## CGH calculation with the ray tracing method for the Fourier transform optical system |

Optics Express, Vol. 21, Issue 26, pp. 32019-32031 (2013)

http://dx.doi.org/10.1364/OE.21.032019

Acrobat PDF (1307 KB)

### Abstract

Computer-generated holograms (CGHs) are usually displayed on electronic devices. However, the resolution of current output devices is not high enough to display CGHs, so the visual field is very narrow. A method using a Fourier transform optical system has been proposed, to enlarge the size of reconstructed images. This paper describes a method of CGH calculations for the Fourier transform optical system to enlarge the visual field and reconstruct realistic images by using the ray tracing method. This method reconstructs images at arbitrary depths and also eliminates unnecessary light including zero-th order light.

© 2013 Optical Society of America

## 1. Introduction

3. D. J. Sandin, E. Sandor, W. T. Cunnally, M. Resch, and T. A. DeFanti, “Computer-generated barrier-strip autostereography,” Proc. SPIE **1083**, 65–75 (1989). [CrossRef]

6. Y. Takaki, “Super multi-view display with 128 viewpoints and viewpoint formation,” Proc. SPIE **7237**, 72371T (2009). [CrossRef]

14. R. Haussler, S. Reichlet, N. Leister, E. Zchau, R. Missbach, and A. Schwerdtner, “Large real-time holographic displays: from prototypes to a consumer product,” Proc. SPIE **7237**, 72370S (1989). [CrossRef]

15. J. Hahn, H. Kim, Y. Lim, G. Park, and B. Lee, “Wide viewing angle dynamic holographic stereogram with a curved array of spatial light modulators,” Opt. Express **16**, 12372–12386 (2008). [CrossRef] [PubMed]

16. T. Senoh, T. Mishima, K. Yamamoto, R. Oi, and T. Kurita, “Viewing-zone-angle-expanded color electronic holography system using ultra-high-definition liquid crystal displays with undesirable light elimination,” J. Display Technol. **7**(7), 382–390 (2011). [CrossRef]

17. T. Ichikawa, K. Yamaguchi, and Y. Sakamoto, “Realistic 3D image reconstruction in CGH with Fourier transform optical sytem,” Proc. SPIE **8644**, 86440D (2013). [CrossRef]

13. T. Ichikawa, K. Yamaguchi, and Y. Sakamoto, “Realistic expression for full-parallax computer-generated holograms with the ray tracing method,” Appl. Opt. **52**, A201–A209 (2013). [CrossRef] [PubMed]

## 2. CGH calculations with ray tracing method

*θ*between rays is 1/60 degrees, because the angular resolution of the human eye is assumed to be 1/60 degrees. Hence, as gaps between intersections are not visible by the observer, the aggregate of points is regarded as a plane. Therefore, elementary holograms have different point light source groups that are visible from each elementary hologram.

*I*is the intensity of ambient light,

_{a}*k*and

_{d}*k*correspond to the ratio of Lambert light and specular light (

_{s}*k*+

_{d}*k*= 1), and values

_{s}*ρ*,

_{a}*ρ*, and

_{d}*ρ*correspond to the reflectance of ambient, Lambert, and specular light. Vectors

_{s}**N**and

**L**are a normal unit vector and a unit vector to the light source. The qualities of material are mostly expressed by changing the reflectance of specular light

*ρ*. For instance, the Phong reflection model is used to express a surface like plastic material. The Phong reflection model is calculated by using the direction of rays and the position of the light source that illuminates the objects as shown in Fig. 2.

