Holographic three-dimensional display and hologram calculation based on liquid crystal on silicon device [Invited] Junchang Li,1,* Han-Yen Tu,2 Wei-Chieh Yeh,3 Jinbin Gui,1 and Chau-Jern Cheng3 1
Faculty of Science, Kunming University of Science and Technology, Kunming 650093, China 2
3
Department of Electrical Engineering, Chinese Culture University, Taipei 11114, Taiwan
Institute of Electro-Optical Science and Technology, National Taiwan Normal University, Taipei 11677, Taiwan *Corresponding author:
[email protected] Received 22 April 2014; revised 7 August 2014; accepted 10 August 2014; posted 12 August 2014 (Doc. ID 210507); published 11 September 2014
Based on scalar diffraction theory and the geometric structure of liquid crystal on silicon (LCoS), we study the impulse responses and image depth of focus in a holographic three-dimensional (3D) display system. Theoretical expressions of the impulse response and the depth of focus of reconstructed 3D images are obtained, and experimental verifications of the imaging properties are performed. The results indicated that the images formed by holographic display based on the LCoS device were periodic image fields surrounding optical axes. The widths of the image fields were directly proportional to the wavelength and diffraction distance, and inversely proportional to the pixel size of the LCoS device. Based on the features of holographic 3D imaging and focal depth, we enhance currently popular hologram calculation methods of 3D objects to improve the computing speed of hologram calculation. © 2014 Optical Society of America OCIS codes: (090.2870) Holographic display; (070.6120) Spatial light modulators; (090.1760) Computer holography. http://dx.doi.org/10.1364/AO.53.00G222
1. Introduction
Recent studies on holographic three-dimensional (3D) display [1–5] have demonstrated that liquid crystal on silicon (LCoS) microdisplay devices [6] that feature phase-only modulation exhibit relatively high diffraction efficiency [7–9]. LCoS is a basic device used extensively in current holographic 3D display systems [10–12]. Because of constant technological advancements in the manufacturing and packaging processes of such device materials, the pixel resolution and number of pixels of commercial LCoS have continued to increase. Because of these features, LCoS-based spatial light modulators (SLMs) can be applied in holographic 3D display. Nevertheless, 1559-128X/14/27G222-10$15.00/0 © 2014 Optical Society of America G222
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LCoS is applied to phase holograms, which consider the amplitude of the object light waves reaching an LCoS surface as a constant. Additionally, rather than saving all information of the object light waves, only information regarding the phase of the waves is retained. To identify the effect of this type of treatment on 3D image quality, the image quality of optical systems must be examined. Furthermore, the phase hologram of 3D objects involves complex and timeconsuming calculations [13–20]. Currently, two types of calculations are commonly used: 1) the point source method [13,21–24], in which the light waves generated from 3D objects are considered as light waves generated by a high number of point sources that constitute a surface to be displayed, and 2) the panel method [25–28], in which the luminous surface of an object is treated as the sum of numerous tiny surface light sources. The point source method is based on
simple geometrical-optics principles, and object surface characteristics can be easily expressed during calculations. However, this method involves a massive amount of calculation. To accelerate the calculations for generating holograms, Lucente from the Massachusetts Institute of Technology proposed the look-up table (LUT) method [13]. Specifically, the contribution that discrete point sources distributed at a particular density in a 3D space make to a holographic plane is calculated in advance. The calculated results are prestored in computer storage devices. When calculations are performed on a given 3D object, the relationship between the object surface location and calculated spatial points is considered, and the LUT method is applied to obtain the diffracted field of the object surface. To obtain rapid and accurate