Abstract

The majority of image-generating computer-generated holograms (CGHs) are calculated on a discrete numerical grid, whose spacing is defined by the desired pixel size. For single-plane CGHs the influence of the pixel shape and the illumination wave on the actual output distribution is minor and can be treated separately from the numerical calculation. We show that in the case of multiplane CGHs this influence is much more severe. We introduce a new method that takes the pixel shape into account during the design and derive conditions to retain an illumination-wave-independent behavior.

© 2008 Optical Society of America

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References

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  1. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237-246 (1972).
  2. S. Borgsmüller, S. Noethe, C. Dietrich, T. Kresse, and R. Männer, “Computer-generated stratified diffractive optical elements,” Appl. Opt. 42, 5274-5283 (2003).
    [CrossRef] [PubMed]
  3. T. Kämpfe, E. B. Kley, and A. Tünnermann, “Creation of multicolor images by diffractive optical elements arranged in a stacked setup,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM, OSA Technical Digest (CD) (Optical Society of America, 2007), paper DTuD8.
    [PubMed]
  4. W. Cai, T. Reber, and R. Piestun, “Computer-generated volume holograms fabricated by femtosecond laser micromachining,” Opt. Lett. 31, 1836-1838 (2006).
    [CrossRef] [PubMed]
  5. D. Chambers, G. Nordin, and S. Kim, “Fabrication and analysis of a three-layer stratified volume diffractive optical element high-efficiency grating,” Opt. Express 11, 27-38 (2003).
    [CrossRef] [PubMed]
  6. R. Johnson and A. Tanguay, “Stratified volume holographic optical elements,” Opt. Lett. 13, 189-191 (1988).
    [CrossRef] [PubMed]
  7. E. Buckley, A. Cable, N. Lawrence, and T. Wilkinson, “Viewing angle enhancement for two- and three-dimensional holographic displays with random superresolution phase masks,” Appl. Opt. 45, 7334-7341 (2006).
    [CrossRef] [PubMed]
  8. T. Kämpfe, E. B. Kley, A. Tünnermann, and P. Dannberg, “Design and fabrication of stacked, computer-generated holograms for multicolor image generation,” Appl. Opt. 46, 5482-5488 (2007).
    [CrossRef] [PubMed]
  9. X. Deng and R. Chen, “Design of cascaded diffractive phase elements for three-dimensional multiwavelength optical interconnects,” Opt. Lett. 25, 1046-1048 (2000).
    [CrossRef]
  10. I. Barton, P. Blair, and M. R. Taghizadeh, “Dual-wavelength operation diffractive phase elements for pattern formation,” Opt. Express 1, 54-59 (1997).
    [CrossRef] [PubMed]
  11. A. Caley, A. Waddie, and M. Taghizadeh, “A novel algorithm for designing diffractive optical elements for two colour far-field pattern formation,” J. Opt. A, Pure Appl. Opt. 7, 276-279 (2005).
    [CrossRef]
  12. J. Bengtsson, “Kinoforms designed to produce different fan-out patterns for two wavelengths,” Appl. Opt. 37, 2011-2020 (1998).
    [CrossRef]
  13. Y. Ogura, N. Shirai, J. Tanida, and Y. Ichioka, “Wavelength-multiplexing diffractive phase elements: design, fabrication, and performance evaluation,” J. Opt. Soc. Am. A 18, 1082-1092 (2001).
    [CrossRef]
  14. P. Refregier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20, 767-769 (1995).
    [CrossRef] [PubMed]
  15. H. Chang, W. Lu, and C. Kuo, “Multiple-phase retrieval for optical security systems by use of random-phase encoding,” Appl. Opt. 41, 4815-4834 (2002).
    [CrossRef]
  16. L. Chen and D. Zhao, “Optical color image encryption by wavelength multiplexing and lensless Fresnel transform holograms,” Opt. Express 14, 8552-8560 (2006).
    [CrossRef] [PubMed]
  17. E. Glytsis, “Two-dimensionally-periodic diffractive optical elements: limitations of scalar analysis,” J. Opt. Soc. Am. A 19, 702-715 (2002).
    [CrossRef]
  18. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).
  19. L. Lesem, P. Hirsch, and J. Jordan, Jr., “Computer synthesis of holograms for 3-D display,” Commun. ACM 11, 661-674 (1968).
    [CrossRef]
  20. F. Wyrowski, R. Hauck, and O. Bryngdahl, “Computer-generated holography: hologram repetition and phase manipulations,” J. Opt. Soc. Am. A 4, 694-698 (1987).
    [CrossRef]