_{s}*h*(

_{m}*x*,

*y*) on an elementary hologram plane are obtained as where parameters

*x*,

*y*and

*z*represent the horizontal, vertical, and depth components, while indices

*m*and

*i*correspond to the number of elementary holograms and point light sources. Here,

*λ*is the wavelength, and

*ϕ*is the initial phase of each point light source. The number of point light sources that create 3D objects in each elementary hologram is given by

_{i}*N*. The amplitude of point light source

_{m}*A*is calculated by taking the square root of light intensity

_{i}*I*of Eq. (1) in the shading process. Finally, whole light waves on a hologram plane are obtained by superimposing all elementary holograms. The method of expressing reflective and refractive objects in CGHs can be found in our recent paper [13

_{r}13. T. Ichikawa, K. Yamaguchi, and Y. Sakamoto, “Realistic expression for full-parallax computer-generated holograms with the ray tracing method,” Appl. Opt. **52**, A201–A209 (2013). [CrossRef] [PubMed]

## 3. Fourier transform optical system

18. G. W. Stroke, “Lensless Fourier-transform method for optical holography,” Appl. Phys. Lett. **6**, 201–203 (1965). [CrossRef]

*O*(

*y*,

_{o}*z*) in the process to calculate CGHs, as shown in Fig. 4. A point light source as a reference light is located at

_{o}*R*(0. −

*f*), and then interference patterns are obtained. The point light source for the observations is located in front of the lens at

*R′*(0,

*f*). At this time, a conjugate image

*Q*(

*y*

_{2},

*z*

_{2}) and a real image

*P*(

*y*

_{1},

*z*

_{1}) are reconstructed. Supposing the gap between the hologram and the lens is illimitably small, the coordinates of a conjugate image

*Q*(

*y*

_{2},

*z*

_{2}) and a real image

*P*(

*y*

_{1},

*z*

_{1}) are given as following equations with optical imaging theory. According to Eq. (4), when virtual object

*O*is located nearby rather than

*z*= −

_{o}*f*/2, a real image

*P*(

*x*

_{1},

*z*

_{1}) is reconstructed at the back of the hologram. However, we observe a real image and a conjugate image simultaneously. Therefore, it is necessary to eliminate conjugate image

*Q*to observe only the real image at the back of the hologram.

*PQ*passing through the focal point in Fig. 5, the hologram is divided into two regions

*R*

_{1}and

*R*

_{2}by line

*PQ*. It is presumed from Fig. 5 that region

*R*

_{1}generates light waves of

*P*that arrive at the viewpoint and region

*R*

_{2}generates light waves of

*Q*. Therefore, we only observe real images by only calculating the region

*R*

_{1}. The border

*y*of

_{th}*R*

_{1}and

*R*

_{2}on the hologram plane is calculated by

*z*= −

_{o}*f*/2 and calculating interference patterns on the hologram above

*y*. At this time, the visual field of the Fourier transform optical system is behind the hologram shown in Fig. 6. Reconstructed light is converged by the lens, and passed through a window with width

_{th}*w*given by where

*p*is the pixel pitch of the output device. When the position of the viewpoint

*z*is equal to

_{e}*z*, maximum viewing angle

_{e_min}*ϕ*is obtained by Where L is the size of the hologram. From Eqs. (7)–(9), we confirmed that viewing angle

_{F}*ϕ*could be expanded by changing the size of the hologram and focal length of the lens.

_{F}## 4. CGH calculations for Fourier transform optical system

### 4.1. Compensation calculation for Fourier transform optical system

17. T. Ichikawa, K. Yamaguchi, and Y. Sakamoto, “Realistic 3D image reconstruction in CGH with Fourier transform optical sytem,” Proc. SPIE **8644**, 86440D (2013). [CrossRef]

*z*= −

_{o}*f*/2, and ray tracing was then conducted to obtain point light source groups for CGH calculations. However, reconstructed images through the Fourier transform optical system were expanded and deformed because the magnifying power increased as the depth increased according to Eq. (4). Therefore, it is necessary to have a method to compensate for the Fourier transform optical system. Distortion by the optical system can be computed. Therefore, the compensation is possible by back calculations of distortion. The method of compensation described here obtains the position from which we want to reconstruct 3-D images by ray tracing method in advance. Then, the coordinates of point light sources reconstructed at the positions are calculated by back calculations of distortion. Moreover the method of compensation takes into account gap

*D*between the hologram and the lens in Fig. 7.