calculations requires a huge storage space. Therefore, to reduce the storage space requirement and accelerate calculations [29–31], researchers have primarily employed various methods for approximating and simplifying the equations used to calculate the diffracted fields of point sources. By contrast, the panel method is based on the fact that analytical solutions exist in the frequency spectrum of diffracted fields between the observation screen and the uniform surface source of a space featuring special geometry. The curved surface is deconstructed into a series of surface elements. The frequency spectrum of the diffracted field that reaches the observation plane can be obtained by performing linear superposition on the frequency spectra of the diffracted fields of all surface elements. Inverse Fourier transform can be performed on the spectra to yield the diffracted field of the curved surface. To accelerate the calculations and enhance the quality of reconstructed images, many researchers have investigated [19–35] the following processes: optimizing surface element deconstruction in objects featuring complex surface structures, using mathematical expressions that involve few calculations to express surface element frequency spectra, exploring the blocks that occur during the propagation of light with various surface elements, and eliminating the effects of surface element boundaries on object image reconstruction. In the panel method, surface elements with uniform amplitude are used to construct object surfaces. Therefore, surface element sizes must be reduced substantially to express object surface materials, thereby substantially increasing the number of calculations involved. Therefore, methods for accelerating the calculations used in the surface elements warrant future investigation. A review of studies [3,20] that have employed the two calculation methods mentioned previously revealed that during calculations, object surface sampling is not affected by the wavelength of the illumination light or the display distance of the reconstructed image, but by the geometry and material of the objects. However, an observation of the physical process of holographic 3D display indicated that the reconstructed images exhibit depth of focus [2]. Based on the physical importance of depth of focus, sampling
in the direction of the optics axis by using the depth of focus as the sampling spacing can reduce the number of samples while accelerating the phase hologram calculations. Therefore, exploring the depth of focus in LCoS-based holographic 3D display systems is vital. Based on the scalar diffraction theory and the geometry of LCoS, the impulse response of 3D imaging systems featuring phase-only modulation was investigated in this study. An impulse response expression [36] for optical systems was obtained. The results indicated that after loading phase holograms to the LCoS device, a disc of confusion centered on an ideal image point can be formed, and the disc is only a very small blur spot caused by the aperture effect of the LCoS device, thereby yielding satisfactory 3D images of objects. However, the images yielded by the optical systems were cyclical image fields surrounding the optics axis, and the energy distribution of the image fields was uneven. A high aperture ratio of the LCoS indicates concentrated energy in the image light fields surrounding the optics axis. In addition, the width of the image light field was proportional to the light wavelength and diffraction distance, and inversely proportional to the pixel width of LCoS. These findings have been confirmed by experimental studies. Although recent works [37–39] have studied the impulse response and frequency analysis of digital holography, this work studies an LCoS device operated in the phase modulation mode for optical reconstruction, which is different from the intensity recording of the holographic fringes on a CCD device. Hence, the theoretical analysis and experimental verification are more suitable for the optical implementation of holographic 3D reconstruction and display. 2. LCoS-Based Holographic Display
The LCoS-based holographic display system can be considered a linear system, the image quality of which is determined by the impulse response of the systems. In this study, the impulse responses of the system were investigated based on scalar diffraction theory. A. Impulse Responses of LCoS-Based Holographic Display System
