## Integral imaging with multiple image planes using a uniaxial crystal plate

Optics Express, Vol. 11, Issue 16, pp. 1862-1875 (2003)

http://dx.doi.org/10.1364/OE.11.001862

Acrobat PDF (1479 KB)

### Abstract

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

## 1. Introduction

2. F. Okano, H. Hoshino, J. Arai, and I. Yuyama, “Real-time pickup method for a three-dimensional image based on integral photography,” Appl. Opt. **36**, 1598–1603 (1997). [CrossRef] [PubMed]

12. S. Jung, J.-H. Park, H. Choi, and B. Lee, “Wide-viewing integral three-dimensional imaging by use of orthogonal polarization switching,” Appl. Opt. **42**, 2513–2520 (2003). [CrossRef] [PubMed]

9. B. Lee, S. Jung, and J. -H. Park, “Viewing-angle-enhanced integral imaging using lens switching,” Opt. Lett. **27**, 818–820 (2002). [CrossRef]

12. S. Jung, J.-H. Park, H. Choi, and B. Lee, “Wide-viewing integral three-dimensional imaging by use of orthogonal polarization switching,” Appl. Opt. **42**, 2513–2520 (2003). [CrossRef] [PubMed]

5. L. Erdmann and K. J. Gabriel, “High-resolution digital integral photography by use of a scanning microlens array,” Appl. Opt. **40**, 5592–5599 (2001). [CrossRef]

4. J.-H. Park, S.-W. Min, S. Jung, and B. Lee, “Analysis of viewing parameters for two display methods based on integral photography,” Appl. Opt. **40**, 5217–5232 (2001). [CrossRef]

6. B. Lee, S. Jung, S.-W. Min, and J.-H. Park, “Three-dimensional display using integral photography with dynamically variable image planes,” Opt. Lett. **26**, 1481–1482 (2001). [CrossRef]

8. S.-W. Min, B. Javidi, and B. Lee, “Enhanced three-dimensional integral imaging system by use of double display devices,” Appl. Opt. **42**, 4186–4195 (2003). [CrossRef] [PubMed]

13. S. Sanyal and A. Ghosh, “High focal depth with a quasi-bifocus birefringent lens,” Appl. Opt. **39**, 2321–2325 (2000). [CrossRef]

14. J. Lesso, A. Duncan, W. Sibbett, and M. Padgett, “Aberrations introduced by a lens made from a birefringent material,” Appl. Opt. **39**, 592–598 (2000). [CrossRef]

15. H. Kikuta and K. Iwata, “First-order aberration of a double-focus lens made of a uniaxial crystal,” J. Opt. Soc. Am. A **9**, 814–819 (1992). [CrossRef]

## 2. Principle of the proposed method

### 2.1 Overview and system configuration

### 2.2 Longitudinal shift by the uniaxial crystal

*n*in the glass plate and are focused at an image point by the lens. According to Snell’s law, the longitudinal shift Δ

_{glass}*z*by the glass plate is given as [16]:

*d*is the thickness of the glass plate and

*θ*the incident angle at the air-glass boundary and is assumed to be small sufficiently so that sin

_{i}*θ*≈0 and cos

_{i}*θ*≈1 hold.

_{i}*n*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

_{o}*n*with

_{glass}*n*, that is,

_{o}*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:

*n*to

_{e}*n*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 [17]:

_{o}*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

*n*(

_{e}*ϕ*)=

*n*. Since the refractive index that the horizontally incident extraordinary rays experience is always

_{e}*n*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

_{e}*n*.

_{e}*ϕ*in Eq. (4) is equal to

*π*/2+

*θ*. By applying Eq. (4) to Snell’s law(sin

_{ev}*θ*=

_{i}*n*(

_{e}*ϕ*)sin

*θ*) with consideration of

_{ev}*ϕ*=

*π*/2+

*θ*, we can then obtain the refracted angle

_{ev}*θ*as:

_{ev}*θ*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 [17]:

_{ev}### 2.3 Function of the sliding slit mask

_{1}H

_{2}, is much smaller than without a vertical slit, H

_{1}′H

_{2}′, 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.

*u*and

*x*are the horizontal positions of the vertical slit and the given image point and

_{v}*l*, and

_{eh}*l*are given by Eqs. (6) and (10). Since the point P

_{ev}_{h}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:

*Δz*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.

