1. Introduction

Incoherent digital holography (IDH) is an attractive 3D imaging technique, as it holographically records and reconstructs objects captured with spatially incoherent lights, including fluorescent lights, light-emitting diodes (LEDs), and sunlight [1–4]. The basic principle of IDH resembles the concept of coherent digital holography [5], which uses coherent laser illumination. However, their main difference is that IDH must rely on self-interference for the creation of holograms with a spatially incoherent light source. Thus, there is no apparent distinction between the object and reference beams in IDH, whereas coherent digital holography generally distinguishes between them. Moreover, IDH has valuable imaging capabilities, such as the violation of the Lagrange invariant [6,7] and an infinite depth of field [8] owing to the nature of the self-interference. These properties are impossible to achieve with coherent digital holography and traditional imaging techniques.

The self-interference holograms in IDH contain undesirable zeroth-order and twin-image terms that deteriorate the quality of the reconstructed images. A solution for this technical difficulty is to detect the complex amplitude distribution with a sequential phase-shifting method [9]. A series of self-interference holograms with temporally different phase shifts is sequentially recorded. The application of the algorithm of the phase-shifting method to these holograms yields the complex amplitude of the object without the undesirable terms. This approach is simple yet powerful and has enabled significant applications, such as fluorescence microscopy [10–14], radiometric temperature measurements [15], and photography under sunlight [3]. However, the sequential phase-shifting method requires that the objects and scenes are static during recording, which renders it impractical for videography.

The implementation of a single-shot technique is of special interest to the field of IDH. Previously, several single-shot techniques have been proposed, such as off-axis recording, compressive sensing, pixelated-mask-based phase sifting, and grating-based phase shifting, as briefly reviewed below. Off-axis recording techniques [16–18] introduce a linear phase carrier onto a single self-interference hologram and reconstruct an image from it without undesirable terms based on Fourier fringe analysis [19]. Because the Fourier spectra of undesirable terms should be spatially separated from the Fourier spectrum of the object in the spatial-frequency domain, the resolution of reconstructed images is significantly poorer than that of the image sensor. The introduction of compressive sensing is an alternative approach for removing the undesirable components [20]. This approach infers a signal component from a single in-line self-interference hologram without undesirable terms under the assumption of the signal’s sparseness in some domains. This can eliminate the need for the linear phase carrier for separating the Fourier spectra. However, the requirement of the sparseness of the objects limits an achievable space-bandwidth product, and compressive-sensing inference requires a long calculation time and large computational memory. In terms of a space-bandwidth product, a pixelated-mask-based phase-shifting method is an attractive approach [21–24]. This method requires an image sensor attached to a pixelated-mask-based phase-shifter mask, such as a linear polarizer array or waveplate array on a pixel-by-pixel basis. Such image sensors can be implemented with, e.g., a polarization-sensitive image sensor, which is currently available commercially [25]. The image recorded using this image sensor contains four holograms with different phase shifts. The four holograms are extracted by demosaicing, in which missing pixels are interpolated and then processed using the phase-shifting algorithm. Alternatively, low-pass filtering of the product of the acquired raw image and the use of a virtual phase-shifter mask can effectively yield a complex amplitude distribution [26]. Since this low-pass filtering process can be incorporated into a diffraction calculation, such as the angular spectrum method, the calculation load can be improved. Although effective demonstrations have been reported by several researchers using polarization-sensitive image sensor [27,28], the selection of the sensor specifications, such as pixel size, number of pixels, quantization level, and quantum efficiency, is currently limited. Because the quality, resolution, and field of view of the reconstructed images of IDH are seriously affected by the image sensor specifications [29,30], it is necessary to carefully select them depending on the requirements of the application, manufacturing cost, and target image quality. Moreover, the resolution of each recorded hologram produced using the pixelated-mask-based phase-shifting method is inherently reduced by 2 $\times$ 2 pixels from the acquired raw image resolution due to the demosaicing process, although some of the high-frequency components may be retrieved by adequately designing the interpolation strategy [22,31]. In contrast to the aforementioned method, we previously proposed a space-division-based phase-shifting method that uses a pair of multiplexed phase gratings [32]. Owing to the grating space-division functionality, four self-interference holograms can be simultaneously created. Each self-interference hologram has a different phase shift due to the adequate alignment of the pair of gratings in accordance with the Fourier-shift theorem. The proposed system eliminates the need for demosaicing and interpolation processes, and the hologram resolution is the same as that of the image sensor. However, the hologram size should be restricted to avoid spatial overlap of the holograms, mainly leading to a reduction in the field of view. Therefore, the space-bandwidth product is equivalent to that of the pixelated-mask-based phase-shifting method assuming that the same image sensor (with the same resolution and pixel pitch) is used. Using our previous method, it is possible to flexibly select image sensors with the desired specifications depending on the application. Moreover, from the viewpoint of engineering, the space-bandwidth product can be improved by simply introducing a sensor array in which a single image sensor records one hologram. A major limitation of our previous method is the need for two gratings and two optical arms. This causes the optical setup to be sensitive to vibrations. Furthermore, the construction of the optical setup requires highly precise alignment so that the optical path difference is less than the coherence length, whereas the relative position between the dual gratings must satisfy the specific condition of the Fourier-shift theorem. The required critical alignment makes the optical setup difficult to implement for many practical applications. It is therefore preferable to develop an in-line geometry. To overcome this limitation, the further multiplexing of the dual multiplexed phase gratings in a random fashion was proposed [33]. Although this approach allows the implementation of single-shot IDH with an in-line geometry, random or background noise due to the random multiplexing is inevitable, leading to a degradation in the quality of the reconstructed images. Moreover, both this method [33] and our previous method [32] rely on a phase-only spatial light modulator (PSLM) to implement the light-splitting function of the phase grating. The PSLM suffers from spatial and temporal phase fluctuations due to its imperfections [34], which also decreases the diffraction efficiency of the phase grating and quality of the reconstructed images.

