## Solution for pseudoscopic problem in integral imaging using phase-conjugated reconstruction of lens-array holographic optical elements |

Optics Express, Vol. 22, Issue 11, pp. 13659-13670 (2014)

http://dx.doi.org/10.1364/OE.22.013659

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### Abstract

We propose an optical pseudoscopic to orthoscopic conversion method for integral imaging using a lens-array holographic optical element (LAHOE), which solves the pseudoscopic problem. The LAHOE reconstructs an array of diverging spherical waves when a probe wave with the phase-conjugated condition is imposed on it, while an array of converging spherical waves is reconstructed in ordinary reconstruction. For given pseudoscopic elemental images, the array of the diverging spherical waves integrates the orthoscopic three-dimensional images without a distortion. The principle of the proposed method is verified by the experiments of displaying the integral imaging on the LAHOE using computer generated and optically acquired elemental images.

© 2014 Optical Society of America

## 1. Introduction

1. B. Lee, “Three-dimensional displays, past and present,” Phys. Today **66**(4), 36–41 (2013). [CrossRef]

5. J.-H. Park, K. Hong, and B. Lee, “Recent progress in three-dimensional information processing based on integral imaging,” Appl. Opt. **48**(34), H77–H94 (2009). [CrossRef] [PubMed]

13. J. Arai, H. Kawai, and F. Okano, “Microlens arrays for integral imaging system,” Appl. Opt. **45**(36), 9066–9078 (2006). [CrossRef] [PubMed]

7. 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**(7), 1598–1603 (1997). [CrossRef] [PubMed]

8. M. Martínez-Corral, B. Javidi, R. Martínez-Cuenca, and G. Saavedra, “Formation of real, orthoscopic integral images by smart pixel mapping,” Opt. Express **13**(23), 9175–9180 (2005). [CrossRef] [PubMed]

9. D. H. Shin, B. G. Lee, and E.-S. Kim, “Modified smart pixel mapping method for displaying orthoscopic 3D images in integral imaging,” Opt. Lasers Eng. **47**(11), 1189–1194 (2009). [CrossRef]

10. J.-H. Jung, J. Kim, and B. Lee, “Solution of pseudoscopic problem in integral imaging for real-time processing,” Opt. Lett. **38**(1), 76–78 (2013). [CrossRef] [PubMed]

11. J. Kim, J.-H. Jung, C. Jang, and B. Lee, “Real-time capturing and 3D visualization method based on integral imaging,” Opt. Express **21**(16), 18742–18753 (2013). [CrossRef] [PubMed]

12. J.-S. Jang and B. Javidi, “Three-dimensional projection integral imaging using micro-convex-mirror arrays,” Opt. Express **12**(6), 1077–1083 (2004). [CrossRef] [PubMed]

*et al*. proposed a gradient index lens-array and multiple layers of convex lens-arrays [13

13. J. Arai, H. Kawai, and F. Okano, “Microlens arrays for integral imaging system,” Appl. Opt. **45**(36), 9066–9078 (2006). [CrossRef] [PubMed]

## 2. Principles

### 2.1. PS problem and methods for converting from PS to OS images in InIm

12. J.-S. Jang and B. Javidi, “Three-dimensional projection integral imaging using micro-convex-mirror arrays,” Opt. Express **12**(6), 1077–1083 (2004). [CrossRef] [PubMed]

### 2.2. Phase-conjugated reconstruction on LAHOE for solving PS problem

14. K. Hong, J. Yeom, C. Jang, J. Hong, and B. Lee, “Full-color lens-array holographic optical element for three-dimensional optical see-through augmented reality,” Opt. Lett. **39**(1), 127–130 (2014). [CrossRef] [PubMed]

12. J.-S. Jang and B. Javidi, “Three-dimensional projection integral imaging using micro-convex-mirror arrays,” Opt. Express **12**(6), 1077–1083 (2004). [CrossRef] [PubMed]

15. Y. Jeong, S. Jung, J.-H. Park, and B. Lee, “Reflection-type integral imaging scheme for displaying three-dimensional images,” Opt. Lett. **27**(9), 704–706 (2002). [CrossRef] [PubMed]

16. R. R. A. Syms and L. Solymar, “Analysis of volume holographic cylindrical lenses,” J. Opt. Soc. Am. **72**(2), 179–186 (1982). [CrossRef]

17. R. R. A. Syms and L. Solymar, “Higher diffraction orders in on-axis holographic lenses,” Appl. Opt. **21**(18), 3263–3268 (1982). [CrossRef] [PubMed]

*θ*

_{HOE}) is identical to the converging angle of the reference lens-array and can be calculated aswhere

*p*

_{LA}and

*f*

_{LA}are a lens pitch and focal length of the reference lens-array, respectively.

