Abstract

The shifted angular spectrum method allows a reduction of the number of samples required for numerical off-axis propagation of scalar wave fields. In this work, a modification of the shifted angular spectrum method is presented. It allows a further reduction of the spatial sampling rate for certain wave fields. We calculate the benefit of this method for spherical waves. Additionally, a working implementation is presented showing the example of a spherical wave propagating through a circular aperture.

© 2014 Optical Society of America

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References

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  1. D. M. Paganin, Coherent X-Ray optics (Oxford University, 2006).
    [Crossref]
  2. K. Matsushima and T. Shimobaba, “Band-limited angular spectrum method for numerical simulation of free-space propagation in far and near fields,” Opt. Express 17(22), 19662–19673 (2009).
    [Crossref] [PubMed]
  3. K. Matsushima, “Shifted angular spectrum method for off-axis numerical propagation,” Opt. Express 18(17), 18453–18463 (2010).
    [Crossref] [PubMed]
  4. C. Guo, Y. Xie, and B. Sha, “Diffraction algorithm suitable for both near and far field with shifted destination window and oblique illumination,” Opt. Lett. 39(8), 2338–2341 (2014).
    [Crossref] [PubMed]
  5. X. Yu, T. xiong, P. Hao, and W. Wei, “Wide-window angular spectrum method for diffraction propagation in far and near field,” Opt. Lett. 37(23), 4943–4945 (2012).
    [Crossref] [PubMed]
  6. T. Kozacki, K. Falaggis, and M. Kujawinska, “Computation of diffracted fields for the case of high numerical aperture using the angular spectrum method,” Appl. Opt. 51(29), 7080–7088 (2012).
    [Crossref] [PubMed]
  7. L. Onural, “Sampling of the diffraction field,” Appl. Opt. 39(32), 5929–5935 (2000).
    [Crossref]
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    [Crossref]
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    [Crossref]
  10. R. P. Muffoletto, J. M. Tyler, and J. E. Tohline, “Shifted Fresnel diffraction for computational holography,” Opt. Express 15(9), 5631–5640 (2007).
    [Crossref] [PubMed]
  11. P. Lobaz, “Memory-efficient reference calculation of light propagation using the convolution method,” Opt. Express 21(3), 2795–2806 (2013).
    [Crossref] [PubMed]
  12. K. Falaggis, T. Kozacki, and M. Kujawinska, “Computation of highly off-axis diffracted fields using the band-limited angular spectrum method with suppressed Gibbs related artifacts,” Appl. Opt. 52(14), 3288–3297 (2013).
    [Crossref] [PubMed]
  13. S. Odate, C. Koike, H. Toba, T. Koike, A. Sugaya, K. Sugisaki, K. Otaki, and K. Uchikawa, “Angular spectrum calculations for arbitrary focal length with a scaled convolution,” Opt. Express 19(15), 14268–14276 (2011).
    [Crossref] [PubMed]
  14. P. Lobaz, “Reference calculation of light propagation between parallel planes of different sizes and sampling rates,” Opt. Express 19(1), 32–39 (2011).
    [Crossref] [PubMed]
  15. K. Yamamoto, Y. Ichihashi, T. Senoh, R. Oi, and T. Kurita, “Calculating the Fresnel diffraction of light from a shifted and tilted plane,” Opt. Express 20(12), 12949–12958 (2012).
    [Crossref] [PubMed]
  16. 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(23), 5130–5133 (2013).
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  18. A. Ritter, P. Bartl, F. Bayer, K. C. Gödel, W. Haas, T. Michel, G. Pelzer, J. Rieger, T. Weber, A. Zang, and G. Anton, “A simulation framework for coherent and incoherent X-ray imaging and its application in Talbot-Lau dark-field imaging,” Opt. Express 22(18), 23276–23289 (2014).

2014 (3)

2013 (3)

2012 (3)

2011 (2)

2010 (1)

2009 (1)

2007 (2)

2006 (1)

2000 (1)

Anton, G.

Bartl, P.

Bayer, F.

Endo, Y.

Falaggis, K.

Gödel, K. C.

Guo, C.

Haas, W.

Hao, P.

Hillenbrand, M.

Hirayama, R.

Ichihashi, Y.