A Cartesian coordinate system o − xyz was established in the space. Figure 1 shows the definitions of the coordinates based on scalar diffraction theory. Specifically, the plane with z 0 is the window plane
Fig. 1. Coordinate definitions for LCoS-based reconstructed imaging systems. LCoS plane. 20 September 2014 / Vol. 53, No. 27 / APPLIED OPTICS
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of LCoS. The diffraction field of light waves [emitted from point source Pξ; η; −d on plane z −d] passing through the LCoS-based reconstructed imaging system was examined. The light field formed by the light waves generated by point sources in plane z 0 can be obtained using the δ function and the Fresnel diffraction integral [36]: Z Z expjkd ∞ ∞ uδ x; y; ξ; η δx0 − ξ; y0 − η jλd −∞ −∞ jk 2 2 x − x0 y − y0 dx0 dy0 : × exp 2d (1) Incorporating the sifting property of the δ function yields expjkd jk 2 2 exp x − ξ y − η : uδ x; y; ξ; η jλd 2d (2) Based on the expression of a phase hologram with unitary amplitude, let expjkd jk 0 2 2 exp x − ξ y − η : uδ x; y; ξ; η j 2d (3) The phase hologram formed in LCoS can be expressed as follows: H δ x; y; ξ; η u0δ x; y; ξ; ηwx; y;
(4)
where wx; y is the window function of LCoS. Based on the pixel structure and coordinate definitions of the LCoS in SLMs (Fig. 2), a rectangular function (rect) and a comb function (comb) were integrated. Thus, the window function can be expressed as
wx; y rect
x y wα xrect w y; N x Δx N y Δy β
(5)
x x − Δx∕2 comb ; wα x rect αΔx Δx
(5a)
y y − Δy∕2 wβ y rect comb ; βΔy Δy
(5b)
where the aperture ratio α; β ∈ 0; 1 was used to describe the nonactive region that exists between neighboring pixels in the LCoS. Let αΔx × βΔy be the size of an individual pixel. The distance between the centers of two neighboring pixels is Δx in the direction of the x axis, and Δy in the direction of the y axis. N x and N y are the numbers of pixels in the directions of the x axis and y axis, respectively. After unit amplitude plane waves reach the LCoS along the z axis, the light field of the reflected waves in plane z −d can be expressed using inverse operation of Fresnel diffraction [36]: Z Z exp−jkd ∞ ∞ H δ x; y; ξ; η hδ xi ; yi ; ξ; η −jλd −∞ −∞ jk 2 2 × exp − xi − x yi − y dxdy: 2d (6) Equation (4) is integrated into Eq. (6), which is rewritten based on the definition of two-dimensional (2D) inverse Fourier transform: hδ xi ; yi ; ξ; η
1 jk 2 exp ξ η2 λd 2d jk 2 2 × exp − xi yi F −1 fwx; yg; (7) 2d
where the coordinates of F −1 fwx; yg following the 2D inverse Fourier transform are fx
xi − ξ ; λd
fy
yi − η : λd
(8)
After computation, Eq. (7) is simplified as follows: 1 jk 2 2 ξ η hδ xi ; yi ; ξ; η exp λd 2d jk 2 2 × exp − xi yi × N x ΔxN y Δy 2d × sincN x Δxf x sincN y Δyf y Φα f x ; ΔxΦβ f y ; Δy; Fig. 2. Pixel structure and coordinate definitions of the LCoS device. G224
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where
(9)
Z Φα f x ; Δx
∞ −∞
hδ xi ; yi ; ξ; η
wα x expj2πf x xdx
αΔx2 sincαΔxf x combΔxf x − 1∕2; (10) Z∞ wβ yexpj2πf y ydy Φβ f y ;Δy −∞
βΔy2 sincβΔyf y combΔyf y − 1∕2: (11) Thus, the impulse responses of the LCoS holographic display system were obtained. To analyze the imaging properties of the LCoSbased display system, the expression of the function following the convolution sign in Eq. (9) was generated: Φα f x ; ΔxΦβ f y ; Δy αβΔx2 Δy2 sincαΔxf x sincβΔyf y combΔxf x − 1∕2combΔyf y − 1∕2;
(12)
where combΔxf x − 1∕2combΔyf y − 1∕2 denotes the sampling array of the δ function when the lateral and longitudinal cycles are 1∕Δx, 1∕Δy, respectively. After using this sampling array to sample the function of sincαΔxf x sincβΔyf y , Φα f x ; ΔxΦβ f y ; Δy becomes a 2D weighted sampling array with a weight of sincαΔxf x sincβΔyf y . The value of this array decreases as Δxf x − 1∕2 nx 0; 1; 2; … and Δyf y − 1∕2 ny 0; 1; 2; … vary. The symbols nx and ny denote the diffraction orders of the sampling array in the spatial frequency plane. The coordinates of the sampling array in the image plane are integrated into Eq. (8): λd ; Δx λd : yi η ny 1∕2 Δy
xi ξ nx 1∕2
(13)
