Abstract

The angular spectrum (AS) method is a popular solution to the Helmholtz Equation without the use of approximations. In this work, new criteria on sampling requirements are derived using the Wigner distribution (WD). It is shown that for the case of high numerical aperture the conventional AS method requires a very large amount of zero-padding, making it impractical due to requirements on memory and computational effort. This work proposes the use of a modified AS algorithm that evaluates only non-zero components of the field. The results obtained from the WD combined with the modified AS algorithm enable an accurate and efficient field computation for cases where the conventional AS method cannot be implemented.

© 2012 Optical Society of America

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References

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  1. 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, 19662–19673 (2009).
    [CrossRef]
  2. F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh–Sommerfeld diffraction formula,” Appl. Opt. 45, 1102–1110 (2006).
    [CrossRef]
  3. A. Wuttig, M. Kanka, H. J. Kreuzer, and R. Riesenberg, “Packed domain Rayleigh–Sommerfeld wavefield propagation for large targets,” Opt. Express 18, 27036–27047 (2010).
    [CrossRef]
  4. V. Nascov and P. C. Logofătu, “Fast computation algorithm for the Rayleigh–Sommerfeld diffraction formula using a type of scaled convolution,” Appl. Opt. 48, 4310–4319 (2009).
    [CrossRef]
  5. K. Matsushima, H. Schimmel, and F. Wyrowski, “Fast calculation method for optical diffraction on tilted planes by use of the angular spectrum of plane waves,” J. Opt. Soc. Am. A 20, 1755–1762 (2003).
    [CrossRef]
  6. 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]
  7. Y. Lim, S.-Y. Lee, and B. Lee, “Transflective digital holographic microscopy and its use for probing plasmonic light beaming,” Opt. Express 19, 5202–5212 (2011).
    [CrossRef]
  8. T. Colomb, N. Pavillon, J. Kühn, E. Cuche, C. Depeursinge, and Y. Emery, “Extended depth-of-focus by digital holographic microscopy,” Opt. Lett. 35, 1840–1842 (2010).
    [CrossRef]
  9. J. Hahn, H. Kim, Y. Lim, G. Park, and B. Lee, “Wide viewing angle dynamic holographic stereogram with a curved array of spatial light modulators,” Opt. Express 16, 12372–12386 (2008).
    [CrossRef]
  10. T. Kozacki, M. Kujawinska, G. Finke, W. Zaperty, and B. M. Hennelly, “Holographic capture and display systems in circular configurations,” J. Display Technol. 8, 225–232 (2012).
    [CrossRef]
  11. A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imag. 4, 336–350 (1982).
    [CrossRef]
  12. T. Kozacki, M. Kujawinska, and P. Kniażewski, “Investigation of limitations of optical diffraction tomography,” Opto-Electron. Rev. 15, 102–109 (2007).
    [CrossRef]
  13. G. S. Khan, K. Mantel, I. Harder, N. Lindlein, and J. Schwider, “Design considerations for the absolute testing approach of aspherics using combined diffractive optical elements,” Appl. Opt. 46, 7040–7048 (2007).
    [CrossRef]
  14. X. Zhou, D. P. Poenar, K. Y. Liu, M. S. Tse, C.-K. Heng, S. N. Tan, and N. Zhang, “Limits of scalar diffraction theory and an iterative angular spectrum algorithm for finite aperture diffractive optical element design,” Opt. Express 8, 705–722 (2001).
    [CrossRef]
  15. P. Langehanenberg, G. Bally, and B. Kemper, “Autofocusing in digital holographic microscopy,” 3D Research 2, 1–11(2011).
    [CrossRef]
  16. T. Kozacki, M. Józwik, and R. Jóźwicki, “Determination of optical field generated by a microlens using digital holographic method,” Opto-Electron. Rev. 17, 211–216 (2009).
    [CrossRef]
  17. J. J. Stamnes, Waves in Focal Regions: Propagation, Diffraction and Focusing of Light, Sound and Water Waves (Taylor & Francis, 1986).
  18. T. Kozacki, “Numerical errors of diffraction computing using plane wave spectrum decomposition,” Opt. Commun. 281, 4219–4223 (2008).
    [CrossRef]
  19. K. Matsushima, “Shifted angular spectrum method for off-axis numerical propagation.,” Opt. Express 18, 18453–18463 (2010).
    [CrossRef]
  20. T. Kozacki, R. Krajewski, and M. Kujawinska, “Reconstruction of refractive-index distribution in off-axis digital holography optical diffraction tomographic system,” Opt. Express 17, 13758–13767 (2009).
    [CrossRef]
  21. M. E. Testorf, B. M. Hennelly, and J. Ojeda-Castaneda, Phase-Space Optics Fundamentals and Applications (McGraw-Hill Professional, 2009).
  22. H. Gross, W. Singer, and M. Totzek, Handbook of Optical Systems, Vol. 2 (Wiley, 2005).
  23. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).
  24. J. G. Proakis and D. G. Manolakis, Digital Signal Processing: Principles, Algorithms and Applications, Vol. 4 (Prentice Hall, 1996), p. 471.
  25. T. Kozacki, “On resolution and viewing of holographic image generated by 3D holographic display,” Opt. Express 18, 27118–27129 (2010).
    [CrossRef]
  26. T. Kozacki, M. Kujawinska, G. Finke, B. Hennelly, and N. Pandey, “Extended viewing angle holographic display system with tilted SLMs in a circular configuration,” Appl. Opt. 51, 1771–1780 (2012).
    [CrossRef]
  27. NVIDIA CUDA Programming Guide,Version 3 (NVIDIA Corporation, 2010), pp. 1–111.