Ito, T.

Kakue, T.

Kelly, D. P.

Koike, C.

Koike, T.

Kozacki, T.

Kujawinska, M.

Kurita, T.

Lobaz, P.

Matsushima, K.

Michel, T.

Muffoletto, R. P.

Odate, S.

Oi, R.

Oikawa, M.

Okada, N.

Onural, L.

Otaki, K.

Paganin, D. M.

D. M. Paganin, Coherent X-Ray optics (Oxford University, 2006).
[Crossref]

Pelzer, G.

Rieger, J.

Ritter, A.

Senoh, T.

Sha, B.

Shen, F.

Shimobaba, T.

Sinzinger, S.

Sugaya, A.

Sugisaki, K.

Toba, H.

Tohline, J. E.

Tyler, J. M.

Uchikawa, K.

Wang, A.

Weber, T.

Wei, W.

Xie, Y.

xiong, T.

Yamamoto, K.

Yu, X.

Zang, A.

Appl. Opt. (4)

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

Opt. Express (8)

R. P. Muffoletto, J. M. Tyler, and J. E. Tohline, “Shifted Fresnel diffraction for computational holography,” Opt. Express 15(9), 5631–5640 (2007).
[Crossref] [PubMed]

P. Lobaz, “Memory-efficient reference calculation of light propagation using the convolution method,” Opt. Express 21(3), 2795–2806 (2013).
[Crossref] [PubMed]

K. Matsushima and T. Shimobaba, “Band-limited angular spectrum method for numerical simulation of free-space propagation in far and near fields,” Opt. Express 17(22), 19662–19673 (2009).
[Crossref] [PubMed]

K. Matsushima, “Shifted angular spectrum method for off-axis numerical propagation,” Opt. Express 18(17), 18453–18463 (2010).
[Crossref] [PubMed]

A. Ritter, P. Bartl, F. Bayer, K. C. Gödel, W. Haas, T. Michel, G. Pelzer, J. Rieger, T. Weber, A. Zang, and G. Anton, “A simulation framework for coherent and incoherent X-ray imaging and its application in Talbot-Lau dark-field imaging,” Opt. Express 22(18), 23276–23289 (2014).

S. Odate, C. Koike, H. Toba, T. Koike, A. Sugaya, K. Sugisaki, K. Otaki, and K. Uchikawa, “Angular spectrum calculations for arbitrary focal length with a scaled convolution,” Opt. Express 19(15), 14268–14276 (2011).
[Crossref] [PubMed]

P. Lobaz, “Reference calculation of light propagation between parallel planes of different sizes and sampling rates,” Opt. Express 19(1), 32–39 (2011).
[Crossref] [PubMed]

K. Yamamoto, Y. Ichihashi, T. Senoh, R. Oi, and T. Kurita, “Calculating the Fresnel diffraction of light from a shifted and tilted plane,” Opt. Express 20(12), 12949–12958 (2012).
[Crossref] [PubMed]

Opt. Lett. (3)

Other (1)

D. M. Paganin, Coherent X-Ray optics (Oxford University, 2006).
[Crossref]

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Figures (4)

