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Abstract
The three-dimensional information of an object can be obtained under low coherent light through incoherent digital holography (IDH). In the spatially parallel phase-shifting method by the geometric phase, the IDH optical setup using a diffractive lens can cause noise due to high-order diffracted and scattered light. Therefore, we constructed an IDH optical setup using a Michelson interferometer without a diffractive lens. We investigated the relationship between the focal length of the concave mirror in the interferometer and the resolution of the reconstructed image. The resolution could be improved by shortening the focal length. Furthermore, we confirmed that the motion blur in the IDH is consistent with the conventional two-dimensional imaging system, and demonstrated that videos could be captured at 60 fps.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Incoherent digital holography (IDH) can generate holograms under incoherent light such as sunlight, light-emitting-diode (LED) lighting, and fluorescence [1]. The applications of IDH include the acquisition of three-dimensional information of an object illuminated with incoherent light [2–11], fluorescence microscopes [12–18], radiation thermometers [19], and wavefront measurements [20]. A camera that can acquire three-dimensional information of the object in the same lighting environment as a conventional two-dimensional camera is an attractive application, and IDH can acquire objects both indoors and outdoors as well as human faces [4,10]. To develop IDH and capture videos, as with conventional two-dimensional cameras, a frame rate of several tens of fps is required.
In IDH, a light from an object (object beam) illuminated by incoherent light sources is divided into two beams, and a hologram is generated through self-interference in which the optical path length of one beam is slightly changed and combined. Various interferometers such as common-path interferometers [2,6,8–11], Michelson interferometers [4,7], Mach–Zehnder interferometers [21], and triangular interferometers [22] have been used as the IDH optical setup. Similar to conventional holography using laser, IDH also uses the phase-shifting method to obtain a highly accurate complex amplitude distribution of the hologram [23]. Generally, in the phase-shifting method, the length of one of two optical paths forming an interferometer is temporally changed to generate a plurality of holograms with different phases. The complex amplitude distribution of the hologram can be calculated from the intensity distribution of these holograms. However, when the object moves while a plurality of holograms is acquired, acquiring holograms with various phases under the same conditions becomes difficult. To solve this problem, a spatially parallel phase-shifting method, in which holograms with various phases are spatially divided and obtained simultaneously, was studied extensively. In a proposed method, a checker-patterned phase grating is placed in each branched optical path in a Michelson interferometer, and a spatially parallel phase-shifting method is realized by the relative positional deviation of phase gratings [24]. However, constructing an optical setup is difficult because aligning the two phase gratings with high precision is essential. For easily constructing an optical setup, the generation of a hologram coaxially using a geometric phase and acquisition of it by a spatially parallel phase-shifting method have been proposed. Figure 1 shows this principle. Two orthogonal linearly polarized light beams are extracted from the object beam, and a birefringent lens with a different focal length for each linearly polarized light makes a slight optical path difference. By passing both the polarization components of the object beam through a 1/4 wave plate, they are converted to right-handed and left-handed circularly polarized light. Holograms are extracted as a geometric phase by a spatially arranged polarizer and captured by a camera. This allows the holograms of various phases to be captured simultaneously.
A spatially parallel phase-shifting method using a transmissive GP lens as a birefringent lens and a polarized camera as a spatially arranged polarizer has been proposed [6,10]. Video capture at 25 fps was demonstrated by simultaneously generating and shooting four holograms with different phases [6]. However, because the GP lens is a diffractive optical element, high-order diffracted light and scattered light are likely to occur, as well as wavelength dependence. Furthermore, the GP lens has a low selectivity in designing the focal length for condensing or diverging two orthogonal linearly polarized light components. These factors affect the quality of reconstructed images. Capturing videos at a frame rate of 1000 fps has been reported by constructing the IDH optical setup with a lens using birefringent materials instead of the GP lens in order not to produce high-order diffracted light [11]. Very high frame rates are required for scientific applications, but general cameras capture videos at 60 fps from the viewpoint of visual characteristics of human eyes and amount of information of videos. In addition, the details of videos such as shooting plural moving objects placed at different depths and comparing motion blur in accordance with frame rate are not described. Furthermore, it is necessary to introduce a liquid crystal phase modulator to adjust the optical path length, which complicates the adjustment of the optical setup to interfere with incoherent light. Liquid crystal lenses are being considered as birefringent lens, but their size and focal length flexibility is limited. Furthermore, because it is also a diffractive optical element, image quality deterioration similar to that of a GP lens occurs. A method of constructing a birefringent lens using a high-resolution reflective liquid crystal panel has been proposed [25,26]. Because it is a diffractive optical element, high-order diffracted light, scattered light, and wavelength dependence occur, but the selectivity in designing the focal length increases. However, because it is a reflective type of liquid crystal panel, inserting a beam splitter (BS) in the optical path is necessary, which reduces the light utilization efficiency to 1/4. The light utilization efficiency is critical for capturing videos because the camera exposure time decreases when the frame rate of the video increases. In IDH, a narrow-band wavelength filter is used to improve time coherency, which results in low light utilization efficiency. Therefore, an optical setup with minimal light loss is necessary. If the object beam is obliquely incident on the liquid crystal panel without using BS, the light utilization efficiency does not decrease [2]. However, because the degree of modulation by liquid crystal panel depends on the incident angle of the object beam, the incident angle is limited and the field of view of the image narrows.
