Integral imaging has been attracting considerable attention recently because of its advantages that include full parallax, continuous view-points and real-time full-color operation. However, the thickness of the displayed three-dimensional image is limited to a relatively small value due to the degradation of image resolution. A method is proposed here to provide observers with an enhanced perception of depth without severe degradation of resolution by taking advantage of the birefringence of a uniaxial crystal plate. The proposed integral imaging system can display images integrated around three central depth planes by dynamically altering the polarization and controlling both the elemental images and the dynamic slit array mask accordingly. The principle of the proposed method is described and is then verified experimentally.
©2003 Optical Society of America
Integral imaging (InIm), which is also referred to as integral photography , is one of the currently popular three-dimensional (3D) display techniques. It can provide bare-eyed observers within a certain viewing angle with full color and real-time 3D images that have both horizontal and vertical parallaxes. These features of InIm make it desirable and have stimulated extensive research [2–12]. Figure 1 shows the basic concept of InIm. In the pickup process, each elemental lens constituting the lens array forms each corresponding elemental image based on its position relative to the object, and these elemental images are then stored. In the display process, the elemental images displayed on the display panel are integrated at the original position of the object forming a 3D image.
Although many recent studies have reported on enhancements in the performance of the 3D display system based on InIm from the viewpoint of the viewing angle [9–12] and resolution , several problems remain that must be solved. Limitations in the thickness of the 3D image displayed is one of those problems. When the thickness of the displayed 3D image increases, the resolution of the marginal depth planes (i.e., the lateral slice planes at both ends of the thickness of the 3D image), apart from the central depth plane (i.e., a focused image plane determined by the gap between the lens array and the display panel), degrades so that the thick 3D image cannot be displayed due to the severe resolution degradation . Some efforts to alleviate this problem by controlling the position of the central depth plane have been proposed [6,8]. One method adopts double imaging devices combined with a beam splitter, thus obtaining two central depth planes simultaneously. However this approach involves two pairs of lens arrays and display panels that are aligned precisely through the beam splitter, making the entire system bulky. Moreover, the optical efficiency is low due to loss at the beam splitter. Another method involves controlling the gap between the lens array and the display panel dynamically, resulting in a change in the position of the central depth plane. This method, however, involves the rapid mechanical movement of the lens array in the direction of the lens axis, which causes some problems such as air resistance and noise. Furthermore, these two methods (the beam splitter method and the moving lens array method) become more inefficient when they are applied to a large 3D display. For example, in the beam splitter method, if we wish to display a 3D image of which lateral dimension is 14″, it becomes necessary to prepare at least double 14″ by 14″ imaging devices. In the moving lens array method, it is not a simple problem to move the 14″ lens array axially fast enough to induce the after-image effect.
In this paper, we propose a novel method that is capable of enhancing the depth of a 3D image using a system that has three central depth planes without any mechanical movement of the lens array. This can be achieved by utilizing the birefringence of a uniaxial crystal plate to alter the optical path lengths. There have been several reports on the use of a double-focus lens made of a uniaxial crystal in various applications [13,14]. However, to focus the extraordinary rays at single point is not possible because of a strong astigmatism, i.e., the horizontally incident extraordinary rays are focused at a position different from that of the vertically incident extraordinary rays . In the method proposed here, two converging points of extraordinary rays in the horizontal direction and the vertical direction are used separately by adopting a sliding slit array mask. As a result, the proposed system can produce three central depth planes which are selected by a dynamic polarizer and the sliding slit array mask. By altering the state of the polarizer and sliding the slit array mask sufficiently fast, observers can see integrated 3D images around three central depth planes. Since the resolution of the integrated image is best at the central depth planes and degrades with distance from the central depth plane, the proposed system can provides resolution that is superior to a conventional InIm system that has only one central depth plane. The principle of the proposed method is described below, along with some experimental results.
