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Abstract
Although digital holography is a powerful technique obtaining a phase image of a transparent object, the image reconstructed by the technique merely expresses phase distribution of the light wave after transmitting through the object. Phase variation of inside of the object is difficult to be obtained. Then, we applied Abel inversion method to the high-speed phase image of a dynamic transparent object assumed axially symmetric. The phase is accurately recorded by phase-shifting method. We experimentally recorded transparent dynamic gas flow, assumed axially symmetric along the direction in which gas flowed, at 3,000 frame/s and reconstructed motion picture of 3D distribution of the refractive index of the gas from the high-speed phase motion picture obtained by parallel phase-shifting digital holography.
© 2017 Optical Society of America
1. Introduction
Digital holography (DH) [1] is a technique that records a hologram with an image sensor and quantitatively reconstructs the amplitude and phase images of an object wave at arbitrary depth by a computer. Because of the capability to obtain three-dimensional (3D) shape of an object, 3D shape measurement by DH has been researched actively [2–4], in-recent years. In particular, phase imaging by DH is important because the capability to measure 3D image of a transparent object [5–7]. In-line DH, whose object wave and reference wave are in-line, has often been used because of the low resolution of an image sensor and the necessity of a large pitch of the interference fringes. However, undesired images, such as the zeroth-order diffraction image and the conjugate image, are superimposed on the desire image of an object and the quality of the reconstructed image is affected in in-line DH. Phase-shifting digital holography (PSDH) [8] was proposed so as to reconstruct complex amplitude image free from the undesired images. In PSDH, multiple interference fringe images are recorded during the phase of the reference wave is shifted. However, recording a moving object by PSDH is quite difficult because of necessity of sequential recording of multiple holograms. Parallel phase-shifting DH (PPSDH) was proposed as a single-shot PSDH that can eliminates the undesired images [9]. PPSDH can provide high-quality 3D and phase motion pictures of dynamic object [10–12], and high-speed imaging of phase variation of transparent gas flow and electrical discharging phenomenon by PPSDH [13,14]. However, the phase image obtained by this technique does no more retrieve the phase variation of inside of the object than other digital holography can retrieve, in general. This becomes a problem when measuring true shape or phase of an object. Therefore, it is worth imaging phase variation of inside of an object, but inside imaging by using PPSDH has not been reported yet.
In this paper, we propose a technique for imaging inside of a transparent object moving at high speed in PPSDH by applied Abel inversion [15], for the first time. The proposed technique enables to calculate the 3D phase distribution including the inside of the object, by the phase distribution of a light wave after transmitting through an object. Then the 3D distribution of the refractive index of the object is obtained. We experimentally demonstrated the proposed technique by recording dynamically-flowing transparent gas, assumed axially symmetric along the direction in which gas flowed, and calculating the 3D distribution of the refractive index from the phase distribution after transmitting through the gas flow. The gas flow was recorded and reconstructed by the optical system of PPSDH we constructed.
2. Parallel phase-shifting digital holography
Parallel phase-shifting digital holography is a technique that can record the multiple holograms required for PSDH with a single-shot exposure. Figure 1 shows the principle of PPSDH. The holograms required for PSDH are recorded as a single hologram using the pixel-by-pixel based space-division multiplexing technique [9]. A recorded single hologram is demultiplexed into multiple phase-shifted holograms. Blanked pixels of each hologram are interpolated by using neighboring pixels. Therefore, holograms required for PSDH are generated. Complex amplitude distribution of the object wave at arbitrary depth is obtained by applying the calculation for the image reconstruction of PSDH to the holograms. Recording holograms with a single-shot exposure, PPSDH has capability of measuring a moving object. Although PPSDH has the capability of high-speed phase imaging, the reconstructed image merely expresses the phase difference between the light wave after transmitting through the object and the light propagated in air, phase variation of the inside of the object is not be obtained.