In this study, we propose a novel in-line geometry for the grating-based phase-shifting IDH. The proposed method uses a single-phase grating as a static diffractive optical element (DOE) to create four self-interference holograms of incident lights with orthogonally circular polarization. Subsequently, by using a segmented linear polarizer, four-step phase shifts can be introduced to the self-interference holograms depending on the polarizer’s transmission angle on the basis of the Pancharatnam–Berry phase, or geometric phase. In the coherent interferometry, a similar concept of the in-line geometry with a segmented birefringent phase mask was proposed [35]. The proposed IDH system is inspired by this work [35], but we utilize the geometric phase of the circular polarization rather than the dynamic phase of the phase mask. Unlike our previous geometry using dual gratings [32], the proposed method enables an in-line optical setup and eliminates the need for highly precise alignment. Moreover, there is no random or background noise, unlike the random multiplexing of two gratings [33], as the proposed method is equivalent to multiplexing the two gratings in the polarization degree. With the proposed method, we implemented snapshot and video recordings of 3D reflective moving objects as a proof-of-principle demonstration.

The remainder of this paper is organized as follows. In Section 2, we describe the principle of the proposed method and discuss possible optical setups. In Section 3, we describe the proof-of-principle experiment used to verify the effectiveness of the proposed method to implement snapshots and video recordings. Finally, we present our conclusions in Section 4.

2. Principle of grating-based in-line geometric-phase-shifting IDH

Figure 1 presents a schematic of the proposed method. The setup consists of several polarization optical elements and a single-phase grating. The main difference between the proposed method and our previous method using dual gratings [32] is the use of the polarization degree. In the proposed method, the in-line geometry with a single-phase grating and application of polarization achieves the self-interference required for IDH, as described below.