2007 (2)

T. Kämpfe, E. B. Kley, and A. Tünnermann, “Creation of multicolor images by diffractive optical elements arranged in a stacked setup,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM, OSA Technical Digest (CD) (Optical Society of America, 2007), paper DTuD8.
[PubMed]

T. Kämpfe, E. B. Kley, A. Tünnermann, and P. Dannberg, “Design and fabrication of stacked, computer-generated holograms for multicolor image generation,” Appl. Opt. 46, 5482-5488 (2007).
[CrossRef] [PubMed]

2006 (3)

2005 (2)

A. Caley, A. Waddie, and M. Taghizadeh, “A novel algorithm for designing diffractive optical elements for two colour far-field pattern formation,” J. Opt. A, Pure Appl. Opt. 7, 276-279 (2005).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

2003 (2)

2002 (2)

E. Glytsis, “Two-dimensionally-periodic diffractive optical elements: limitations of scalar analysis,” J. Opt. Soc. Am. A 19, 702-715 (2002).
[CrossRef]

H. Chang, W. Lu, and C. Kuo, “Multiple-phase retrieval for optical security systems by use of random-phase encoding,” Appl. Opt. 41, 4815-4834 (2002).
[CrossRef]

2001 (1)

2000 (1)

1998 (1)

1997 (1)

1995 (1)

1988 (1)

1987 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237-246 (1972).

1968 (1)

L. Lesem, P. Hirsch, and J. Jordan, Jr., “Computer synthesis of holograms for 3-D display,” Commun. ACM 11, 661-674 (1968).
[CrossRef]

Barton, I.

Bengtsson, J.

Blair, P.

Borgsmüller, S.

Bryngdahl, O.

Buckley, E.

Cable, A.

Cai, W.

Caley, A.

A. Caley, A. Waddie, and M. Taghizadeh, “A novel algorithm for designing diffractive optical elements for two colour far-field pattern formation,” J. Opt. A, Pure Appl. Opt. 7, 276-279 (2005).
[CrossRef]

Chambers, D.

Chang, H.

H. Chang, W. Lu, and C. Kuo, “Multiple-phase retrieval for optical security systems by use of random-phase encoding,” Appl. Opt. 41, 4815-4834 (2002).
[CrossRef]

Chen, L.

Chen, R.

Dannberg, P.

Deng, X.

Dietrich, C.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237-246 (1972).

Glytsis, E.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

Hauck, R.

Hirsch, P.

L. Lesem, P. Hirsch, and J. Jordan, Jr., “Computer synthesis of holograms for 3-D display,” Commun. ACM 11, 661-674 (1968).
[CrossRef]

Ichioka, Y.

Javidi, B.

Johnson, R.

Jordan, J.

L. Lesem, P. Hirsch, and J. Jordan, Jr., “Computer synthesis of holograms for 3-D display,” Commun. ACM 11, 661-674 (1968).
[CrossRef]

Kämpfe, T.

T. Kämpfe, E. B. Kley, and A. Tünnermann, “Creation of multicolor images by diffractive optical elements arranged in a stacked setup,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM, OSA Technical Digest (CD) (Optical Society of America, 2007), paper DTuD8.
[PubMed]

T. Kämpfe, E. B. Kley, A. Tünnermann, and P. Dannberg, “Design and fabrication of stacked, computer-generated holograms for multicolor image generation,” Appl. Opt. 46, 5482-5488 (2007).
[CrossRef] [PubMed]

Kim, S.

Kley, E. B.

T. Kämpfe, E. B. Kley, and A. Tünnermann, “Creation of multicolor images by diffractive optical elements arranged in a stacked setup,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM, OSA Technical Digest (CD) (Optical Society of America, 2007), paper DTuD8.
[PubMed]

T. Kämpfe, E. B. Kley, A. Tünnermann, and P. Dannberg, “Design and fabrication of stacked, computer-generated holograms for multicolor image generation,” Appl. Opt. 46, 5482-5488 (2007).
[CrossRef] [PubMed]

Kresse, T.