18. Y. Luo, J. Castro, J. K. Barton, R. K. Kostuk, and G. Barbastathis, “Simulations and experiments of aperiodic and multiplexed gratings in volume holographic imaging systems,” Opt. Express **18**(18), 19273–19285 (2010). [CrossRef] [PubMed]

*n*number of sub-regions. In each sub-region a wave vector of signal wave can be approximated by a single plane wave vector. In Figs. 4(a)-4(c), we note topmost and bottommost regions of the volume hologram as the 1st and the

*n*

^{th}sub-regions, and represent the reference wave and signal wave in the recording process as

*R*and

*S*. Also, the probe waves and reconstructed waves are represented as

*R*and

*S*for the ordinary reconstruction, and

*R** and

*S** for the phase-conjugated reconstruction, respectively. The angle between the wave vector of signal wave and

*z*axis varies from -

*θ*

_{HOE}/2 to

*θ*

_{HOE}/2 as the position changes, while an incidence angle of the reference wave is fixed to

*θ*

_{r}.

*n*

^{th}sub-regions. Considering the interference between two plane waves, a relationship among the grating vector (

*K*), a wave vector of reference wave (

*k*

_{r}), and signal wave (

*k*

_{s}) is given by

*k*

_{d}) is represented as the sum of the grating vector and wave vector of the probe wave:which means the reconstructed wave has the same wave vector as that of the signal wave, as shown in Fig. 4(d). On the other hand, if the probe wave is the phase-conjugation of the reference wave in the reocrding process as shown in Fig. 4(e), the wave vector of reconstructed wave is represented aswhere the superscipt * means the phase-conjugation, and the phase-conjugation of wave vector is represented by using a minus sign for the original wave vector:

*K** = -

*K, k*

_{r}*

*= -k*

_{r}, and

*k*

_{s}*

*= -k*

_{s}[19

19. Y. Lim, J. Hahn, and B. Lee, “Phase-conjugate holographic lithography based on micromirror array recording,” Appl. Opt. **50**(34), H68–H74 (2011). [CrossRef] [PubMed]

*k*

_{s}* =

*-k*

_{s}, when the probe wave with the wavevector

*k*

_{r}* is imposed. As a result, the whole reconstructed wave, which is superposition of

*n*number of plane waves in every sub-region, becomes a diverging spherical wave propagating along the -

*z*direction as shown in Fig. 4(c). From the wave vector analaysis, it is verified that the volume hologram recorded with the converging spherical wave as the signal wave reconstructs the diverging spherical wave traveling backwards when we impose the phase-conjugatged probe wave.

## 3. Experimental results

*z*direction to capture intensity images on imaging planes at different distances from the LAHOE. The LAHOE shown in Fig. 5(a) is optically recorded on a photopolymer as the holographic material using 532 nm laser. The reference lens-array used in the recording process has focal length of 41.9 mm, and lens pitch of 5.4 mm and 7 mm in the horizontal and vertical directions, respectively. A collimated plane wave is used for the reference wave in the recording process with an incidence angle (

*θ*

_{r}) of 45º. The intensity profiles at the plane of 20 mm and 40 mm in front of the LAHOE with the ordinary and phase-conjugated reconstruction are presented in Figs. 5(b) and 5(c), respectively. In the ordinary reconstruction of Fig. 5(b), the captured image around 40 mm in front of the LAHOE shows the array of focused points as expected, because the focal length of the reference lens-array is 41.9 mm. On the other hand, when we flip the LAHOE and impose the same probe wave for the phase-conjugated illumination, the reconstructed wavefront changes into the diverging spherical waves as shown in Fig. 5(c).

