Abstract

We propose a technique for calculating the diffraction of light in the Fresnel region from a plane that is the light source (source plane) to a plane at which the diffracted light is to be calculated (destination plane). When the wavefield of the source plane is described by a group of points on a grid, this technique can be used to calculate the wavefield of the group of points on a grid on the destination plane. The positions of both planes may be shifted, and the plane normal vectors of both planes may have different directions. Since a scaled Fourier transform is used for the calculation, it can be calculated faster than calculating the diffraction by a Fresnel transform at each point. This technique can be used to calculate and generate planar holograms from computer graphics data.

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References

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  1. 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]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
  7. R. P. Muffoletto, “Numerical techniques for fresnel diffraction in computational holography,” PhD thesis (Louisiana State University, 2006).
  8. R. P. Muffoletto, J. M. Tyler, and J. E. Tohline, “Shifted Fresnel diffraction for computational holography,” Opt. Express15(9), 5631–5640 (2007).
    [CrossRef] [PubMed]
  9. L. Yu, U. An, and L. Cai, “Numerical reconstruction of digital holograms with variable viewing angles,” Opt. Express10(22), 1250–1257 (2002).
    [PubMed]
  10. J. Miura and T. Shimobaba, “Shifted-fresnel diffraction between two tilted planes (in Japanese),” Watake Seminar in TohokuYS–6–52 (2008).
  11. L. Onural, “Exact solution for scalar diffraction between tilted and translated planes using impulse functions over a surface,” J. Opt. Soc. Am. A28(3), 290–295 (2011).
    [CrossRef]
  12. J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company Publishers, Englewood, CO, 2005), Chap. 4.
  13. D. H. Bailey and P. N. Swarztrauber, “The fractional fourier transform and applications,” SIAM Rev. 33(3), 389–404 (1991).
    [CrossRef]

2011

2010

2008

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]

J. Miura and T. Shimobaba, “Shifted-fresnel diffraction between two tilted planes (in Japanese),” Watake Seminar in TohokuYS–6–52 (2008).

2007

2005

2003

2002

1998

1993

1991

D. H. Bailey and P. N. Swarztrauber, “The fractional fourier transform and applications,” SIAM Rev. 33(3), 389–404 (1991).
[CrossRef]

Ahrenberg, L.

An, U.

Bailey, D. H.

D. H. Bailey and P. N. Swarztrauber, “The fractional fourier transform and applications,” SIAM Rev. 33(3), 389–404 (1991).
[CrossRef]

Benzie, P.

Bianco, B.

Cai, L.

Delen, N.

Finizio, A.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company Publishers, Englewood, CO, 2005), Chap. 4.

Hooker, B.

Magnor, M.

Matsushima, K.

Miura, J.

J. Miura and T. Shimobaba, “Shifted-fresnel diffraction between two tilted planes (in Japanese),” Watake Seminar in TohokuYS–6–52 (2008).

Muffoletto, R. P.

R. P. Muffoletto, J. M. Tyler, and J. E. Tohline, “Shifted Fresnel diffraction for computational holography,” Opt. Express15(9), 5631–5640 (2007).
[CrossRef] [PubMed]

R. P. Muffoletto, “Numerical techniques for fresnel diffraction in computational holography,” PhD thesis (Louisiana State University, 2006).

Nicola, S. D.

Onural, L.

Pierattini, G.

Schimmel, H.

Shimobaba, T.

J. Miura and T. Shimobaba, “Shifted-fresnel diffraction between two tilted planes (in Japanese),” Watake Seminar in TohokuYS–6–52 (2008).

Swarztrauber, P. N.

D. H. Bailey and P. N. Swarztrauber, “The fractional fourier transform and applications,” SIAM Rev. 33(3), 389–404 (1991).
[CrossRef]

Tohline, J. E.

Tommasi, T.

Tyler, J. M.

Watson, J.

Wyrowski, F.

Yu, L.