We proposed an IDH optical setup composed of a Michelson interferometer with a polarization beam splitter (PBS) and a polarized camera to capture video with frame rate of 60 fps. This IDH optical setup is an interferometer using a geometric phase similar to an optical setup using a birefringent lens. By using PBS instead of BS, it is possible to prevent a decrease in light utilization efficiency. The Michelson interferometer of similar configuration have been used for laser-based interferometric measurements [27], atmospheric wind velocity and temperature measurements [28], and three-dimensional imaging of objects illuminated by incoherent light [29–31]. Because a diffractive optical element is not used in the Michelson interferometer, high-order diffracted light and scattered light originating from the diffractive optical element are not generated. Moreover, the resolution of the reconstructed image should be improved because of various options for the focal length of the concave mirror for creating interference fringes. Focusing on the parameters of the IDH optical setup to shoot objects in the same lighting environment as a conventional two-dimensional camera, we investigated the focal length of the concave mirror and found the conditions for obtaining high resolution and subsequently constructed the optical setup. Since this optical setup that prevents a decrease in light utilization efficiency and a noise due to high-order diffracted and scattered light, and obtains high resolution is suitable for capturing and evaluating videos, by using it, we captured videos of plural moving objects placed at different depths at a frame rate of 60 fps, and evaluated the motion blur of the videos.
2. IDH optical setup using Michelson interferometer with PBS
2.1 Optical setup
Figure 2 shows an IDH optical setup with a Michelson interferometer for improving the efficiency of light utilization [29–31]. The principle of generating a hologram is the same as the spatially parallel phase-shifting method using the geometric phase depicted in Fig. 1. Here, an object-side lens and a camera-side lens are placed before and after the Michelson interferometer to adjust the imaging magnification of the object. The object beam passes through the object-side lens, a wavelength filter, and a polarizer at an angle of 45° to the horizontal plane and enters a PBS. The s-polarized component of the object beam is reflected by the PBS, passes through the 1/4 wave plate and becomes circularly polarized. This component is reflected by the flat mirror, passes through the 1/4 wave plate again, and becomes p-polarized, and subsequently passes through PBS. Conversely, the p-polarized component of the object beam passes through the PBS, passes through the 1/4 wave plate, and becomes circularly polarized and is reflected by the concave mirror, passes through the 1/4 wave plate again and becomes s-polarized and is reflected by PBS. The s-polarized and p-polarized object beams that enter the PBS is emitted to the camera side, and the object beam does not return to the object side. The two object beams that pass through the camera-side lens become circularly polarized in opposite directions after passing through the 1/4 wave plate. Interference fringes are generated because of the small optical path length difference between the two object beams and are captured by a polarized camera with 0°, 45°, 90°, and 135° polarizers at each pixel. Consequently, four holograms I1 to I4, whose phases are different every 90° can be obtained simultaneously.
Fig. 2. Optical setup of IDH consisting of Michelson interferometry with PBS using the geometric phase.
2.2 Reconstruction calculation and resolution of image
The complex amplitude distribution of hologram H is expressed as follows using the acquired four holograms I1to I4.