Thus, the impulse responses expressed in Eq. (9) indicate the cyclical distribution of function sincN x Δxf x sincN y Δyf y , which is centered on the ideal image point. The coordinates of the center in each cycle are nx 1∕2λd∕Δx; ny 1∕2λd∕Δy. For a given object point ξ; η, the impulse responses in each image cycle should form a disc of confusion centered on the ideal image point. When the LCoS array size is large, the disc of confusion or image point size is small and the quality of the reconstructed image is high. In addition, an analysis of Eq. (11) revealed that the LCoS-based holographic display system is not a linear spatially invariant system [36]. This is because impulse responses hδ xi ; yi ; ξ; η are related to the value of ξ; η. However, if the focus is only the image intensity distribution of a particular cycle near the origin, such as the cycle of nx ny 0, Eq. (8) can be integrated into Eq. (9) and the phase factors that are irrelevant to the intensity distribution can be ignored. Thus, Eq. (9) can be further simplified and written as
1 αβN x Δx3 N y Δy3 sincα∕2 λd Δx λd xi − ξ − × sincβ∕2sinc N x λd 2Δx Δy λd y −η− : (14) × sinc N y λd i 2Δx
The relationship between the sinc and δ functions indicated that the image quality is high when the LCoS array size N x Δx, N y Δy is large and Eq. (14) approaches the δ function. When the surface of a 3D object is considered as a set of numerous point sources, various point sources form discs of confusion centered on the respective ideal image points in corresponding image planes. When phase holograms are generated, the amplitude of the diffracted field is considered as a constant. Consequently, the amplitude distribution in object spatial reconstruction results in distorted 3D object images. Notably, Eq. (14) is only the result of Eq. (9) in the cycle where nx ny 0. Different nx and ny can result in varying expressions. However, the reconstructed images are modulated by the 2D function sincN x Δxf x sincN y Δyf y . Therefore, the images formed in the image cycles in the four quadrants surrounding the optics axis are relatively intense. B. Experimental Verification of the Reconstructed Image Distribution
To verify the theoretical analysis, the following empirical results are provided. Figure 3 shows the simplified experimental setup of the LCoS-based holographic display. A green laser beam with a wavelength of 532 nm was projected vertically onto a polarizing beam splitter (PS). After being expanded and collimated, the incident light wave was partially reflected from the PS and emitted onto LCoS, thereby yielding reconstruction waves. The light waves emitted from the LCoS penetrate the PS and reach the observation screen, which shows the reconstruction images. During the experiment, phase holograms of the 3D object were loaded into the LCoS. The diffraction distance between the LCoS and the observation screen was d 1200 mm. The distance between the pixels in the LCoS was Δx Δy 6.4 μm, the aperture ratio was α β 0.93, and the array number was N x 1920 and N y 1080. Figure 4(a) shows an image of the observation
Fig. 3. Simplified experimental setup of the LCoS-based holographic display. 20 September 2014 / Vol. 53, No. 27 / APPLIED OPTICS
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screen taken using a camera during the experiment. The image shows four reconstructed image cycles surrounding the optics axis. At the four corners of each cycle are bright rectangular spots, which are diffraction images of the 2D periodic gratings formed by LCoS pixel structures that had existed before signals were loaded. The actual measurement results indicated that the cycle of the rectangular spots (or reconstructed image cycle) approached the result predicted based on theories, that is, λd∕Δx λd∕ Δy 99.75 mm. In addition, Fig. 4 shows the experimental results of the 3D reconstruction image, where Fig. 4(a) is a virtual 3D object to be calculated into a phase hologram and Fig. 4(b) is the reconstructed image in the observation screen. The result shows that despite the amplitude distortion that theoretically occurs during reconstruction, the reconstructed image yielded by the experiment can satisfactorily represent the original morphology of 3D objects. C.
Fig. 4. Experimental results of the 3D reconstruction image. (a) Virtual 3D object and (b) 3D reconstructed image.
