Abstract

Scalar diffraction calculations, such as the angular spectrum method (ASM) and Fresnel diffraction, are widely used in the research fields of optics, x rays, electron beams, and ultrasonics. It is possible to accelerate the calculation using fast Fourier transform (FFT); unfortunately, acceleration of the calculation of nonuniform sampled planes is limited due to the property of the FFT that imposes uniform sampling. In addition, it gives rise to wasteful sampling data if we calculate a plane having locally low and high spatial frequencies. In this Letter, we developed nonuniform sampled ASM and Fresnel diffraction to improve the problem using the nonuniform FFT.

© 2013 Optical Society of America

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References

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  11. L. Greengard and J. Y. Lee, SIAM Rev. 46, 443 (2004).
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    [CrossRef]
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    [CrossRef]
  15. J. Weng, T. Shimobaba, N. Okada, H. Nakayama, M. Oikawa, N. Masuda, and T. Ito, Opt. Express 20, 4018 (2012).
    [CrossRef]
  16. X. Yu, T. Xiahui, Q. Y. Xiong, P. Hao, and W. Wei, Opt. Lett. 37, 4943 (2012).
    [CrossRef]
  17. M. Frigo and S. G. Johnson, in Proceedings of 1998 IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1998), pp. 1381–1384.
  18. T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, Comput. Phys. Commun. 183, 1124 (2012).
    [CrossRef]

2013

T. Shimobaba, T. Kakue, N. Okada, M. Oikawa, Y. Yamaguchi, and T. Ito, J. Opt. 15, 075405 (2013).
[CrossRef]

T. Shimobaba, M. Makowski, T. Kakue, M. Oikawa, N. Okada, Y. Endo, R. Hirayama, and T. Ito, Opt. Express 21, 25285–25290 (2013).

2012

2011

2010

K. Matsushima, Opt. Express 18, 18453 (2010).
[CrossRef]

J. F. Restrepo and J. G. Sucerquia, Appl. Opt. 49, 6430 (2010).
[CrossRef]

L. Bilevich and L. Yaroslavsky, Proc. SPIE 7724, 77240N (2010).
[CrossRef]

T. Shimobaba, N. Masuda, Y. Ichihashi, and T. Ito, J. Opt. 12, 065402 (2010).
[CrossRef]

2007

2006

L. Yaroslavsky, Proc. SPIE 6252, 625216 (2006).
[CrossRef]

2004

L. Greengard and J. Y. Lee, SIAM Rev. 46, 443 (2004).
[CrossRef]

Bilevich, L.

L. Bilevich and L. Yaroslavsky, Proc. SPIE 7724, 77240N (2010).
[CrossRef]

Endo, Y.

Frigo, M.

M. Frigo and S. G. Johnson, in Proceedings of 1998 IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1998), pp. 1381–1384.

Goodman, J. W.

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

Greengard, L.

L. Greengard and J. Y. Lee, SIAM Rev. 46, 443 (2004).
[CrossRef]

Hao, P.

Hirayama, R.

Ichihashi, Y.

T. Shimobaba, N. Masuda, Y. Ichihashi, and T. Ito, J. Opt. 12, 065402 (2010).
[CrossRef]

Ito, T.

T. Shimobaba, M. Makowski, T. Kakue, M. Oikawa, N. Okada, Y. Endo, R. Hirayama, and T. Ito, Opt. Express 21, 25285–25290 (2013).

T. Shimobaba, T. Kakue, N. Okada, M. Oikawa, Y. Yamaguchi, and T. Ito, J. Opt. 15, 075405 (2013).
[CrossRef]

T. Shimobaba, K. Matsushima, T. Kakue, N. Masuda, and T. Ito, Opt. Lett. 37, 4128 (2012).
[CrossRef]

T. Shimobaba, T. Kakue, N. Masuda, and T. Ito, J. Soc. Inf. Disp. 20, 533 (2012).
[CrossRef]

J. Weng, T. Shimobaba, N. Okada, H. Nakayama, M. Oikawa, N. Masuda, and T. Ito, Opt. Express 20, 4018 (2012).
[CrossRef]

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, Comput. Phys. Commun. 183, 1124 (2012).
[CrossRef]

T. Shimobaba, N. Masuda, Y. Ichihashi, and T. Ito, J. Opt. 12, 065402 (2010).
[CrossRef]

Johnson, S. G.

M. Frigo and S. G. Johnson, in Proceedings of 1998 IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1998), pp. 1381–1384.