 

Fig. 1. Schematic of grating-based in-line geometric-phase-shifting IDH. (a) Optical setup. (b) Phase grating for light-splitting function. (c) Fourier spectrum of modulated light with (a). (d) Segmented linear polarizer to introduce geometric phases.

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The spatially incoherent light on the object can be regarded as a collection of infinitesimal point sources. For simplicity without loss of generality, we describe a process of creating self-interference holograms from a single point source located at the optical axis on the object with a quasimonochromatic wavelength. Note that we assume that the temporal coherence of the point source is sufficiently high for the creation of the holograms with high spatial resolution [36,37]. A spherical wave from the point source propagates with linear polarization. The angle of the linear polarizer is set to $+45^{\circ }$ or $-45^{\circ }$ with respect to the fast axis of the subsequent birefringent lens. The birefringent lens has two focal lengths $f_{d1}$ and $f_{d2}$ along the slow and fast axes, which act as a bifocal lens. This function is important for creating self-interference holograms containing 3D information about the object in the in-line geometry. Note that instead of a birefringent lens, the use of a liquid-crystal lens [38] or PSLM [1,2] is also effective. In a plane immediately behind the birefringent lens, a complicated and spatially variant vector optical field $\overrightarrow {u_B}(x,y)$ is obtained. This inherently corresponds to the vector superimposition of horizontally and vertically linear polarization with different spherical phases, which may be mathematically represented by the Jones formulation:

Note that the above equation describes the vector field when the distance between the birefringent lens and the quarter-wave plate is sufficiently small. However, regardless of the gap distance between them, the following description and the resulting holograms remain unchanged. This vector optical field is subsequently incident on a phase grating to create its four copies that propagate in different directions. The phase grating consists of a checkerboard phase distribution with a binary phase value 0 and $\delta$ and a pitch of $p$, as presented in Fig. 1(b). Although $\delta$ is arbitrary except for 0 for the light-splitting function, $\pi$ is desirable for maximizing the light utilization efficiency. A discussion of the optimal phase value and the tolerance is provided in Appendix A. When $\delta =\pi$, the checkerboard phase distribution can be decomposed into orthogonal 1D binary phase gratings with a phase step $\pi$. Thus, the Fourier spectrum of the modulated light with the phase grating is the convolution of the two orthogonal 1D binary phase gratings under the assumption of scalar diffraction theory. As presented in Fig. 1(c), the Fourier spectrum contains four intense diffraction orders without the zeroth-order component. Note that high diffraction orders are generated due to the binary structure. The theoretical diffraction efficiency of each intense main diffraction order is therefore the square of that of a single 1D binary phase grating, or $\{\rm {sinc}^2 (0.5)\}^2$, giving an efficiency of 16.4$\%$ [39]. The four copies propagate along the angle determined by the Fourier spectrum presented in Fig. 1(c) and pass through a segmented linear polarizer. The segmented linear polarizer consists of four subdivision linear polarizers with different transmission angles, $\theta$ = 0, $\pi /4$, $\pi /2$ and $3\pi /4$, as presented in Fig. 1(d). After passing through the linear polarizers, orthogonal circular polarizations are converted to a common linear polarization state, resulting in a self-interference hologram on an image sensor. Moreover, different phase shifts, $\phi$ = 0, $\pi /2$, $\pi$ and $3\pi /2$, are introduced into the self-interference holograms according to the geometric phase of the circular polarization, or $\phi = 2 \theta$. Therefore, the four holograms are recorded in a single exposure with the image sensor. By cropping individual holograms from the recorded image, the phase-shifting algorithm can be applied to them. Because the four holograms $I_n(x,y)$ ($n$= 1, 2, 3, and 4) can be simply regarded as laterally shifted versions of a self-interference hologram without the phase grating in an $x$-$y$ plane under the paraxial approximation, they may be represented as follows:

