Abstract

Diffraction calculations, such as Fresnel diffraction and the angular spectrum method, are essential in optical simulations. These methods are numerically calculated by discrete Fourier transform or fast Fourier transform (FFT). Although these diffraction calculations require FFT-based convolution, FFT-based convolution becomes circular convolution without using zero padding. Zero padding helps to avoid circular convolution, but increases calculation time and memory usage. This paper proposes efficient diffraction calculations using implicit convolution. The proposed diffraction scheme accelerates calculation time, reduces memory usage, and can be applied to any convolution-based diffraction.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (Roberts and Company Publishers, 2005).
  2. T. Shimobaba and T. Ito, Computer Holography—Acceleration Algorithms and Hardware Implementations (CRC Press, in press).
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  12. Y. Xiao, X. Tang, Y. Qin, H. Peng, W. Wang, and L. Zhong, “Nonuniform fast Fourier transform method for numerical diffraction simulation on tilted planes,” J. Opt. Soc. Am. A 33, 2027–2033 (2016).
    [Crossref]
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    [Crossref]
  14. H. V. Sorensen and C. S. Burrus, “Efficient computation of the DFT with only a subset of input or output points,” IEEE Trans. Signal Process. 41, 1184–1200 (1993).
    [Crossref]
  15. T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
    [Crossref]

2016 (1)

2015 (1)

Y.-H. Kim, C.-W. Byun, H. Oh, J.-E. Pi, J.-H. Choi, G. H. Kim, M.-L. Lee, H. Ryu, and C.-S. Hwang, “Off-axis angular spectrum method with variable sampling interval,” Opt. Commun. 348, 31–37 (2015).
[Crossref]

2014 (1)

2013 (2)

T. Shimobaba, T. Kakue, M. Oikawa, N. Okada, Y. Endo, R. Hirayama, and T. Ito, “Nonuniform sampled scalar diffraction calculation using nonuniform fast Fourier transform,” Opt. Lett. 38, 5130–5133 (2013).
[Crossref] [PubMed]

T. Shimobaba, T. Kakue, N. Okada, M. Oikawa, Y. Yamaguchi, and T. Ito, “Aliasing-reduced Fresnel diffraction with scale and shift operations,” J. Opt.  15, 075405 (2013).
[Crossref]

2012 (2)

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

T. Shimobaba, K. Matsushima, T. Kakue, N. Masuda, and T. Ito, “Scaled angular spectrum method,” Opt. Lett. 37, 4128–4130 (2012).
[Crossref] [PubMed]

2011 (2)

2010 (1)

2007 (1)

2003 (1)

1993 (1)

H. V. Sorensen and C. S. Burrus, “Efficient computation of the DFT with only a subset of input or output points,” IEEE Trans. Signal Process. 41, 1184–1200 (1993).
[Crossref]

Bowman, J. C.

J. C. Bowman and M. Roberts, “Efficient dealiased convolutions without padding,” SIAM J. Sci. Comput. 33, 386–406 (2011).
[Crossref]

Burrus, C. S.

H. V. Sorensen and C. S. Burrus, “Efficient computation of the DFT with only a subset of input or output points,” IEEE Trans. Signal Process. 41, 1184–1200 (1993).
[Crossref]

Byun, C.-W.

Y.-H. Kim, C.-W. Byun, H. Oh, J.-E. Pi, J.-H. Choi, G. H. Kim, M.-L. Lee, H. Ryu, and C.-S. Hwang, “Off-axis angular spectrum method with variable sampling interval,” Opt. Commun. 348, 31–37 (2015).
[Crossref]

Choi, J.-H.

Y.-H. Kim, C.-W. Byun, H. Oh, J.-E. Pi, J.-H. Choi, G. H. Kim, M.-L. Lee, H. Ryu, and C.-S. Hwang, “Off-axis angular spectrum method with variable sampling interval,” Opt. Commun. 348, 31–37 (2015).
[Crossref]

Endo, Y.

Garcia-Sucerquia, J.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (Roberts and Company Publishers, 2005).

Hillenbrand, M.

Hirayama, R.

Hwang, C.-S.

Y.-H. Kim, C.-W. Byun, H. Oh, J.-E. Pi, J.-H. Choi, G. H. Kim, M.-L. Lee, H. Ryu, and C.-S. Hwang, “Off-axis angular spectrum method with variable sampling interval,” Opt. Commun. 348, 31–37 (2015).
[Crossref]

Ito, T.

