## Abstract

The modulation function of holographic functional screen (HFS) in the real-time, large-size full-color (RLF), three-dimensional (3D) display system is derived from angular spectrum analysis. The directional laser speckle (DLS) method to realize the HFS is proposed. A HFS by the DLS method was fabricated and used in the experiment. Experimental results show that the HFS is valid in the RLF 3D display, and that the derived modulation function is valuable for the design of the HFS. The research results are important to realize the RLF 3D display system which will find many applications such as holographic video.

©2010 Optical Society of America

## 1. Introduction

Three-Dimensional (3D) display has attracted much attention because of its many applications, such as scientific research, industry, medical operation, military, architecture design, and virtual reality. Several 3D display techniques have been demonstrated, such as binocular parallax (including parallax barrier) [1], the integral photography (including multi-view imaging) by lens-board [2], and holography [3–8]. The binocular parallax and integral photography belong to parallax imaging. Both their image resolution and brightness are much lower, and there are blind areas between the vision regions of eyes. Recently, digital technology-based holography has captured people’s wide interest, such as computer-generated holography [9–11], 3D holographic video with space light modulator (SLM) [12–14], and others [15–19]. However, it is necessary to improve image qualities, such as resolution, image depth, chromatic aberration, and fidelity. For the conventional holography, it is difficult to realize real-time, full-color, large-size 3D display due to the limitations of recording materials and techniques. Volume displays for realizing 3D images are based on rotation of mechanical components and duration in vision, but they cannot provide a fully convincing 3D experience because of their limited color reproduction and small body display space. In order to overcome above shortages and reduce the system complexity, we proposed and developed a practical real-time, large-size full-color (RLF), 3D display system [20,21]. However, the problems associated with the RLF 3D display system are not fully solved, especially the operating principle of the holographic function screen (HFS). To further investigate the proposed technique, we study the HFS’ modulation function and its realizing method.

## 2. Modulation function of the HFS

As shown in Fig. 1 , the RLF 3D display system is mainly composed of a color digital cameras array (CA), a video server (VS), a color projector array (PA), and a HFS. The 3D information of the object is picked up and recorded by the CA. The VS connects the CA and the PA through conventional AV cables. The information transmitted by AV cables is processed in the VS and is then projected by the PA. 3D information is modulated and displayed by the HFS. Here we analyze both the original and recovered 3D information of the 3D object in the system with angular spectrum which is a representation form of space frequency. The light waves scattered from the object are illustrated by Eq. (1)

*α*

_{mn},

*β*

_{mn}], and can be represented as Eq. (2)

*z.*The $\frac{{{\displaystyle \alpha}}_{mn}}{{{\displaystyle \lambda}}_{l}},\frac{{{\displaystyle \beta}}_{mn}}{{{\displaystyle \lambda}}_{l}}$ are the separate angular spectrum components and the ${{\displaystyle \varphi}}_{mn}(\frac{{{\displaystyle \alpha}}_{mn}}{{{\displaystyle \lambda}}_{l}},\frac{{{\displaystyle \beta}}_{mn}}{{{\displaystyle \lambda}}_{l}};t)$is the angular spectrum distribution at the vision angle [

*α*

_{mn},

*β*

_{mn}]. The relation between them can be described as

*x*,

*y*,

*z*are simply used for representing the coordinated of the image space. The angular spectrum information is along the direction [

*α*

_{mn},

*β*

_{mn}], and the carried${{\displaystyle f({{\displaystyle x}}^{\text{'}},{{\displaystyle y}}^{\text{'}};{\lambda}_{l};t)}}_{mn}$ can be respectively observed with a diffusion screen. However, this superposed 3D-image can’t be recognized with the same screen. Thus, it is necessary to expand each angular spectrum and to make all angular spectra synthesized as continuous spectra – that is to recover the original spatial angular spectrum distribution ${{\displaystyle \varphi}}_{0}(\frac{\alpha}{{{\displaystyle \lambda}}_{l}},\frac{\beta}{{{\displaystyle \lambda}}_{l}};t)$. We designed a HFS which has a special modulation function

*T*(

*x*,

*y*, 0). The angular spectrum expansion in a small solid angle

*ω*

_{mn}can be represented by integral operation ${\iint}_{{\omega}_{mn}}\mathrm{exp}[i2\pi (\frac{\alpha}{{{\displaystyle \lambda}}_{l}}x+\frac{\beta}{{{\displaystyle \lambda}}_{l}}y)]}d(\frac{\alpha}{{{\displaystyle \lambda}}_{l}})d(\frac{\beta}{{{\displaystyle \lambda}}_{l}})$. The convolution operation (*) of the integral and $\delta (\frac{\alpha}{{{\displaystyle \lambda}}_{l}}-\frac{{{\displaystyle \alpha}}_{mn}}{{{\displaystyle \lambda}}_{l}},\frac{\beta}{{{\displaystyle \lambda}}_{l}}-\frac{{{\displaystyle \beta}}_{mn}}{{{\displaystyle \lambda}}_{l}})$ function can represent all expanded angular spectra in the reconstructed image. So the

*T*(

*x*,

*y*, 0) represented by angular spectrum is shown in Eq. (5).

