Abstract

Fresnel diffraction calculation on an arbitrary shape surface is proposed. This method is capable of calculating Fresnel diffraction from a source surface with an arbitrary shape to a planar destination surface. Although such calculation can be readily calculated by the direct integral of a diffraction calculation, the calculation cost is proportional to O(N2) in one dimensional or O(N4) in two dimensional cases, where N is the number of sampling points. However, the calculation cost of the proposed method is O(N log N) in one dimensional or O(N2 log N) in two dimensional cases using non-uniform fast Fourier transform.

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References

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    [CrossRef]
  17. Q. H. Liu, N. Nguyen, and X. Y. Tang, “Accurate algorithms for nonuniform fast forward and inverse Fourier transforms and their applications,” IEEE Trans. Geosci. Remote Sens. 1, 288–290 (1998).
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    [CrossRef]

2009 (1)

2008 (1)

2006 (1)

G. B. Esmer and L. Onural, “Computation of holographic patterns between tilted planes,” Proc. SPIE 6252, 62521K (2006).
[CrossRef]

2004 (1)

L. Greengard and J. Y. Lee, “Accelerating the Nonuniform Fast Fourier Transform,” SIAM Rev. 46, 443–454 (2004).
[CrossRef]

2003 (1)

2002 (1)

1998 (3)

N. Delen and B. Hooker, “Free-space beam propagation between arbitrarily oriented planes based on full diffraction theory: a fast Fourier transform approach,” J. Opt. Soc. Am. A 15, 857–867 (1998).
[CrossRef]

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

Q. H. Liu, N. Nguyen, and X. Y. Tang, “Accurate algorithms for nonuniform fast forward and inverse Fourier transforms and their applications,” IEEE Trans. Geosci. Remote Sens. 1, 288–290 (1998).

1993 (1)

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

1992 (1)

1989 (1)

1988 (1)

Ahrenberg, L.

An, Y.

Benzie, P.

Bianco, B.

Cai, L.

Delen, N.

Dutt, A.

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

Ersoy, Okan K.

Okan K. Ersoy, Diffraction, Fourier Optics And Imaging (Wiley-Interscience, 2006).

Esmer, G. B.

G. B. Esmer and L. Onural, “Computation of holographic patterns between tilted planes,” Proc. SPIE 6252, 62521K (2006).
[CrossRef]

Frere, C.

Frére, C.

Goodman, J. W.

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

Greengard, L.

L. Greengard and J. Y. Lee, “Accelerating the Nonuniform Fast Fourier Transform,” SIAM Rev. 46, 443–454 (2004).
[CrossRef]

Hooker, B.

Lee, J. Y.

L. Greengard and J. Y. Lee, “Accelerating the Nonuniform Fast Fourier Transform,” SIAM Rev. 46, 443–454 (2004).
[CrossRef]

Leseberg, D.

Liu, Q. H.

Q. H. Liu, N. Nguyen, and X. Y. Tang, “Accurate algorithms for nonuniform fast forward and inverse Fourier transforms and their applications,” IEEE Trans. Geosci. Remote Sens. 1, 288–290 (1998).

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

Magnor, M.

Matsushima, K.

Nguyen, N.

Q. H. Liu, N. Nguyen, and X. Y. Tang, “Accurate algorithms for nonuniform fast forward and inverse Fourier transforms and their applications,” IEEE Trans. Geosci. Remote Sens. 1, 288–290 (1998).

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

Onural, L.

G. B. Esmer and L. Onural, “Computation of holographic patterns between tilted planes,” Proc. SPIE 6252, 62521K (2006).
[CrossRef]

Paganin, D. M.

D. M. Paganin, Coherent X-Ray Optics (Oxford University Press, 2006).
[CrossRef]

Rokhlin, V.

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

Sakamoto, Y.

Sakata, H.

Schimmel, H.

Tang, X. Y.

Q. H. Liu, N. Nguyen, and X. Y. Tang, “Accurate algorithms for nonuniform fast forward and inverse Fourier transforms and their applications,” IEEE Trans. Geosci. Remote Sens. 1, 288–290 (1998).

Tommasi, T.

Watson, J.

Williams, E. G.

E. G. Williams, Fourier Acoustics – Sound Radiation and Nearfield Acoustical Holography (Academic Press, 1999).
[PubMed]

Wyrowski, F.