2. Principle of the proposed method
2.1 Overview and system configuration
The configuration based on the proposed method is shown in Fig. 2. A dynamic polarizer, a sliding slit array mask and a transparent uniaxial crystal plate, the optic axis of which is parallel to the aperture, are inserted between the display panel and the lens array. The sliding slit array mask contains vertical or horizontal slit arrays where the spacing between the slits is equal to the elemental lens pitch and the width (aperture) of each slit is narrower than the elemental lens pitch, while still sufficiently wide to prevent significant diffraction. The proposed system operates in three modes, the ordinary mode, the extraordinary-horizontal mode, and the extraordinary-vertical mode, which correspond to three central depth planes respectively. In the ordinary mode, the polarizer selects the ordinary polarization and the sliding slit array mask becomes transparent. In the two extraordinary modes, the polarizer selects the extraordinary polarizations and the mask acquires a vertical slit array pattern for the extraordinary-vertical mode or a horizontal slit array pattern for the extraordinary-horizontal mode. The slit array pattern of the mask slides by the amount of the pitch of the elemental lens for each slit to sweep corresponding elemental lens, and the elemental images on the display panel change so as to form a 3D image around the corresponding central depth plane for the extraordinary mode.
In the following sections, we explain the imaging properties of the optical system that consists of a uniaxial crystal plate and a single lens, and discuss the function of the mask in the proposed system. Finally, we return to the proposed system and explain its operation.
2.2 Longitudinal shift by the uniaxial crystal
The key point of the proposed method is to change the effective gap (optical path length) between the lens array and the display panel by using the birefringence of the uniaxial crystal, the dynamic polarizer and the sliding slit array mask. Before discussing the proposed InIm system, suppose that rays from one point source on the display panel pass through a glass plate and are focused by a single lens, as shown in Fig 3.
The rays from the point source experience a constant refractive index nglass in the glass plate and are focused at an image point by the lens. According to Snell’s law, the longitudinal shift Δz by the glass plate is given as :
where d is the thickness of the glass plate and θi the incident angle at the air-glass boundary and is assumed to be small sufficiently so that sinθi≈0 and cosθi≈1 hold.
Next, suppose that the uniaxial crystal is inserted in place of the glass plate, as shown in Fig. 4. The optic axis of the uniaxial crystal is aligned vertically, parallel to the aperture. In this case, the rays from the point source experience different refractive indexes according to their polarizations and propagation directions and, hence are not focused at one image point by the lens. Let us investigate this point in detail.
The rays from the point source can be classified as ordinary rays and extraordinary rays, based on their polarizations. The ordinary rays which are polarized perpendicularly to the optic axis of the uniaxial crystal experience a constant refractive index no regardless of the direction of propagation so that they are focused at one image point. The longitudinal shift which the ordinary rays experience in the uniaxial crystal is given by the same form as Eq. (1) with the replacement of nglass with no, that is,
where d is the thickness of the uniaxial crystal. By the lens equation and Eq. (2), the corresponding image plane for the ordinary rays is located at:
from the lens, where g is shown in Fig. 4 and f is the focal length of the elemental lens.
On the other hand, the extraordinary rays, which are polarized in parallel to the optic axis of the uniaxial crystal, experience various refractive indexes from ne to no according to the direction of propagation so that they cannot be focused at one image point. The refractive index which the extraordinary ray experiences is given as :
where ϕ is the angle between the optic axis of the uniaxial crystal and the wave vector of the extraordinary ray. At this point, let us concentrate on two components of the extraordinary rays, that are horizontally incident extraordinary rays whose plane of incidence is perpendicular to the optic axis of the uniaxial crystal and the vertically incident extraordinary rays whose plane of incidence is parallel to the optic axis of the uniaxial crystal. As shown in Fig. 5(a), in the case of the horizontally incident extraordinary rays, ϕ is always 90° so that ne(ϕ)=ne. Since the refractive index that the horizontally incident extraordinary rays experience is always ne regardless of the direction of propagation as shown in Fig. 5(b), the horizontally incident extraordinary rays experience a longitudinal shift by the uniaxial crystal and are focused by the lens at one image point in the same way as the ordinary rays except that the refractive index is now ne.