Then, the reconstructed image O at an arbitrary depth distance rZ from the hologram plane can be obtained through back propagation calculation of light.3. Resolution of reconstructed image depending on focal length of concaved mirror
3.1 Simulation
In the optical setup of Fig. 2, when the focal lengths of the object-side lens and the camera-side lens change, both the field of view and the resolution of the reconstructed image change, and generally, the wider the field of view, the worse the resolution is. In addition, the higher resolution is suited for evaluating motion blur described later. Therefore, we investigated an optical setup that improves the resolution when the field of view is fixed. Figure 3 shows the field of view FOV and the resolution δ of the reconstructed image with respect to the focal length of the concave mirror, respectively. Table 1 shows each parameter except for the focal length of the concave mirror of the optical setup. In this simulation, we investigated the focal length of the concave mirror from 300 to 1000 mm because the focal length of approximately 600 mm was used in the IDH optical setup for photographing outdoor landscapes [4]. Furthermore, we studied the same focal length from the perspective of the resolution of reconstructed images. The relation between the resolution and the focal length has been investigated, and the larger the focal length difference between the two lenses in the interferometer, the higher the resolution of the reconstructed image is [13,32]. However, if the difference in the focal length becomes too large, the optical path difference between the two beams becomes large, and the cycle of the interference fringes becomes narrower than the pixel pitch of the camera. This indicates that the higher frequency of the hologram cannot be captured. Therefore, the focal length of the concave mirror was set from 300 to 1000 mm. As shown in Fig. 3, the field of view of the reconstructed image FOV was constant at 110 mm regardless of the focal length of the concave mirror. By contrast, the resolution δ became smaller as the focal length shortened. As for the parameters of this optical setup, the resolution could be improved by decreasing the focal length of the concave mirror without changing the field of view of the reconstructed image.
Table 1. Parameters of Optical Setup
3.2 Experimental result
Figure 4 shows the prototype IDH optical setup based on Fig. 2. Table 1 presents each parameter of the optical setup. The wavelength filter had a center wavelength of 630 nm, and a wavelength width of 10 nm. The size of the lenses, mirrors, and filters was 1 inch. The polarized camera had a pixel pitch of 3.45 µm and used 2048 × 2048 pixels out of 2448 × 2048 pixels. The angles of the 1/4 wave plates between the mirrors and the PBS could in principle be either ±45°. Therefore, we adjusted the angle to maximize the light intensity passing through the PBS by rotating the 1/4 wave plate. Furthermore, phase shift at the mirror surface does not need to be considered in this interferometer because the object lights are reflected by flat and concave mirrors, respectively. A transmissive USAF target was used as the object, and a red LED was illuminated from the back. Figure 4 displays the captured area in red rectangle. Figure 5 shows the reconstructed image of the object. The vertical lines indicated by blue rectangle have been captured clearly. This area was group 1 and element 2 in the USAF target, and the spatial frequency was 2.24 lp/mm. Figure 6 shows the contrast between bright and dark area of vertical lines with respect to the spatial frequency in Fig. 5. By using the brightness of bright and dark areas Bbright and Bdark, the contrast Cs is represented as follows:
As shown in Fig. 6, the contrast does not decrease less than 2.24 lp/mm of spatial frequency. When spatial frequency was 2.24 lp/mm, the width of each line was 0.22 mm. As shown in Fig. 3, the theoretical resolution of this optical setup was 0.22 mm, and the theoretical resolution could be obtained. In general, because the Michelson interferometer is susceptible to vibration, we configured the IDH optical setup on a vibration-isolating table in order to obtain data with high accuracy in this experiment. However, in our previous study [29,30], we confirmed that the IDH optical setup can be placed on a heavy desk without vibration-isolation ability to take images and videos.4. Capturing moving objects
4.1 Capturing video at 60 fps
We captured a video using the IDH optical setup shown in Fig. 4. Currently, it takes several seconds for the calculation to reconstruct an image of one frame. Therefore, first, an image with polarization information taken by a polarized camera was recorded as a video file, and subsequently each frame was extracted from the recorded video file to reconstruct an image with a desired depth distance. Then, we created a video file based on the reconstructed images of each frame. Although the exposure time varies with frame rate, the brightness levels of each video were normalized by using maximum and minimum brightness when calculating reconstruction. As objects, two dice were placed 200-mm apart in the depth direction, with the die in the front rotating and the die in the back moving linearly to the left and right, as shown in Fig. 7. They were illuminated with red LEDs. Videos shot at 10, 30, and 60 fps are presented in Videos 1 to 3. In each video, (a) is focused on the front dice, and (b) is the back dice. Videos 1(a), 1(b), 2(a), 2(b), (3a), and 3(b) are associated with Visualization 1, Visualization 2, Visualization 3, Visualization 4, Visualization 5, Visualization 6, respectively. Table 2 shows the exposure time per frame at each frame rate. With the increase in the frame rate, the motion blur of the video decreased, but noise was superimposed and the image quality deteriorated because the exposure time reduced. However, we demonstrated that videos can be acquired at 60 fps.