Fig. 1
Fig. 1 Ratio of reduced sampling rate r* of the modified angular spectrum method to the sampling rate r for the shifted angular spectrum method.
Fig. 2
Fig. 2 Numerical off-axis propagation of a spherical wave through a circular aperture. Each subfigure shows the intensity in the destination sampling window of the propagation. The sampling window in each subfigure is 120 μm × 120 μm with 968×2420 (Ny × Nx) samples (a), (c), (e) and 968×968 samples (b), (d), (f). Where Nx, Ny mean the number of samples in x and respectively y dimension. The propagation methods used are: (a), (b) angular spectrum; (c), (d) shifted angular spectrum; (e), (f) modified shifted angular spectrum. The subfigures (a) and (b) are centered around the origin of the xy-plane the remaining subfigures are centered around (−30 μm, 0 μm) due to the spatial shift applied by the shifted methods.
Fig. 3
Fig. 3 Numerical off-axis propagation of a spherical wave through a circular aperture. The distance between origin of the spherical wave and the aperture and the distance between aperture and destination plane is 10 mm. The origin of the spherical wave is shifted by 100 μm perpendicular to the optical axis relative to the center of the aperture. Each subfigure shows the intensity in the destination sampling window of the propagation. The sampling window in each subfigure is 120 μm × 120 μm. (a) shifted angular spectrum method (sasm) with 968×5711 (Ny × Nx) samples. (b) shifted angular spectrum method with 968×5227 samples. (c) modified shifted angular spectrum method (msasm) with 968×968 samples.
Fig. 4
Fig. 4 Numerical off-axis propagation of a spherical wave through a circular aperture. The distance between origin of the spherical wave and the aperture and the distance between aperture and destination plane is 10 mm. The origin of the spherical wave is shifted by 1000 μm perpendicular to the optical axis relative to the center of the aperture. Each subfigure shows the intensity in the destination sampling window of the propagation. The sampling window in each subfigure is 120 μm × 120 μm. (a) shifted angular spectrum method (sasm) with 968×49263 (Ny × Nx) samples. (b) shifted angular spectrum method with 968×48779 samples. (c) modified shifted angular spectrum method (msasm) with 968×968 samples.

Equations (26)

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φ out ( x , y ) = F 1 { F { φ in ( x , y ) } p k , x 0 , y 0 ( k x , k y ) }
p k , x 0 , y 0 ( k x , k y ) = exp ( i z k 2 k x 2 k y 2 + i k x x 0 + i k y y 0 )
φ ^ ( k x , k y ) = F { φ ( x , y ) } = φ ( x , y ) exp ( i k x x i k y y ) d x d y
φ ( x , y ) = F 1 { φ ^ ( k x , k y ) } = 1 ( 2 π ) 2 φ ^ ( k x , k y ) exp ( i k x x + i k y y ) d k x d k y .
φ ( x , y ) = φ * ( x , y ) exp ( i q x x + i q y y ) .
φ ^ ( k x , k y ) = φ ^ * ( k x q x , k y q y )
φ ^ out * ( k x q x , k y q y ) = φ ^ in * ( k x q x , k y q y ) p k , x 0 , y 0 ( k x , k y ) .
k x q x = k x k x = k x + q x .
φ ^ out * ( k x , k y ) = φ ^ in * ( k x , k y ) p k , x 0 , y 0 ( k x + q x , k y + q y ) .
φ out * ( x , y ) = F 1 { F { φ in * ( x , y ) } p k , x 0 , y 0 ( k x + q x , k y + q y ) }
φ * ( x , y ) = φ ( x , y ) exp ( i q x x i q y y ) .
φ out ( x , y ) = F 1 { F { φ in ( x , y ) exp ( i q x x i q y y ) } p k , x 0 , y 0 ( k x + q x , k y + q y ) } exp ( i q x x + i q y y ) .
Δ k x π L x
r k x = 2 π Δ k x = 2 L x .
ω x ( k x ) = k x 1 i log ( p k , x 0 , y 0 ( k x + q x , k y + q y ) ) ,
ω x = z ( k x + q x ) k 2 ( k x + q x ) 2 ( k y + q y ) 2 + x 0
ω y = z ( k y + q y ) k 2 ( k x + q x ) 2 ( k y + q y ) 2 + y 0
| ω x ( k x ) | 1 2 r k x = L x and | ω y ( k y ) | 1 2 r k y = L y
r x = 1 π max ( | k x min | , | k x max | ) and r y = 1 π max ( | k y min | , | k y max | ) .
φ in * ( x , y ) = φ in ( x , y ) exp ( i q x x i q y y )
q x = k x min + k x max 2 and q y = k y min + k y max 2
r x * = 1 π max ( | k x min q x | , | k x max q x | ) and r y * = 1 π max ( | k y min q y | , | k y max q y | ) .
φ in ( x , y ) exp ( i ϕ ( x , y ) )
ϕ ( x , y ) = k ( x Δ x ) 2 + ( y Δ y ) 2 + Δ z 2 q x x q y y
k x ( x , y ) = x ϕ ( x , y ) = k 2 ( x Δ x ) ϕ ( x , y ) q x
k y ( x , y ) = y ϕ ( x , y ) = k 2 ( y Δ y ) ϕ ( x , y ) q y .

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