waves are adjusted to low and high, respectively. Such an adjustment creates a recorded image including two types of information: one is the hologram that is used to reconstruct the object image, and the other is the interference fringes that are used to detect the phase-shifting amount, as indicated in the left-hand side of Fig. 1.
Fig. 1. Principle of the calibrated phase-shifting digital holography based on the space-division multiplexing technique.
The phase-shifting amount only depends on the change in the optical length of the reference wave. Consequently, the phase-shifting amounts of the image-reconstruction and calibration areas will be changed synchronously. Then the phase-shifting amount between two consecutive phase-shifted holograms can be calculated from the calibration areas by certain image-processing methods, such as the sampling Moiré method [19–21] and windowed Fourier transform method [22]. The accuracy of the sampling Moiré method can theoretically reach 1/1000 of the interference fringe pitch [21], and recently, we confirmed that the sampling Moiré method has the powerful ability to resist random noise [23]. Therefore, we applied the sampling Moiré method to detect the phase-shifting amount from the interference fringes. The maximum measurable displacement by the sampling Moiré method is half the pitch of the fringe patterns. Therefore, the phase-shifting amount between any two consecutive images should be small than π. First, the interference fringes in the calibration areas are extracted. Next, multiple phase-shifted Moiré fringes can be obtained by down-sampling and intensity interpolation processing. Then three phase distributions that are required for Eq. (4) can be obtained through the discrete Fourier transform algorithm [19].
3. Numerical simulation
A numerical simulation was conducted to verify the effectiveness of the proposed method. In the general phase-shifting digital holography, the phase-shifting amount is set as a constant. To achieve a convenient calculation, we set the phase-shifting amount to π/2, and the phase value of the reference wave in the first image was set as zero. It means that the theoretical phase-shifting amounts in the second and third images were π/2 and π, respectively. However, the phase-shifting amount is formidable to achieve precisely due to the precision of the phase shifter. Then the phase-shifting errors at a maximum of 20% were randomly introduced in the holograms and the interferograms. Here, the introduced true phase-shifting amount of the reference wave in the second and third images were 1.8585 and 2.9927 radians, respectively.
Besides, some random noise at a maximum of 20% was also added in the calibration area, because it is inevitable for a part of the object wave to enter the calibration area. The images with 1024×1024 pixels of 3.45µm pixel pitch shown in Figs. 2(a) and 2(b) were treated as the object. The pixel value of the amplitude image was from 10 (black area) to 255 (white area), and the phase distribution was a random pattern. The phase value was ranged from -π to π, and the histogram of the phase image is indicated in Fig. 2(c). The wavelength of the laser and the distance between the object and image sensor were assumed to be 473 nm and 100 mm, respectively. Then three phase-shifted images at the image sensor plane were generated, as described in Figs. 2(d)–2(f).
Fig. 2. Simulated object. (a) and (b) are amplitude and phase distributions, (c) is the histogram of the phase distribution, (d)-(f) are the phase-shifted images, respectively.
The detected phase values of the reference wave by the sampling Moiré technique were 1.8581 and 2.9919. Here, we call the method that calculates the object image by the Eq. (4) with the assumed the phase-shifting amounts π/2 and π, as the conventional method. On the other hand, the method that reconstructs the object image by the Eq. (4) with the detected the phase-shifting amounts is called as the proposed method. Then the object images were reconstructed by the conventional method and the proposed method. The numerical simulation results are represented in Fig. 3.
Fig. 3. Simulation results. (a) and (b) are the amplitude and phase distributions obtained by the conventional method; (c) and (d) are the amplitude and phase distributions reconstructed by the proposed method; (e)-(h) are the magnified images of the areas indicated in (a)-(d), respectively.
Obviously, the quality of the reconstructed images by the proposed method is much better than that obtained by the conventional method. In addition, the root-mean-square errors (RMSEs) of the reconstructed images were calculated and the results are presented in Table 1. The effectiveness of the proposed method is numerically confirmed by the simulation.
Table 1. RMSE results of the amplitude and phase obtained by the conventional and the proposed methods.
4. Experiment
4.1 Optical setup
The experimental setup of the random phase-shifting digital holography measurement system is presented in Fig. 4. The collimated beam was divided into two beams by a beam splitter (BS) 1, one of which passed through BS 2 for illuminating the object. The other beam worked as the reference wave whose optical length was shifted by a PZT (including a mirror). The beam reflected from BS 2 worked as a calibration wave that passed through a slit to make it impinge a small area of the image sensor. By adjusting the intensity of the object wave, the reference wave, and the calibration wave, an image data which included the hologram and interference fringes were recorded, as described in Fig. 5. In the experiment, a Nd:YAG laser with a wavelength of 473 nm and output power of 50 mW was utilized as the light source. A complementary metal-oxide semiconductor camera (VCXU-50, Baumer, Inc.) with a resolution of 2448 × 2048 pixels and a 3.45-µm pixel pitch was used for recording three phase-shifted holograms and interference fringes. To compare with the conventional method, the phase-shifting amount was set to π/2.