Let
Φα f 0x ; Δx
Depth of Focus
To simplify the calculations for obtaining phase holograms, the distribution of point sources in a defocused image when the image deviates from the ideal image plane was explored. In Eq. (6), let the reconstruction distance d0 ≠ d; thus, Z∞Z∞ exp−jkd0 − d 0 × wx; y hδ xi ; yi ; ξ; η λd0 −∞ −∞ jk 2 2 × exp x − ξ y − η 2d jk × exp − 0 xi − x2 yi − y2 dxdy: 2d (15) Convert Eq. (15) into Fourier transform form: exp−jkd0 − d jk 2 0 2 ξ η exp hδ xi ; yi ; ξ; η λd0 2d Z Z ∞ ∞ jk wx; y × exp − 0 x2i y2i 2d −∞ −∞ jk × exp − 0 x2 y2 2d x ξ × exp j2π x i 0 − λd λd y η dxdy: (16) y i0 − λd λd
xi ξ ; − λd0 λd
f 0x
Z
∞ −∞
yi η ; − λd0 λd
f 0y
wα x expj2πf 0x xdx
αΔx2 sincαΔxf 0x combΔxf 0x − 1∕2; (18) Φβ f 0β ; Δy
Z
∞ −∞
wβ y expj2πf 0y ydy
βΔy2 sincβΔyf 0y combΔyf 0y − 1∕2: (19) Rewrite Eq. (16) as h0δ xi ; yi ; ξ; η
exp−jkd0 − d jk 2 2 ξ η exp λd0 2d jk × exp − 0 x2i y2i F −1 2d x y rect × rect N x Δx N y Δy jk × exp − 0 x2 y2 2d Φα f 0x ; ΔxΦβ f 0y ; Δy:
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Z
∞
Z
∞
(20)
Combine the phase factors in the Fourier transform integral in Eq. (20). After computation, Eq. (21) was obtained as h0δ xi ; yi ; ξ; η Ψxi ; yi ; ξ; η Φα f 0x ; ΔxΦβ f 0y ; Δy; (21) where
X Y rect exp 1 − d0 ∕dN x Δx 1 − d0 ∕dN y Δy −∞ −∞ 8 h i2 h i2 9 > > 0 0 > d d = < Y − yi − d η > X − xi − d ξ dXdY × jk 0 > > > > 2d0 1 − dd ; :
Ψxi ; yi ; ξ; η Θ0 xi ; yi ; ξ; η
(17)
rect
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(22)
and Θ0 xi ; yi ; ξ; η is a complex constant related to xi ; yi ; ξ; η. Thus, the complex amplitude of the defocused point source image yielded by the LCoS-based display system was obtained. Equations (21) and (22) show that the amplitude distribution of the defocused image is a Fresnel diffraction image in which the rectangular hole passes through distance d0 1 − d0 ∕d. The widths of the rectangular hole were 1 − d0 ∕dN x Δx and 1 − d0 ∕dN y Δy, respectively. Based on the characteristics of Fresnel diffraction, when d0 1 − d0 ∕d is relatively small, the size of the Fresnel diffraction spots can be considered as the approximate geometric projection of the rectangular hole. Therefore, when the difference between d0 and d is small or the defocus distance is short, the defocused image of the point source concentrates on a small area. To obtain the depth of focus for the reconstructed image, Eq. (14), the equation for obtaining the impulse response, was examined. In Eq. (14), let T x λd∕N x Δx, T y λd∕N y Δy, x0i xi − ξ − λd∕2Δx, and y0i yi − η − λd∕2Δy. Thus, the light intensity distribution at the image point was proportional to sincx0i ∕T x sincy0i ∕T y 2 . Let T T x T y for Fig. 5, which shows the image of sincx0i ∕T2 . Noticeably, the light field energy of the reconstructed image of the point source was primarily confined within a square area (width: 2T) surrounding the image point coordinates. Observing the shape of the reconstructed image from the perspective of human eyes, if the width of the defocused image remains 2T when the image is far from the ideal image plane, the defocused image can be considered a sufficiently satisfactory point source image. Based on this assumption, the following was obtained:
d − d0
λd
N Δx: 2T 2 N x Δx d x
(23)
Define dh jd − d0 j as the depth of focus of the reconstructed image. Thus,
Fig. 5. Curve of sincx0i ∕T2 .
Fig. 6. Four letters object used in a phase hologram.
dh
2λd2 : N x Δx2
(24)
This indicates that for a given LCoS, the depth of focus of reconstructed images increases as the diffraction distance increases. For example, let λ 532 nm, N x 1024, and Δx 6.4 μm. When d 400 mm, dh ≈ 3.96 mm; when d 1200 mm, dh ≈ 35.67 mm. D.