*D*away from the hologram as seen in Fig 7, the hologram is magnified by the lens and forms an image at

*z*in depth. Hence, the position of the reconstructed image (

_{h}*x*,

_{i}*y*,

_{i}*z*) is restricted to

_{i}*z*<

_{i}*z*. When the coordinates of a virtual object point that should be reconstructed are (

_{h}*x*,

_{i}*y*,

_{i}*z*), the coordinates of the point light source (

_{i}*x*,

_{o}*y*,

_{o}*z*) in the light wave calculations are obtained by where

_{o}*A*and

*B*are expressed as Where

*z*is given by Therefore, light waves on the hologram plane for the Fourier transform optical system are calculated by translating the distance

_{h}*r*in Eq. (2) to

_{i}*r*expressed as Only light waves that arrived above

_{o}*y*are calculated to observe only the real image at the back of the hologram. The same image that has been rendered with ray tracing can be observed at position (

_{th}*x*,

_{i}*y*,

_{i}*z*) without distortion by doing so.

_{i}### 4.2. Calculation process

*y*was calculated at every point light source to eliminate the conjugate images described in Section 3. Light waves on each elementary hologram above

_{th}*y*from the point light source (

_{th}*x*,

_{o}*y*,

_{o}*z*) were then calculated. Entire light waves were obtained by summing up all elementary holograms. Interference patterns on the hologram were obtained by summing up spherical waves as reference light from the point light source at the focal point of the lens.

_{o}## 5. Experiment

### 5.1. Experimental setup

19. T. Yoneyama, C. Yang, Y. Sakamoto, and F. Okuyama, “Eyepiece-type full-color electro-holographic binocular display with see-through vision,” in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest (online) (Optical Society of America, 2013), paper DW2A.11. [CrossRef]

19. T. Yoneyama, C. Yang, Y. Sakamoto, and F. Okuyama, “Eyepiece-type full-color electro-holographic binocular display with see-through vision,” in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest (online) (Optical Society of America, 2013), paper DW2A.11. [CrossRef]

*ϕ*is calculated by Eq. (9). On the other hand, the maximum viewing angle

_{F}*ϕ*of the Fresnel hologram is calculated by the following equation. According to Eq. (17), the viewing angle

_{max}*ϕ*of the Fresnel hologram by the SLM used in this experiment is 2.78 [degrees]. Therefore, we succeeded in expanding a viewing angle

_{max}*ϕ*of the Fourier transform optical system about 3.5 times compared with the Fresnel hologram. Moreover it is possible to extend a viewing angle by using a lens with a shorter focal length.

_{F}### 5.2. Measurement of the size of reconstructed images

17. T. Ichikawa, K. Yamaguchi, and Y. Sakamoto, “Realistic 3D image reconstruction in CGH with Fourier transform optical sytem,” Proc. SPIE **8644**, 86440D (2013). [CrossRef]

*z*= −2/

*f*in the previous research. And the 3D images are reconstructed at the position which differs from the position of arranged virtual objects by the magnification effect of the Fourier transform optical system. So it was difficult to reconstruct the virtual objects at the objective position. Moreover, since magnifying power becomes large as the depth becomes deep, reconstructed images deform. Therefore, the depth perception of the reconstructed images is no longer felt. These problems are solved by using the compensation calculation described in section 4.1 in this paper. Furthermore, restriction of the virtual object arrangement in a previous study does not also exist, and it becomes possible to reconstruct 3D images at the arbitrary positions behind a hologram.

### 5.3. Optical reconstructions

*z*= −400[mm] was defined as a transparent object. In this experiment, the refractive index of glass is 1.3. Figure 12(b) shows the reconstructed image of a glass sphere and a checker board. It turned out that the checker patterns through the glass sphere were magnified because the checker patterns were refracted on a transparent object. This result indicates that refraction by the transparent object was also achieved by using the proposed method. As results of these optical reconstructions, it was confirmed that we succeeded in expressing a complex scene that required multiple rendering techniques simultaneously without distortion by the proposed method. Conventional methods are unable to express a complex scene, which requires the use of several rendering techniques to be used at a time.