T. Shimobaba, T. Kakue, N. Okada, M. Oikawa, Y. Yamaguchi, and T. Ito, “Aliasing-reduced Fresnel diffraction with scale and shift operations,” J. Opt.  15, 075405 (2013).
[Crossref]

T. Shimobaba, T. Kakue, M. Oikawa, N. Okada, Y. Endo, R. Hirayama, and T. Ito, “Nonuniform sampled scalar diffraction calculation using nonuniform fast Fourier transform,” Opt. Lett. 38, 5130–5133 (2013).
[Crossref] [PubMed]

T. Shimobaba, K. Matsushima, T. Kakue, N. Masuda, and T. Ito, “Scaled angular spectrum method,” Opt. Lett. 37, 4128–4130 (2012).
[Crossref] [PubMed]

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

T. Shimobaba and T. Ito, Computer Holography—Acceleration Algorithms and Hardware Implementations (CRC Press, in press).

Kakue, T.

Kelly, D. P.

Kim, G. H.

Y.-H. Kim, C.-W. Byun, H. Oh, J.-E. Pi, J.-H. Choi, G. H. Kim, M.-L. Lee, H. Ryu, and C.-S. Hwang, “Off-axis angular spectrum method with variable sampling interval,” Opt. Commun. 348, 31–37 (2015).
[Crossref]

Kim, Y.-H.

Y.-H. Kim, C.-W. Byun, H. Oh, J.-E. Pi, J.-H. Choi, G. H. Kim, M.-L. Lee, H. Ryu, and C.-S. Hwang, “Off-axis angular spectrum method with variable sampling interval,” Opt. Commun. 348, 31–37 (2015).
[Crossref]

Koike, C.

Koike, T.

Lee, M.-L.

Y.-H. Kim, C.-W. Byun, H. Oh, J.-E. Pi, J.-H. Choi, G. H. Kim, M.-L. Lee, H. Ryu, and C.-S. Hwang, “Off-axis angular spectrum method with variable sampling interval,” Opt. Commun. 348, 31–37 (2015).
[Crossref]

Masuda, N.

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

T. Shimobaba, K. Matsushima, T. Kakue, N. Masuda, and T. Ito, “Scaled angular spectrum method,” Opt. Lett. 37, 4128–4130 (2012).
[Crossref] [PubMed]

Matsushima, K.

Muffoletto, R. P.

Nishitsuji, T.

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

Odate, S.

Oh, H.

Y.-H. Kim, C.-W. Byun, H. Oh, J.-E. Pi, J.-H. Choi, G. H. Kim, M.-L. Lee, H. Ryu, and C.-S. Hwang, “Off-axis angular spectrum method with variable sampling interval,” Opt. Commun. 348, 31–37 (2015).
[Crossref]

Oikawa, M.

T. Shimobaba, T. Kakue, N. Okada, M. Oikawa, Y. Yamaguchi, and T. Ito, “Aliasing-reduced Fresnel diffraction with scale and shift operations,” J. Opt.  15, 075405 (2013).
[Crossref]

T. Shimobaba, T. Kakue, M. Oikawa, N. Okada, Y. Endo, R. Hirayama, and T. Ito, “Nonuniform sampled scalar diffraction calculation using nonuniform fast Fourier transform,” Opt. Lett. 38, 5130–5133 (2013).
[Crossref] [PubMed]

Okada, N.

T. Shimobaba, T. Kakue, M. Oikawa, N. Okada, Y. Endo, R. Hirayama, and T. Ito, “Nonuniform sampled scalar diffraction calculation using nonuniform fast Fourier transform,” Opt. Lett. 38, 5130–5133 (2013).
[Crossref] [PubMed]

T. Shimobaba, T. Kakue, N. Okada, M. Oikawa, Y. Yamaguchi, and T. Ito, “Aliasing-reduced Fresnel diffraction with scale and shift operations,” J. Opt.  15, 075405 (2013).
[Crossref]

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

Otaki, K.

Peng, H.

Pi, J.-E.

Y.-H. Kim, C.-W. Byun, H. Oh, J.-E. Pi, J.-H. Choi, G. H. Kim, M.-L. Lee, H. Ryu, and C.-S. Hwang, “Off-axis angular spectrum method with variable sampling interval,” Opt. Commun. 348, 31–37 (2015).
[Crossref]

Qin, Y.

Restrepo, J. F.

Roberts, M.

J. C. Bowman and M. Roberts, “Efficient dealiased convolutions without padding,” SIAM J. Sci. Comput. 33, 386–406 (2011).
[Crossref]

Ryu, H.

Y.-H. Kim, C.-W. Byun, H. Oh, J.-E. Pi, J.-H. Choi, G. H. Kim, M.-L. Lee, H. Ryu, and C.-S. Hwang, “Off-axis angular spectrum method with variable sampling interval,” Opt. Commun. 348, 31–37 (2015).
[Crossref]

Sakurai, T.