To implement our proposed method, there are potentially many variations in the optical setup. Figure 2 presents alternative optical setups to implement the proposed method, inspired by previous studies on IDH [1,2]. The drawback of the aforementioned setup presented in Fig. 1 is the small difference between the two focal lengths of the birefringent lens and liquid-crystal lens. In IDH, the use of a bifocal lens with significantly different focal lengths is useful in the flexible design of its setup to provide a high spatial resolution, large utilization efficiency, and compact system. However, the achievable difference between the two focal lengths is limited by the intrinsic birefringence of the materials and its ability to be fabricated into a lens. Alternatively, Fig. 2(a) presents a system based on a geometric phase lens, which has chirality and enables spin-dependent focusing. The geometric phase lens can be fabricated using liquid-crystal molecules or photonic crystal based on polarization holography [27,40,41]. In addition, a metalens with chirality focusing is effective [42–44]. Because the geometric phase lens plays the same role as the birefringent lens and the quarter-wave plate in Fig. 1, the setup in Fig. 2(a) can reduce the number of optical components. However, a side effect of this setup is the generation of undesired concentric diffraction orders from the geometric phase lens, because its working principle is based on diffraction, which potentially reduces the quality of the reconstructed images. Alternatively, it is effective to use a reflective PSLM instead of the birefringent lens, as presented in Fig. 2(b). This is similar to the initial invention of the Fresnel incoherent correlation holography (FINCH) system [45]. The advantage of this architecture is the easy modulation provided by the PSLM, which allows changing the focal length of the lens. Although high-order diffraction components are generated by the pixelated structure, they can be spatially removed by filtering. However, to prevent mechanical interference between optical components, smaller incident angles onto the PSLM generally require larger systems. The geometry presented in Fig. 2(c) enables easy modulation owing to the PSLM; it is also easy to downsize the system by introducing a beam splitter.

 

Fig. 2. Alternative setups for grating-based in-line geometric-phase-shifting IDH.

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Although the light utilization efficiency, aberrations, and manufacturing costs depend on the specific setup, they are similar in terms of the basic operation. Depending on the target properties and applications, the most appropriate setup should be selected. The essence of the proposed method described here is the use and combination of the phase grating, segmented linear polarizer, and geometric phase.

3. Proof-of-principle experiment

3.1 Experimental setup

On the basis of Fig. 2(c), we constructed the optical setup using a reflective PSLM for the proof-of-principle experiment. This setup was mainly selected as it is easy to calibrate the optical setup and align the hologram positions by displaying a computer-generated hologram (CGH) on the PSLM, as described later in Appendix B. Although the light utilization efficiency is lower than that of Fig. 2(b), the optical setup in Fig. 2(c) enables us to easily shorten the distances between optical components, which is important for achieving a compact setup.