The longitudinal shift and the position of the image plane for the horizontally incident extraordinary rays are given as:
Finally, in the case of the vertically incident extraordinary rays, the refractive index that the vertically incident extraordinary rays experience, varies according to the direction of propagation by Eq. (4). From the geometry shown in Figs. 5(c) and (d), it can be seen that ϕ in Eq. (4) is equal to π/2+θev. By applying Eq. (4) to Snell’s law(sinθi=ne(ϕ)sinθev) with consideration of ϕ=π/2+θev, we can then obtain the refracted angle θev as:
Actually, the refracted angle θev represents not the direction of the rays but the direction of the wave vector in the uniaxial crystal. The direction of the rays is the same as the direction of the Poynting vector. In the case of vertically incident extraordinary rays, as shown in Fig. 5(d), the direction of the Poynting vector which is perpendicular to the k-surface is not the same as that of the wave vector since the k-surface is not circular in this case. By simple geometrical calculation, the direction of the ray which is the same as that of the Poynting vector is given as :
and the position of the image plane is given by:
In short, three types of rays from the point source, which are ordinary rays, horizontally incident extraordinary rays, and vertically incident extraordinary rays, experience different longitudinal shifts, as given by Eqs. (2), (5) and (9) respectively, and are focused at their corresponding image planes which are called the ordinary mode image plane, extraordinary-horizontal mode image plane, and extraordinary-vertical mode image plane and are given by Eqs. (3), (6) and (10) respectively. Figure 6 shows this point.
2.3 Function of the sliding slit mask
In the previous section, we investigated the imaging properties of a system that consists of a uniaxial crystal plate and a lens. One image plane is formed by ordinary rays and two image planes are formed by the horizontally incident extraordinary rays and the vertically incident extraordinary rays. Among three image planes, two image planes of the extraordinary rays, however, are not useful by themselves since, in general, the rays have both horizontal and vertical directions. To make them useful, we need to restrict one of the two directional components in turn while permitting the other directional components to pass without restriction. A slit mask whose slit position slides to cover whole lens aperture, can be used to perform this function. Figure 7 shows the configuration required to locate a point image at the extraordinary-vertical mode image plane. As shown in Fig. 7, a vertical slit mask placed just behind the lens is used to restrict the horizontal component of the incident extraordinary rays. The vertically incident extraordinary rays pass through the vertical slit without any restriction and are focused at the extraordinary-vertical mode image plane. On the contrary, most of the horizontally incident extraordinary rays from the point source are blocked by the vertical slit so that, approximately, only rays directing the desired image point at the extraordinary-vertical mode image plane can pass the vertical slit (note that the horizontal dispersion with a vertical slit, H1H2, is much smaller than without a vertical slit, H1′H2′, and can be negligible if the slit width is sufficiently narrow). Namely, the role of the vertical slit is to sample the horizontal directional component of the incident extraordinary rays. Therefore the width of the slit should be much smaller than that of the lens, so as to reduce the horizontal dispersion while it must be sufficiently wide to prevent any significant diffraction.
In addition to the use of the vertical slit mask, the horizontal position of the point source for a certain given image point at the extraordinary-vertical mode image plane should be modified according to the position of the slit. Figure 8 shows a top-view of Fig. 7 and shows the paths of the extraordinary rays radiated horizontally from a point source on the display panel. In Fig. 8, the uniaxial crystal is hidden and instead, is replaced by an equivalent longitudinal shift Δzeh (note that we should introduce the longitudinal shift for the horizontally incident extraordinary ray given by Eq. (5) since we concentrate on the horizontal directional component). If we assume that the slit width is sufficiently narrow, we can see that most of the horizontal extraordinary rays (dotted line) are blocked by the vertical slit except one horizontal extraordinary ray (solid line) which is determined by the horizontal position of the vertical slit and the point source. The unblocked horizontal extraordinary ray extends to the given position on the extraordinary-vertical mode image plane and serves to form the given image point. To find the horizontal position x of the point source for the given image point and slit position, let us firstly obtain the horizontal position xh of point Ph which is the point of intersection between the line extending from slit to the given image point and the extraordinary-horizontal mode image plane (see Fig. 8). From the similarity of the triangles, xh should satisfy:
where u and xv are the horizontal positions of the vertical slit and the given image point and leh, and lev are given by Eqs. (6) and (10). Since the point Ph is also a focus for the horizontal extraordinary rays, we can find the horizontal position of the point source from lens equation and Eq. (11) as:
where Δzeh is given by Eq. (5). Equation (12) indicates that the horizontal position of the point source is dependent on the horizontal position of the slit. Therefore it is necessary to modify the horizontal position of the point source according to the slit position.