Table 2. Exposure Time at Each Frame Rate
4.2 Evaluation of video blur
We captured videos at 10 to 60 fps by moving the transmissive USAF target horizontally at a constant speed of 20 mm/s at a position 800 mm from the object-side lens of the IDH optical setup. The relation between the speed and the position is suitable for evaluating motion blur among these frame rates. Table 2 presents frame rates at which we captured video and its exposure time per frame. Figure 8 shows an enlarged image of one frame extracted from the video. Figure 8 reveals that the motion blur occurs at a low frame rate, and it decreases with the increase in the frame rate. In this experiment, a theoretical field of view of 110 mm (as shown in blue line of Fig. 3) was captured with an image sensor pixel count of 2048 pixels. Therefore, the estimated speed of the object movement on the image sensor plane is 370 pixels/s. The amount of object movement per frame at each frame rate can be estimated to be 33.3, 22.2, 16.7, 11.1, 7.4, and 5.5 pixels. Figure 9 shows brightness levels for horizontal pixels in the lower three vertical lines in Fig. 8 (group-2, element 4 in the USAF target) and its spatial frequency characteristics that is normalized with the DC component to evaluate the signal intensity against the DC component. The brightness level is the average value of 10 pixels in the vertical direction. To evaluate the motion blur, we investigated the relationship between the frame rate and the contrast Cd (Fig. 10). By using the maximum brightness Bmax, and the minimum brightness Bmin in Fig. 9(a), the contrast Cd is calculated as follows:
As shown in Fig. 10, the contrast increased as the frame rate was higher. When the frame rate was 10 fps, the contrast was calculated to be larger than at 15 fps. However, as shown in Fig. 8(a) and from the green line in Fig. 9(a), the regular striped pattern could not be recognized at 10 fps and the accidental appearance of the striped pattern in which the light and dark phases were reversed, which caused an increase in contrast. Therefore, the actual contrast at 10 fps can be regarded as 0. Furthermore, the contrast was saturated above 30 fps. However, as shown in Figs. 8(d) to (f), when the frame rate was 60 fps, the boundary between light and dark became clear, the contrast near the boundary improved, and the motion blur reduced compared with that at the frame rate of 30 fps. This is clear from the increase in the spectral intensity of the spatial frequency around 0.015 in accordance with increase of frame rate, as shown in Fig. 9(b). Motion blur could be reduced as the frame rate increased in the IDH as well as in the imaging system.5. Conclusions
In this study, the IDH optical setup composed of a Michelson interferometer with a PBS and a polarized camera was proposed to realize a spatially parallel phase-shifting method using a geometric phase. Because this optical setup does not use a birefringent lens consisting of diffractive optical element, high-order diffracted light or scattered light originating from the diffractive optical element are not generated, and wavelength dependency does not occur. Furthermore, the focal length of the concave mirror can be suitably chosen for creating interference fringes, which enables an optical design for improving the resolution of the reconstructed image easily. Thus, we investigated the field of view and resolution of the reconstructed image with respect to the focal length of the concave mirror in the IDH optical setup parameters for photographing in the same environment as a conventional two-dimensional camera. As a result, the field of view was constant, but the resolution improved as the focal length was shortened. Based on this investigation, we fabricated the IDH optical setup and confirmed that the reconstructed image was acquired with the theoretical resolution. Furthermore, we demonstrated that video acquisition at 60 fps was possible in IDH. We also captured videos from 10 to 60 fps and revealed that the moving blur of the video reduced as the frame rate increased, which was similar to the conventional imaging system. This method is effective for capturing high-resolution videos in IDH.
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.
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