Figure 3 presents the optical setup in detail. Reflected light from the object was first incident on a lens with a 750-mm focal length. The use of the lens is optional, but it is useful for controlling the image magnification and light utilization efficiency [29]. The collected light from the lens was subsequently incident on a 10-nm bandpass filter centered at 632.8 nm to enhance the temporal coherence. The use of the bandpass filter is necessary to enhance the temporal coherence, which improves the spatial resolution of reconstructed images [36,37]. Moreover, it prevents smearing the holograms along diagonal directions due to the diffraction dispersion of the phase grating. An aperture was used to restrict the illumination area onto the subsequent PSLM and prevent any undesirable diffraction between the boundaries of active and inactive areas of the PSLM. Moreover, it is useful for adjusting the size of the holograms on the following sensor plane to avoid overlap and ensure that they fit within the sensor area. A linear polarizer ($45^{\circ }$) was used to provide horizontal and vertical linear polarizations. Instead of a birefringent lens, we used a PSLM with 1920 $\times$ 1200 pixels and a pixel pitch of 8 $\mu$m. The PSLM exhibited a spherical phase pattern with a 1000-mm focal length at the center position. Based on the anisotropy of the liquid-crystal molecules of the PSLM, one of the linear polarization components acquired the spherical phase, and the other was reflected from the PSLM without spatial-variant phase modulation. Both polarization components were redirected with a beam splitter and converted into orthogonal circular polarizations through a quarter-wave plate. The lights were incident on the phase grating. The phase grating with $\pi$ phase step and a 5-$\mu$m pitch was fabricated as a glass DOE by one-step photolithography and etching processes. The details of the design and characterization of the phase grating are described in Appendices A and C, respectively. The orthogonal circular polarizations were divided into four copies by passing through the phase grating. The segmented linear polarizer contains four polarizers with different transmission angles (Fig. 1(d)). This causes the orthogonal circular polarizations to be in a common linear polarization with a geometric phase depending on the transmission angle. This results in the creation of four self-interference holograms. The holograms were recorded as a single image with a cooled CCD camera with 11600 $\times$ 8700 pixels and a pixel pitch of 4.6 $\mu$m. The temperature of the CCD camera was maintained at $-5^{\circ }$C, and the quantization level was set to 12 bits for the following experiments. The recorded holograms were cropped from the single recorded image. The misalignment of the cropping position of the images reduces the reconstruction quality. To correctly crop the images, we investigated and calibrated the positions of the four generated optical fields from the grating in advance using a CGH technology and correlation calculations. The details of this calibration step are presented in Appendix B. The cropped holograms were processed using the four-step phase-shifting algorithm to retrieve the complex amplitude $u_r (x,y)$ on the image sensor plane, or

Finally, by applying a diffraction calculation such as the angular spectrum method, the object image can be reconstructed. The detailed geometric parameters of the optical setup are described in Appendix D. With the above experimental setup and procedures, we demonstrated snapshot and video recording as shown below.

 

Fig. 3. Experimental setup for the demonstration of the proposed method.

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3.2 Snapshot recording

With the setup presented in Fig. 3, we experimentally implemented a snapshot recording of 3D reflective objects. The objects to be captured were two dice with dimensions of 5 $\times$ 5 $\times$ 5 mm$^3$ located at different axial planes, as presented in Fig. 4(a). The axial separation distance was 200 mm. The objects were incoherently illuminated by a red LED light source with a center wavelength of 630 nm. The recorded image is presented in Fig. 4(b), which was cropped from the original raw image size (11600 $\times$ 8700 pixels) to eliminate the outer redundant regions. Note that the brightness of the image was adjusted to enhance its visibility. The four individual holograms were observed. There was a dim cross-shaped black region at the center of the image, which was caused by the shadow of the mount of the segmented linear polarizer. We applied a series of processing steps to the single image by cropping the holograms with 2048 $\times$ 2048 pixels according to the calibration results, implementing the phase-shifting method, and finally applying the angular spectrum method. Figures 4(c) and 4(d) present two reconstructed images with the focus on the back and front objects, respectively. A series of the reconstructed images is provided in Fig. 4(e). From the results, we verified the feasibility of the proposed method that enables recording of the 3D reflective objects at a single exposure and reconstruction of the images at arbitrary depth planes.

 

Fig. 4. Experimental results of snapshot recording. (a) Illustration of objects being captured. (b) The recorded image contains four self-interference holograms. Reconstructed images focused at (c) back and (d) front objects. (e) Series of reconstructed images between (c) and (d).