2012 (2)

2011 (2)

Y. Lim, S.-Y. Lee, and B. Lee, “Transflective digital holographic microscopy and its use for probing plasmonic light beaming,” Opt. Express 19, 5202–5212 (2011).
[CrossRef]

P. Langehanenberg, G. Bally, and B. Kemper, “Autofocusing in digital holographic microscopy,” 3D Research 2, 1–11(2011).
[CrossRef]

2010 (4)

2009 (4)

2008 (2)

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

J. Hahn, H. Kim, Y. Lim, G. Park, and B. Lee, “Wide viewing angle dynamic holographic stereogram with a curved array of spatial light modulators,” Opt. Express 16, 12372–12386 (2008).
[CrossRef]

2007 (2)

T. Kozacki, M. Kujawinska, and P. Kniażewski, “Investigation of limitations of optical diffraction tomography,” Opto-Electron. Rev. 15, 102–109 (2007).
[CrossRef]

G. S. Khan, K. Mantel, I. Harder, N. Lindlein, and J. Schwider, “Design considerations for the absolute testing approach of aspherics using combined diffractive optical elements,” Appl. Opt. 46, 7040–7048 (2007).
[CrossRef]

2006 (1)

2003 (1)

2001 (1)

1998 (1)

1982 (1)

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imag. 4, 336–350 (1982).
[CrossRef]

Bally, G.

P. Langehanenberg, G. Bally, and B. Kemper, “Autofocusing in digital holographic microscopy,” 3D Research 2, 1–11(2011).
[CrossRef]

Colomb, T.

Cuche, E.

Delen, N.

Depeursinge, C.

Devaney, A. J.

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imag. 4, 336–350 (1982).
[CrossRef]

Emery, Y.

Finke, G.

Goodman, J. W.

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

Gross, H.

H. Gross, W. Singer, and M. Totzek, Handbook of Optical Systems, Vol. 2 (Wiley, 2005).

Hahn, J.

Harder, I.

Heng, C.-K.

Hennelly, B.

Hennelly, B. M.

T. Kozacki, M. Kujawinska, G. Finke, W. Zaperty, and B. M. Hennelly, “Holographic capture and display systems in circular configurations,” J. Display Technol. 8, 225–232 (2012).
[CrossRef]

M. E. Testorf, B. M. Hennelly, and J. Ojeda-Castaneda, Phase-Space Optics Fundamentals and Applications (McGraw-Hill Professional, 2009).

Hooker, B.

Józwicki, R.

T. Kozacki, M. Józwik, and R. Jóźwicki, “Determination of optical field generated by a microlens using digital holographic method,” Opto-Electron. Rev. 17, 211–216 (2009).
[CrossRef]

Józwik, M.

T. Kozacki, M. Józwik, and R. Jóźwicki, “Determination of optical field generated by a microlens using digital holographic method,” Opto-Electron. Rev. 17, 211–216 (2009).
[CrossRef]

Kanka, M.

Kemper, B.

P. Langehanenberg, G. Bally, and B. Kemper, “Autofocusing in digital holographic microscopy,” 3D Research 2, 1–11(2011).
[CrossRef]

Khan, G. S.

Kim, H.

Kniazewski, P.

T. Kozacki, M. Kujawinska, and P. Kniażewski, “Investigation of limitations of optical diffraction tomography,” Opto-Electron. Rev. 15, 102–109 (2007).
[CrossRef]

Kozacki, T.

Krajewski, R.

Kreuzer, H. J.

