Abstract

Fourier-based approaches to calculate the Fresnel diffraction of light provide one of the most efficient algorithms for holographic computations because this permits the use of the fast Fourier transform (FFT). This research overcomes the limitations on sampling imposed by Fourier-based algorithms by the development of a fast shifted Fresnel transform. This fast shifted Fresnel transform is used to develop a tiling approach to hologram construction and reconstruction, which computes the Fresnel propagation of light between parallel planes having different resolutions.

© 2007 Optical Society of America

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References

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  1. J. Ludman, H. J. Caulfield, and J. Riccobono, Holography for the New Millennium (Springer-Verlag, 2002).
    [CrossRef]
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  5. O. Nishikawa, T. Okada, H. Yoshikawa, K. Sato, and T. Honda, "High-Speed Holographic-Stereogram Calculation Method Using 2D FFT," in Proc. SPIE Vol. 3010, Diffractive and Holographic Device Technologies and Applications IV, I. Cindrich and S. H. Lee, eds., pp. 49-57 (1997).
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  15. K. Matsushima and A. Kondoh, "Wave optical algorithm for creating digitally synthetic holograms of threedimensional surface objects," in Proc. SPIE Vol. 5005, Practical Holography XVII and Holographic Materials IX, T. H. Jeong and S. H. Stevenson, eds., pp. 190-197 (2003).
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2005

2004

2003

2000

1999

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, "Fast algorithms for free-space diffraction patterns calculation," Opt. Commun. 164,233-245 (1999).
[CrossRef]

1996

1991

D. H. Bailey and P. N. Swarztrauber, "The Fractional Fourier Transform and Applications," SIAM Review 33, 389-404 (1991).
[CrossRef]

1988

Abookasis, D.

Alfieri, D.

Bailey, D. H.

D. H. Bailey and P. N. Swarztrauber, "The Fractional Fourier Transform and Applications," SIAM Review 33, 389-404 (1991).
[CrossRef]

Bernardo, L. M.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, "Fast algorithms for free-space diffraction patterns calculation," Opt. Commun. 164,233-245 (1999).
[CrossRef]

De Nicola, S.

Dorsch, R. G.

Ferraro, P.

Ferreira, C.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, "Fast algorithms for free-space diffraction patterns calculation," Opt. Commun. 164,233-245 (1999).
[CrossRef]

Finizio, A.

Frere, C.

Garcia, J.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, "Fast algorithms for free-space diffraction patterns calculation," Opt. Commun. 164,233-245 (1999).
[CrossRef]

J. Garcia, D. Mas, and R. G. Dorsch, "Fractional-Fourier-transform calculation through the fast-Fouriertransform algorithm," Appl. Opt. 35,7013-7018 (1996).
[CrossRef] [PubMed]

Ito, T.

Leseberg, D.

Lucente, M.

M. Lucente, "Holographic bandwidth compression using spatial subsampling," Opt. Eng. 35,1529-1537 (1996).
[CrossRef]

Marinho, F.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, "Fast algorithms for free-space diffraction patterns calculation," Opt. Commun. 164,233-245 (1999).
[CrossRef]

Mas, D.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, "Fast algorithms for free-space diffraction patterns calculation," Opt. Commun. 164,233-245 (1999).
[CrossRef]

J. Garcia, D. Mas, and R. G. Dorsch, "Fractional-Fourier-transform calculation through the fast-Fouriertransform algorithm," Appl. Opt. 35,7013-7018 (1996).
[CrossRef] [PubMed]

Matsushima, K.

Pierattini, G.

Rosen, J.

Shimobaba, T.

Swarztrauber, P. N.

D. H. Bailey and P. N. Swarztrauber, "The Fractional Fourier Transform and Applications," SIAM Review 33, 389-404 (1991).
[CrossRef]

Takai, M.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Commun.

D. Mas, J. Garcia, C. Ferreira, L. M. Bernardo, and F. Marinho, "Fast algorithms for free-space diffraction patterns calculation," Opt. Commun. 164,233-245 (1999).
[CrossRef]

Opt. Eng.

M. Lucente, "Holographic bandwidth compression using spatial subsampling," Opt. Eng. 35,1529-1537 (1996).
[CrossRef]

Opt. Express

SIAM Review

D. H. Bailey and P. N. Swarztrauber, "The Fractional Fourier Transform and Applications," SIAM Review 33, 389-404 (1991).
[CrossRef]

Other

G. B. Esmer and L. Onural, "Simulation of scalar optical diffraction between arbitrarily oriented planes," in First International Symposium on Control, Communications and Signal Processing, pp. 225-228 (2004).

K. Matsushima, H. Schimmel, and F. Wyrowski, "New Creation Algorithm for Digitally Synthesized Holograms in Surface Model by Diffraction from Tilted Planes," in Proc. SPIE Vol. 4659, Practical Holography XVI and Holographic Materials VIII, S. A. Benton, S. H. Stevenson, and T. J. Trout, eds., pp. 53-60 (2002).

K. Matsushima and A. Kondoh, "Wave optical algorithm for creating digitally synthetic holograms of threedimensional surface objects," in Proc. SPIE Vol. 5005, Practical Holography XVII and Holographic Materials IX, T. H. Jeong and S. H. Stevenson, eds., pp. 190-197 (2003).

