Abstract: We propose a hogel overlapping method for the holographic printer to enhance the lateral resolution of holographic stereograms. The hogel size is directly related to the lateral resolution of the holographic stereogram. Our analysis by computer simulation shows that there is a limit to decreasing the hogel size while printing holographic stereograms. Instead of reducing the size of hogel, the lateral resolution of holographic stereograms can be enhanced by printing overlapped hogels, which makes it possible to take advantage of multiplexing property of the volume hologram. We built a holographic printer, and recorded two holographic stereograms using the conventional and proposed overlapping methods. The images and movies of the holographic stereograms experimentally captured were compared between the conventional and proposed methods. The experimental results confirm that the proposed hogel overlapping method improves the lateral resolution of holographic stereograms compared to the conventional holographic printing method.
© 2013 OSA
The holographic stereogram is one of the prominent methods for displaying autostereoscopic three-dimensional (3D) images . In this technique, 3D images are synthesized from sequences of closely spaced two-dimensional (2D) perspective views and are discretely recorded as a hologram on a holographic material like photopolymer. These series of 2D perspective images are stored in small holographic elements, called hogels . With an appropriate illumination, each hogel in the holographic stereogram diffracts a fraction of perspective images within a certain viewing angle. The diffracted lights are fractions of the perspective image recorded on the hogels in the holographic stereogram. On the observing plane, relatively large numbers of viewing points are formed by the merged lights diffracted from the hogels. An observer can see a 3D image from these perspective views at the corresponding viewing point. When the observer’s left and right eyes are located at different viewing points, the observer sees stereoscopic images and perceives 3D images .
The holographic stereogram comprises a large number of hogels. To build a large number of hogels on the holographic material, a holographic printer sequentially records the hogels until the whole area of the holographic material is recorded like a conventional dot printer.
Figure 1 illustrates the principle of a reflective-type holographic printer. For a signal beam, the perspective object images displayed on a spatial light modulator (SLM) are illuminated by an expanded laser beam and converged on the surface of the holographic material with a predefined area. This predefined area determines the size of the hogel as p. A reference laser beam is incident on the same area from the opposite side to generate the reflective-type hologram. The information about the perspective images on the SLM is recoded as an interference pattern formed by the signal and reference beams in the hogel. After a single hogel is recorded, a motorized 2-axis linear stage translates the holographic material horizontally and vertically to the next recording position, and consecutive perspective images are displayed on the SLM for the next hogel recording. The distance of translations, or shifting distance, along horizontal and vertical directions of the motorized stage is indicated as δ in Fig. 1. In the conventional holographic printer, in general, translations of the holographic material (δ) are equal to the size of the hogel (p).
The recording characteristics of the hogel are closely related to the quality of 3D images in the holographic stereogram. For example, viewing image qualities, such as lateral and angular resolutions, color representation, viewing angle, are directly affected by the characteristics of hogels. Each hogel should provide a homogenous spectrum to provide a natural 3D image, and the size of the hogel is directly related to the lateral resolution of the holographic stereogram . In view of the importance of hogels in the holographic stereogram, methods for recording hogels in holographic printing have been actively studied in recent years in both academic and commercial fields [4–15]. Yamaguchi et al. reported a holographic printer using an SLM for exposing perspective images to the corresponding hogels in 1990s [4–7]. Peyghambarian group worked on rewritable holographic printing by using a photorefractive polymer as holographic material [8–10]. Some holographic printers have been used commercially by several companies like Zebra Imaging and Geola. Zebra Imaging commercialized a full-color and large format holographic printer [11, 12], and Geola suggested a fast recordable holographic printer using a pulsed laser and reported an even faster printing method using a hologram copier [13–15]. However, the enhancement of lateral resolution for holographic printing has not been actively investigated, though the lateral resolution of the printer is an important quality factor.
In this paper, we propose a hogel overlapping method to enhance the resolution of the holographic printer. We computer-simulate the hogel recording in the holographic printer to explain why the recordable hogel size, obviously affecting the lateral resolution, cannot be reduced infinitesimally. The proposed method assumes that the holographic printer using photopolymer as a holographic material can record hogels as a volume hologram. According to the volume hologram theory, multiple holograms can be superimposed with several multiplexing methods including angle, wavelength, spatial, and so on [16–18]. With the multiplexing property of the photopolymer, the proposed method can increase the lateral resolution by recording spatially multiplexed hogels with overlap to avoid decreasing the hogel size. In addition, to demonstrate the validity of the proposed hogel overlapping method on the holographic printer, we compare the lateral resolutions of the two holographic stereograms optically recorded by the conventional and proposed methods.
