Abstract

We propose a full-color complex holographic display system design comprised of three R/G/B amplitude-only spatial light modulators and an achromatic Fourier filter. A key feature of the design is a single achromatic Fourier bandpass filter for robust axial R/G/B color matching, whereby the R/G/B holographic image light fields can be three-dimensionally aligned. The synthesis algorithm producing the full-color computer-generated holograms for this system is described and a full-color optical reconstruction of the designed holographic three-dimensional images is experimentally demonstrated.

© 2017 Optical Society of America

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References

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    [Crossref] [PubMed]
  4. 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(16), 12372–12386 (2008).
    [Crossref] [PubMed]
  5. S. Kim, J. Choi, J. Hahn, and H. Kim, “Viewing-window extension of holographic display using high-order diffraction,” in Proceedings of Digital Holography and Three-dimensional Imaging Conference 2017 (Optical Society of America, 2017), Th3A.6.
  6. R. Stahl, V. Rochus, X. Rottenberg, S. Cosemans, L. Haspeslagh, S. Severi, G. V. Plas, G. Lafruit, and S. Donnay, “Modulator sub-wavelength diffractive light modulator for high definition holographic displays,” J. Phys. Conf. Ser. 415, 012057 (2013), doi:.
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  17. A. Maimone, A. Georgiou, and J. S. Kollin, “Holographic near-eye displays for virtual and augmented reality,” ACM Trans. Graph. 36(85), 1–16 (2017).
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  20. S.-C. Kim and E.-S. Kim, “Effective generation of digital holograms of three-dimensional objects using a novel look-up table method,” Appl. Opt. 47(19), D55–D62 (2008).
    [Crossref] [PubMed]
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2017 (1)

A. Maimone, A. Georgiou, and J. S. Kollin, “Holographic near-eye displays for virtual and augmented reality,” ACM Trans. Graph. 36(85), 1–16 (2017).

2016 (1)

H. Niwase, N. Takada, H. Araki, Y. Maeda, M. Fujiwara, H. Nakayama, T. Kakue, T. Shimobaba, and T. Ito, “Real-time electroholography using a multiple-graphics processing unit cluster system with a single spatial light modulator and the InfiniBand network,” Opt. Eng. 55(9), 093108 (2016).
[Crossref]

2015 (4)

2014 (5)

2013 (3)

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

R. Stahl, V. Rochus, X. Rottenberg, S. Cosemans, L. Haspeslagh, S. Severi, G. V. Plas, G. Lafruit, and S. Donnay, “Modulator sub-wavelength diffractive light modulator for high definition holographic displays,” J. Phys. Conf. Ser. 415, 012057 (2013), doi:.
[Crossref]

T. Ichikawa, K. Yamaguchi, and Y. Sakamoto, “Realistic expression for full-parallax computer-generated holograms with the ray-tracing method,” Appl. Opt. 52(1), A201–A209 (2013).
[Crossref] [PubMed]

2012 (2)

2011 (2)

2010 (1)

K.-I. Aoshima, N. Funabashi, K. Machida, Y. Miyamoto, K. Kuga, T. Ishibashi, N. Shimidzu, and F. Sato, “Submicron magneto-optical spatial light modulation device for holographic displays driven by spin-polarized electrons,” J. Displ. Technol. 6(9), 374–380 (2010).
[Crossref]

2008 (2)

Aoshima, K.-I.

K.-I. Aoshima, N. Funabashi, K. Machida, Y. Miyamoto, K. Kuga, T. Ishibashi, N. Shimidzu, and F. Sato, “Submicron magneto-optical spatial light modulation device for holographic displays driven by spin-polarized electrons,” J. Displ. Technol. 6(9), 374–380 (2010).
[Crossref]

Araki, H.

Chen, J.-S.

Chen, N.

Cho, J.

Choi, H. J.

Choi, J.

S. Kim, J. Choi, J. Hahn, and H. Kim, “Viewing-window extension of holographic display using high-order diffraction,” in Proceedings of Digital Holography and Three-dimensional Imaging Conference 2017 (Optical Society of America, 2017), Th3A.6.

Chu, D. P.

Cosemans, S.

R. Stahl, V. Rochus, X. Rottenberg, S. Cosemans, L. Haspeslagh, S. Severi, G. V. Plas, G. Lafruit, and S. Donnay, “Modulator sub-wavelength diffractive light modulator for high definition holographic displays,” J. Phys. Conf. Ser. 415, 012057 (2013), doi:.
[Crossref]

Donnay, S.