Kakue, T.

T. Shimobaba, M. Makowski, T. Kakue, M. Oikawa, N. Okada, Y. Endo, R. Hirayama, and T. Ito, Opt. Express 21, 25285–25290 (2013).

T. Shimobaba, T. Kakue, N. Okada, M. Oikawa, Y. Yamaguchi, and T. Ito, J. Opt. 15, 075405 (2013).
[CrossRef]

T. Shimobaba, K. Matsushima, T. Kakue, N. Masuda, and T. Ito, Opt. Lett. 37, 4128 (2012).
[CrossRef]

T. Shimobaba, T. Kakue, N. Masuda, and T. Ito, J. Soc. Inf. Disp. 20, 533 (2012).
[CrossRef]

Koike, C.

Koike, T.

Lee, J. Y.

L. Greengard and J. Y. Lee, SIAM Rev. 46, 443 (2004).
[CrossRef]

Makowski, M.

Masuda, N.

T. Shimobaba, T. Kakue, N. Masuda, and T. Ito, J. Soc. Inf. Disp. 20, 533 (2012).
[CrossRef]

J. Weng, T. Shimobaba, N. Okada, H. Nakayama, M. Oikawa, N. Masuda, and T. Ito, Opt. Express 20, 4018 (2012).
[CrossRef]

T. Shimobaba, K. Matsushima, T. Kakue, N. Masuda, and T. Ito, Opt. Lett. 37, 4128 (2012).
[CrossRef]

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, Comput. Phys. Commun. 183, 1124 (2012).
[CrossRef]

T. Shimobaba, N. Masuda, Y. Ichihashi, and T. Ito, J. Opt. 12, 065402 (2010).
[CrossRef]

Matsushima, K.

Muffoletto, R. P.

Nakayama, H.

Nishitsuji, T.

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, Comput. Phys. Commun. 183, 1124 (2012).
[CrossRef]

Odate, S.

Oikawa, M.

Okada, N.

T. Shimobaba, M. Makowski, T. Kakue, M. Oikawa, N. Okada, Y. Endo, R. Hirayama, and T. Ito, Opt. Express 21, 25285–25290 (2013).

T. Shimobaba, T. Kakue, N. Okada, M. Oikawa, Y. Yamaguchi, and T. Ito, J. Opt. 15, 075405 (2013).
[CrossRef]

J. Weng, T. Shimobaba, N. Okada, H. Nakayama, M. Oikawa, N. Masuda, and T. Ito, Opt. Express 20, 4018 (2012).
[CrossRef]

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, Comput. Phys. Commun. 183, 1124 (2012).
[CrossRef]

Otaki, K.

Restrepo, J. F.

Sakurai, T.

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, Comput. Phys. Commun. 183, 1124 (2012).
[CrossRef]

Shimobaba, T.

T. Shimobaba, M. Makowski, T. Kakue, M. Oikawa, N. Okada, Y. Endo, R. Hirayama, and T. Ito, Opt. Express 21, 25285–25290 (2013).

T. Shimobaba, T. Kakue, N. Okada, M. Oikawa, Y. Yamaguchi, and T. Ito, J. Opt. 15, 075405 (2013).
[CrossRef]

T. Shimobaba, K. Matsushima, T. Kakue, N. Masuda, and T. Ito, Opt. Lett. 37, 4128 (2012).
[CrossRef]

T. Shimobaba, T. Kakue, N. Masuda, and T. Ito, J. Soc. Inf. Disp. 20, 533 (2012).
[CrossRef]

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, Comput. Phys. Commun. 183, 1124 (2012).
[CrossRef]

J. Weng, T. Shimobaba, N. Okada, H. Nakayama, M. Oikawa, N. Masuda, and T. Ito, Opt. Express 20, 4018 (2012).
[CrossRef]

T. Shimobaba, N. Masuda, Y. Ichihashi, and T. Ito, J. Opt. 12, 065402 (2010).
[CrossRef]

Shiraki, A.

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, Comput. Phys. Commun. 183, 1124 (2012).
[CrossRef]

Sucerquia, J. G.

Sugaya, A.

Sugisaki, K.

Takada, N.

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, Comput. Phys. Commun. 183, 1124 (2012).
[CrossRef]

Toba, H.

Tohline, J. E.

Tyler, J. M.

Uchikawa, K.

Wei, W.

Weng, J.