Kuo, C.

H. Chang, W. Lu, and C. Kuo, “Multiple-phase retrieval for optical security systems by use of random-phase encoding,” Appl. Opt. 41, 4815-4834 (2002).
[CrossRef]

Lawrence, N.

Lesem, L.

L. Lesem, P. Hirsch, and J. Jordan, Jr., “Computer synthesis of holograms for 3-D display,” Commun. ACM 11, 661-674 (1968).
[CrossRef]

Lu, W.

H. Chang, W. Lu, and C. Kuo, “Multiple-phase retrieval for optical security systems by use of random-phase encoding,” Appl. Opt. 41, 4815-4834 (2002).
[CrossRef]

Männer, R.

Noethe, S.

Nordin, G.

Ogura, Y.

Piestun, R.

Reber, T.

Refregier, P.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237-246 (1972).

Shirai, N.

Taghizadeh, M.

A. Caley, A. Waddie, and M. Taghizadeh, “A novel algorithm for designing diffractive optical elements for two colour far-field pattern formation,” J. Opt. A, Pure Appl. Opt. 7, 276-279 (2005).
[CrossRef]

Taghizadeh, M. R.

Tanguay, A.

Tanida, J.

Tünnermann, A.

T. Kämpfe, E. B. Kley, A. Tünnermann, and P. Dannberg, “Design and fabrication of stacked, computer-generated holograms for multicolor image generation,” Appl. Opt. 46, 5482-5488 (2007).
[CrossRef] [PubMed]

T. Kämpfe, E. B. Kley, and A. Tünnermann, “Creation of multicolor images by diffractive optical elements arranged in a stacked setup,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM, OSA Technical Digest (CD) (Optical Society of America, 2007), paper DTuD8.
[PubMed]

Waddie, A.

A. Caley, A. Waddie, and M. Taghizadeh, “A novel algorithm for designing diffractive optical elements for two colour far-field pattern formation,” J. Opt. A, Pure Appl. Opt. 7, 276-279 (2005).
[CrossRef]

Wilkinson, T.

Wyrowski, F.

Zhao, D.

Appl. Opt. (5)

Commun. ACM (1)

L. Lesem, P. Hirsch, and J. Jordan, Jr., “Computer synthesis of holograms for 3-D display,” Commun. ACM 11, 661-674 (1968).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

A. Caley, A. Waddie, and M. Taghizadeh, “A novel algorithm for designing diffractive optical elements for two colour far-field pattern formation,” J. Opt. A, Pure Appl. Opt. 7, 276-279 (2005).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Express (3)

Opt. Lett. (4)

Optik (Stuttgart) (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Stuttgart) 35, 237-246 (1972).

Other (2)

T. Kämpfe, E. B. Kley, and A. Tünnermann, “Creation of multicolor images by diffractive optical elements arranged in a stacked setup,” in Adaptive Optics: Analysis and Methods/Computational Optical Sensing and Imaging/Information Photonics/Signal Recovery and Synthesis Topical Meetings on CD-ROM, OSA Technical Digest (CD) (Optical Society of America, 2007), paper DTuD8.
[PubMed]

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

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Figures (11)

Fig. 1
Fig. 1

Scheme of the optical setup of a multiplane CGH for far-field pattern generation.

Fig. 2
Fig. 2

Visualization of the formation of the output image for a single-plane CGH.

Fig. 3
Fig. 3

Flow diagram of a multiplane IFTA for multiple pages of information, in this case three signals for three different wavelengths.

Fig. 4
Fig. 4

Flow diagram of a general IFTA, including the pixel shape regeneration during the element constraint application step.

Fig. 5
Fig. 5

Simulated output for a single-plane and a two-plane CGH with different values of β [Eq. (19)]. The design was done without supersampling. For further explanation, see text.

Fig. 6
Fig. 6

Convergence behavior of the IFTA for different supersampling factors q D .

Fig. 7
Fig. 7

Analysis of the effect of the design supersampling factor q D on the SNR and the efficiency η.

Fig. 8
Fig. 8

Analysis of the effect of the setup parameters β [Eq. (19)] and κ [Eq. (7)] on the SNR and the efficiency η. The bold dashed line at β = 1 indicates the analytically derived threshold (Subsection 3B).