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3.3 Video recording

To further verify the feasibility of the proposed method, we experimentally implemented a video recording of 3D reflective moving objects. The objects to be captured were three vertically stacked dice located on a rotating table and a watch with the diameter of 40 mm, which were axially separated at a 300-mm distance (Fig. 5(a)). The light source and the experimental setup were the same as used in the previous experiment. We recorded 70 frames at a frame rate of approximately 1 fps. Figure 5(b) presents the first recorded image of the 70 frames. The image brightness was also adjusted for clear visibility. Following the processes to record images, we reconstructed two images focused on either the dice or watch, as presented in Figs. 5(c) and 5(d), respectively. Although the two reconstructed images were retrieved from the first recorded frame, we provided a video of the reconstructed images from all the recorded frames in Visualization 1. To verify the refocusing capability, we provided a series of reconstructed images in Fig. 5(e). These results indicate that we successfully acquired 4D data including 3D-spatial and 1D-temporal information using the proposed method. This 4D data enables arbitrary changes to the focal position at each frame, as presented in Visualization 2. These results prove that the proposed method is effective for recording 3D reflective moving objects and performing refocusing for each frame, which enables the recording of 3D videos.

 

Fig. 5. Experimental results of video recordings. (a) Illustration of objects being captured. (b) First recorded frame containing four self-interference holograms. Reconstructed images focused at the (c) back (watch) and (d) front (dice) objects from the first frame of (a). (e) Series of reconstructed images between (c) and (d). A video of the reconstructed images focused at the back and front objects from all frames is provided in Visualization 1. In addition, a video of arbitrary changes to the focus position at each frame is provided in Visualization 2.

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4. Conclusion

We proposed and demonstrated grating-based in-line geometric-phase-shifting IDH. The proposed method is based on a single-phase grating and the geometric phase of orthogonal circular polarizations. With the proposed method, we implemented snapshot and video recordings of 3D reflective objects. We acquired the 1fps-video of 3D reflective moving objects, and reconstructed it with the refocusing function at each frame. This low frame rate was mainly a requirement of the data transfer time of large image files from the CCD camera and the low light utilization efficiency of the experimental setup presented in Fig. 3. Although the use of the beam splitter in Fig. 3 is effective for achieving a compact optical setup, the light utilization efficiency was reduced to 25$\%$. Furthermore, there was a loss due to the binary phase grating. This technical limitation could be overcome by modifying the setup to those presented in Figs. 1, 2(a) and 2(b). Regarding the phase grating, it is possible to design and fabricate a multilevel grating [39]. However, such optimizations are beyond the scope of this study.

Because the proposed system is based on an in-line geometry and single-exposure recording, it is robust to mechanical vibrations. Compared with the off-axis geometry, the proposed method can achieve a high spatial bandwidth product by using the phase grating. Moreover, there is no laterally spatial displacement between the two interference lights in the proposed method owing to its in-line geometry.

Unlike the conventional single-shot method with a polarization-sensitive sensor, the proposed method enables a wider selection of image sensors with the desired pixel pitch, resolution, quantization level, quantum efficiency, noise level, and frame rate. Although in this study we used a cooled CCD camera for demonstration purposes, it is possible to use various image sensors from amateur sensors to high-end ones, such as scientific CMOS cameras. Moreover, in theory, it is possible to use four parallelly aligned image sensors. From an engineering perspective, the broad choice of image sensors is advantageous for designing the optical setup depending on the target image quality, manufacturing cost, and applications.

Appendix A: Design of phase grating

The phase grating was designed as a DOE. The binary phase values 0 and $\delta$ are used with a one-step structure. When the DOE is surrounded by air, the height difference $h$ of the step structure for introducing a phase step $\delta$ is given by

(5)$$h = \frac{\delta \lambda}{ 2 \pi (n_s - 1) },$$

where $n_s$ denotes the refractive index of a material. When the material is a glass, its refractive index is 1.457 at the operation wavelength of 632.8 nm. The proposed method described in Section 2 uses the first diffraction orders of the DOE. For maximizing the light utilization efficiency with the glass, $\delta$ should be $\pi$, i.e., $h$ is 692 nm. In fact, the fabricated DOE would contain a fabrication error in $h$, resulting in a phase error in $\delta$. Figure 6(a) presents typical Fourier spectra of the phase grating with phase errors. Note that the fabrication error was defined as (actual height)/692$\times$100 ($\%$). Figure 6(b) shows the change in the diffraction efficiency against the fabrication (phase) error. The fabrication error redistributes the light energy between the main diffraction and zeroth orders. A decrease in the light utilization efficiency leads to a decrease in the signal-to-noise ratio in photon detection and results in a degradation in the reconstructed image quality. Although the increase in the phase error increases the zeroth-order power and decreases the first-order power, the four main diffraction orders are still available. Thus, it is applicable to our proposed method in theory regardless of $\delta$. If there is a target image quality, Fig. 6(b) provides the guideline for determining the fabrication tolerance of the phase grating.