3. Calculation of 3D distribution of refractive index by using Abel inversion and parallel phase-shifting digital holography
Figure 2 shows principle of calculation of 3D distribution of refractive index of a transparent object by using Abel inversion method and PPSDH. A light wave propagating along the z-axis and transmitting through an object is recorded by PPSDH, as a transparent object whose refractive index is axially symmetric. Then the two-dimensional (2D) phase lag distribution φ(x, y) of the light wave after transmitting through the object is reconstructed. Next, Abel inversion is applied to the obtained phase lag distribution and the 2D radius distribution of refractive index n(r, y) is calculated. Finally, the 3D distribution of refractive index n(x, y, z) is calculated by rotating the 2D radius distribution around the axis.
Fig. 2 Calculation of 3D distribution of refractive index by Abel inversion and parallel phase-shifting digital holography. The transparent object has axially symmetric refractive index around y-axis, the light wave propagates along z-axis and transmits through an object.
A light wave propagates along z-axis, then the phase lag distribution of the light wave after transmitting through the object having the refractive index n(x, y, z) is given as
where Δn(x, y, z) = n(x, y, z) - n0 and n0 is the refractive index of air. If the refractive index n(x, y, z) is axially symmetric along the y-axis, the relationship between the phase lag distribution and the refractive index is given asThe relationship between φ(x, y) and (2π/λ)Δn(r, y) is called the Abel transform. The 3D distribution of refractive index of the object is calculated by the inverse of the Abel transform or Abel inversion of φ(x, y):where the radius r = (x2 + z2)1/2 and R is an arbitrary maximum value of r.The 2D distribution of the object wave is reconstructed by PPSDH. Therefore, 3D distribution of the refractive index of moving transparent object is retrieved.
4. Experiment
We constructed a parallel phase-shifting digital holographic system and experimentally demonstrated availability of the proposed technique reconstructing 3D distribution of the refractive index of gas flow.
4.1 Experimental setup
We constructed the experimental system shown in Fig. 3(a). This system is based on polarization-based PPSDH [16]. The four phase-shifted holograms are simultaneously recorded with a single-shot exposure by using a polarization imaging-camera (PIC). A PIC is a camera having a 2 × 2 micro-polarizer array in front of the image sensor of the PIC as shown in Fig. 3(a) and capable of capturing the 2D distribution of the polarization state of an incident light to the image sensor plane with a single-shot exposure. Intensities of the four linear polarizations contained in the incident light are detected by each 2 × 2 pixels of the image sensor of PIC. A light wave emitted from the laser is divided into two light waves by the polarizing beam splitter (PBS). Both polarizations of the waves are orthogonal to each other. One wave transmits through the object and becomes the object wave, and the other light wave is the reference wave. The two waves become the clockwise and the counter clockwise circularly polarized light waves after passing through the quarter wave plate (QWP), respectively. These light waves are incident to the image sensor of the PIC. The same linear polarization components of the two waves are interfered. Then, the four phase-shifted holograms required for PSDH are simultaneously recorded by 2 × 2 pixels of the PIC based on the space-division multiplexing of the holograms.
Fig. 3 Experimental set up of the system of parallel phase-shifting digital holography. (a) Schematic of the system. PBS, polarizing beam splitter; QWP, quarter wave plate; PIC, polarization-imaging camera. (b) Motion pictures of the transparent gas flow as the object (see Visualization 1). To visualize the flow, a black paper was used and bent in the motion pictures.
A Nd:YVO4 laser operated at 532 nm was used as a light source. A Photron FASTCAM-SA2-P whose pixel pitch is 10 μm was used as a PIC, which recorded motion picture of holograms at 3,000 fps. The number of the pixels in the holograms was 1024 × 1024. Transparent gas flowed from a nozzle of a gas duster with a diameter of about 1 mm, as shown in Fig. 3(b), was recorded. A commercial gas duster emits compressed gas consists of carbon dioxide and dimethyl ether was used.