Fig. 9
Fig. 9

Simulated output distributions for a multicolor design, calculated with different supersampling parameters q D and different distances between the element planes.

Fig. 10
Fig. 10

Detail of phase distribution in the first element plane for multifunctional designs with q D = 1 and q D = 4 , showing the different use of spatial frequencies.

Fig. 11
Fig. 11

Photographs of the experimentally realized output distributions and measured values for the figures of merit.

Tables (1)

Tables Icon

Table 1 Merit Figures for the Multifunctional Two-Plane CGH of Fig. 9

Equations (45)

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u o u t m = FT N ( u e l m ) U e l m .
u e l ( r ) = [ ( m = ( 1 , 1 ) M u e l m δ ( r m p ) ) rect ( r ÷ p ) m = ( , ) ( , ) δ ( r m p M ) ] ,
rect ( r ) = rect ( r x ) rect ( r y ) and δ ( r ) = δ ( r x ) δ ( r y ) .
u o u t ( f ) = FT [ u e l ( r ) u i l l ( r ) ] ,
u o u t ( f ) = { [ ( m = ( 1 , 1 ) M U e l m δ ( f m ÷ ( p M ) ) ) n = ( , ) ( , ) δ ( f n ÷ p ) ] sinc ( f p ) } FT ( u i l l ( r ) ) ,
u g ( r ) = exp ( 4 r x 2 ( S x κ x ) 2 ) exp ( 4 r y 2 ( S y κ y ) 2 ) ,
κ > ( ln ( 1 4 SNR o u t r e l ) 16 π 2 ) ,
u o u t ( f ) FT [ NFT z 1 ( u e l , 1 ( r ) u i l l ( r ) ) u e l , 2 ( r ) ] ,
NFT z ( u ( r ) ) = FT 1 [ FT ( u ( r ) ) exp ( i 2 π z 1 λ 2 f 2 ) ] .
u o u t ( f ) FT [ u i l l , 2 ( r ) u e l , 2 ( r ) ] .
u o u t ( f ) = FT [ FT 1 [ FT ( u e l , 1 ( r ) ) exp ( i 2 π z 1 λ 2 f 2 ) ] u e l , 2 ( r ) ] .
α ( f ) FT ( u e l , 1 ( r ) ) exp ( i 2 π z 1 λ 2 f 2 ) ,
u o u t ( f ) = FT [ FT 1 ( α ( f ) ) u e l , 2 ( r ) ] ,
α ( f ) [ ( m = ( 1 , 1 ) M U e l , 1 m δ ( f m ÷ ( p M ) ) ) n = ( , ) ( , ) δ ( f n ÷ p ) ] sinc ( p f ) exp ( i 2 π z 1 λ 2 f 2 ) .
u o u t ( f ) [ ( m = ( 1 , 1 ) q M U ̂ o u t m δ ( f m ÷ ( p M ) ) ) n = ( , ) ( , ) δ ( f n ÷ p q ) ] sinc ( p f ) ,
α ( f ) [ FT ( u e l , 1 ( r ) u i l l ( r ) ) exp ( i 2 π z 1 λ 2 f 2 ) ] .
α ( f ) [ m = ( , ) ( , ) U ̃ e l , 1 m δ ( f m ÷ p ) FT ( u i l l ( r ) ) ] exp ( i 2 π z 1 λ 2 f 2 ) ,
α ( f ) FT ( u i l l ( r ) ) [ m = ( , ) ( , ) U ̃ e l , 1 m δ ( f m ÷ p ) exp ( i π z λ f 2 ) ]
β z λ pS 1 .
u o u t ( f ) FT ( u i l l ( r ) ) FT [ FT 1 [ FT ( u e l , 1 ( r ) ) exp ( i π z λ f 2 ) ] u e l , 2 ( r ) ] .
SNR ( u o u t , u r e f ) = A s i g u r e f 2 A s i g ( u r e f γ ( u o u t , u r e f ) u o u t ) 2 ,