In summary, by sliding the vertical slit to cover the width of the lens sufficiently rapidly to achieve the afterimage effect and appropriately changing the horizontal position of the point source given by Eq. (12), we can see as if the point source is focused at the extraordinary-vertical mode image plane. In the same manner, the image of the point source can be placed at the extraordinary-horizontal mode image plane using the horizontal slit mask and controlling the vertical position of the point source. Consequently, it is possible to locate images at three image planes via the use of the polarizer (to select between the ordinary and extraordinary rays), uniaxial crystal and the sliding slit mask.
2.4 Proposed system
If the lens array and slit array is used instead of the single lens and the single slit, by the principle of InIm a 3D image can be displayed around the three central depth planes whose positions are given by Eqs. (3), (6), and (10). Based on this principle, a novel InIm system with three central depth planes is configured as shown in Fig. 2. The proposed system operates in three modes. In the ordinary mode, the polarizer selects the ordinary rays and the sliding slit array mask becomes transparent so that an integrated image is formed around the ordinary central depth plane, the position of which is given by Eq. (3). In the extraordinary-vertical mode, the polarizer selects the extraordinary rays and the sliding slit array mask becomes a vertical slit array pattern sliding for each slit to sweep the width of the corresponding elemental lens. The horizontal positions of the elemental images are modified by Eq. (11) in synchronization with the sliding slit array mask and consequently, an integrated image is formed around the extraordinary-vertical central depth plane. The extraordinary-horizontal mode operates in the same manner as the extraordinary-vertical mode. In one operation cycle of two extraordinary modes, the slit array slides, so as to maintain the spacing between the slits so that each slit sweeps corresponding elemental lens. Ideally, an electrically controllable mask such as a liquid crystal (LC) shutter or a polymer-dispersed LC would be appropriate for this purpose since it can change the direction of the slit between the horizontal and vertical directions for two extraordinary modes and can be completely transparent for the ordinary mode.
In the proposed method, the 3D images around three central depth planes have different viewing angles. The viewing angle in the integral imaging system is given by ψ=2tan-1(φ/2g) where φ is the diameter of the elemental lens and g is the gap between the lens array and the display panel [2, 4]. In the proposed method, g should be replaced by g-Δzo for the ordinary mode and g-Δzev and g-Δzeh for the two extraordinary modes. Therefore the viewing angles of the three modes are different from each other. In fact, for a given location of the central depth plane, g in the conventional InIm system and g-Δz of each mode in the proposed method should be the same. Consequently, the viewing angle of the 3D image integrated around the central depth plane of each mode in the proposed method is equal to that of the 3D image integrated by the conventional InIm system of the same central depth plane.
The brightness of the 3D image is also different for the three modes. In the ordinary mode, the sliding slit array mask becomes totally transparent, and in the two extraordinary modes, it becomes vertical or horizontal slit array shapes. This results in a difference in the brightness of the 3D images integrated by the three modes. Therefore the brightness of the display panel for three modes must be controlled to achieve constant brightness for the 3D images.
The adoption of the uniaxial crystal to the InIm increases the number of the central depth planes, but it also can involve accompanying aberrations. The main aberration caused by the unaxial crystal plate is chromatic aberration and errors in the assumption used in the derivation of Eqs. (2), (5), and (9), i.e. sinθi≈0 and cosθi≈1. However these aberrations can be controlled to small values by selecting appropriate materials and system parameters. For example, the refractive index of calcite, the material used in our experiment, varies with wavelength from no=1.68014, ne=1.49640 at λ=410nm to no=1.65207, ne=1.48353 at λ=706nm. By comparing the variation in the refractive index of BK7 which is a widely used glass in optic systems (from n=1.53024 at λ=410nm to n=1.51289 at λ=706nm), we can see that chromatic aberration by calcite is not much more severe than that by BK7. The experimental results in the following section do not show any noticeable chromatic aberration. Errors caused by the assumption of sinθi≈0 and cosθi≈1 also can be minimized by configuring the proposed system with proper parameters as presented in the following section. In fact, the dominant aberration of the InIm system is the spherical aberration of the lens array which causes cracks in the integrated 3D image. Other aberrations such as chromatic aberration by the lens array and the uniaxial crystal are negligible.
