Abstract

We have derived the basic spectral relation between a 3-D object and its 2-D diffracted wavefront by interpreting the diffraction calculation in the 3-D Fourier domain. Information on the 3-D object, which is inherent in the diffracted wavefront, becomes clear by using this relation. After the derivation, a method for obtaining the Fourier spectrum that is required to synthesize a hologram with a realistic sampling number for visible light is described. Finally, to verify the validity and the practicality of the above-mentioned spectral relation, fast calculation of a series of wavefronts radially diffracted from a 3-D voxel-based object is demonstrated.

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References

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2010

2008

2003

1997

1993

M. Lucente, “Interactive computation of hologram using a look-up table,” J. Electron. Imaging2, 28–34 (1993).
[CrossRef]

T. Tommasi and B. Bianco, “Computer-generated holograms of tilted planes by a spatial frequency approach,” J. Opt. Soc. Am. A10, 299–305 (1993).
[CrossRef]

1988

1970

1967

1948

D. Gabor, “A new microscopic principle,” Nature161, 777–778 (1948).
[CrossRef] [PubMed]

Beauchamp, H. L.

Bianco, B.

Chen, J.

Dalsgaard, E.

Frère, C.

Fujii, T.

Gabor, D.

D. Gabor, “A new microscopic principle,” Nature161, 777–778 (1948).
[CrossRef] [PubMed]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Ichihashi, Y.

Ito, T.

Kak, A. C.

A. C. Kak and M. Slaney, “Tomographic Imaging with Diffracting Sources,” in Principles of Computerized Tomographic Imaging, R. F. Cotellessa, J. K. Aggarwal, and G. Wade, eds. (Institute of Electrical and Electronics Engineers, 1988), pp. 203–273.

Lai, H.

Leseberg, D.

Lin, L H.

Liu, S.

Lohmann, A. W.

Lucente, M.

M. Lucente, “Interactive computation of hologram using a look-up table,” J. Electron. Imaging2, 28–34 (1993).
[CrossRef]

Masuda, N.

Matsushima, K.

Paris, D. P.

Schimmel, H.

Shimobaba, T.

Slaney, M.

A. C. Kak and M. Slaney, “Tomographic Imaging with Diffracting Sources,” in Principles of Computerized Tomographic Imaging, R. F. Cotellessa, J. K. Aggarwal, and G. Wade, eds. (Institute of Electrical and Electronics Engineers, 1988), pp. 203–273.

Takada, N.

Tommasi, T.

Wyrowski, F.

Yamaguchi, T.

Yoshikawa, H.

Zhang, X.

Appl. Opt.

J. Electron. Imaging

M. Lucente, “Interactive computation of hologram using a look-up table,” J. Electron. Imaging2, 28–34 (1993).
[CrossRef]

J. Opt. Soc. Am. A

Nature

D. Gabor, “A new microscopic principle,” Nature161, 777–778 (1948).
[CrossRef] [PubMed]

Opt. Express

Other

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

A. C. Kak and M. Slaney, “Tomographic Imaging with Diffracting Sources,” in Principles of Computerized Tomographic Imaging, R. F. Cotellessa, J. K. Aggarwal, and G. Wade, eds. (Institute of Electrical and Electronics Engineers, 1988), pp. 203–273.

Supplementary Material (2)

» Media 1: MPG (3730 KB)     
» Media 2: MPG (3730 KB)     

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Figures (7)

Fig. 1
Fig. 1

Schematic of the virtual optical system.

Fig. 2
Fig. 2

The path of the line integral (blue solid line).

Fig. 3
Fig. 3

Spectrum on the hemispherical surface corresponding to diffracted wavefronts in different directions.

Fig. 4
Fig. 4

Schematic of the virtual optical system for the observation of radially diffracted wavefronts: (a) the perspective view and (b) top view.

Fig. 5
Fig. 5

World map that is spherically mapped onto the object surface.

Fig. 6
Fig. 6

Diffraction patterns in several directions: (a) θ = 46° and ϕ = 135°, (b) θ = 55° and ϕ = 0°, and (c) θ = −55° and ϕ = 180°.

Fig. 7
Fig. 7

Images of movies showing 360 diffraction patterns; the patterns are obtained by varying the (a) diffraction angle θ ( Media 1) and (b) diffraction angle ϕ ( Media 2).

Equations (9)

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f ( x , y , z ) = o ( x 0 , y 0 , z 0 ) g ( x x 0 , y y 0 , z z 0 ) d x 0 d y 0 d z 0
= o ( x , y , z ) * g ( x 0 , y 0 , z 0 ) ,
g ( x , y , z ) = exp ( i 2 π λ x 2 + y 2 + z 2 ) x 2 + y 2 + z 2 .
G ( u , v , w ) = F [ g ( x , y , z ) ] = 1 4 π 2 ( u 2 + v 2 + w 2 1 / λ 2 ) ,
f ( x , y , z ) = 1 4 π 2 O ( u , v , w ) u 2 + v 2 + w 2 1 / λ 2 exp { i 2 π ( u x + u y + w z ) } d u d v d w ,
f ( x , y ) | z = R = 1 4 π 2 O ( u , v , w ) u 2 + v 2 + w 2 1 / λ 2 exp { i 2 π ( u x + u y + w R ) } d u d v d w .
f ( x , y ) | z = R = i 4 π O ( u , v 1 / λ 2 u 2 v 2 ) 1 / λ 2 u 2 v 2 exp ( i 2 π R 1 / λ 2 u 2 v 2 ) × exp { i 2 π ( u x + u y ) } d u d v .
F ( u , v ) | z = R = i 4 π O ( u , v 1 / λ 2 u 2 v 2 ) 1 / λ 2 u 2 v 2 exp ( i 2 π R 1 / λ 2 u 2 v 2 )
O ( u , v , w ) = F [ o ( x , y , z ) comb ( x δ , y δ , z δ ) ] = 1 W O ( u , v , w ) * comb ( u W , v W , w W ) ,

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