Fig. 4. Optical setup of the proposed system. BS1–5: beam splitter, BE: beam expander, M1-4: Mirror, L: lens.
4.2 Experiment results and discussion
A Japanese 100-yen coin was set as the object at the position 280 mm away from the image sensor, and fixed on a rotation stage, as presented in Fig. 6. A PZT (PAZ005, Thorlabs, Inc.) with a resolution of 5 nm in the closed-loop mode was used as the phase shifter. The shift amount of the PZT was set to 60 nm, equivalent to 1.594 radian in theory. Then the phase-shifting of ${\delta _2},{\delta _3}$ were 1.594 and 3.188 radians. We recorded three phase-shifted images and reconstructed the object images by using the conventional method and the proposed method. The detected ${\delta _2},{\delta _3}$ were 1.982 and 3.917 radians. The reason for causing the difference between the set values and detected values mainly has two respects. On the one hand caused by the precision of the PZT, on the other hand was that the surface of the mirror mounted on the PZT was not strictly perpendicular to the incoming beam. The amplitude images of the entire object are represented in Figs. 7(a) and 7(b). Figures 7(c) and 7(d) show the magnified images of the areas indicated in Figs. 7(a) and 7(b). The right side of the reconstructed image is the conjugate image area. In theory, the conjugate image should be eliminated completely by phase-shifting calculation. However, the signal of the conjugate image was residual on the right side of the image that reconstructed by the conventional method. The profiles of the lines located in the conjugate image area in Figs. 7(c) and 7(d) were compared and the results are indicated in Fig. 7(e). Furthermore, we reconstructed the object image in the opposite direction (−280 mm), as described in Fig. 8. As can be seen, the conjugate image which symmetrically appears with the object image in the opposite direction could be observed clearly in the conventional reconstruction. On the other hand, the conjugate image was completely removed because of the correct phase-shifting calculation. The experiment provides a reasonable capability for high-quality image reconstruction by the proposed method.
Fig. 7. Experiment results: (a) and (b) are the amplitude images reconstructed by the conventional method and the proposed method, respectively; (c) and (d) are the magnified images in the areas indicated in (a) and (b), respectively; (e) is the profiles of amplitude values along lines indicated in (c) and (d), respectively.
Fig. 8. Inverse reconstruction of the amplitude image: (a) the conventional method, (b) the proposed method.
It is well known that the phase measurement is an important application of digital holography in experimental mechanics. Here, we simply manually rotated the stage to give a bitty rigid-body displacement to the object, and recorded phase-shifted holograms before and after rotations. The amplitude images reconstructed by the conventional method and the proposed method, and the contrast of the reconstructed amplitude images was adjusted under the same conditions to observe the conjugate image clearly, as shown in Figs. 9(a) and 9(b). The conjugate image can be observed clearly in the reconstructed image by the conventional method due to the phase-shifting errors. An example of the reconstructed phase image is shown in Fig. 9(c). The phase of the conjugate image is arduous to observe directly from the phase image because of the rough surface of the object and speckle noise. To reduce the speckle noise, we utilized a windowed phase-shifting digital holography [24] to calculate the phase difference between before and after rotations by conventional method and the proposed method. The extracted phase difference images obtained by each method are displayed in Figs. 9(d) and 9(f). Besides, the magnified images in the areas indicated in Figs. 9(d) and 9(f) are indicated in Figs. 9(e) and 9(g), respectively. The phase difference image obtained by the proposed method is more precise and has little phase variation than the conventional method. There results demonstrated the effectiveness of the proposed method in both the image reconstruction and the phase distribution calculation.
Fig. 9. Experiment results of phase difference measurement: (a) and (b) are the amplitude images reconstructed by the conventional method and the proposed method before the object rotated; (c) is the phase image reconstructed proposed method before the object rotated; (d) and (f) are the phase difference images obtained by the conventional method and the proposed method, respectively; (e) and (g) are the magnified images in the areas indicated in (d) and (f).
5. Conclusions
A random phase-shifting digital holography based on a self-calibrated system was firstly presented to record a reflective object. In the proposed method, only three randomly phase-shifted holograms and interference fringes were required, and the phase-shifting amount could be detected precisely without utilizing any additional electronic devices such as a single photodiode or camera. By the proposed system, high-quality image reconstruction and phase measurement can be achieved conveniently even if using an inexpensive PZT that cannot perform accurate phase-shifting. Therefore, it is expected to contribute to various fields such as micro-displacement measurement, 3D object recognition, cell measurement, and 3D surface measurement, etc.
Funding
Japan Society for the Promotion of Science (19K15003, 20H02038).
Disclosures
The authors declare no conflicts of interest.
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