Kühn, J.

Kujawinska, M.

Langehanenberg, P.

P. Langehanenberg, G. Bally, and B. Kemper, “Autofocusing in digital holographic microscopy,” 3D Research 2, 1–11(2011).
[CrossRef]

Lee, B.

Lee, S.-Y.

Lim, Y.

Lindlein, N.

Liu, K. Y.

Logofatu, P. C.

Manolakis, D. G.

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

Mantel, K.

Matsushima, K.

Nascov, V.

Ojeda-Castaneda, J.

M. E. Testorf, B. M. Hennelly, and J. Ojeda-Castaneda, Phase-Space Optics Fundamentals and Applications (McGraw-Hill Professional, 2009).

Pandey, N.

Park, G.

Pavillon, N.

Poenar, D. P.

Proakis, J. G.

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

Riesenberg, R.

Schimmel, H.

Schwider, J.

Shen, F.

Shimobaba, T.

Singer, W.

H. Gross, W. Singer, and M. Totzek, Handbook of Optical Systems, Vol. 2 (Wiley, 2005).

Stamnes, J. J.

J. J. Stamnes, Waves in Focal Regions: Propagation, Diffraction and Focusing of Light, Sound and Water Waves (Taylor & Francis, 1986).

Tan, S. N.

Testorf, M. E.

M. E. Testorf, B. M. Hennelly, and J. Ojeda-Castaneda, Phase-Space Optics Fundamentals and Applications (McGraw-Hill Professional, 2009).

Totzek, M.

H. Gross, W. Singer, and M. Totzek, Handbook of Optical Systems, Vol. 2 (Wiley, 2005).

Tse, M. S.

Wang, A.

Wuttig, A.

Wyrowski, F.

Zaperty, W.

Zhang, N.

Zhou, X.

3D Research (1)

P. Langehanenberg, G. Bally, and B. Kemper, “Autofocusing in digital holographic microscopy,” 3D Research 2, 1–11(2011).
[CrossRef]

Appl. Opt. (4)

J. Display Technol. (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 (8)

X. Zhou, D. P. Poenar, K. Y. Liu, M. S. Tse, C.-K. Heng, S. N. Tan, and N. Zhang, “Limits of scalar diffraction theory and an iterative angular spectrum algorithm for finite aperture diffractive optical element design,” Opt. Express 8, 705–722 (2001).
[CrossRef]

J. Hahn, H. Kim, Y. Lim, G. Park, and B. Lee, “Wide viewing angle dynamic holographic stereogram with a curved array of spatial light modulators,” Opt. Express 16, 12372–12386 (2008).
[CrossRef]

T. Kozacki, R. Krajewski, and M. Kujawinska, “Reconstruction of refractive-index distribution in off-axis digital holography optical diffraction tomographic system,” Opt. Express 17, 13758–13767 (2009).
[CrossRef]

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, 19662–19673 (2009).
[CrossRef]

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

A. Wuttig, M. Kanka, H. J. Kreuzer, and R. Riesenberg, “Packed domain Rayleigh–Sommerfeld wavefield propagation for large targets,” Opt. Express 18, 27036–27047 (2010).
[CrossRef]

T. Kozacki, “On resolution and viewing of holographic image generated by 3D holographic display,” Opt. Express 18, 27118–27129 (2010).
[CrossRef]

Y. Lim, S.-Y. Lee, and B. Lee, “Transflective digital holographic microscopy and its use for probing plasmonic light beaming,” Opt. Express 19, 5202–5212 (2011).
[CrossRef]

Opt. Lett. (1)

Opto-Electron. Rev. (2)

T. Kozacki, M. Józwik, and R. Jóźwicki, “Determination of optical field generated by a microlens using digital holographic method,” Opto-Electron. Rev. 17, 211–216 (2009).
[CrossRef]

T. Kozacki, M. Kujawinska, and P. Kniażewski, “Investigation of limitations of optical diffraction tomography,” Opto-Electron. Rev. 15, 102–109 (2007).
[CrossRef]

Ultrason. Imag. (1)

A. J. Devaney, “A filtered backpropagation algorithm for diffraction tomography,” Ultrason. Imag. 4, 336–350 (1982).
[CrossRef]

Other (6)

J. J. Stamnes, Waves in Focal Regions: Propagation, Diffraction and Focusing of Light, Sound and Water Waves (Taylor & Francis, 1986).

M. E. Testorf, B. M. Hennelly, and J. Ojeda-Castaneda, Phase-Space Optics Fundamentals and Applications (McGraw-Hill Professional, 2009).