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

Schimmel, H.

Shimobaba, T.

T. Shimobaba, T. Kakue, N. Okada, M. Oikawa, Y. Yamaguchi, and T. Ito, “Aliasing-reduced Fresnel diffraction with scale and shift operations,” J. Opt.  15, 075405 (2013).
[Crossref]

T. Shimobaba, T. Kakue, M. Oikawa, N. Okada, Y. Endo, R. Hirayama, and T. Ito, “Nonuniform sampled scalar diffraction calculation using nonuniform fast Fourier transform,” Opt. Lett. 38, 5130–5133 (2013).
[Crossref] [PubMed]

T. Shimobaba, K. Matsushima, T. Kakue, N. Masuda, and T. Ito, “Scaled angular spectrum method,” Opt. Lett. 37, 4128–4130 (2012).
[Crossref] [PubMed]

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

T. Shimobaba and T. Ito, Computer Holography—Acceleration Algorithms and Hardware Implementations (CRC Press, in press).

Shiraki, A.

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

Sinzinger, S.

Sorensen, H. V.

H. V. Sorensen and C. S. Burrus, “Efficient computation of the DFT with only a subset of input or output points,” IEEE Trans. Signal Process. 41, 1184–1200 (1993).
[Crossref]

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, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

Tang, X.

Toba, H.

Tohline, J. E.

Tyler, J. M.

Uchikawa, K.

Wang, W.

Weng, J.

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

Wyrowski, F.

Xiao, Y.

Yamaguchi, Y.

T. Shimobaba, T. Kakue, N. Okada, M. Oikawa, Y. Yamaguchi, and T. Ito, “Aliasing-reduced Fresnel diffraction with scale and shift operations,” J. Opt.  15, 075405 (2013).
[Crossref]

Zhong, L.

Appl. Opt. (1)

Comput. Phys. Commun. (1)

T. Shimobaba, J. Weng, T. Sakurai, N. Okada, T. Nishitsuji, N. Takada, A. Shiraki, N. Masuda, and T. Ito, “Computational wave optics library for C++: CWO++ library,” Comput. Phys. Commun. 183, 1124–1138 (2012).
[Crossref]

IEEE Trans. Signal Process. (1)

H. V. Sorensen and C. S. Burrus, “Efficient computation of the DFT with only a subset of input or output points,” IEEE Trans. Signal Process. 41, 1184–1200 (1993).
[Crossref]

J. Opt (1)

T. Shimobaba, T. Kakue, N. Okada, M. Oikawa, Y. Yamaguchi, and T. Ito, “Aliasing-reduced Fresnel diffraction with scale and shift operations,” J. Opt.  15, 075405 (2013).
[Crossref]

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

Y.-H. Kim, C.-W. Byun, H. Oh, J.-E. Pi, J.-H. Choi, G. H. Kim, M.-L. Lee, H. Ryu, and C.-S. Hwang, “Off-axis angular spectrum method with variable sampling interval,” Opt. Commun. 348, 31–37 (2015).
[Crossref]

Opt. Express (2)

Opt. Lett. (2)

SIAM J. Sci. Comput. (1)

J. C. Bowman and M. Roberts, “Efficient dealiased convolutions without padding,” SIAM J. Sci. Comput. 33, 386–406 (2011).
[Crossref]

Other (2)

J. W. Goodman, Introduction to Fourier Optics (Roberts and Company Publishers, 2005).

T. Shimobaba and T. Ito, Computer Holography—Acceleration Algorithms and Hardware Implementations (CRC Press, in press).

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Figures (6)

Fig. 1
Fig. 1 Wraparound noise in diffracted result. The top and bottom figures show the diffracted results obtained with and without zero padding, respectively. The wavelength, sampling pitch and propagation distance are 633 nm, 10µm and 0.1m, respectively. The calculation window has 512×512 pixels. In the zero padding case, the calculation window size is expanded to 1,024×1,024 pixels during the calculation.
Fig. 2
Fig. 2 Scaled diffraction results obtained using a natural image with and without zero padding. The sampling pitches of the source and destination planes are 10µm and 8µm, respectively. The wavelength and propagation distance are 633 nm and 0.3m, respectively. The calculation window has 2,048×2,048 pixels. In the zero padding case, the calculation window size is expanded to 4,096×4,096 pixels during the calculation.
Fig. 3
Fig. 3 Diffracted results using angular spectrum method: (a) and (b) are intensity and phase using the explicit angular spectrum method, and (c) and (d) are intensity and phase using the implicit angular spectrum method.
Fig. 4
Fig. 4 Diffracted results obtained using scaled diffraction calculation: (a) and (b) are intensity and phase using explicit scaled diffraction, and (c) and (d) are those of implicit scaled diffraction. The sampling pitches on the source and destination planes are 10µm and 15µm, respectively.
Fig. 5
Fig. 5 Diffracted results using scaled diffraction: (a) and (b) are intensity and phase using explicit scaled diffraction, and (c) and (d) are intensity and phase of the implicit scaled diffraction. The sampling pitches on the source and destination planes are 10µm and 5µm, respectively.
Fig. 6
Fig. 6 Reconstructed images from a kinoform using the explicit and implicit angular spectrum methods: (a) is the kinoform, (b) and (c) are the reconstructed images using the explicit and implicit angular spectrum methods, respectively.