*ω*

_{mn}are combined into a bigger solid visual angle

*Ω*which is a sum of

*M*×

*N ω*

_{mn.},

*ω*

_{mn}is determined by the distance z

_{0}from the recorded object to CA and the space

*Λ*between two neighborhood color digital cameras in the system. Thus, it can be approximately described as

*ω*

_{mn}= arctan(

*Λ*/ z

_{0}). The light waves modulated by the HFS are given by Eq. (7)

*T*(

*x*,

*y*, 0). Substituting Eq. (3) and (6) into Eq. (7), we can obtain the Eq. (8) as

*Ω*

From the above, the HFS is one of the key devices to realize the RLF 3D-display and the Eq. (5) present the HFS’ functions of spreading beam (or transferring wavefronts) and synthesizing light waves.

## 3. Implementation of the HFS

As the human eyes are capable of acclimatizing to the horizontal parallax only (HPO) 3D display, we study the HFS suited for this display. A method to realize the HFS, called as the directional laser speckle (DLS), is given as follows.

When a diffusing plate with the size *a* × *b* and diffusing grain size of *δu* and *δv* as shown in Fig. 2(a)
is illuminated by a laser, a laser speckle field appears behind the diffusing plate. The average size of the speckle at *X*-*Y* plane is *δx* = *λz*
_{0}/*a*, *δy* = *λz*
_{0}/*b*. Assuming *a*«*b* and *δx*»*δy*, a narrow strip-shaped speckle pattern at *X*-*Y* plane can be achieved. We record the speckle pattern at a distance *z* = *z*
_{0}, and the overlay area of the speckles being$\mathrm{\Delta}x\times \mathrm{\Delta}y$, where *Δx* = *λz*
_{0}/*δu*, *Δy* = *λz*
_{0}/*δv*. When the speckle pattern is illuminated by a light wave, it diffuses and limits the light in a solid diffused angle *θ*
_{‖} = *λ*/*δx* = *a*/*z*
_{0} and *θ*
_{⊥} = *λ*/*δy* = *b*/*z*
_{0}, as shown in Fig. 2(b). It is evident that the diffused angle *θ*
_{⊥}is much bigger than *θ*
_{‖,} so we can consider that the diffused light wave is scattered only along a direction perpendicular to the strip speckles. The speckle screen can facilitate direction selectivity and certain diffusing angle for diffusing the light waves, so it is called as the DLS screen and can spread beam and synchronize light waves in the region of its diffusing angle.

With the analysis above, we propose the DLS method to realize the HFS. According to Eq. (5), the designed HFS is composed of the *J* × *K* volume pixels (VP)s, as shown in Fig. 3(a)
. Each VP is a small-square DLS screen with a side *D*
_{n} and the horizontal diffusing angle *θ*
_{‖} = *ω*
_{n}/2 as shown in Fig. 3(b) (here *ω*
_{n} is 1D form of *ω*
_{mn}). Its vertical diffused angle *θ*
_{⊥} satisfies the requirements of viewing by human eyes. When each VP receives the light waves from all projectors, it can spread these light waves to a spatial angle *Ω* = *Nω*
_{n} in the horizontal direction. Thus the intensity and color information from all projectors can be modulated and recovered with the HFS.

In addition, the holographic micro-cylinder lens (HML) and the holographic micro-cylinder lens (HMCL) can spread beam and synchronize light waves, so we considered the implement of the HFS with them. Through theoretical analysis and primary experimental research, we have shown that the HML method is suitable for the full-parallax 3D display, and the HMCL method can be applied on the HPO 3D display.

## 4. Experiment and analysis

In order to demonstrate the proposed scheme and to simplify the experimental system, a 3D HPO display system with 64 color digital cameras and color projectors is constructed. Here each digital camera or projector has the resolution of 480 × 640 pixels. An HFS is designed according to Eq. (5), Eq. (6), etc., and realized with DLS method due to its advantages of simple design and fabrication over the HML method and HMCL method. The setup of manufacturing the VP is with the digital holographic technology and shown in Fig. 4
[22]. The VP parameters (*D*
_{n}, *ω*
_{n}/2, *f’*) are realized by selecting proper diaphragm aperture at the *D*, adjusting devices in the setup. With the computer control, the VP master using the photo-resist recording medium is first fabricated, and then the HFS is fabricated by embossing holographic technology. The different magnitude of the HFS can also be realized based on the requirements of the 3D display reconstructed.

The HFS fabricated by us is with the total horizontal visual angle of 45°and the size of 1.8 × 1.3 m^{2}. When it is placed at the plane near the point *R*’, two group pictures of 3D-image displayed of a kangaroo and an object recorded are taken as shown in Fig. 5
. And the pictures (a) and (b) are respectively composed of five pictures, which are separately taken at five visual angles from left to right. We can see that the recovered 3D images have high definition, plenty of color, high brightness and high diffractive efficiency, which is may be seen as a reasonable approximation to full-parallax 3D display. The experiment results show that the DLS method can effectively implement large size HFS and the established display system is effectual to the RLF 3D display.

## 5. Conclusions

In summary, the HFS in the RLF 3D display system is analyzed in detail. The modulation function of the HFS is firstly derived, which reveals the basic principle of its modulation waves. The DLS method to realize the HFS is proposed, and used to design and fabricate the HFS. The 3D display experiments with the fabricated HFS are successfully demonstrated. The experimental results show that the 3D information of the recorded object can be well recovered. Therefore, the methods of analyzing and realizing the HFS are effectual, and the derived modulation function Eq. (5) is valuable for the design of the HFS. Moreover, the presented results help to implement the RLF 3D display system, and will find many applications such as the holographic video system.

## Acknowledgments

The research was supported by the Fundamental Research Funds for the Central Universities under no. 2009RC0414. The authors thank all the staff from AFC Technology Co. Ltd. for their support.

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