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Abstract
Motionless optical scanning holography (MOSH) has been proposed for three-dimensional incoherent imaging in single-pixel holography with a simple optical setup. To reduce the measurement time in MOSH, a spatially divided phase-shifting technique is introduced. The proposed method realizes measurements four times faster than the original MOSH, owing to the simultaneous lateral and phase shifts of a time-varying Fresnel zone plate. A hologram reproduced by the proposed method forms a spatially multiplexed phase-shifting hologram similar to parallel phase-shifting digital holography. The effectiveness of the proposed method is numerically and experimentally verified.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Digital holography is a fundamental three-dimensional (3D) imaging technique, various applications of which have been proposed [1–8]. In general, a two-dimensional (2D) image sensor is used to capture an interferometric fringe pattern. Recently, single-pixel imaging (SPI) has attracted much attention because of its applicability to low-photon [9,10] and broader spectral range conditions [11–13]. SPI has also been applied to digital holography, which is called single-pixel holography (SPH). One example of an SPH technique based on ghost imaging [14–16], digital ghost holography [17], detects a complex amplitude distribution through a correlation function. Basis-transformed imaging-based SPH [18] has also been proposed [19] and applied to various applications [20,21]. As an alternative method to two-axis SPH, single-point holography [22,23] has been proposed, which mitigates the problem of a complicated optical configuration for SPH. However, these methods cannot be applied to incoherent imaging, owing to their requirement of a coherent light source. One solution to realize incoherent imaging by SPH is optical scanning holography (OSH) [24,25], which can reproduce an incoherent hologram from detected signals. Therefore, 3D fluorescent imaging has been achieved by OSH [26–28]. In addition, spatial coherence can be varied by locating an aperture directly in front of a photodetector [29,30]. In this way, quantitative phase imaging has already been realized [31]. Owing to its functional effectiveness, OSH has been applied to various techniques [32–35]. However, OSH has the problem of a complicated optical setup, owing to the requirements of an interferometer, phase-shifter, and 2D mechanical scanning architecture. To reduce this complexity, we have proposed a motionless OSH (MOSH) technique [36]. MOSH utilizes the polarization property of a liquid crystal on a silicon spatial light modulator (LCoS-SLM), which acts as an interferometer, a phase-shifter, and a 2D mechanical scanning architecture. Although the optical setup of MOSH is simple and compact because of its in-line configuration, MOSH has the problem of a longer measurement time compared with other SPI approaches, because phase-shifted signals are required to remove unnecessary terms in a hologram. Time-consuming measurements is also a problem for other SPH methods.
In this paper, to suppress the problem of long measurement times, a spatially divided phase-shifting technique is introduced to MOSH. This method is termed spatially divided phase-shifting MOSH (SP-MOSH). In the reproduction process of a hologram in the original MOSH, laterally shifted spherical phase distributions are sequentially displayed on an LCoS-SLM. Thereafter, the same process is performed with other bias phases of the spherical phase distributions. Therefore, the measurement time is proportional to $N\times S$, where $N$ and $S$ are the number of pixels of a reproduced hologram and the number of phase-shifting steps, respectively. When the general four-step phase-shifting method [37] is applied, $S$ is equal to four. In contrast, in SP-MOSH, the measurement time is simply proportional to $N$, because the spatial and phase shifts of the spherical phase distributions are implemented simultaneously. A reproduced hologram by SP-MOSH forms a parallel phase-shifted digital hologram. Therefore, a complex hologram can be obtained by using the reconstruction process of parallel phase-shifting digital holography [38–41].
2. Principle
A schematic diagram of the proposed SP-MOSH is shown in Fig. 1, which is exhibited as a transmitted configuration for simplicity. A diagonally polarized plane wave illuminates an LCoS-SLM displaying spherical phase distributions. Because an LCoS-SLM can modulate only horizontal or vertical polarization components, a modulated beam has a spherical phase distribution and the other remains a plane wave. To project a time-varying Fresnel zone plate (FZP) to an object, the polarization components of these beams are aligned by a polarizer. The transmitted beam from an illuminated object is detected by a photodetector and is described as
Fig. 2. Relationship between the spherical phase distributions on the SLM and the incoherent hologram. The colored squares in the incoherent hologram indicate the intensity values using the phase distribution surrounded by these colors.