2. Limitation on enhancing lateral resolution of holographic printer by decreasing the hogel size
The lateral resolution of the holographic printer can be increased by decreasing the hogel size, because it is inversely proportional to the hogel size. However, the size of hogel cannot decrease infinitesimally owing to the recording principle of the holographic printer. Figure 2 shows an optical arrangement on the signal (object) beam path in the holographic printer to explain the recording principle. The perspective image on the SLM is Fourier transformed by a lens to the Fourier plane where the hogel mask is located. After passing the hogel mask, the signal beam is then demagnified and relayed through 4-f optics to the holographic material of photopolymer. On the plane of the holographic material, the relayed signal beam is interfered with the incident reference beam from the opposite side, and the information on the SLM is recorded on the holographic material with the resultant interference pattern.
Although the hogel mask is designed to set the size and shape of the hogel, it also plays a role of a spatial filter due to its position on the Fourier plane. The hogel mask performs as a high-frequency cutoff spatial filter, and blocks the high-frequency components of the image on the SLM. As a result, if the hogel mask is too small, the perspective image on the SLM cannot be perfectly transferred to the holographic material, and thus a blurred image is recorded. For this reason, the recording small-sized hogel for a high lateral resolution of the holographic stereogram causes loss of high-frequency components of the perspective image, which means a quality degradation of the recorded image.
In order to find out an impact of the size of hogel mask on the quality of the image reconstructed on the holographic stereogram, we have performed a computer simulation. In this computer simulation, we assume that the hogel mask is a perfect square which has the length of a side W. Then, the hogel mask function H is expressed in coordinates (u, v) on the Fourier plane with a rectangular function rect
The input image Iin on the SLM in coordinates (x, y) is Fourier transformed on the Fourier plane by the first lens in front of the SLM. The spatial filtering of the hogel mask on the Fourier plane can be expressed by multiplying the Fourier transformed input image by the hogel mask function. In order to visualize the degradation on the reconstructed hogel image due to the size of hogel mask, we reconstruct the spatially-filtered Fourier transformed input image by inverse Fourier transform. The reconstructed perspective image Ire is represented by19].
In our computer simulation, SLM has a 720 x 720 lateral resolution with a 14-μm pixel pitch, and the focal lengths for the first lens (f1) and the 4-f optics (f2 and f3) are 150 mm, 75 mm, and 4 mm, respectively. The specifications of the SLM and lenses assumed in the computer simulation are the same as those used in the experiments explained in Section 4. We measured the peak signal to noise ratio (PSNR) between the input image (Iin) displayed on the SLM and the reconstructed perspective image (Ire) by varying the size of the hogel mask from 0 mm to 8 mm at intervals of 0.1 mm. The actual hogel sizes recorded on the holographic material plane are 0 μm to 427 μm resulted from the demagnification in the 4-f optics.
We used the USAF resolution chart for the input image on the SLM which has a resolution of 720 × 720 as shown in Fig. 3(a). The reconstructed hogel images for the hogel sizes of 100 μm, 200 μm, and 400 μm are shown in Figs. 3(b)-3(d), respectively. The graph in Fig. 3(e) illustrates the dependency of PSNR on the size of hogel mask, which indicates the PSNR decrease with reducing the size of recorded hogel. The numerical values of PNSR are 40.79 dB, 47.87 dB, and 55.12 dB for the hogel images reconstructed with hogel sizes of 100 μm, 200 μm, and 400 μm, respectively. As expected, the reconstructed hogel images show quality degradation, such as image blur, and the degradation is getting worse when the size of hogel mask decreases.
3. Proposed hogel overlapping method for enhancing lateral resolution of holographic printer
We employ a volume holographic multiplexing technique to enhance the lateral resolution of the holographic printer without image degradation due to its small hogel size. Since a photopolymer is used as the holographic material for the volume hologram, every perspective image in the hogels is recorded as a volume hologram. Bragg selectivity of the volume hologram allows multiple holograms to be recorded in the same area of the holographic material .
A hogel is the smallest addressable element in a holographic stereogram, and the interval among the hogels, shifting distance in the holographic printer, is inversely proportional to the lateral resolution of the holographic stereogram. Since the shifting distance δ is equal to the hogel size p in the conventional holographic printer, the lateral resolution of the conventional holographic printer is decided by the hogel size. As explained in the previous section, however, the hogel size cannot be arranged to be infinitesimally small for increasing the lateral resolution.
In order to improve the lateral resolution while maintaining a constant hogel size, we propose a hogel overlapping method for the holographic printer. In the proposed hogel overlapping method, the shifting distance is smaller than the hogel size, which makes adjacent hogels overlapped. When the hogel is a perfect square and shifting distances in vertical and horizontal directions are equal, the maximum number M of overlapped hogels is expressed as
The maximum number of recordable overlapped hogels depends on the dynamic range of holographic material used in holographic printing. The dynamic range of holographic material is a measure of the capacity of the media, i.e., how many holograms can be stored at a given diffraction efficiency. And it is determined by the maximum refractive index modulation and the thickness of holographic material .
To get a uniform image, the ratio (α) of p to δ should be a natural number so that the overall hogel is identically overlapped except for the hogels at the edges. When this condition is satisfied, the lateral resolution of the proposed hogel overlapping method can be uniformly increased by a factor of α2.
