Abstract

A method of diffraction calculation between tilted planes with variable sampling rates is proposed. The proposed method is based on the Fourier spectrum rotation from a tilted plane to a parallel plane. The nonuniform fast Fourier transformation (NUFFT) is used to calculate the nonuniform sampled Fourier spectrum on the tilted plane with variable sampling rates, which overcomes the sampling restriction of FFT in the conventional method. Both of the computer simulation and the optical experiment shows the feasibility of our method in calculating the hologram of polygon-based object with scalable size, which can be considered as an important application in the holographic three-dimensional display.

© 2014 Optical Society of America

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References

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  1. C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38(8), 46–53 (2005).
    [Crossref]
  2. M. Lucente, “Interactive computation of holograms using a look-up table,” J. Electron. Imaging 2(1), 28–34 (1993).
    [Crossref]
  3. K. Matsushima, “Computer-generated holograms for three-dimensional surface objects with shade and texture,” Appl. Opt. 44(22), 4607–4614 (2005).
    [Crossref] [PubMed]
  4. K. Matsushima and S. Nakahara, “Extremely high-definition full-parallax computer-generated hologram created by the polygon-based method,” Appl. Opt. 48(34), H54–H63 (2009).
    [Crossref] [PubMed]
  5. Y. Z. Liu, J. W. Dong, Y. Y. Pu, B. C. Chen, H. X. He, and H. Z. Wang, “High-speed full analytical holographic computations for true-life scenes,” Opt. Express 18(4), 3345–3351 (2010).
    [Crossref] [PubMed]
  6. L. Ahrenberg, P. Benzie, M. Magnor, and J. Watson, “Computer generated holograms from three dimensional meshes using an analytic light transport model,” Appl. Opt. 47(10), 1567–1574 (2008).
    [Crossref] [PubMed]
  7. Y. Pan, Y. Wang, J. Liu, X. Li, and J. Jia, “Fast polygon-based method for calculating computer-generated holograms in three-dimensional display,” Appl. Opt. 52(1), A290–A299 (2013).
    [Crossref] [PubMed]
  8. D. Leseberg and C. Frère, “Computer-generated holograms of 3-D objects composed of tilted planar segments,” Appl. Opt. 27(14), 3020–3024 (1988).
    [Crossref] [PubMed]
  9. K. Matsushima, H. Schimmel, and F. Wyrowski, “Fast calculation method for optical diffraction on tilted planes by use of the angular spectrum of plane waves,” J. Opt. Soc. Am. A 20(9), 1755–1762 (2003).
    [Crossref] [PubMed]
  10. S. De Nicola, A. Finizio, G. Pierattini, P. Ferraro, and D. Alfieri, “Angular spectrum method with correction of anamorphism for numerical reconstruction of digital holograms on tilted planes,” Opt. Express 13(24), 9935–9940 (2005).
    [Crossref] [PubMed]
  11. A. Dutt and V. Rokhlin, “Fast Fourier transforms for nonequispaced data,” SIAM J. Sci. Comput. (USA) 14(6), 1368–1393 (1993).
    [Crossref]
  12. Q. H. Liu and N. Nguyen, “An accurate algorithm for nonuniform fast Fourier transforms (NUFFTs),” IEEE Microw. Guid. Wave Lett. 8(1), 18–20 (1998).
    [Crossref]
  13. L. Greengard and J. Y. Lee, “Accelerating the nonuniform fast Fourier transform,” SIAM Rev. 46(3), 443–454 (2004).
    [Crossref]
  14. J. W. Goodman, Introduction to Fourier Optics (Roberts & Company Publishers, 2004).

2013 (1)

2010 (1)

2009 (1)

2008 (1)

2005 (3)

2004 (1)

L. Greengard and J. Y. Lee, “Accelerating the nonuniform fast Fourier transform,” SIAM Rev. 46(3), 443–454 (2004).
[Crossref]

2003 (1)

1998 (1)

Q. H. Liu and N. Nguyen, “An accurate algorithm for nonuniform fast Fourier transforms (NUFFTs),” IEEE Microw. Guid. Wave Lett. 8(1), 18–20 (1998).
[Crossref]

1993 (2)

M. Lucente, “Interactive computation of holograms using a look-up table,” J. Electron. Imaging 2(1), 28–34 (1993).
[Crossref]

A. Dutt and V. Rokhlin, “Fast Fourier transforms for nonequispaced data,” SIAM J. Sci. Comput. (USA) 14(6), 1368–1393 (1993).
[Crossref]

1988 (1)

Ahrenberg, L.

Alfieri, D.

Benzie, P.

Cameron, C.

C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38(8), 46–53 (2005).
[Crossref]

Chen, B. C.

De Nicola, S.

Dong, J. W.

Dutt, A.

A. Dutt and V. Rokhlin, “Fast Fourier transforms for nonequispaced data,” SIAM J. Sci. Comput. (USA) 14(6), 1368–1393 (1993).
[Crossref]

Ferraro, P.

Finizio, A.

Frère, C.

Greengard, L.

L. Greengard and J. Y. Lee, “Accelerating the nonuniform fast Fourier transform,” SIAM Rev. 46(3), 443–454 (2004).
[Crossref]

He, H. X.

Jia, J.

Lee, J. Y.

L. Greengard and J. Y. Lee, “Accelerating the nonuniform fast Fourier transform,” SIAM Rev. 46(3), 443–454 (2004).
[Crossref]

Leseberg, D.

Li, X.

Liu, J.

Liu, Q. H.

Q. H. Liu and N. Nguyen, “An accurate algorithm for nonuniform fast Fourier transforms (NUFFTs),” IEEE Microw. Guid. Wave Lett. 8(1), 18–20 (1998).
[Crossref]

Liu, Y. Z.