We reconstructed the 2D phase distribution of the light wave after transmitting through the flowed transparent gas. Then, we assumed that the gas is axially symmetric along the direction in which the gas flowed, and then calculated the 3D distribution of refractive index by applying Abel inversion. n0 = 1.0003 was assumed as the refractive index of air.
4.2 Experimental results
Figure 4 shows the experimental results. From the flow start of the gas, the results after 3 ms are shown in Figs. 4(a) and 4(c), and the results after 120 ms are shown in (b) and (d). Figures 4(a) and (b) show the unwrapped 2D phase lag distributions of the light wave, after transmitting through the gas, reconstructed by PPSDH, respectively. Here, the phase was unwrapped [17]. Figures 4(c) and 4(d) show the 3D distribution of the refractive index distributions calculated from the reconstructed 2D phase distributions as shown in Figs. 4(a) and 4(b), the phase distributions are approximately linearly symmetric in 2D, and thus the refractive index distributions are assumed to be approximately axially symmetric along the direction in which gas flowed. Therefore, Abel inversion can be applied to the 2D distribution, and the 3D distribution of refractive the index is calculated, as shown in Figs. 4(c) and 4(d). Interestingly, periodic phase distributions appeared as shown in Fig. 4(b). Also, the periodic pattern was also retrieved in the reconstructed refractive index distributions as shown in Fig. 4(d). Thus, the 3D distribution of refractive index of the flowed transparent gas was retrieved.
Fig. 4 Reconstructed 2D phase distribution and 3D distribution of refractive index. These distribution (a) 3 and (b) 120 ms after the gas flow started were obtained.
Motion pictures of the reconstructed 3D distribution of the refractive index from the flow start of the gas to 10 ms after and from 90 ms to 140 ms after the flow start are shown in Figs. 5(a) and 5(b), respectively. As shown in Fig. 5(b), periodic distribution of the refractive index appeared from 90ms after the flow start, which does not appear just after the flow start as shown in Fig. 5(a). Thus, we succeeded in retrieving the motion-pictures of 3D distribution of the refractive index of flowed transparent gas.
Fig. 5 Motion pictures of the reconstructed 3D distribution of the refractive index. These motion pictures show the distribution (a) from the flow start of the gas to 10 ms after (see Visualization 2), (b) from 90 ms to 140 ms after the flow start (see Visualization 3).
Figure 6 shows the line profiles of the reconstructed radius distribution of the refractive index 120 ms after the flow start of the gas. Figure 6(a) shows the radius distribution. The gas was flowed along the y-axis. Figures 6(b)–(e) show the distributions of the refractive index along the four lines in Fig. 6 (a). Along both the lines through the ridge and the valley of the radius distribution, as shown in Figs. 6(b) and 6(c), the distribution had maximum value not on the central axis but outside of the axis symmetrically. As shown in Fig. 6(d), the period of the distribution along the direction in which gas flowed was about 1 mm. As shown in Fig. 6(e), periodic distribution did not appear around the border between air and the gas.
Fig. 6 Line profiles of the radial distribution of the refractive index. (a) The radius distribution of the refractive index at 120 ms after the gas flow started. Line profiles is along (b) line 1, (c) line 2, (d) line 3, and (e) line 4.
5. Conclusion
We have proposed a technique reconstructing 3D distribution of the refractive index of a high-speed transparent object. The technique applies Abel inversion method to phase distribution reconstructed by parallel phase-shifting digital holography. We have experimentally retrieved the motion picture of 3D distribution of the refractive index of high-speed and transparent gas flow, thus we demonstrated availability of the proposed technique. Indeed, the proposed technique has been applied to an object having approximately axial symmetric refractive index. However, imaging phase variation of inside of an object, which has been difficult conventionally, has been achieved by the proposed technique. Thus, the technique is useful for measurement of true shape or phase of an object.
Funding
This study was partially supported by Japan Society for the Promotion of Science (JSPS) KAKENHI Grant-in-Aid for Challenging Exploratory Research, No. 16K14274.
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