Figure 4 illustrates a typical example of recording schemes for the conventional holographic printing method (M = 1 and α = 1) and the proposed hogel overlapping method (M = 4 and α = 2). In this figure, a dashed box and dots in it represent the recorded hogel and the center points of recorded hogels, respectively. Both methods have the same hogel size of 400 μm. In contrast to the conventional method which has the shifting distance identical to the hogel size, the proposed new method has a shifting distance of 200 μm which is the half of the hogel size. Though the overlap region exists in four adjacent hogels in the proposed method, image information on each of four perspective images can be recorded on the same area of the photopolymer thanks to the multiplexing property of the volume hologram. In this example, the proposed method has a lateral resolution enhanced by 4 times compared to the conventional method.
4. Experiments on holographic printing with hogel overlapping
An experimental setup for holographic printing has been built up to verify the resolution enhancement by the proposed method. Figure 5 shows a schematic diagram of the experimental setup of holographic printer system. A diode-pumped solid-state (DPSS) green (532 nm) laser is used for a light source, and an electric shutter is used to control the time of a light exposure. The polarization state of the laser is adjusted by a pair of a λ/2 wave-plate and a polarizer. After a beam is expanded by a spatial filter and a 50 mm focal length collimating lens (L1), a power ratio of the signal to reference beam is adjusted by a λ/2 wave-plate, a polarizing beam splitter, and polarizers. On the signal beam path, a series of perspective images are loaded on the SLM with a diffuser, and focused on the photopolymer by an objective lens (OL2) with 4-f relay lenses (L2 and L3) whose focal lengths are 150 mm and 75 mm, respectively. The SLM used in the experimental setup is a transparent liquid crystal (LC) display which has a 1280 × 720 resolution with a 14 μm pixel pitch. The displayed perspective image on the SLM has a resolution of 720 × 720 pixels. The reference beam is incident on the photopolymer through two 250 mm focal length lenses (L4 and L5) with an incidence angle of 60°. To record rectangle-shaped hogels on the photopolymer, hogel masks are located on both signal and reference beam paths. In the intersection of signal and reference beams, a photopolymer is located on a motorized 2-axis linear stage. The linear stage has a 0.25 μm precision and a 205 mm travel range on both horizontal and vertical directions. The photograph of the experimental setup for holographic printing is shown in Fig. 6.
We recorded two holographic stereograms to compare the lateral resolutions between the conventional and proposed holographic printing methods as described in Fig. 4 and Table 1. We used the computer generated perspective images having a scene of two objects, a cube and a sphere, located in different depths. To record their holographic stereograms in the same area for both methods, 50 × 50 and 100 × 100 perspective images are printed in hogels for the conventional and proposed method, respectively. Compared to the conventional method, total number of hogels in the proposed method is four times larger than that in the conventional method, which means that the lateral resolution of the proposed method is increased by four times.
The resultant holographic stereograms captured from four different viewing points for the conventional and proposed methods specified in Table 1 are compared in Fig. 7. Both holographic stereograms are captured from right, left, top, and bottom viewing points, and the captured images are focused on the photopolymer plane. It is clearly confirmed in the experiments that the captured holographic stereogram images in Fig. 7(b) using the proposed hogel overlapping method have much higher lateral resolution compared to the images in Fig. 7(a) captured from the conventional method. We also recorded movies for both methods and single-frame excerpts from video recordings are compared in Fig. 8. The movies in Media 1 and Media 2 are recorded by varying the focused point from the cube to the sphere. Edges of the cube and sphere are much clearly observed in the movie recorded using the proposed method (Media 1) compared to the movie from the conventional method (Media 2). The experimental results shown in Figs. 7 and 8 support the validity of the proposed method of hogel overlapping holographic printing for the enhancement of the lateral resolution.
In this paper, we have proposed a new method for enhancing the lateral resolution of holographic stereogram by hogel overlapping in the holographic printer. Since the high-frequency components of perspective images are lost when the size of hogel is decreased, the enhancement of the lateral resolution by decreasing hogel size has limitations. By taking advantage of multiplexing property of the volume hologram, hogels can be recorded in the overlapped region. When we record hogels in the overlapped region, the interval among hogels is smaller than the size of hogel, and thus the lateral resolution of holographic stereogram can be enhanced. The maximum number of overlapped hogels is discussed in conjunction with the dynamic range of the holographic material. We set the holographic printing system which can control the shifting distance by the motorized 2-axis linear stage. Comparison of the two holographic stereograms, optically recorded by using conventional and proposed holographic printing methods, reveals the feasibility of the lateral resolution enhancement by the proposed method. The perspective images from different viewing points and the movies recorded by varying the focus were demonstrated to compare the lateral resolutions between two holographic stereograms. The experimental results for the proposed hogel overlapping holographic stereogram confirmed the enhancement of lateral resolution compared to the conventional holographic stereogram.
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