J. Ludman, H. J. Caulfield, and J. Riccobono, Holography for the New Millennium (Springer-Verlag, 2002).
[CrossRef]

M. L. Huebschman, B. Munjuluri, J. Hunt, and H. R. Garner, "Holographic video display using digital micromirrors," in Proc. SPIE Vol. 5742, Emerging Liquid Crystal Technologies, L.-C. Chien, ed., pp. 1-14 (2005).

O. Nishikawa, T. Okada, H. Yoshikawa, K. Sato, and T. Honda, "High-Speed Holographic-Stereogram Calculation Method Using 2D FFT," in Proc. SPIE Vol. 3010, Diffractive and Holographic Device Technologies and Applications IV, I. Cindrich and S. H. Lee, eds., pp. 49-57 (1997).

M. Lucente, "Diffraction-Specific Fringe Computation for Electro-Holography," Ph.D. thesis, Massachusetts Institute of Technology (1994).

L. Onural and H. Ozaktas, "Signal processing issues in diffraction and holographic 3DTV," in Proc. EURASIP 13th European Signal Processing Conference (2005).

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Figures (8)

Fig. 1.
Fig. 1.

Shifted Fresnel diffraction geometry.

Fig. 2.
Fig. 2.

Lower resolution source wavefront for holographic tiling.

Fig. 3.
Fig. 3.

Lower resolution target wavefront for holographic tiling.

Fig. 4.
Fig. 4.

Source images used for demonstration having various depths.

Fig. 5.
Fig. 5.

Hologram computed.

Fig. 6.
Fig. 6.

Reconstruction of image A.

Fig. 7.
Fig. 7.

Reconstruction of image B.

Fig. 8.
Fig. 8.

Reconstruction of image C.

Tables (1)

Tables Icon

Table 1. Properties of images for tiling demonstration (λ = 633 nm; Δx = Δy = 8 μm).

Equations (22)

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Ψ d ( x ) = ( iλd ) 1 exp ( i 2 πd λ ) exp ( iπx 2 λd ) ψ ( x ) exp ( iπx 2 λd ) exp ( i 2 x λd ) d x ,
U d ( m ) Ψ d ( m Δ x ) = ( iλd ) 1 exp ( ikd ) exp [ ( m Δ x ) 2 λd ] p { ψ ( p Δ x ) exp [ ( p Δ x ) 2 λd ] exp ( i 2 πp Δ x m Δ x λd ) } ,
Δ x Δ x λd = 1 N
U d ( m ) = ( iλd ) 1 exp ( ikd ) exp [ ( m Δ x ) 2 λd ] p ( { ψ ( p Δ x ) exp [ ( p Δ x ) 2 λd ] } exp ( i 2 πp m N ) ) ,
G ( m ) = p g ( p ) exp ( i 2 πpm N ) .
x m = x 0 + m Δ x x p = x 0 + p Δ x
y n = y 0 + n Δ y y q = y 0 + q Δ y ,
0 m M 1 0 p P 1
0 n N 1 0 q Q 1 .
U d m n Ψ d ( x 0 + m Δ x , y 0 + n Δ y )
u p q ψ ( x 0 + p Δ x , y 0 + q Δ y ' ) ,
U d m n = ( iλd ) 1 exp ( ikd ) exp [ ( x m 2 + y n 2 ) λd ] p = 0 P 1 q = 0 Q 1 { u p q exp [ ( x p 2 + y q 2 ) λd ] exp [ i 2 π ( x p x m + y q y n ) λd ] }
= ( iλd ) 1 exp ( ikd ) exp [ ( x m 2 + y n 2 ) λd ] exp [ i 2 π ( x 0 m Δ x + y 0 n Δ y ) λd ] × p = 0 P 1 q = 0 Q 1 ( u p q exp [ ( x p 2 + y q 2 ) λd ] exp [ i 2 π ( x p x 0 + y q y 0 ) λd ] × exp [ i 2 π ( Δ x Δ xpm + Δ y Δ yqn ) λd ] ) .
H m n = p q h p q exp [ i 2 π ( spm tqn ) ] ,
H ( k ) = j h ( j ) exp ( i 2 πsjk ) = j h ( j ) exp { iπs [ j 2 + k 2 ( k j ) 2 ] }
= exp ( iπsk 2 ) j h ( j ) exp ( iπsj 2 ) exp [ iπs ( k j ) 2 ]
= exp ( iπsk 2 ) j χ ( j ) ζ ( k j ) ,
χ ( j ) h ( j ) exp ( iπsj 2 ) ,
ζ ( k j ) exp [ iπs ( k j ) 2 ] .
O ( 3 ( 2 N ) 2 log 2 ( 2 N ) ) = O ( 12 N 2 ( log 2 2 + log 2 N ) ) = O ( 12 N 2 log 2 N ) = O ( N 2 log 2 N ) .
( 2 K ) 2 Q 2 log 2 Q = ( 2 K ) 2 ( N 2 K ) 2 log 2 ( N 2 K ) = N 2 ( log 2 N K ) .
( 2 K ) 2 ( Q 2 log 2 Q + Q 2 ) = ( 2 K ) 2 ( N 2 K ) 2 [ log 2 ( N 2 K + 1 ) ] = N 2 ( log 2 N K + 1 ) .

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