η ( u o u t ) = A ̂ s i g u o u t 2 A d i f f u o u t 2 ,
u o u t ( f ) = FT { [ ( m = ( 1 , 1 ) M u e l m δ ( r m p ) ) rect ( r ÷ p ) n = ( , ) ( , ) δ ( r n p M ) ] u i l l ( r ) } .
u o u t ( f ) = { ( m = ( 1 , 1 ) M u e l m FT [ δ ( r m p ) ] ) FT [ rect ( r ÷ p ) ] FT [ n = ( , ) ( , ) δ ( r n p M ) ] } FT [ u i l l ( r ) ] .
u o u t ( f ) = { ( m = ( 1 , 1 ) M u e l m e i 2 π ( m p ) f ) sinc ( f p ) n = ( , ) ( , ) δ ( f n ÷ ( p M ) ) } FT [ u i l l ( r ) ] .
u o u t ( f ) = { [ ( m = ( 1 , 1 ) M U e l m δ ( f m ÷ ( p M ) ) ) n = ( , ) ( , ) δ ( f n ÷ p ) ] sinc ( f p ) } FT ( u i l l ( r ) ) .
FT ( u g ( r ) ) = U g ( f ) = exp ( π 2 S x 2 κ x 2 4 f x 2 ) exp ( π 2 S y 2 κ y 2 4 f y 2 ) .
u b o m = ( u o u t m + u o u t m + 1 ) exp ( π 2 S x κ x 2 4 ( 1 2 S x ) 2 ) .
I b o 4 I m a x exp ( π 2 κ x 2 16 ) 2 < 1 SNR o u t ,
κ x > ( ln ( 1 4 SNR o u t r e l ) 16 π 2 ) .
α ( f ) [ ( m = ( 1 , 1 ) q M U ̂ e l , 1 m δ ( f m ÷ ( p q q M ) ) ) n = ( , ) ( , ) δ ( f n ÷ p q ) ] sinc ( p q f ) exp ( i 2 π z 1 λ 2 f 2 ) .
α ( f ) [ ( m = ( 1 , 1 ) q M U ̂ e l , 1 m exp ( i 2 π z 1 λ 2 f m 2 ) δ ( f m ÷ ( p M ) ) ) n = ( , ) ( , ) δ ( f n ÷ p q ) ] sinc ( f p q ) .
u ̂ i l l , 2 m FT N 1 [ FT N ( u ̂ e l , 1 m ) exp ( i 2 π z 1 λ 2 f m 2 ) ] .
u o u t ( f ) FT ( { [ ( m = ( 1 , 1 ) q M u ̂ i l l , 2 m δ ( r m p q ) ) n = ( , ) ( , ) δ ( r n p M ) ] rect ( r ÷ p q ) } u e l , 2 ( r ) ) .
u o u t ( f ) FT { [ ( m = ( 1 , 1 ) q M u ̂ i l l , 2 m u ̂ e l , 2 m δ ( r m p q ) ) n = ( , ) ( , ) δ ( r n p M ) ] rect ( r ÷ p q ) } .
U o u t m = FT N ( u ̂ i l l , 2 m u ̂ e l , 2 m ) ,
u o u t ( f ) [ ( m = ( 1 , 1 ) q M U ̂ o u t m δ ( f m ÷ ( p M ) ) ) n = ( , ) ( , ) δ ( f n ÷ p q ) ] sinc ( f p ) .
α ( f ) = [ U ̃ e l , 1 m δ ( f m ÷ p ) FT ( u i l l ( r ) ) ] exp [ i π z λ f 2 ] .
α ( f m + Δ f ) = U ̃ e l , 1 m U i l l ( Δ f ) exp [ i π z λ ( f m + Δ f ) 2 ] ,
α ( f m + Δ f ) = U ̃ e l , 1 m U i l l ( Δ f ) exp ( i π z λ f m 2 ) exp [ i π z λ ( 2 f m Δ f + Δ f 2 ) ] .
π z λ ( 2 f m Δ f + Δ f 2 ) 2 π .
β z λ pS 1 .
β z λ 2 p S 1 .
α ( f m + Δ f ) U i l l ( Δ f ) U ̃ e l , 1 m exp ( i π z λ f m 2 ) ,
α ( f ) FT ( u i l l ( r ) ) [ m = ( , ) ( , ) U ̃ e l , 1 m δ ( f m ÷ p ) exp ( i π z λ f 2 ) ] .

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