 

Fig. 6. Effect of height or phase error on phase grating. (a) Fourier spectra with height (phase) errors. (b) Change in the diffraction efficiency of one of the main diffraction orders as a function of the height (phase) error.

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Appendix B: Calibration with computer-generated holograms

The proposed method divides the light into four copies to create four self-reference holograms at spatially different regions and records them in a single image. The four self-reference holograms should be cropped from the recorded single image. The misalignment during cropping potentially leads to phase-shifting algorithm errors. To correctly identify the four hologram positions, we calibrated the optical setup by displaying a CGH in the PSLM in advance. The light propagation between the PSLM and CCD camera was regarded as free-space propagation at the Fresnel region, although some optical elements were present. If we design a Fresnel-based CGH and input it into the PSLM, four target intensity patterns should be generated on the CCD camera plane. This enables easy calibration of the hologram positions. We therefore designed a CGH using the Gerchberg–Saxton (GS) algorithm, or iterative Fourier-transform method, at the Fresnel region to generate the calibration pattern presented in Fig. 7(a). The iteration number was 500 for the GS algorithm, and the propagation distance was set to 180 mm. The resulting CGH pattern is presented in Fig. 7(b). The numerical reconstruction of Fig. 7(b) is shown in Fig. 7(c). Although there was a speckle due to the lack of amplitude information, the numerical reconstruction result is reasonably consistent with the original image presented in Fig. 7(a). During the experimental calibration process, we used a temporally and spatially coherent light source propagating from a coupled optical fiber with a He–Ne laser to reconstruct the CGH without low-coherence blur. In addition, we rotated the linear polarizer to align the operation angle of the PSLM shown in Fig. 3 without any other modifications. The PSLM displayed the designed CGH shown in Fig. 7(b). The resulting image was captured using the CCD camera. Figure 8(a) shows the captured image containing the four calibration patterns. We cropped one of them without vignetting and used it as a template for calculating the cross-correlation and phase-only correlation between the captured raw image and the template image. Figure 8(b) presents the cross-correlation result. Note that because the phase-only correlation result had many peaks and it is difficult to correlation signal from the correlation image, we provided only the cross-correlation result. From these results, we identified the peak positions of the correlation results as the relative positions of four copies. Note that the correlation results between cross-correlation and phase-only correlation are slightly different as their performance and robustness against noise differed. By referring to both results, we empirically selected the best positions for cropping holograms. We referred to the information on the relative position when cropping the four holograms of 3D reflective objects in the proof-of-principle experiments discussed in Section 3.

 

Fig. 7. Design of the CGH pattern for calibration purpose. (a) Target calibration pattern. (b) Phase CGH designed using the GS algorithm with 500 iterations. (c) Numerical reconstruction of (b).

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Fig. 8. Experimental calibration results. (a) Captured single image with CCD camera. (b) Correlation result of cross-correlation.