Another possible problem associated with the proposed method is that the front 3D image integrated around the front central depth plane cannot occlude 3D images behind, i.e. it is transparent. This originates from the fact that the proposed method displays 3D images around different central depth planes using time-multiplexing. In fact, this problem exists, not only for the proposed method, but is common for many volumetric display methods. Although further research is needed, in this paper, this problem is avoided by displaying 3D images with different lateral offsets so that they can be observed without any overlap within the viewing angle.
3. Experimental results
Based on the principles described above, we performed two experiments to display the integrated image around the three central depth planes. In the first experiment, we verify that an integrated image is formed around the three central depth planes with a manually-sliding slit mask. The specifications of the experimental setup are listed in Table 1 (Experiment 1). A calcite plate was used as a uniaxial plate. Calcite is an appropriate material, in that it has a fairly large birefringence (0.172) and can be produced as a large crystal relatively easily. As the mask, a transparent film patterned with equally spaced vertical or horizontal slits is used. Since the pitch of the elemental lens is 10 mm, the spacing between slits on the transparent film is also fixed at 10 mm. The aperture of each slit is 1 mm. Finally, a static polarizer is used to alter the polarizations between the ordinary and extraordinary polarization. Although we cannot see the integrated images around the three central depth planes simultaneously by an afterimage effect due to the static nature of the polarizer and mask used in our experiment, the feasibility of the proposed method can be verified.
The elemental images are generated by a computer. The horizontal positions of the elemental images in the extraordinary-vertical mode and the vertical positions of the elemental images in the extraordinary-horizontal mode are modified according to Eq. (11). Since the slit aperture is 1 mm and the elemental lens pitch is 10 mm, 10 elemental images are prepared for each of the two extraordinary modes. In this experimental specification, when the gap between the lens array and the display panel is adjusted to locate the position of the ordinary central depth plane 130 mm from the lens array, the central depth planes of the extraordinary-vertical mode and the extraordinary-horizontal mode are placed at 210 mm and 95 mm, respectively, based on Eqs. (3), (6), and (10). In addition, the assumption used in the derivation of Eqs. (2), (5), and (9), i.e. sinθi≈0 and cosθi≈1, is validated in our experimental setup, since the maximum errors on the longitudinal shift caused by that assumption do not exceed 0.8 mm.
Figures 9 (a) and (b) show the integrated images displayed by conventional InIm. Two apples whose locations are 95 mm (left image in each of Figs. 9(a) and (b)) and 210 mm (right image in each of Figs. 9(a) and (b)) in front of the lens array are displayed at the same time. In conventional InIm, only one central depth plane exists whose location is determined by the gap between the lens array and the display panel. By adjusting the gap between the lens array and the display panel, the central depth plane is placed at 95 mm for Fig. 9(a) and 210 mm for Fig. 9(b). In each figure we can see that, although an apple image that is placed at the central depth plane (left apple image in Fig. 9(a) and right apple image in Fig. 9(b)) is integrated clearly, the other apple image, far from the central depth plane (right apple image in Fig. 9(a) and the left apple image in Fig. 9(b)) suffers from severe distortion. This indicates that conventional InIm cannot simultaneously display a 3D image of a large depth difference.