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, Comput. Phys. Commun. 183, 1124 (2012).
[CrossRef]

J. Weng, T. Shimobaba, N. Okada, H. Nakayama, M. Oikawa, N. Masuda, and T. Ito, Opt. Express 20, 4018 (2012).
[CrossRef]

Xiahui, T.

Xiong, Q. Y.

Yamaguchi, Y.

T. Shimobaba, T. Kakue, N. Okada, M. Oikawa, Y. Yamaguchi, and T. Ito, J. Opt. 15, 075405 (2013).
[CrossRef]

Yaroslavsky, L.

L. Bilevich and L. Yaroslavsky, Proc. SPIE 7724, 77240N (2010).
[CrossRef]

L. Yaroslavsky, Proc. SPIE 6252, 625216 (2006).
[CrossRef]

Yingxiong, Q.

Yu, X.

Appl. Opt.

Comput. Phys. Commun.

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, Comput. Phys. Commun. 183, 1124 (2012).
[CrossRef]

J. Opt.

T. Shimobaba, T. Kakue, N. Okada, M. Oikawa, Y. Yamaguchi, and T. Ito, J. Opt. 15, 075405 (2013).
[CrossRef]

T. Shimobaba, N. Masuda, Y. Ichihashi, and T. Ito, J. Opt. 12, 065402 (2010).
[CrossRef]

J. Opt. Soc. Am. A

J. Soc. Inf. Disp.

T. Shimobaba, T. Kakue, N. Masuda, and T. Ito, J. Soc. Inf. Disp. 20, 533 (2012).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. SPIE

L. Bilevich and L. Yaroslavsky, Proc. SPIE 7724, 77240N (2010).
[CrossRef]

L. Yaroslavsky, Proc. SPIE 6252, 625216 (2006).
[CrossRef]

SIAM Rev.

L. Greengard and J. Y. Lee, SIAM Rev. 46, 443 (2004).
[CrossRef]

Other

M. Frigo and S. G. Johnson, in Proceedings of 1998 IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1998), pp. 1381–1384.

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

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Figures (5)

Fig. 1.
Fig. 1.

Coordinate conversion for NUFFT. (a) Coordinate system of the original plane. (b) Coordinate system of the plane for NUFFT. Blue circles indicate nonuniform sampled points.

Fig. 2.
Fig. 2.

Source plane with 256×256 pixels. “Lenna” as the amplitude distribution of the source plane, and “Fresnel zone plate” as the phase distribution where the pixel value 0 corresponds to the phase value π/2 and the pixel value 255 corresponds to the phase value +π/2.

Fig. 3.
Fig. 3.

Diffracted results of the amplitude and phase distribution by RS diffraction and NU-ASM. (a) Amplitude by RS diffraction. (b) Phase by RS diffraction. (c) Amplitude by NU-ASM. (d) Phase by NU-ASM.

Fig. 4.
Fig. 4.

Errors of NU-ASM1 and NU-FRE1 to RS diffraction by the signal-to-noise ratio (SNR).

Fig. 5.
Fig. 5.

Diffracted results of the amplitude and phase distribution by RS diffraction and NU-ASM. (a) Amplitude by RS diffraction. (b) Phase by RS diffraction. (c) Amplitude by NU-ASM. (d) Phase by NU-ASM.

Equations (10)

Equations on this page are rendered with MathJax. Learn more.

u2(x2)=u1(x1)hz(x2x1)dx1=F1[F[u1(x1)]Hz(f)],
u2(x2)=F[u1(x1)]exp(2πiz1λ2|f|2)×exp(2πifx2)df=F1[F[u1(x1)]Hz(f)],
u2(m2)=FFT1[FFT[u1(m1)]Hz,Δf(mf)],
Hz,Δ(mf)=exp(2πi(omf+z1λ2|mfΔ|2))×Rect(mfcmwm,nfcnwn),
Hz,Δ(mf)=FFT[exp(ikz)exp(i(m1Δ+o)2λz)iλz].
u2(m2)=NUASM1[u1(x1)]=FFT1[NUFFT1[u1(x1)]Hz,Δ(mf)],
F(mf)=NUFFT1[f(x1)]=x1f(x1)exp(iπmfx1),
f(x2)=NUFFT2[F(mf)]=mfF(mf)exp(iπmfx2),
u2(x2)=NUASM2[u1(m1)],=NUFFT2[FFT[u1(m1)]Hz,Δ(mf)],
u2(x2)=12πu1(x1){zr(1ikr)exp(ikr)r2}dx1,

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