In the case of SP-MOSH, the spatial position and bias phase of a spherical phase distribution are sequentially shifted. As a result, a spatially multiplexed phase-shifting hologram obtained by parallel phase-shifting digital holography is reproduced, as shown in Fig. 3. In this study, the nearest-neighbor interpolation method was employed because the implementation is simpler than that of other interpolation methods. Therefore, local areas of $2\times 2$ pixels are interpolated with the sampling intensities. In addition, the quality of the reconstructed object can be improved by changing to another interpolation method [45].
Fig. 3. Schematic of the reproduction processes of phase-shifted holograms from the parallel phase-shifted hologram for (a) PSM and (b) RSM.
In general, a spatially multiplexed phase-shifting hologram is periodically sampled, because parallel phase-shifting digital holography utilizes a camera with a periodically arranged phase retarder array. Because the phase retarder array is fixed to an image sensor, the periodic sampling effect for parallel phase-shifting digital holography cannot be readily evaluated. On the contrary, because the SP-MOSH system does not require a phase retarder array, the sampling position for a spatially divided hologram can be changed flexibly. In this study, two spatially divided sampling methods for the hologram are compared. One is a periodic sampling method (PSM), and the other is a random sampling method (RSM). In the PSM, a bias phase of spherical phase distributions is periodically changed pixel-by-pixel, as in a phase retarder array, which is generally used in parallel phase-shifting digital holography. In the RSM, sampling positions for a phase-shifted hologram are randomly selected in local areas of $2\times 2$ pixels. The reproduction processes by the PSM and the RSM are shown in Fig. 3(a) and (b), respectively. The sampling rate of the RSM is the same as that of the PSM. In the next section, the effects of periodic and random sampling are numerically evaluated.
Because the sampling rate of each hologram is 25%, the spatial resolution of holograms constructed by SP-MOSH is lower than that for the original MOSH. In spite of this fact, it has been shown that comparable reconstruction results can be obtained by parallel phase-shifting digital holography [38–41].
3. Numerical simulation
To verify the feasibility of the proposed method, the imaging properties of SP-MOSH are numerically evaluated. For simplicity, it is assumed that spherical phase distributions at an LCoS-SLM plane are projected onto an object through a $4$-$f$ optical setup, as shown in Fig. 4(a). The wavelength of the light source is set to 532 nm. This situation assumes that the reproduced hologram is described by Eq. (2): The 2D plane object shown in Fig. 4(b) was used as a measurement target. The two types of spherical phase distributions shown in Fig. 4(c) and (d) were used to evaluate the effect of their PSFs, and their focal lengths are $150$ and $60$ mm, respectively. The green rectangles in Fig. 4(c) and (d) indicate the region of an LCoS-SLM. The number of pixels of these spherical phase distributions is $128\times 128$. From Fig. 4(c), the spherical phase distribution is correctly sampled, because it satisfies the sampling condition described in Appendix A. On the contrary, the spherical phase distribution in Fig. 4(d) has an aliasing error, because it does not satisfy the sampling condition. Although the aliasing error causes image quality degradation, high spatial frequency components can be reconstructed by using the phase in Fig. 4(d), because it has a sharper PSF than that of the phase in Fig. 4(c).
Fig. 4. Conditions for numerical simulation: (a) Optical setup, (b) amplitude distribution of measurement target, (c) and (d) are spherical phase distributions with $f=150$ mm and $f=60$ mm on an SLM plane. SF, spatial filter; L, lens; SLM, spatial light modulator; P, polarizer; O, object; PD, photodiode.
Reconstruction results are shown in Fig. 5. Figure 5(a)–(c) show intensity distributions by MOSH with the sequential phase-shifting method, SP-MOSH with the PSM, and the RSM with $f=150$ mm, respectively. Additionally, Fig. 5(d)–(f) show results for MOSH with the sequential phase-shifting method, SP-MOSH with the PSM, and the RSM with $f=60$ mm, respectively. From Fig. 5, comparable distributions can be acquired by SP-MOSH. To evaluate the difference between MOSH and SP-MOSH quantitatively, a coefficient of variation (CV) is used. The evaluation region is the forehead of the panda, that is, the interior of the blue rectangle in Fig. 4(b). A CV is defined as
Fig. 5. Reconstructed intensity distributions: (a) Original MOSH ($f=150$ mm), (b) SP-MOSH of PSM ($f=150$ mm) and (c) RSM ($f=150$ mm), (d) original MOSH ($f=60$ mm), and (e) SP-MOSH of PSM ($f=60$ mm) and (f) RSM ($f=60$ mm).