The enhanced results by the proposed method are shown in Figs. 10(a)–(c). Figure 10(a) shows the extraordinary-horizontal mode, Fig. 10(b) shows the ordinary mode and Fig. 10(c) shows the extraordinary-vertical mode, whose corresponding central depth planes are 95 mm, 130 mm, and 210 mm respectively. In Figs. 10(a)–(c), the integrated apple images are located at these three central depth planes by controlling the polarizer, mask, and elemental images without adjusting the gap between the lens array and the display panel. The integrated images of the two extraordinary modes shown in Figs. 10(a) and (c) are obtained by capturing 10 partial integrated images with a fixed charge-coupled-device camera position, and overlapping them digitally without any other image processing. It can be seen that the integrated images shown in Figs. 10(a)–(c) are displayed without distortion. We should compare here two pairs of integrated images: one is the apple in Fig. 10(a) and the left apple in Fig. 9(b) that are integrated at 95mm, and the other is the apple in Fig. 10(c) and the right apple in Fig. 9(a) that are integrated at 210mm. From this comparison, it is clear that the image quality of the apples in Figs. 10(a) and (c) is superior to that of the left apple in Fig. 9(b) and the right apple in Fig. 9(a), respectively. This result shows that in the proposed method, it is possible to integrate 3D images with a large depth difference without any mechanical movement and with minimal image distortion (see Fig. 10(a)–(c)) since the proposed method has three central depth planes, while in conventional InIm with a given position of the central depth plane, it is not possible to integrate 3D images with large depth differences at the same time due to the degradation of image quality (see right apple image in Fig. 9(a) and left apple image in Fig. 9(b)).
The vertical seam-like cracks shown in Figs. 10(a) and (c) are the result of the subtle misalignment of the sliding slit array mask caused by the manual operation in our experiment. By using an electrically controllable mask such as a polymer-dispersed LC, these cracks can be removed. Moreover, the resolution of the integrated image can be enhanced by adjusting the moving step of the slit array pattern under the sub-slit-aperture level, which can be easily achieved by the use of an electrically controllable mask(Note that in our experiment, the moving step of the slit array is the same as the slit aperture). Narrowing the moving step increases the sampling rate and, as a result, enhances the resolution of the integrated image by the principle similar to that of the moving lens array technique reported recently [5,18].
In the second experiment, we performed a preliminary experiment for real-time operation. We used a spatial light modulator (SLM) instead of a sliding slit array mask with a pair of linear polarizers. Two computers, one to display the slit pattern on the SLM and the other to display elemental images on the display panel, were synchronized by an RS-232C serial cable. Although the integrated image can be displayed around only two central depth planes of extraordinary modes due to the static nature of the linear polarizer fabricated on the SLM, they can be integrated without any mechanical movement by realizing a moving mask pattern on the SLM. Details of the experimental setup are listed in Table 1 (Experiment 2) and the experimental setup is shown in Fig. 11.
The gap between the lens array and the display panel is adjusted to be located two central depth planes at 140mm for the extraordinary horizontal mode and 243mm for the extraordinary vertical mode. The experimental results are shown in Fig. 12. Figure 12(a) shows the images integrated by the conventional integral imaging method with the gap (between the lens array and the display panel) adjusted so the central depth plane is located at 190mm. Figure 12(b) shows images integrated by the proposed method. Since our experimental setup was not optimized for the real-time operation at the time of this experiment, 0.5s was required to complete one operation cycle, which is the time required for each slit pattern on the SLM to cover the corresponding elemental lens aperture so that one 3D image can be displayed. Due to the fairly long operation cycle, Fig. 12(b) was obtained by superimposing the partial images taken for the different slit positions of the sliding slit array mask, instead of the after-image effect. The integrated images shown in Fig. 12 are of lesser quality, on the whole, compared to those in Figs. 9 and 10, since the SLM used in our experiment is too small(see Table 1) to display an integrated image of sufficient size for reasonable resolution. However, the improved result of the proposed method over that of the conventional one can be easily seen in Fig. 12.
Although the integrated images in Figs. 10, 12 are located at the exact central depth planes, in general, we can locate the integrated image around each of the three central depth planes. Consequently, the proposed method can provide observers an integrated image with an enhanced perception of depth.
We proposed and experimentally verified a novel method for enhancing the depth of an integrated image utilizing the birefringence of a uniaxial crystal plate. The proposed method creates three central depth planes such that the integrated images can be displayed around them without significant distortion.
This work was supported by the Ministry of Science and Technology of Korea through the National Research Laboratory Program.
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