H. Gross, W. Singer, and M. Totzek, Handbook of Optical Systems, Vol. 2 (Wiley, 2005).

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

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

NVIDIA CUDA Programming Guide,Version 3 (NVIDIA Corporation, 2010), pp. 1–111.

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Figures (5)

Fig. 1.
Fig. 1.

(a) Geometry of the wave propagation problem: a portion of a field located within the boundary d0D/2xd0+D/2 at the plane z=z0 is calculated from a field distribution within the boundary s0S/2xs0+S/2 at the location z=0. (b) Wigner diagram representation of the propagation problem for an input field with λ=1μm, s0=d0=0, S=160μm, fs=(5/3)μm1, D=200μm at the propagation distance for z0=200μm. The frequency range that is highlighted in red represents the frequencies that cannot be represented for discrete signals.

Fig. 2.
Fig. 2.

(a) Wigner diagram representation of a source with S=4000μm, D=400μm, s0=d0=0, fs=1.6393μm1, λ=1μm, and z0=200μm. (b) Wigner diagram representation of a field with S=1300μm, D=800μm, s0=d0=0, fs=1.25μm1, λ=1μm, z0=2400μm, and Wx=2971.5μm.

Fig. 3.
Fig. 3.

Local spatial frequency spectrum of the kernel for a detector window of size D=512Δx, λ=1μm, Δx=1.5λ, s0=d0=0, and z0=5mm for various sizes of the computational window Wx.

Fig. 4.
Fig. 4.

Wigner diagram representation for the case of an on-axis propagation of a field with λ=1μm, fs=(5/3)μm1, s0=d0=0, S=D=160μm, z0=350μm, and Wx=320μm.

Fig. 5.
Fig. 5.

Calculated diffracted field (log10 scale) of a plane wave diffracted by circular aperture (diameter of 137.5 μm) at a propagation distance of z0=5mm, with fs=1/(0.65λ), N=1024, and λ=0.5μm, using (a) a rectangular filter with P=Q=2 and (b) the modified algorithms with of Section 3.

Equations (30)

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

fx=1λxdxs(z0)2+(xdxs)2,
xdxs=z0fx1/λ2(fx)2.
WD(x,fx)=f(x+s2)f*(xs2)exp(j2πsfx)ds,
WD(x,fx)=F(fx+ν2)F*(fxv2)exp(j2πxν)dν,
u[z=0]=u[mΔx,nΔy,z=0],
A[z]=m=0PM1n=0QN1u[z]ej2π(pΔfxmΔx+qΔfynΔy),
A[z]=FFT{[ςrx,ryςrx,ryςrx,ryςrx,ryu[z]ςrx,ryςrx,ryςrx,ryςrx,ry]},
A[z0]=G[z0]A[z=0],
G[z0]=ej2πz0λ2(pΔfx)2(qΔfy)2.
ŭ[z0]=[%%%%u[z0]%%%%],
G[z0]=ej2πz01/λ2(pΔfx)2(qΔfy)2×ej2π(pΔfx(s0xd0x)+qΔfy(s0yd0y)),
Wxxupxlow2|δ|,
PM|2δΔx1|.
WxS+2|δ|.
WxD+2|δ|.
WxS2+D2+δ.
WDg(x,fx)=WDf(x,fx)+WDfa(x,fx)+2Re{WDf,fa(x,fx)}
WDerror(x,fx)=WDfa(x,fx)+2Re{WDf,fa(x,fx)}
WDf,fa(x,fx)=f(x+s2)fa*(xs2)exp(j2πsfx)ds,
G(fx)=D/2D/2WDg(x,fx)dx
Gerror(fx)=D/2D/2WDerror(x,fx)dx.
Perror=fs/2Bf/2Gerror(fx)dfx+Bf/2fs/2Gerror(fx)dfx,
fx,lower|xD2fx,upper|xWx+D2,fx,upper|x+D2fx,lower|x+WxD2.
WxS+D,
Wx2S.
IFFT[F((pp0)Δfx)]=f(mΔx)exp(j2πmΔxp0Δfx),
A[z]=A[(p+αP)Δfx,(q+βQ)Δfy,z]=α=0P1β=0Q1(Ha,β[p,q,z]),
Ha,β[p,q,z]=FFT{u[z]exp[j2π(mαPM+nβQN)]}.
FFT[f(mm0)Δx)]=FFT[f(mΔx)]exp(j2πm0ΔxpΔfx),
u[z0]=α=0P1β=0Q1IFFT{A[(p+αP)Δfx,(q+βQ)Δfy,z0]}×exp[j2π(mαPM+nβQN)].

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