Experimental Verification of the Defocused Image
A 3D Cartesian coordinate system o − xyz was established. Figure 6 shows the definitions of the coordinates that constitute a phase hologram. Plane z 0 is the plane where LCoS is located. In this experiment, the objects were four 2D luminous letters, namely, A, B, C, and D, which were placed in planes z −d, z −d Δd, z −d 2Δd, and z −d 3Δd, respectively. The locations are shown in Fig. 6. Let the letters (A, B, C, and D) propagate plane waves in the direction of the optics axis at a wavelength of λ 532 nm. The light field reaching plane z 0 can be regarded as a superposed light wave generated by each letter. To compare the ideal image and the defocused image within the depth of focus, d 400 mm, d 1200 mm, and Δd dh∕3 were established before two sets of experiments were conducted. The LCoS pixel width used in the experiments was Δx Δy 6.4 μm. The width of the planes in Fig. 6 was set at λd∕Δx before calculations were conducted to obtain the phase holograms. After the phase holograms were obtained by conducting theoretical calculations, the experiment was performed by following the optical setup shown in Fig. 3. Figures 7 and 8 show the reconstructed images of four cycles surrounding the optics axis.
Fig. 7. Comparison of the reconstructed image for the observation distance d 400 mm. 20 September 2014 / Vol. 53, No. 27 / APPLIED OPTICS
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effectively. This section presents a discussion on the improvement methods that are applicable to the point source and panel methods. A. Improving the Calculations in the Point Source Method
Fig. 8. Comparison of the reconstructed image for the observation distance d 1200 mm.
The images were taken from the observation screen captured by a digital camera. The images to be observed were marked in the red-dashed block. Among the two sets of images, at Δd 0 mm, the reconstructed images of the four letters (A, B, C, and D) in the plane z −d as shown on the right of Figs. 7 and 8, respectively, appear to have the same image quality with different intensity levels. The slopes of the boundary of each letter image were checked to verify the sharpness of the image. Therefore, these were ideal reconstructed images without blurring. Images in which Δd 0 were reconstructed images within the range of the depth of focus. No perceivable difference was observed between the ideal images and the reconstructed images within the range of the depth of focus. By contrast, Δd 1.3 mm and Δd 13 mm, the reconstructed images for d 400 and 1200 mm, respectively, as shown in the right of Figs. 7 and 8, showed the focused letter image of A and the slightly defocused image of D. The slope of the boundary of letter A was larger than that of letter D, which appears slightly blurring in the boundary of the letter. Hence, the depth information of 3D reconstruction of the LCoS device was observed in the Fresnel plane and the experimental image quality can be further improved by using a highdefinition camera.
Figure 9 is a schematic diagram of the theoretical research. Specifically, plane z 0 was defined as the LCoS plane, and plane z −d was defined as the plane tangent to 3D objects. An object was divided into layers with a thickness identical to the depth of focus based on Eq. (24) and in a direction opposite to that of the optics axis. According to the discussion regarding the depth of focus for imaging, the light waves that a sampled point set P0 (sampled from the surface of a layer of the object) emitted in the direction of the optics axis can be considered as the light waves emitted by the projected point set PT sampled from the front split plane. By integrating the phase variations resulting from the optical path changes that occurred during the projection into the calculations, the diffraction field on curved surfaces can be replaced with the point source diffraction field on the split plane. Although in LUT calculations, point sources are distributed at a particular density in a predesigned 3D space, according to the findings of this study, the model can be modified to feature fewer point sources on a 2D surface. Thus, the calculations and storage space necessary for generating a table can be reduced [40]. B. Improving the Calculations in the Panel Method
The calculations used in studies on the depth of focus of images reconstructed using the LCoS-based holographic 3D display system can be simplified
Figure 10 shows a schematic diagram of the simplified panel method used in studies on the depth of focus in LCoS-based holographic imaging. Similarly, plane z 0 was defined as the LCoS plane, and plane z −d was defined as the plane tangent to 3D objects. The objects were divided into layers with a thickness identical to the depth of focus based on Eq. (24) and in a direction opposite to that of the optics axis. What triggers human eyes to perceive images is the intensity distribution of images. The light waves that the tilted microsurface light sources emit from the surfaces of each layer of the optics axis can be considered as the light waves that the tilted microsurface light sources emit from the projection light source in the front split plane. Therefore, given a 3D object and an imaging distance, the object can
Fig. 9. Schematic diagram of the simplified point source method.
Fig. 10. Schematic diagram of the simplified panel method.