Yu, L.

Appl. Opt. (4)

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, 18–20 (1998).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

Q. H. Liu, N. Nguyen, and X. Y. Tang, “Accurate algorithms for nonuniform fast forward and inverse Fourier transforms and their applications,” IEEE Trans. Geosci. Remote Sens. 1, 288–290 (1998).

J. Opt. Soc. Am. A (2)

Opt. Express (1)

Opt. Lett. (1)

Proc. SPIE (1)

G. B. Esmer and L. Onural, “Computation of holographic patterns between tilted planes,” Proc. SPIE 6252, 62521K (2006).
[CrossRef]

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

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

SIAM Rev. (1)

L. Greengard and J. Y. Lee, “Accelerating the Nonuniform Fast Fourier Transform,” SIAM Rev. 46, 443–454 (2004).
[CrossRef]

Other (5)

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

Okan K. Ersoy, Diffraction, Fourier Optics And Imaging (Wiley-Interscience, 2006).

E. G. Williams, Fourier Acoustics – Sound Radiation and Nearfield Acoustical Holography (Academic Press, 1999).
[PubMed]

D. M. Paganin, Coherent X-Ray Optics (Oxford University Press, 2006).
[CrossRef]

T. C. Poon (ed.), Digital Holography and Three-Dimensional Display (Springer, 2006).
[CrossRef]

Supplementary Material (2)

» Media 1: AVI (740 KB)     
» Media 2: AVI (795 KB)     

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Figures (3)

Fig. 1
Fig. 1

Diffraction calculation between source surface with arbitrary and planar destination surface.

Fig. 2
Fig. 2

Intensity profiles of diffraction results from a source surface composed of four small planar surfaces with 128 points, which are tilted −30°, −50°, 0° and +50° to x1, respectively. (a) source surface (b) diffraction result by Eq.(2) (c) diffraction result by Eq.(5) (d) diffraction result by Eq.(8) (e) Absolute error between (b) and (d). ( Media 1 shows the diffraction result while changing z0=1.0 m to 2.4m)

Fig. 3
Fig. 3

Intensity profiles of diffraction results from quadratic curve surface. (a) source surface (b) diffraction result by Eq.(2) (c) diffraction result by Eq.(5) (d) diffraction result by Eq.(8) (e) absolute error between (b) and (d). ( Media 2 shows the diffraction result while changing z0=1.0 m to 2.4m)

Equations (9)

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u 2 ( x 2 ) = z 0 i λ u I ( x 1 ) u 1 ( x 1 ) exp ( i k r ) r 2 d x 1 ,
u ( x 2 ) = z 0 i λ u I ( x 1 , d 1 ) u 1 ( x 1 , d 1 ) exp ( i k r ) r 2 d x 1 ,
r = | x 2 x 1 | 2 + ( z 0 d 1 ) 2 .
r r 0 + x 1 2 2 r 0 x 1 x 2 r 0 + x 2 2 2 r 0 .
u ( x 2 ) = exp ( i k z 0 ) i λ z 0 u I ( x 1 , d 1 ) u 1 ( x 1 , d 1 ) exp ( i k ( d 1 + x 1 2 2 r 0 ) ) exp ( i k x 1 x 2 r 0 ) exp ( i k x 2 2 2 r 0 ) d x 1
exp ( i k x 2 2 2 r 0 ) exp ( i k x 2 2 2 z 0 ) .
u ( x 2 ) = exp ( i k ( z 0 + x 2 2 2 z 0 ) ) i λ z 0 u I ( x 1 , d 1 ) u 1 ( x 1 , d 1 ) exp ( i k ( d 1 + x 1 2 2 ( z 0 d 1 ) ) ) exp ( i k x 1 x 2 z 0 d 1 ) d x 1 .
u ( x 2 ) = exp ( i k ( z 0 + x 2 2 2 z 0 ) ) i λ z 0 NUF [ u I ( x 1 , d 1 ) u 1 ( x 1 , d 1 ) exp ( i k ( d 1 + x 1 2 2 ( z 0 d 1 ) ) ) ] ,
F ( x 2 ) = NUF [ f ( x 1 ) ] = f ( x 1 ) exp ( i π x 1 x 2 ) d x 1 .

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