R. Stahl, V. Rochus, X. Rottenberg, S. Cosemans, L. Haspeslagh, S. Severi, G. V. Plas, G. Lafruit, and S. Donnay, “Modulator sub-wavelength diffractive light modulator for high definition holographic displays,” J. Phys. Conf. Ser. 415, 012057 (2013), doi:.
[Crossref]

Fujiwara, M.

H. Niwase, N. Takada, H. Araki, Y. Maeda, M. Fujiwara, H. Nakayama, T. Kakue, T. Shimobaba, and T. Ito, “Real-time electroholography using a multiple-graphics processing unit cluster system with a single spatial light modulator and the InfiniBand network,” Opt. Eng. 55(9), 093108 (2016).
[Crossref]

H. Araki, N. Takada, H. Niwase, S. Ikawa, M. Fujiwara, H. Nakayama, T. Kakue, T. Shimobaba, and T. Ito, “Real-time time-division color electroholography using a single GPU and a USB module for synchronizing reference light,” Appl. Opt. 54(34), 10029–10034 (2015).
[Crossref] [PubMed]

Funabashi, N.

K.-I. Aoshima, N. Funabashi, K. Machida, Y. Miyamoto, K. Kuga, T. Ishibashi, N. Shimidzu, and F. Sato, “Submicron magneto-optical spatial light modulation device for holographic displays driven by spin-polarized electrons,” J. Displ. Technol. 6(9), 374–380 (2010).
[Crossref]

Fütterer, G.

Georgiou, A.

A. Maimone, A. Georgiou, and J. S. Kollin, “Holographic near-eye displays for virtual and augmented reality,” ACM Trans. Graph. 36(85), 1–16 (2017).

Hahn, J.

D. Im, J. Cho, J. Hahn, B. Lee, and H. Kim, “Accelerated synthesis algorithm of polygon computer-generated holograms,” Opt. Express 23(3), 2863–2871 (2015).
[Crossref] [PubMed]

H. Kim, C.-Y. Hwang, K.-S. Kim, J. Roh, W. Moon, S. Kim, B.-R. Lee, S. Oh, and J. Hahn, “Anamorphic optical transformation of an amplitude spatial light modulator to a complex spatial light modulator with square pixels [invited],” Appl. Opt. 53(27), G139–G146 (2014).
[Crossref] [PubMed]

D. Im, E. Moon, Y. Park, D. Lee, J. Hahn, and H. Kim, “Phase-regularized polygon computer-generated holograms,” Opt. Lett. 39(12), 3642–3645 (2014).
[Crossref] [PubMed]

E. Moon, M. Kim, J. Roh, H. Kim, and J. Hahn, “Holographic head-mounted display with RGB light emitting diode light source,” Opt. Express 22(6), 6526–6534 (2014).
[Crossref] [PubMed]

J. Hong, Y. Kim, H. J. Choi, J. Hahn, J.-H. Park, H. Kim, S.-W. Min, N. Chen, and B. Lee, “Three-dimensional display technologies of recent interest: principles, status, and issues [Invited],” Appl. Opt. 50(34), H87–H115 (2011).
[Crossref] [PubMed]

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(16), 12372–12386 (2008).
[Crossref] [PubMed]

S. Kim, J. Choi, J. Hahn, and H. Kim, “Viewing-window extension of holographic display using high-order diffraction,” in Proceedings of Digital Holography and Three-dimensional Imaging Conference 2017 (Optical Society of America, 2017), Th3A.6.

Haspeslagh, L.

R. Stahl, V. Rochus, X. Rottenberg, S. Cosemans, L. Haspeslagh, S. Severi, G. V. Plas, G. Lafruit, and S. Donnay, “Modulator sub-wavelength diffractive light modulator for high definition holographic displays,” J. Phys. Conf. Ser. 415, 012057 (2013), doi:.
[Crossref]

Häussler, R.

Hong, J.

Hwang, C.-Y.

Ichihashi, Y.

H. Sasaki, K. Yamamoto, K. Wakunami, Y. Ichihashi, R. Oi, and T. Senoh, “Large size three-dimensional video by electronic holography using multiple spatial light modulators,” Sci. Rep. 4(1), 6177 (2015), doi:.
[Crossref] [PubMed]

Ichikawa, T.