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Express

SIAM Rev

D. H. Bailey and P. N. Swarztrauber, “The fractional fourier transform and applications,” SIAM Rev. 33(3), 389–404 (1991).
[CrossRef]

Watake Seminar in Tohoku

J. Miura and T. Shimobaba, “Shifted-fresnel diffraction between two tilted planes (in Japanese),” Watake Seminar in TohokuYS–6–52 (2008).

Other

J. W. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts and Company Publishers, Englewood, CO, 2005), Chap. 4.

R. P. Muffoletto, “Numerical techniques for fresnel diffraction in computational holography,” PhD thesis (Louisiana State University, 2006).

Supplementary Material (1)

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Figures (8)

Fig. 1
Fig. 1

Setup of source plane and destination plane

Fig. 2
Fig. 2

Optical system used in numerical experiments

Fig. 3
Fig. 3

Intensity distribution of surface plane

Fig. 4
Fig. 4

Experiment 1: experimental results for rotation. (a) Movement of surface plane. (b) Experimental results.

Fig. 5
Fig. 5

Experiment 2: experimental results for translation with no rotation ( Media 1). (a) Position of surface plane and movement of image plane. (b) Experimental results.

Fig. 6
Fig. 6

Experiment 2: experimental results for translation with y-axis positive rotation ( Media 1). (a) Position of surface plane and movement of image plane. (b) Experimental results.

Fig. 7
Fig. 7

Experiment 2: experimental results of translation with x-axis negative rotation and y-axis positive rotation ( Media 1). (a) Position of surface plane and movement of image plane. (b) Experimental results.

Fig. 8
Fig. 8

Computational complexity. (a) Index at Fresnel transform. (b) Index at proposed technique.

Tables (2)

Tables Icon

Table 1 Numerical experiment setup

Tables Icon

Table 2 Calculation time tN

Equations (18)

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[ x s t y s t z s t ] = P 0 + s Δ s + t Δ t = [ x 0 y 0 z 0 ] + s [ Δ s x Δ s y Δ s z ] + t [ Δ t x Δ t y Δ t z ]
[ x u v y u v z u v ] = P 1 + u Δ u + v Δ v = [ x 1 y 1 z 1 ] + u [ Δ u 0 0 ] + v [ 0 Δ v 0 ]
U 1 ( P u v ) = U 0 ( P s t ) j λ | z u v z s t | exp { j k ( x u v x s t ) 2 + ( y u v y s t ) 2 + ( z u v z s t ) 2 } d s d t
( z u v z s t ) 3 π 4 λ [ ( x u v x s t ) 2 + ( y u v y s t ) 2 ] 2
z 01 = z 1 z 0
Δ z = s Δ s z t Δ t z
x 01 = x 1 x 0
x s t = ( s Δ s x ) 2 + ( t Δ t x ) 2 2 x 01 s Δ s x 2 x 01 t Δ t x + 2 Δ s x Δ t x s t
x u v = x 01 2 + ( u Δ u ) 2 + 2 x 01 u Δ u
y 01 = y 1 y 0
y s t = ( s Δ s y ) 2 + ( t Δ t y ) 2 2 y 01 s Δ s y 2 y 01 t Δ t y + 2 Δ s y Δ t y s t
y u v = y 01 2 + ( v Δ v ) 2 + 2 y 01 v Δ v
( Δ z ) 2 z 01 2
( Δ z ) 2 z 01
U 1 ( P u v ) = 1 j λ exp { j k x u v + y u v 2 z 01 } U 0 ( P s t ) | z 01 Δ z | z 01 2 exp { j k ( z 01 + Δ z ) } exp { j k x s t + y s t 2 z 01 } exp { j k ( Δ u Δ s x u + Δ v Δ s y v ) s + ( Δ u Δ t x u + Δ v Δ t y v ) t z 01 } d s d t
F ( u ) = f ( s ) exp { j 2 π a u s } d s
a N = t N / N 4
b N = t N / N 2 log 2 N

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