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Appendix C: Evaluation of the fabricated phase grating

The DOE for the phase grating with binary phase values was fabricated from a glass substrate via one-step photolithography and etching processes. The structure of the fabricated grating was measured via profilometry using a laser scanning microscope, and the result is presented in Fig. 9. We measured the heights of five regions of the fabricated sample: middle, upper right, upper left, lower right, and lower left. The average height was 697.6 nm, with a fabrication error of 0.8$\%$. In addition, we optically evaluated the diffraction pattern of the DOE. The DOE was illuminated with a beam from the He–Ne laser. The intensity of the diffracted light was measured using a power meter. The diffraction efficiency of the zeroth-order beam was 4.6$\%$, which was mainly due to the fabrication error of the DOE. The diffraction efficiencies of the four main diffraction orders were 14.9$\%$, 14.3$\%$, 15.5$\%$, and 14.6$\%$. The difference between the theoretical value (16.4$\%$) and the measured values was attributed to the measurement uncertainty, loss of reflection at the boundaries between the DOE and air, and small fabrication error. Moreover, the theoretical value was derived under the assumption of scalar diffraction theory, which tends to deviate when the grating pitch is small. This is also the reason for the difference between the numerical and experimental results. The intensity fluctuation between the main orders was compensated for by normalizing the intensity level of the holograms. Although there was a fabrication error, it is acceptable for the light-splitting function, as discussed in Appendix A. As presented in Section 3, we verified the feasibility of the proposed method using the DOE.

 

Fig. 9. Profilometry result of fabricated phase grating. (a) Top and (b) bird’s-eye view.

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Appendix D: Detailed geometry of experimental setup

The geometric parameters of optics for constructing IDH systems affect the relationship between recording distance $z_s$ and reconstruction distance $z_r$. Moreover, they dictate image magnification $M_T$ and spatial resolution in a highly complex manner [8,29]. In the experimental setup shown in Fig. 3, the distance between the lens and PSLM was $z_l=$ 100 mm. Although the lights are redirected with the beam splitter, the effective distance between the PSLM and image sensor was $z_h=$ 200 mm. The focal length of the lens again was $f_o=$ 750 mm. The PSLM acted as a bifocal lens with focal lengths $f_{d1}=$ 1000 mm and $f_{d2}\rightarrow \infty$. The relationship between recording distance $z_s$ and reconstruction distance $z_r$ is given by

(6)$$z_r = \frac{\{z_d z_h-f_{d1}(z_d + z_h) \} \{z_d z_h-f_{d2}(z_d + z_h) \}}{ z^2_d( f_{d1}-f_{d2} ) },$$

(7)$$z_d = \frac{z_s f_o + z_l f_o - z_l z_s}{ f_o - z_s }.$$

The derivation of these equations can be found in [29]. The numerical plot of Eqs. (6) and (7) using the aforementioned is shown in Fig. 10(a). There is obviously the nonlinear relationship between the recording and reconstruction distances in IDH, which is attributed to the self-interference process of IDH. In Fig. 5(a), the recording distance between the vertically stacked dice and the lens was roughly 830 mm, and the focused image can be obtained at the reconstruction distance $z_r=$ 754 mm with the numerical propagation using the angular spectrum method. Similarly, although the actual distance between the watch and lens was 1130 mm, the focused image can be obtained at the reconstruction distance $z_r=$ 640 mm. This is because of the nature of the self-interference. To correctly retrieve the 3D information of the object from a series of numerical focused images with IDH, it is required to adequately convert the reconstruction distance $z_r$ into the recording distance $z_s$ according to the inverse function of Eqs. (6) and (7) [46]. The image magnification of the experimental setup is a function of the recording distance $z_s$ and is mathematically represented by the following equation [8]:

(8)$$M_T = \frac{z_h f_o}{ z_l f_o - z_l z_s + z_s f_o }.$$

 

Fig. 10. Characteristics of experimental setup. (a) Relationship between recording and reconstruction distances. (b) Change in image magnification.

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Figure 10(b) shows the theoretical change in the image magnification against recording distance $z_s$. The image magnification decreases with the increase in the recording distance. The field of view is determined by the product of the pixel pitch, reciprocal image magnification, and the pixels of cropped holograms under that there is no effect of vignetting and aberrations due to the optics. Thus, the recordable object size is determined by the geometric parameters of optics in the complex manner.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

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