_{eh}### 2.4 Proposed system

*ψ*=2tan

^{-1}(

*φ*/2

*g*) where

*φ*is the diameter of the elemental lens and

*g*is the gap between the lens array and the display panel [2

2. F. Okano, H. Hoshino, J. Arai, and I. Yuyama, “Real-time pickup method for a three-dimensional image based on integral photography,” Appl. Opt. **36**, 1598–1603 (1997). [CrossRef] [PubMed]

4. J.-H. Park, S.-W. Min, S. Jung, and B. Lee, “Analysis of viewing parameters for two display methods based on integral photography,” Appl. Opt. **40**, 5217–5232 (2001). [CrossRef]

*g*should be replaced by

*g*-Δ

*z*for the ordinary mode and

_{o}*g*-Δ

*z*and

_{ev}*g*-Δ

*z*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,

_{eh}*g*in the conventional InIm system and

*g*-Δ

*z*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.

*θ*≈0 and cos

_{i}*θ*≈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

_{i}*n*=1.68014,

_{o}*n*=1.49640 at λ=410nm to

_{e}*n*=1.65207,

_{o}*n*=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

_{e}*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 sin

*θ*≈0 and cos

_{i}*θ*≈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.

_{i}## 3. Experimental results

*θ*≈0 and cos

_{i}*θ*≈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.

_{i}5. L. Erdmann and K. J. Gabriel, “High-resolution digital integral photography by use of a scanning microlens array,” Appl. Opt. **40**, 5592–5599 (2001). [CrossRef]

18. J.-S. Jang and B. Javidi, “Improved viewing resolution of three-dimensional integral imaging with nonstationary micro-optics,” Opt. Lett. **27**, 324–326 (2002). [CrossRef]

## 4. Conclusion

## Acknowledgment

## References and links

1. | G. Lippmann, “La photographie integrale,” Comptes-Rendus Acad. Sci. |

2. | F. Okano, H. Hoshino, J. Arai, and I. Yuyama, “Real-time pickup method for a three-dimensional image based on integral photography,” Appl. Opt. |

3. | S. Manolache, A. Aggoun, M. McCormick, N. Davies, and S. Y. Kung, “Analytical model of a three-dimensional integral image recording system that uses circular and hexagonal-based spherical surface microlenses,” J. Opt. Soc. Am. A. |

4. | J.-H. Park, S.-W. Min, S. Jung, and B. Lee, “Analysis of viewing parameters for two display methods based on integral photography,” Appl. Opt. |

5. | L. Erdmann and K. J. Gabriel, “High-resolution digital integral photography by use of a scanning microlens array,” Appl. Opt. |

6. | B. Lee, S. Jung, S.-W. Min, and J.-H. Park, “Three-dimensional display using integral photography with dynamically variable image planes,” Opt. Lett. |

7. | T. Naemura, T. Yoshida, and H. Harashima, “3-D computer graphics based on integral photography,” Opt. Express |

8. | S.-W. Min, B. Javidi, and B. Lee, “Enhanced three-dimensional integral imaging system by use of double display devices,” Appl. Opt. |

9. | B. Lee, S. Jung, and J. -H. Park, “Viewing-angle-enhanced integral imaging using lens switching,” Opt. Lett. |

10. | S.-W Min, S. Jung, J.-H. Park, and B. Lee, “Study for wide-viewing integral photography using an aspheric Fresnel-lens array,” Opt. Eng. |

11. | H. Choi, S.-W. Min, S. Jung, J.-H. Park, and B. Lee, “Multiple-viewing-zone integral imaging using a dynamic barrier array for three-dimensional displays,” Opt. Express. |

12. | S. Jung, J.-H. Park, H. Choi, and B. Lee, “Wide-viewing integral three-dimensional imaging by use of orthogonal polarization switching,” Appl. Opt. |

13. | S. Sanyal and A. Ghosh, “High focal depth with a quasi-bifocus birefringent lens,” Appl. Opt. |

14. | J. Lesso, A. Duncan, W. Sibbett, and M. Padgett, “Aberrations introduced by a lens made from a birefringent material,” Appl. Opt. |

15. | H. Kikuta and K. Iwata, “First-order aberration of a double-focus lens made of a uniaxial crystal,” J. Opt. Soc. Am. A |

16. | W. J. Smith, Modern Optical Engineering, 2nd ed. (McGraw-Hill, New York,1990). |

17. | A. Yariv and P. Yeh, Optical Waves in Crystals, (Wiley, Yew York, 1983). |

18. | J.-S. Jang and B. Javidi, “Improved viewing resolution of three-dimensional integral imaging with nonstationary micro-optics,” Opt. Lett. |