## 6. Conclusion

## Acknowledgments

## References and links

1. | H. Isono and M. Yasuda, “Flicker-free field sequential stereo-scopic TV system and measurement of human depth perception,” SMPTE J. |

2. | T. Motoki, I. Yayuma, H. Isono, and S. Komiyama, “Research on 3-D television system at NHK,” ABU Tech. Rev. |

3. | D. J. Sandin, E. Sandor, W. T. Cunnally, M. Resch, and T. A. DeFanti, “Computer-generated barrier-strip autostereography,” Proc. SPIE |

4. | S. Ichinose, “Fullcolor stereoscopic video pickup and display technique without special glasses,” Proc.SID |

5. | M. G. Lippmann, “Epreuves reversible donnant la sensation du relief,” J. de Phys. |

6. | Y. Takaki, “Super multi-view display with 128 viewpoints and viewpoint formation,” Proc. SPIE |

7. | D. Gabor, “A new microscopic principle,” Nature |

8. | K. Matsushima and S. Nakahara, “Extremely high-defintion full-parallax computer-generated hologram created by the polygon-based method,” Appl. Opt. |

9. | R. H.-Y. Chen and T. D. Wilkinson, “Computer generated hologram with geometric occlusion using GPU-accelerated depth buffer rasterization for three-dimensional display,” Appl. Opt. |

10. | H. Kim, J. Hahn, and B. Lee, “Mathematical modeling of digital holography,” Appl. Opt. |

11. | K. Matsushima, “Computer-generated holograms for three-dimensional surface objects with shade and texture,” Appl. Opt. |

12. | K. Yamaguchi and Y. Sakamoto, “Computer generated hologram with characteristics of reflection: reflectance distributions and reflected images,” Appl. Opt. |

13. | T. Ichikawa, K. Yamaguchi, and Y. Sakamoto, “Realistic expression for full-parallax computer-generated holograms with the ray tracing method,” Appl. Opt. |

14. | R. Haussler, S. Reichlet, N. Leister, E. Zchau, R. Missbach, and A. Schwerdtner, “Large real-time holographic displays: from prototypes to a consumer product,” Proc. SPIE |

15. | J. Hahn, H. Kim, Y. Lim, G. Park, and B. Lee, “Wide viewing angle dynamic holographic stereogram with a curved array of spatial light modulators,” Opt. Express |

16. | T. Senoh, T. Mishima, K. Yamamoto, R. Oi, and T. Kurita, “Viewing-zone-angle-expanded color electronic holography system using ultra-high-definition liquid crystal displays with undesirable light elimination,” J. Display Technol. |

17. | T. Ichikawa, K. Yamaguchi, and Y. Sakamoto, “Realistic 3D image reconstruction in CGH with Fourier transform optical sytem,” Proc. SPIE |

18. | G. W. Stroke, “Lensless Fourier-transform method for optical holography,” Appl. Phys. Lett. |

19. | T. Yoneyama, C. Yang, Y. Sakamoto, and F. Okuyama, “Eyepiece-type full-color electro-holographic binocular display with see-through vision,” in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest (online) (Optical Society of America, 2013), paper DW2A.11. [CrossRef] |

**OCIS Codes**

(090.1760) Holography : Computer holography

(090.2870) Holography : Holographic display

(090.1705) Holography : Color holography

**ToC Category:**

Holography

**History**

Original Manuscript: October 16, 2013

Revised Manuscript: December 8, 2013

Manuscript Accepted: December 8, 2013

Published: December 17, 2013

**Citation**

Tsubasa Ichikawa, Takuo Yoneyama, and Yuji Sakamoto, "CGH calculation with the ray tracing method for the Fourier transform optical system," Opt. Express **21**, 32019-32031 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-26-32019