Ikawa, S.

Im, D.

Ishibashi, T.

K.-I. Aoshima, N. Funabashi, K. Machida, Y. Miyamoto, K. Kuga, T. Ishibashi, N. Shimidzu, and F. Sato, “Submicron magneto-optical spatial light modulation device for holographic displays driven by spin-polarized electrons,” J. Displ. Technol. 6(9), 374–380 (2010).
[Crossref]

Ito, T.

Kakue, T.

Kanbayashi, Y.

Kang, H.

L. Onural, F. Yaras, and H. Kang, “Digital holographic three-dimensional video displays,” Proc. IEEE 99(4), 576–589 (2011).
[Crossref]

Kato, H.

Kim, E.-S.

Kim, H.

D. Im, J. Cho, J. Hahn, B. Lee, and H. Kim, “Accelerated synthesis algorithm of polygon computer-generated holograms,” Opt. Express 23(3), 2863–2871 (2015).
[Crossref] [PubMed]

D. Im, E. Moon, Y. Park, D. Lee, J. Hahn, and H. Kim, “Phase-regularized polygon computer-generated holograms,” Opt. Lett. 39(12), 3642–3645 (2014).
[Crossref] [PubMed]

H. Kim, C.-Y. Hwang, K.-S. Kim, J. Roh, W. Moon, S. Kim, B.-R. Lee, S. Oh, and J. Hahn, “Anamorphic optical transformation of an amplitude spatial light modulator to a complex spatial light modulator with square pixels [invited],” Appl. Opt. 53(27), G139–G146 (2014).
[Crossref] [PubMed]

E. Moon, M. Kim, J. Roh, H. Kim, and J. Hahn, “Holographic head-mounted display with RGB light emitting diode light source,” Opt. Express 22(6), 6526–6534 (2014).
[Crossref] [PubMed]

J. Hong, Y. Kim, H. J. Choi, J. Hahn, J.-H. Park, H. Kim, S.-W. Min, N. Chen, and B. Lee, “Three-dimensional display technologies of recent interest: principles, status, and issues [Invited],” Appl. Opt. 50(34), H87–H115 (2011).
[Crossref] [PubMed]

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(16), 12372–12386 (2008).
[Crossref] [PubMed]

S. Kim, J. Choi, J. Hahn, and H. Kim, “Viewing-window extension of holographic display using high-order diffraction,” in Proceedings of Digital Holography and Three-dimensional Imaging Conference 2017 (Optical Society of America, 2017), Th3A.6.

Kim, K.-S.

Kim, M.

Kim, S.

H. Kim, C.-Y. Hwang, K.-S. Kim, J. Roh, W. Moon, S. Kim, B.-R. Lee, S. Oh, and J. Hahn, “Anamorphic optical transformation of an amplitude spatial light modulator to a complex spatial light modulator with square pixels [invited],” Appl. Opt. 53(27), G139–G146 (2014).
[Crossref] [PubMed]

S. Kim, J. Choi, J. Hahn, and H. Kim, “Viewing-window extension of holographic display using high-order diffraction,” in Proceedings of Digital Holography and Three-dimensional Imaging Conference 2017 (Optical Society of America, 2017), Th3A.6.

Kim, S.-C.

Kim, Y.

Kollin, J. S.

A. Maimone, A. Georgiou, and J. S. Kollin, “Holographic near-eye displays for virtual and augmented reality,” ACM Trans. Graph. 36(85), 1–16 (2017).

Kuga, K.

K.-I. Aoshima, N. Funabashi, K. Machida, Y. Miyamoto, K. Kuga, T. Ishibashi, N. Shimidzu, and F. Sato, “Submicron magneto-optical spatial light modulation device for holographic displays driven by spin-polarized electrons,” J. Displ. Technol. 6(9), 374–380 (2010).
[Crossref]

Lafruit, G.

R. Stahl, V. Rochus, X. Rottenberg, S. Cosemans, L. Haspeslagh, S. Severi, G. V. Plas, G. Lafruit, and S. Donnay, “Modulator sub-wavelength diffractive light modulator for high definition holographic displays,” J. Phys. Conf. Ser. 415, 012057 (2013), doi:.
[Crossref]

Lee, B.

Lee, B.-R.

Lee, D.

Leister, N.

Lim, Y.

Machida, K.