**OCIS Codes**

(100.6890) Image processing : Three-dimensional image processing

(110.2990) Imaging systems : Image formation theory

(160.1190) Materials : Anisotropic optical materials

(220.2740) Optical design and fabrication : Geometric optical design

**ToC Category:**

Research Papers

**History**

Original Manuscript: June 12, 2003

Revised Manuscript: July 25, 2003

Published: August 11, 2003

**Citation**

Jae-Hyeung Park, Sungyong Jung, Heejin Choi, and Byoungho Lee, "Integral imaging with multiple image planes using a uniaxial crystal plate," Opt. Express **11**, 1862-1875 (2003)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-11-16-1862

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### References

- G. Lippmann, �??La photographie integrale,�?? Comptes-Rendus Acad. Sci. 146, 446-451 (1908).
- F. Okano, H. Hoshino, J. Arai, and I. Yuyama, �??Real-time pickup method for a three-dimensional image based on integral photography,�?? Appl. Opt. 36, 1598-1603 (1997). [CrossRef] [PubMed]
- S. Manolache, A. Aggoun, M. McCormick, N. Davies, and S. Y. Kung, �??Analytical model of a threedimensional integral image recording system that uses circular and hexagonal-based spherical surface microlenses,�?? J. Opt. Soc. Am. A. 18, 1814-1821 (2001). [CrossRef]
- J.-H. Park, S.-W. Min, S. Jung, and B. Lee, "Analysis of viewing parameters for two display methods based on integral photography," Appl. Opt. 40, 5217-5232 (2001). [CrossRef]
- L. Erdmann and K. J. Gabriel, �??High-resolution digital integral photography by use of a scanning microlens array,�?? Appl. Opt. 40, 5592-5599 (2001). [CrossRef]
- B. Lee, S. Jung, S.-W. Min, and J.-H. Park, "Three-dimensional display using integral photography with dynamically variable image planes," Opt. Lett. 26, 1481-1482 (2001). [CrossRef]
- T. Naemura, T. Yoshida, and H. Harashima, �??3-D computer graphics based on integral photography,�?? Opt. Express 8, 255-262 (2001), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-4-255.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-4-255.</a> [CrossRef] [PubMed]
- S.-W. Min, B. Javidi, and B. Lee, "Enhanced three-dimensional integral imaging system by use of double display devices," Appl. Opt. 42, 4186-4195 (2003). [CrossRef] [PubMed]
- B. Lee, S. Jung, and J. -H. Park, �??Viewing-angle-enhanced integral imaging using lens switching,�?? Opt. Lett. 27, 818-820 (2002). [CrossRef]
- S.-W, Min, S. Jung, J.-H. Park, and B. Lee, "Study for wide-viewing integral photography using an aspheric Fresnel-lens array," Opt. Eng. 41, 2572-2576 (2002). [CrossRef]
- H. Choi, S.-W. Min, S. Jung, J.-H. Park, and B. Lee, "Multiple-viewing-zone integral imaging using a dynamic barrier array for three-dimensional displays," Opt. Express. 11, 927-932 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-8-927.">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-8-927.</a> [CrossRef] [PubMed]
- S. Jung, J.-H. Park, H. Choi, and B. Lee, "Wide-viewing integral three-dimensional imaging by use of orthogonal polarization switching," Appl. Opt. 42, 2513-2520 (2003). [CrossRef] [PubMed]
- S. Sanyal, and A. Ghosh, �??High focal depth with a quasi-bifocus birefringent lens,�?? Appl. Opt. 39, 2321-2325 (2000). [CrossRef]
- J. Lesso, A. Duncan, W. Sibbett, and M. Padgett, �??Aberrations introduced by a lens made from a birefringent material,�?? Appl. Opt. 39, 592-598 (2000). [CrossRef]
- H. Kikuta and K. Iwata, �??First-order aberration of a double-focus lens made of a uniaxial crystal,�?? J. Opt. Soc. Am. A 9, 814-819 (1992). [CrossRef]
- W. J. Smith, Modern Optical Engineering, 2nd ed. (McGraw-Hill, New York,1990).
- A. Yariv and P. Yeh, Optical Waves in Crystals, (Wiley, Yew York, 1983).
- J.-S. Jang and B. Javidi, �??Improved viewing resolution of three-dimensional integral imaging with nonstationary micro-optics,�?? Opt. Lett. 27, 324-326 (2002). [CrossRef]

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