Table 1. Comparison of coefficient of variation
In addition, we evaluate the effect of the PSF. Sectional profiles of the reconstruction results along the red line in Fig. 4(b) are shown in Fig. 6. When a spherical phase distribution with a shorter focal length is used, higher spatial frequency components can be reconstructed. Moreover, Fig. 6 indicates that SP-MOSH can reconstruct a comparable distribution to that acquired by the original MOSH.
Fig. 6. Cross-sectional profiles of reconstructed intensity distributions in Fig. 5 at the red line in Fig. 4(b).
4. Optical experiment
The feasibility of the proposed method is experimentally evaluated. The optical setup for the experiment is the same as that for the numerical simulation. A green laser (MPB Communications, Inc. VFL-P-500) with a wavelength of 532.1 nm was used as a light source. A collimated beam illuminated an LCoS-SLM (Hamamatsu K.K. X13138-01) with a pixel pitch of 12.5$\mu$m. A beam modulated by the SLM was magnified by a 4-$f$ optical setup with a magnification factor of 2. The polarization direction of the beam was aligned by a polarizer, and then the beam illuminated an object. The “3" of a USAF target in group 0 was used as a measurement target. The transmitted beam was collected by a lens to a photodiode (Hamamatsu K.K. C10439-01). The measured currents were quantized by a 16-bit analog-to-digital converter (Hamamatsu K.K. C10475).
The holograms produced by the original MOSH and SP-MOSH are shown in Fig. 7(a) and (b), respectively. Figure 7(c) shows four phase-shifted holograms from the spatially divided phase-shifting hologram shown in Fig. 7(b). In this experiment, the PSM was employed for SP-MOSH, because a comparable distribution to that of the RSM could be obtained. The reconstructed intensity distributions from the original MOSH and SP-MOSH are shown in Fig. 8(a) and (b), respectively. From the reconstruction results, SP-MOSH can measure a comparable distribution to that of the original MOSH.
Fig. 7. Reproduced holograms: (a) By original MOSH and (b) by SP-MOSH. (c) Four phase-shifted holograms from (b).
Moreover, a 3D object was also measured by SP-MOSH. Two circular apertures placed at different $z$ positions were used as 3D objects, as shown in Fig. 9(a). The distance between the planes containing the apertures was set to 60 mm. The magnification factor of the 4-$f$ setup Was 5/2. A reproduced hologram and four phase-shifted holograms from the reproduced hologram are shown in Fig. 9(b) and (c), respectively. In addition, the reconstructed intensity distributions at the two object planes are shown in Fig. 9(d) and (e). The cross-sectional profile is shown in Fig. 9(f). These reconstructed intensity distributions demonstrate that 3D information can be obtained using SP-MOSH.
Fig. 9. Measurements for a 3D object. (a) 3D object, (b) spatially divided hologram, (c) four phase-shifted holograms, (d) in-focus intensity distribution of bottom-left circle, and (e) in-focus intensity distribution of upper-right circle.
5. Conclusion
To reduce the measurement time for MOSH, parallel phase-shifting digital holography is applied to MOSH. SP-MOSH can reproduce a spatially divided phase-shifted hologram. A complex hologram is obtained by applying the parallel phase-shifting algorithm to the hologram reproduced using SP-MOSH. Therefore, the proposed method realizes measurements four times faster than the original MOSH.
Through numerical simulations, the effect of periodic sampling is evaluated using the PSM and the RSM. From the simulation results, the effect of sampling methods is found to be negligible. In addition, the simulated and experimental results show that the distribution reconstructed by SP-MOSH has comparable image quality to that by the original MOSH. In the optical experiment, a 3D object was measured by SP-MOSH. The experimental results show that SP-MOSH can image a 3D object in the same measurement time as that required by the general SPI. In this study, we used the four-step phase-shifting method. To reduce the measurement time, other phase-shifting methods such as the three-step method [28] could be applied to MOSH. In OSH, fast measurement techniques have already been proposed [33,46,47]. Further reduction in the measurement time could be realized by combining these methods with SP-MOSH in the future.
APPENDIX A
From Eq. (4), the PSF of the MOSH can be described as
On the contrary, in general SPI, the spatial resolution is determined by the pixel pitch of the SLM. Therefore, when the diffraction limit (PSF) is smaller than the pixel pitch, the sampling condition at the object plane is satisfied. This condition is express as
Funding
Japan Society for the Promotion of Science (JSPS) (20J10441).
Disclosures
The authors declare no conflicts of interest.
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