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### References

- H. Isono and M. Yasuda, “Flicker-free field sequential stereo-scopic TV system and measurement of human depth perception,” SMPTE J.99(2), 138–141(1990).
- T. Motoki, I. Yayuma, H. Isono, and S. Komiyama, “Research on 3-D television system at NHK,” ABU Tech. Rev.150, 14–18 (1991).
- D. J. Sandin, E. Sandor, W. T. Cunnally, M. Resch, and T. A. DeFanti, “Computer-generated barrier-strip autostereography,” Proc. SPIE1083, 65–75 (1989). [CrossRef]
- S. Ichinose, “Fullcolor stereoscopic video pickup and display technique without special glasses,” Proc.SID30-4, 319–323 (1989).
- M. G. Lippmann, “Epreuves reversible donnant la sensation du relief,” J. de Phys.7(4), 821–825 (1908).
- Y. Takaki, “Super multi-view display with 128 viewpoints and viewpoint formation,” Proc. SPIE7237, 72371T (2009). [CrossRef]
- D. Gabor, “A new microscopic principle,” Nature161, 777–778 (1948). [CrossRef] [PubMed]
- K. Matsushima and S. Nakahara, “Extremely high-defintion full-parallax computer-generated hologram created by the polygon-based method,” Appl. Opt.48, H54–H63 (2009). [CrossRef] [PubMed]
- R. H.-Y. Chen and T. D. Wilkinson, “Computer generated hologram with geometric occlusion using GPU-accelerated depth buffer rasterization for three-dimensional display,” Appl. Opt.48, 4246–4255 (2009). [CrossRef] [PubMed]
- H. Kim, J. Hahn, and B. Lee, “Mathematical modeling of digital holography,” Appl. Opt.48, D117–D127 (2008). [CrossRef]
- K. Matsushima, “Computer-generated holograms for three-dimensional surface objects with shade and texture,” Appl. Opt.44, 4607–4614 (2005). [CrossRef] [PubMed]
- K. Yamaguchi and Y. Sakamoto, “Computer generated hologram with characteristics of reflection: reflectance distributions and reflected images,” Appl. Opt.48, H203–H211 (2005). [CrossRef]
- T. Ichikawa, K. Yamaguchi, and Y. Sakamoto, “Realistic expression for full-parallax computer-generated holograms with the ray tracing method,” Appl. Opt.52, A201–A209 (2013). [CrossRef] [PubMed]
- R. Haussler, S. Reichlet, N. Leister, E. Zchau, R. Missbach, and A. Schwerdtner, “Large real-time holographic displays: from prototypes to a consumer product,” Proc. SPIE7237, 72370S (1989). [CrossRef]
- J. Hahn, H. Kim, Y. Lim, G. Park, and B. Lee, “Wide viewing angle dynamic holographic stereogram with a curved array of spatial light modulators,” Opt. Express16, 12372–12386 (2008). [CrossRef] [PubMed]
- T. Senoh, T. Mishima, K. Yamamoto, R. Oi, and T. Kurita, “Viewing-zone-angle-expanded color electronic holography system using ultra-high-definition liquid crystal displays with undesirable light elimination,” J. Display Technol.7(7), 382–390 (2011). [CrossRef]
- T. Ichikawa, K. Yamaguchi, and Y. Sakamoto, “Realistic 3D image reconstruction in CGH with Fourier transform optical sytem,” Proc. SPIE8644, 86440D (2013). [CrossRef]
- G. W. Stroke, “Lensless Fourier-transform method for optical holography,” Appl. Phys. Lett.6, 201–203 (1965). [CrossRef]
- T. Yoneyama, C. Yang, Y. Sakamoto, and F. Okuyama, “Eyepiece-type full-color electro-holographic binocular display with see-through vision,” in Digital Holography and Three-Dimensional Imaging, OSA Technical Digest (online) (Optical Society of America, 2013), paper DW2A.11. [CrossRef]

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