Abstract

The angular spectrum (AS) method is a popular solution to the Helmholtz equation without the use of approximations. Modified band-limited AS methods are of particular interest for the cases of high-off-axis and large distance propagation problems, because conventional AS methods are impractical due to requirements regarding memory and computational effort. However, these techniques make use of rectangular-shaped filters that introduce ringing artifacts in the calculated field that are related to the Gibbs phenomenon. This work proposes AS algorithms based on a smooth band-limiting filter for accurate field computation as well as techniques that evaluate only nonzero components of the field. This enables accurate field calculations with an acceptable level of computational effort that cannot be offered by current AS methods reported in the scientific literature.

© 2013 Optical Society of America

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References

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2012 (2)

K. Falaggis, T. Kozacki, M. Jozwik, and M. Kujawinska, “Accurate and quantitative phase retrieval methods for a series of defocused images with application to in-line Gabor microscopy,” Proc. SPIE 8493, 849335 (2012).
[CrossRef]

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, 7080–7088 (2012).
[CrossRef]

2010 (1)

2009 (1)

2008 (1)

T. Kozacki, “Numerical errors of diffraction computing using plane wave spectrum decomposition,” Opt. Commun. 281, 4219–4223 (2008).
[CrossRef]

2006 (1)

2003 (1)

2001 (1)

1998 (1)

1967 (1)

Delen, N.

Falaggis, K.

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, 7080–7088 (2012).
[CrossRef]

K. Falaggis, T. Kozacki, M. Jozwik, and M. Kujawinska, “Accurate and quantitative phase retrieval methods for a series of defocused images with application to in-line Gabor microscopy,” Proc. SPIE 8493, 849335 (2012).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), Vol. 164.

Hooker, B.

Jozwik, M.

K. Falaggis, T. Kozacki, M. Jozwik, and M. Kujawinska, “Accurate and quantitative phase retrieval methods for a series of defocused images with application to in-line Gabor microscopy,” Proc. SPIE 8493, 849335 (2012).
[CrossRef]

Kozacki, T.

K. Falaggis, T. Kozacki, M. Jozwik, and M. Kujawinska, “Accurate and quantitative phase retrieval methods for a series of defocused images with application to in-line Gabor microscopy,” Proc. SPIE 8493, 849335 (2012).
[CrossRef]

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, 7080–7088 (2012).
[CrossRef]

T. Kozacki, “Numerical errors of diffraction computing using plane wave spectrum decomposition,” Opt. Commun. 281, 4219–4223 (2008).
[CrossRef]

Kujawinska, M.

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, 7080–7088 (2012).
[CrossRef]

K. Falaggis, T. Kozacki, M. Jozwik, and M. Kujawinska, “Accurate and quantitative phase retrieval methods for a series of defocused images with application to in-line Gabor microscopy,” Proc. SPIE 8493, 849335 (2012).
[CrossRef]

Manolakis, D. G.

J. G. Proakis and D. G. Manolakis, Digital Signal Processing: Principles, Algorithms, and Applications, Vol. 4 (Prentice-Hall, 1996), p. 471.

Matsushima, K.

Proakis, J. G.

J. G. Proakis and D. G. Manolakis, Digital Signal Processing: Principles, Algorithms, and Applications, Vol. 4 (Prentice-Hall, 1996), p. 471.

Schimmel, H.

Shen, F.

Sherman, G. C.

Shimobaba, T.

Wang, A.

Wyrowski, F.

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

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

Opt. Commun. (1)

T. Kozacki, “Numerical errors of diffraction computing using plane wave spectrum decomposition,” Opt. Commun. 281, 4219–4223 (2008).
[CrossRef]

Opt. Express (2)

Proc. SPIE (1)

K. Falaggis, T. Kozacki, M. Jozwik, and M. Kujawinska, “Accurate and quantitative phase retrieval methods for a series of defocused images with application to in-line Gabor microscopy,” Proc. SPIE 8493, 849335 (2012).
[CrossRef]

Other (2)

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), Vol. 164.

J. G. Proakis and D. G. Manolakis, Digital Signal Processing: Principles, Algorithms, and Applications, Vol. 4 (Prentice-Hall, 1996), p. 471.

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Figures (3)

Fig. 1.
Fig. 1.

WD for the case of off-axis propagation of an input field with λ=1μm, fs=1.961μm1, Δsx=600μm, Sx=Dx=400μm, z0=850μm, and Wx=800μm.

Fig. 2.
Fig. 2.

(a) Calculated amplitude and (b) error of the field u(x)=R[(x+90Δx)/(20Δx)]+R[(x90Δx)/(20Δx)] after propagation over the distance of z0=17.721mm for the case of AS with a sufficient amount of zero padding (black curve, aliasing free) and BLAS with various filter types, where M=1536, Wx=P1536Δx, λ=0.5μm, and Δx=0.5001μm.

Fig. 3.
Fig. 3.

Calculated diffracted field (log10 scale) of a plane wave diffracted by a circular aperture (diameter of 137.5 μm) at a propagation distance of z0=5mm, with fs=6.2832μm1, N=1024, and λ=0.5μm, using (a) the conventional AS method with P=Q=32, (b) R-BLAS (P=Q=2), (c) H-BLAS (P=Q=2b=4), (d) B-BLAS (P=Q=2b=8), and (e) the convolution-based RS approach.

Equations (6)

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A[z]=FFT{[ςl,tςl,Nςl,tςM,tu[z]ςM,tςl,tςl,Nςl,t]},
fx=1λxdxs(z0)2+(xdxs)2,
A[z]=A[(p+αP)Δfx,(q+βQ)Δfy,z]=α=0P1β=0Q1Ha,β[p,q,z],
u[z0]=α=0P1β=0Q1u^[z0]exp[j2π(mαM+qβN)].
u^[z]=p=p1p2p=p1p2Ha,β[p,q,z0]ej2π(pΔfxmΔx+qΔfynΔy)=ej2π(p1mM+q1nyN)×p=0p2p1q=0q2q1Ha,β[p+p1,q+q1,z0]ej2π(pmM+qnN),
u^[m,n,z0]size=(Kx)×(Ky)=Φ[m,n]RxRyIFFT{A[z0]}size=(Kx)×(Ky),

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