Tables (2)

Tables Icon

Table 1 Calculation Times of the Explicit and Implicit Scaled Diffraction

Tables Icon

Table 2 Calculation Times of the Explicit and Implicit Angular Spectrum Method

Equations (17)

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u 2 ( m , n ) = FFT 1 [ FFT [ u 1 ( m , n ) ] FFT [ h ( m , n ) ] ] ,
F ( k , ) = FFT [ f ( m , n ) ] = n = 0 N 1 m = 0 M 1 f ( m , n ) W N k m W M n ,
f ( m , n ) = FFT 1 [ F ( k , ) ] = 1 N M = 0 N 1 m = 0 M 1 f ( m , n ) W N k m W M n ,
U 1 ( k , ) = n = 0 2 N 1 m = 0 2 M 1 u 1 ( m , n ) W 2 N 2 k m W 2 M 2 n .
U 1 ( 2 k , 2 ) = n = 0 2 N 1 m = 0 2 M 1 u 1 ( m , n ) W 2 N 2 k m W 2 M 2 n ,
= n = 0 N 1 m = 0 M 1 u 1 ( m , n ) W N k m W M n ,
= FFT [ u 1 ( m , n ) ] .
U 1 ( 2 k + 1 , 2 ) = FFT [ u 1 ( m , n ) W 2 M m ] ,
U 1 ( 2 k , 2 + 1 ) = FFT [ u 1 ( m , n ) W 2 N n ] ,
U 1 ( 2 k + 1 , 2 + 1 ) = FFT [ u 1 ( m , n ) W 2 M m W 2 N n ] .
H ( 2 k , 2 ) = FFT [ h ( m , n ) ] ,
H ( 2 k + 1 , 2 ) = FFT [ h ( m , n ) W 2 M m ] ,
H ( 2 k , 2 + 1 ) = FFT [ h ( m , n ) W 2 N n ] ,
H ( 2 k + 1 , 2 + 1 ) = FFT [ h ( m , n ) W 2 M m W 2 N n ] .
G ( 2 k , 2 ) = U 1 ( 2 k , 2 ) H ( 2 k , 2 ) , G ( 2 k + 1 , 2 ) = U 1 ( 2 k + 1 , 2 ) H ( 2 k + 1 , 2 ) , G ( 2 k , 2 + 1 ) = U 1 ( 2 k , 2 + 1 ) H ( 2 k , 2 + 1 ) , G ( 2 k + 1 , 2 + 1 ) = U 1 ( 2 k + 1 , 2 + 1 ) H ( 2 k + 1 , 2 + 1 ) .
u 2 ( m 2 , n 2 ) = FFT 1 [ G ( k , ) ] = = 0 2 N 1 k = 0 2 M 1 G ( k , ) W 2 N 2 k m W 2 M 2 n = = 0 N 1 k = 0 M 1 G ( 2 k , 2 ) W 2 N 2 k m W M n + = 0 N 1 k = 0 M 1 G ( 2 k + 1 , 2 ) W N k m W M n W 2 N n + = 0 N 1 k = 0 M 1 G ( 2 k , 2 + 1 ) W N k m W M n + = 0 N 1 k = 0 M 1 G ( 2 k + 1 , 2 + 1 ) W N k m W M n W 2 M m W 2 N n = FFT 1 [ G ( 2 k , 2 ) ] + W 2 N n FFT 1 [ G ( 2 k , 2 + 1 ) ] + W 2 M m FFT 1 [ G ( 2 k + 1 , 2 ) ] + W 2 M m W 2 N n FFT 1 [ G ( 2 k + 1 , 2 + 1 ) ] .
h ( m 1 , n 1 ) = FFT 1 [ exp ( i 2 π λ 1 ( λ f x ) 2 λ f y ) 2 ) ] ,