K.-I. Aoshima, N. Funabashi, K. Machida, Y. Miyamoto, K. Kuga, T. Ishibashi, N. Shimidzu, and F. Sato, “Submicron magneto-optical spatial light modulation device for holographic displays driven by spin-polarized electrons,” J. Displ. Technol. 6(9), 374–380 (2010).
[Crossref]

Maeda, Y.

H. Niwase, N. Takada, H. Araki, Y. Maeda, M. Fujiwara, H. Nakayama, T. Kakue, T. Shimobaba, and T. Ito, “Real-time electroholography using a multiple-graphics processing unit cluster system with a single spatial light modulator and the InfiniBand network,” Opt. Eng. 55(9), 093108 (2016).
[Crossref]

Maimone, A.

A. Maimone, A. Georgiou, and J. S. Kollin, “Holographic near-eye displays for virtual and augmented reality,” ACM Trans. Graph. 36(85), 1–16 (2017).

Masuda, N.

Matsushima, K.

Min, S.-W.

Miyamoto, Y.

K.-I. Aoshima, N. Funabashi, K. Machida, Y. Miyamoto, K. Kuga, T. Ishibashi, N. Shimidzu, and F. Sato, “Submicron magneto-optical spatial light modulation device for holographic displays driven by spin-polarized electrons,” J. Displ. Technol. 6(9), 374–380 (2010).
[Crossref]

Moon, E.

Moon, W.

Nakahara, S.

Nakamura, M.

Nakayama, H.

Niwase, H.

Oh, S.

Oi, R.

H. Sasaki, K. Yamamoto, K. Wakunami, Y. Ichihashi, R. Oi, and T. Senoh, “Large size three-dimensional video by electronic holography using multiple spatial light modulators,” Sci. Rep. 4(1), 6177 (2015), doi:.
[Crossref] [PubMed]

Oikawa, M.

Okada, N.

Onural, L.

L. Onural, F. Yaras, and H. Kang, “Digital holographic three-dimensional video displays,” Proc. IEEE 99(4), 576–589 (2011).
[Crossref]

Park, G.

Park, J.-H.

Park, Y.

Plas, G. V.

R. Stahl, V. Rochus, X. Rottenberg, S. Cosemans, L. Haspeslagh, S. Severi, G. V. Plas, G. Lafruit, and S. Donnay, “Modulator sub-wavelength diffractive light modulator for high definition holographic displays,” J. Phys. Conf. Ser. 415, 012057 (2013), doi:.
[Crossref]

Reichelt, S.

Rochus, V.

R. Stahl, V. Rochus, X. Rottenberg, S. Cosemans, L. Haspeslagh, S. Severi, G. V. Plas, G. Lafruit, and S. Donnay, “Modulator sub-wavelength diffractive light modulator for high definition holographic displays,” J. Phys. Conf. Ser. 415, 012057 (2013), doi:.
[Crossref]

Roh, J.

Rottenberg, X.

R. Stahl, V. Rochus, X. Rottenberg, S. Cosemans, L. Haspeslagh, S. Severi, G. V. Plas, G. Lafruit, and S. Donnay, “Modulator sub-wavelength diffractive light modulator for high definition holographic displays,” J. Phys. Conf. Ser. 415, 012057 (2013), doi:.
[Crossref]

Sakamoto, Y.

Sasaki, H.

H. Sasaki, K. Yamamoto, K. Wakunami, Y. Ichihashi, R. Oi, and T. Senoh, “Large size three-dimensional video by electronic holography using multiple spatial light modulators,” Sci. Rep. 4(1), 6177 (2015), doi:.
[Crossref] [PubMed]

Sato, F.

K.-I. Aoshima, N. Funabashi, K. Machida, Y. Miyamoto, K. Kuga, T. Ishibashi, N. Shimidzu, and F. Sato, “Submicron magneto-optical spatial light modulation device for holographic displays driven by spin-polarized electrons,” J. Displ. Technol. 6(9), 374–380 (2010).
[Crossref]

Senoh, T.

H. Sasaki, K. Yamamoto, K. Wakunami, Y. Ichihashi, R. Oi, and T. Senoh, “Large size three-dimensional video by electronic holography using multiple spatial light modulators,” Sci. Rep. 4(1), 6177 (2015), doi:.
[Crossref] [PubMed]

Severi, S.