11. J. Kim, J.-H. Jung, C. Jang, and B. Lee, “Real-time capturing and 3D visualization method based on integral imaging,” Opt. Express **21**(16), 18742–18753 (2013). [CrossRef] [PubMed]

14. K. Hong, J. Yeom, C. Jang, J. Hong, and B. Lee, “Full-color lens-array holographic optical element for three-dimensional optical see-through augmented reality,” Opt. Lett. **39**(1), 127–130 (2014). [CrossRef] [PubMed]

21. J.-S. Jang and B. Javidi, “Formation of orthoscopic three-dimensional real images in direct pickup one-step integral imaging,” Opt. Eng. **42**(7), 1869–1870 (2003). [CrossRef]

## 4. Conclusion

## Acknowledgment

## References and links

1. | B. Lee, “Three-dimensional displays, past and present,” Phys. Today |

2. | B. Javidi and F. Okano, eds., |

3. | S.- Park, J. Yeom, Y. Jeong, N. Chen, J.-Y. Hong, and B. Lee, “Recent issues on integral imaging and its applications,” J. Inf. Disp. |

4. | G. Lippmann, “La photograhie integrale,” Comptes Rendus Acad. Sci., Paris, CR (East Lansing, Mich.) |

5. | J.-H. Park, K. Hong, and B. Lee, “Recent progress in three-dimensional information processing based on integral imaging,” Appl. Opt. |

6. | H. E. Ives, “Optical properties of a Lippmann lenticulated sheet,” J. Opt. Soc. Am. |

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

8. | M. Martínez-Corral, B. Javidi, R. Martínez-Cuenca, and G. Saavedra, “Formation of real, orthoscopic integral images by smart pixel mapping,” Opt. Express |

9. | D. H. Shin, B. G. Lee, and E.-S. Kim, “Modified smart pixel mapping method for displaying orthoscopic 3D images in integral imaging,” Opt. Lasers Eng. |

10. | J.-H. Jung, J. Kim, and B. Lee, “Solution of pseudoscopic problem in integral imaging for real-time processing,” Opt. Lett. |

11. | J. Kim, J.-H. Jung, C. Jang, and B. Lee, “Real-time capturing and 3D visualization method based on integral imaging,” Opt. Express |

12. | J.-S. Jang and B. Javidi, “Three-dimensional projection integral imaging using micro-convex-mirror arrays,” Opt. Express |

13. | J. Arai, H. Kawai, and F. Okano, “Microlens arrays for integral imaging system,” Appl. Opt. |

14. | K. Hong, J. Yeom, C. Jang, J. Hong, and B. Lee, “Full-color lens-array holographic optical element for three-dimensional optical see-through augmented reality,” Opt. Lett. |

15. | Y. Jeong, S. Jung, J.-H. Park, and B. Lee, “Reflection-type integral imaging scheme for displaying three-dimensional images,” Opt. Lett. |

16. | R. R. A. Syms and L. Solymar, “Analysis of volume holographic cylindrical lenses,” J. Opt. Soc. Am. |

17. | R. R. A. Syms and L. Solymar, “Higher diffraction orders in on-axis holographic lenses,” Appl. Opt. |

18. | Y. Luo, J. Castro, J. K. Barton, R. K. Kostuk, and G. Barbastathis, “Simulations and experiments of aperiodic and multiplexed gratings in volume holographic imaging systems,” Opt. Express |

19. | Y. Lim, J. Hahn, and B. Lee, “Phase-conjugate holographic lithography based on micromirror array recording,” Appl. Opt. |

20. | R. R. A. Syms, |

21. | J.-S. Jang and B. Javidi, “Formation of orthoscopic three-dimensional real images in direct pickup one-step integral imaging,” Opt. Eng. |

**OCIS Codes**

(100.6890) Image processing : Three-dimensional image processing

(110.2990) Imaging systems : Image formation theory

**ToC Category:**

Imaging Systems

**History**

Original Manuscript: April 28, 2014

Revised Manuscript: May 18, 2014

Manuscript Accepted: May 18, 2014

Published: May 29, 2014

**Citation**

Jiwoon Yeom, Keehoon Hong, Youngmo Jeong, Changwon Jang, and Byoungho Lee, "Solution for pseudoscopic problem in integral imaging using phase-conjugated reconstruction of lens-array holographic optical elements," Opt. Express **22**, 13659-13670 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-11-13659