3. Improving the Methods for Calculating Phase Holograms
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be divided using parallel planes by the depth of focus in a direction vertical to the optics axis. Thus, the surface of curved objects can be divided into a series of curved bands of light. From the perspective of human eyes, the light waves emitted from a band of light can be regarded as the light waves that the projection of the band of light in the direction of the optics axis emits from the light source in the split plane. Thus, the diffraction calculations in the surface element segmentation method [17,41] can be converted to calculations based on the light waves that the projections (PT , surrounding the split plane) of surface elements P0 (of various shapes and sampled from a curved band of light) emit. Because the numerical calculations are typically conducted based on points sampled from the split planes, complex calculations that involve dividing projection light sources into microsurface light sources are unnecessary. Therefore, common diffraction calculations suffice for directly obtaining the diffraction fields [42] of the projection surface elements in each split plane. In other words, the simplified panel method involves calculations similar to those used in the simplified point source method. A comparison with the panel method [17], this study conducted the hologram calculation for a plain 3D object with various numbers of surface elements. The calculation times of the hologram for 1600 surface elements are 420 and 30 s (simplified panel method), respectively, for fair image reconstruction. The results revealed that the simplified panel method has better speed-up on hologram calculation and can dramatically accelerate the calculation speed of the hologram particularly for a complicated 3D object requiring large numbers of surface elements. C. Experimental Verification of the Improved Calculation Methods
To provide an experimental verification of the improved calculations conducted to obtain phase holograms, the spatial coordinates of the surfaces of the 3D letters in Fig. 4 were used as the sampled point groups. A comparison between the improved calculations and the layered model was conducted. The layered model is a calculation method that can
Fig. 12. Comparison of the images reconstructed using the two methods when d 1200 mm. (a) The improved calculation method. (b) The layered model.
theoretically yield satisfactory results regarding the diffraction fields of 3D object surfaces [43]. This method involves dividing the surface of an object by using planes that are vertical to the optics axis and are spaced at short intervals. The points on the intersecting lines between a plane and the object constitute 2D point sources. Using conventional diffraction calculation methods to superpose the diffraction fields of each plane in plane z 0 yields the diffraction field of the 3D object surface. When the intervals of the split planes and the sampled points in the split planes satisfy the sampling theorem, the layered model can be employed to obtain images that can replace ideal reconstructed images. Let the illumination light wavelength λ 532 nm, N x 1024, and Δx 6.4 μm. When d 400 mm, dh ≈ 3.96 mm; when d 1200 mm, dh ≈ 35.67 mm. To simplify the calculations, the space between the split planes in the layered model remained dh 1 mm. The calculations are time consuming primarily because the process involves calculations for each 2D diffraction field. The results of phase hologram calculations performed using a computer indicated the ratio of the time consumed in performing the improved calculations and the layered model approached ds ∕dh . In other words, the improved calculation method features a considerably improved calculation speed. To compare the image quality yielded by the two calculation methods, experiments were conducted using the method employed in Figs. 6 and 7. The experimental results are shown in Figs. 11 and 12, respectively. No difference between the qualities of the images reconstructed based on the improved calculation method and the ideal reconstructed image was observed. 4. Conclusions
Fig. 11. Comparison of the images reconstructed using the two methods when d 400 mm. (a) The improved calculation method. (b) The layered model.
This study describes the holographic 3D display and computation methods based on the phase-modulated LCoS device. We have investigated the impulse responses in an LCoS-based holographic 3D display system and the depth of focus of the display images. Theoretical expressions of the impulse response and the depth of focus of reconstructed 3D images were obtained, and experimental verification was 20 September 2014 / Vol. 53, No. 27 / APPLIED OPTICS
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conducted. Furthermore, based on the theoretical results of the depth of focus, the point source calculations and microsurface element calculations conducted for obtaining phase holograms of 3D objects were improved, yielding a more efficient phase hologram calculation method that is applicable to holographic 3D displays. The results of the theoretical analysis and experiments confirmed that incorporating the depth of focus into the LUT method can reduce necessary storage space effectively. This work is supported by the National Natural Science Foundation of China (60977007), the personal development Foundation of Kunming University of Science and Technology (KKSY201307134), and the National Science Council Taiwan (1022221-E-003-025-MY3).
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