Lucente, M.

M. Lucente, “Interactive computation of holograms using a look-up table,” J. Electron. Imaging 2(1), 28–34 (1993).
[Crossref]

Magnor, M.

Matsushima, K.

Nakahara, S.

Nguyen, N.

Q. H. Liu and N. Nguyen, “An accurate algorithm for nonuniform fast Fourier transforms (NUFFTs),” IEEE Microw. Guid. Wave Lett. 8(1), 18–20 (1998).
[Crossref]

Pan, Y.

Pierattini, G.

Pu, Y. Y.

Rokhlin, V.

A. Dutt and V. Rokhlin, “Fast Fourier transforms for nonequispaced data,” SIAM J. Sci. Comput. (USA) 14(6), 1368–1393 (1993).
[Crossref]

Schimmel, H.

Slinger, C.

C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38(8), 46–53 (2005).
[Crossref]

Stanley, M.

C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38(8), 46–53 (2005).
[Crossref]

Wang, H. Z.

Wang, Y.

Watson, J.

Wyrowski, F.

Appl. Opt. (5)

Computer (1)

C. Slinger, C. Cameron, and M. Stanley, “Computer-generated holography as a generic display technology,” Computer 38(8), 46–53 (2005).
[Crossref]

IEEE Microw. Guid. Wave Lett. (1)

Q. H. Liu and N. Nguyen, “An accurate algorithm for nonuniform fast Fourier transforms (NUFFTs),” IEEE Microw. Guid. Wave Lett. 8(1), 18–20 (1998).
[Crossref]

J. Electron. Imaging (1)

M. Lucente, “Interactive computation of holograms using a look-up table,” J. Electron. Imaging 2(1), 28–34 (1993).
[Crossref]

J. Opt. Soc. Am. A (1)

Opt. Express (2)

SIAM J. Sci. Comput. (USA) (1)

A. Dutt and V. Rokhlin, “Fast Fourier transforms for nonequispaced data,” SIAM J. Sci. Comput. (USA) 14(6), 1368–1393 (1993).
[Crossref]

SIAM Rev. (1)

L. Greengard and J. Y. Lee, “Accelerating the nonuniform fast Fourier transform,” SIAM Rev. 46(3), 443–454 (2004).
[Crossref]

Other (1)

J. W. Goodman, Introduction to Fourier Optics (Roberts & Company Publishers, 2004).

Supplementary Material (1)

» Media 1: MOV (2264 KB)     

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Figures (7)

Fig. 1
Fig. 1 Schematic diagram of diffraction calculation of tilted plane. (a) Optical locations of tilted plane, reference plane and hologram plane. (b) Conventional calculation between tilted plane and reference plane. (c) NUFFT based method of calculation between tilted plane and reference plane.
Fig. 2
Fig. 2 Schematic diagrams of scaled diffraction calculation between tilted planes. (a) Conventional calculation process based on angular spectrum and interpolation between tilted planes with the constraint sampling rates. (b) Scaled diffraction calculation based on NUFFT from tilted plane with variable sampling rate when R<1. (c) Scaled diffraction calculation based on NUFFT from tilted plane with variable sampling rate when R>1.
Fig. 3
Fig. 3 Schematic diagram of coordinate grids on each domain.
Fig. 4
Fig. 4 Comparison of calculation time.
Fig. 5
Fig. 5 Simulation results of the reconstruction on tilted plane from CGH.
Fig. 6
Fig. 6 Reconstructions of the polygon-based object. (a) The object (b) R = 0.6. (c) R = 0.8. (d) R = 1. (e) R = 1.6. (e) and (f) are reconstructions at front and back plane respectively.
Fig. 7
Fig. 7 Optical reconstructions of polygon-based object. (a) Optical setup for holographic display. (b)-(e) Reconstructions of polygon-based object with different scale factors. (f) Reconstruction of object on the back focus plane (Media 1).

Equations (10)

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[ u t v t w t ( u t , v t ) ]=T[ u r v r w r ( u r , v r ) ]=[ a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 ][ u r v r w r ( u r , v r ) ]
u t =α( u r , v r )= a 1 u r + a 2 v r + a 3 w r ( u r , v r ) v t =β( u r , v r )= a 4 u r + a 5 v r + a 6 w r ( u r , v r )
G t ( u t , v t )= G t [ m u Δ( u t ), m v Δ( v t ) ] =NUFFT2[ f t ( m x Δx, m y Δy) ] =NUFFT2[ f t ( x t , y t ) ]
G r ( u r , v r )= G r ( α 1 ( u t , v t ), β 1 ( u t , v t ))= G t ( u t , v t )
G h ( u h , v h )= G r ( u r , v r )exp( 2πiz 1 λ 2 u r 2 v r 2 )
f h ( x h )=NUFFT1{ G r ( u r )exp( 2πiz 1 λ 2 u r 2 ) } =NUFFT1{ G t ( u t )exp( 2πiz 1 λ 2 u r 2 ) } =NUFFT1{ NUFFT2[ f t ( x t ) ]exp( 2πiz 1 λ 2 u r 2 ) }
F(n)=NUFFT1[ f( x m ) ]= m f( x m )exp( in x m )
F( x n )=NUFFT2[ f(m) ]= m f(m)exp( i x n m )
G t ( u n )= f t ( x m )exp( i2πmRΔT u n ) = f t ( x t )exp( 2iπRm u n L t )
f h (m)= G h ( u h )exp( i2πmΔHnΔ u h ' ) = G h ( u h )exp( i2πmΔHn L h ' N ) = G h ( u h )exp( i2πmΔHn L h RN ) = G h ( u h )exp( i2πmn RN )

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