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### References

- B. Lee, “Three-dimensional displays, past and present,” Phys. Today 66(4), 36–41 (2013). [CrossRef]
- B. Javidi and F. Okano, eds., Three Dimensional Television, Video, and Display Technology (Springer, 2002).
- S.- Park, J. Yeom, Y. Jeong, N. Chen, J.-Y. Hong, B. Lee, “Recent issues on integral imaging and its applications,” J. Inf. Disp. 15(1), 37–46 (2014). [CrossRef]
- G. Lippmann, “La photograhie integrale,” Comptes Rendus Acad. Sci., Paris, CR (East Lansing, Mich.) 146, 446–451 (1908).
- J.-H. Park, K. Hong, B. Lee, “Recent progress in three-dimensional information processing based on integral imaging,” Appl. Opt. 48(34), H77–H94 (2009). [CrossRef] [PubMed]
- H. E. Ives, “Optical properties of a Lippmann lenticulated sheet,” J. Opt. Soc. Am. 21(3), 171–179 (1931). [CrossRef]
- F. Okano, H. Hoshino, J. Arai, I. Yuyama, “Real-time pickup method for a three-dimensional image based on integral photography,” Appl. Opt. 36(7), 1598–1603 (1997). [CrossRef] [PubMed]
- M. Martínez-Corral, B. Javidi, R. Martínez-Cuenca, G. Saavedra, “Formation of real, orthoscopic integral images by smart pixel mapping,” Opt. Express 13(23), 9175–9180 (2005). [CrossRef] [PubMed]
- D. H. Shin, B. G. Lee, E.-S. Kim, “Modified smart pixel mapping method for displaying orthoscopic 3D images in integral imaging,” Opt. Lasers Eng. 47(11), 1189–1194 (2009). [CrossRef]
- J.-H. Jung, J. Kim, B. Lee, “Solution of pseudoscopic problem in integral imaging for real-time processing,” Opt. Lett. 38(1), 76–78 (2013). [CrossRef] [PubMed]
- J. Kim, J.-H. Jung, C. Jang, B. Lee, “Real-time capturing and 3D visualization method based on integral imaging,” Opt. Express 21(16), 18742–18753 (2013). [CrossRef] [PubMed]
- J.-S. Jang, B. Javidi, “Three-dimensional projection integral imaging using micro-convex-mirror arrays,” Opt. Express 12(6), 1077–1083 (2004). [CrossRef] [PubMed]
- J. Arai, H. Kawai, F. Okano, “Microlens arrays for integral imaging system,” Appl. Opt. 45(36), 9066–9078 (2006). [CrossRef] [PubMed]
- K. Hong, J. Yeom, C. Jang, J. Hong, B. Lee, “Full-color lens-array holographic optical element for three-dimensional optical see-through augmented reality,” Opt. Lett. 39(1), 127–130 (2014). [CrossRef] [PubMed]
- Y. Jeong, S. Jung, J.-H. Park, B. Lee, “Reflection-type integral imaging scheme for displaying three-dimensional images,” Opt. Lett. 27(9), 704–706 (2002). [CrossRef] [PubMed]
- R. R. A. Syms, L. Solymar, “Analysis of volume holographic cylindrical lenses,” J. Opt. Soc. Am. 72(2), 179–186 (1982). [CrossRef]
- R. R. A. Syms, L. Solymar, “Higher diffraction orders in on-axis holographic lenses,” Appl. Opt. 21(18), 3263–3268 (1982). [CrossRef] [PubMed]
- Y. Luo, J. Castro, J. K. Barton, R. K. Kostuk, G. Barbastathis, “Simulations and experiments of aperiodic and multiplexed gratings in volume holographic imaging systems,” Opt. Express 18(18), 19273–19285 (2010). [CrossRef] [PubMed]
- Y. Lim, J. Hahn, B. Lee, “Phase-conjugate holographic lithography based on micromirror array recording,” Appl. Opt. 50(34), H68–H74 (2011). [CrossRef] [PubMed]
- R. R. A. Syms, Practical Volume Holography (Clarendon, 1990).
- J.-S. Jang, B. Javidi, “Formation of orthoscopic three-dimensional real images in direct pickup one-step integral imaging,” Opt. Eng. 42(7), 1869–1870 (2003). [CrossRef]

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