R. Stahl, V. Rochus, X. Rottenberg, S. Cosemans, L. Haspeslagh, S. Severi, G. V. Plas, G. Lafruit, and S. Donnay, “Modulator sub-wavelength diffractive light modulator for high definition holographic displays,” J. Phys. Conf. Ser. 415, 012057 (2013), doi:.
[Crossref]

Shimidzu, N.

K.-I. Aoshima, N. Funabashi, K. Machida, Y. Miyamoto, K. Kuga, T. Ishibashi, N. Shimidzu, and F. Sato, “Submicron magneto-optical spatial light modulation device for holographic displays driven by spin-polarized electrons,” J. Displ. Technol. 6(9), 374–380 (2010).
[Crossref]

Shimobaba, T.

Shiraki, A.

Stahl, R.

R. Stahl, V. Rochus, X. Rottenberg, S. Cosemans, L. Haspeslagh, S. Severi, G. V. Plas, G. Lafruit, and S. Donnay, “Modulator sub-wavelength diffractive light modulator for high definition holographic displays,” J. Phys. Conf. Ser. 415, 012057 (2013), doi:.
[Crossref]

Sugiyama, A.

Takada, N.

Usukura, N.

Wakunami, K.

H. Sasaki, K. Yamamoto, K. Wakunami, Y. Ichihashi, R. Oi, and T. Senoh, “Large size three-dimensional video by electronic holography using multiple spatial light modulators,” Sci. Rep. 4(1), 6177 (2015), doi:.
[Crossref] [PubMed]

Yamaguchi, K.

Yamamoto, K.

H. Sasaki, K. Yamamoto, K. Wakunami, Y. Ichihashi, R. Oi, and T. Senoh, “Large size three-dimensional video by electronic holography using multiple spatial light modulators,” Sci. Rep. 4(1), 6177 (2015), doi:.
[Crossref] [PubMed]

Yaras, F.

L. Onural, F. Yaras, and H. Kang, “Digital holographic three-dimensional video displays,” Proc. IEEE 99(4), 576–589 (2011).
[Crossref]

ACM Trans. Graph. (1)

A. Maimone, A. Georgiou, and J. S. Kollin, “Holographic near-eye displays for virtual and augmented reality,” ACM Trans. Graph. 36(85), 1–16 (2017).

Appl. Opt. (6)

H. Kim, C.-Y. Hwang, K.-S. Kim, J. Roh, W. Moon, S. Kim, B.-R. Lee, S. Oh, and J. Hahn, “Anamorphic optical transformation of an amplitude spatial light modulator to a complex spatial light modulator with square pixels [invited],” Appl. Opt. 53(27), G139–G146 (2014).
[Crossref] [PubMed]

T. Ichikawa, K. Yamaguchi, and Y. Sakamoto, “Realistic expression for full-parallax computer-generated holograms with the ray-tracing method,” Appl. Opt. 52(1), A201–A209 (2013).
[Crossref] [PubMed]

H. Araki, N. Takada, H. Niwase, S. Ikawa, M. Fujiwara, H. Nakayama, T. Kakue, T. Shimobaba, and T. Ito, “Real-time time-division color electroholography using a single GPU and a USB module for synchronizing reference light,” Appl. Opt. 54(34), 10029–10034 (2015).
[Crossref] [PubMed]

N. Takada, T. Shimobaba, H. Nakayama, A. Shiraki, N. Okada, M. Oikawa, N. Masuda, and T. Ito, “Fast high-resolution computer-generated hologram computation using multiple graphics processing unit cluster system,” Appl. Opt. 51(30), 7303–7307 (2012).
[Crossref] [PubMed]

J. Hong, Y. Kim, H. J. Choi, J. Hahn, J.-H. Park, H. Kim, S.-W. Min, N. Chen, and B. Lee, “Three-dimensional display technologies of recent interest: principles, status, and issues [Invited],” Appl. Opt. 50(34), H87–H115 (2011).
[Crossref] [PubMed]

S.-C. Kim and E.-S. Kim, “Effective generation of digital holograms of three-dimensional objects using a novel look-up table method,” Appl. Opt. 47(19), D55–D62 (2008).
[Crossref] [PubMed]

J. Displ. Technol. (1)

K.-I. Aoshima, N. Funabashi, K. Machida, Y. Miyamoto, K. Kuga, T. Ishibashi, N. Shimidzu, and F. Sato, “Submicron magneto-optical spatial light modulation device for holographic displays driven by spin-polarized electrons,” J. Displ. Technol. 6(9), 374–380 (2010).
[Crossref]

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R. Stahl, V. Rochus, X. Rottenberg, S. Cosemans, L. Haspeslagh, S. Severi, G. V. Plas, G. Lafruit, and S. Donnay, “Modulator sub-wavelength diffractive light modulator for high definition holographic displays,” J. Phys. Conf. Ser. 415, 012057 (2013), doi:.
[Crossref]

Opt. Eng. (1)

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[Crossref]

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Phys. Today (1)

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[Crossref]

Proc. IEEE (1)

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[Crossref]

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H. Sasaki, K. Yamamoto, K. Wakunami, Y. Ichihashi, R. Oi, and T. Senoh, “Large size three-dimensional video by electronic holography using multiple spatial light modulators,” Sci. Rep. 4(1), 6177 (2015), doi:.
[Crossref] [PubMed]

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[Crossref]

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Figures (4)

Fig. 1
Fig. 1

CGH synthesis framework ( x 1 y 1 : complex CGH plane, uv : eye lens plane, x 2 y 2 : retinal plane)

Fig. 2
Fig. 2

System configurations of (a) the conventional full-color holographic 3D display with separated Fourier filters and (b) the proposed system with an achromatic Fourier filter. Single sideband Fourier filtering configurations showing (c) the mismatched axial-direction alignment of the R/G/B image signal components in the conventional system and (d) axial-direction alignment of the R/G/B signal components in the proposed system. Numerical results of optical field distributions in the Fourier filter plane of (e) the conventional system and (f) the proposed system.

Fig. 3
Fig. 3

Numerical observation results of color CGH images at different focal planes; images observed through the conventional projection system at (a) d m =0 , (b) d m =80mm , (c) d m =160mm , and those observed through the proposed projection system at (d) d m =0 , (b) d m =80mm ,(c) d m =160mm .

Fig. 4
Fig. 4

(a) System setup of the proposed full-color holographic 3D CGH display system. Numerical observation of the CGH image focusing on (b) the background (chess board) and (c) the foreground object (yellow square box). Experimental observation of the CGH image focusing on (d) the background and (e) the foreground object.

Equations (13)

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Q( x 2 , y 2 )=CdFr{ P( x 1 , y 1 );F, d eye , f eye ,ρ }
P( x 1 , y 1 )=ICdFr{ Q( x 2 , y 2 );F, d eye , f eye },
W( u,v )=Fr T 1 { P( x 1 , y 1 ) }
Q( x 2 , y 2 )=Fr T 2 { t( u,v )W( u,v ) }.
W( u,v )= e jkF jλF { P( x 1 , y 1 ) e j π λF ( x 1 2 + y 1 2 ) } e j 2π λF ( x 1 u+ y 1 v ) d x 1 d y 1 ,
Q( x 2 , y 2 )= e jk d eye e j π λ d eye ( x 2 2 + y 2 2 ) jλ d eye circ( [ u 2 + v 2 ]/ ρ 2 )t( u,v )W( u,v ) e j 2π λ d eye ( u x 2 +v y 2 ) dudv
t( u,v )= e j π λ ( 1 F + 1 d eye 1 f eye )( u 2 + v 2 ) ,
f eye = d eye (F d m )/( d eye +(F d m ) ),
P( x 1 , y 1 ;λ )= j e jkF λF e j π λF ( x 1 2 + y 1 2 ) ( W( u,v;λ ) t( u,v;λ ) ) e j 2π λF ( x 1 u+ y 1 v ) dudv ,
W( u,v )=IFr T 2 ( Q( x 2 , y 2 ) )= j e jk d eye λ d eye { Q( x 2 , y 2 ) e j π λ d eye ( x 2 2 + y 2 2 ) } e j 2π λ d eye ( x 2 u+ y 2 v ) d x 2 d y 2 .
( Δu,Δv )=( λF/( Δ x 1 N ),λF/( Δ y 1 N ) ).
( Δ x 2 ,Δ y 2 )=( λ d eye /( ΔuN ),λ d eye /( ΔvN ) )=( d eye Δ x 1 /F, d eye Δ y 1 /F ).
P( x 1 , y 1 ;λ )=IFr T 1 ( IFr T 2 ( Q( x 2 , y 2 ;λ ) ) t( u,v;λ ) ).

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