Abstract

A novel method is proposed for simulating free-space propagation. This method is an improvement of the angular spectrum method (AS). The AS does not include any approximation of the propagation distance, because the formula thereof is derived directly from the Rayleigh-Sommerfeld equation. However, the AS is not an all-round method, because it produces severe numerical errors due to a sampling problem of the transfer function even in Fresnel regions. The proposed method resolves this problem by limiting the bandwidth of the propagation field and also expands the region in which exact fields can be calculated by the AS. A discussion on the validity of limiting the bandwidth is also presented.

©2009 Optical Society of America

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References

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  1. J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11(3), 77–79 (1967).
    [Crossref]
  2. U. Schnars and W. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33(2), 179–181 (1994).
    [Crossref] [PubMed]
  3. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22(16), 1268–1270 (1997).
    [Crossref] [PubMed]
  4. T. Nakatsuji and K. Matsushima, “Free-viewpoint images captured using phase-shifting synthetic aperture digital holography,” Appl. Opt. 47(19), D136–D143 (2008).
    [Crossref] [PubMed]
  5. K. Matsushima, “Formulation of the rotational transformation of wave fields and their application to digital holography,” Appl. Opt. 47(19), D110–D116 (2008).
    [Crossref] [PubMed]
  6. K. Matsushima, “Computer-generated holograms for three-dimensional surface objects with shade and texture,” Appl. Opt. 44(22), 4607–4614 (2005).
    [Crossref] [PubMed]
  7. L. Ahrenberg, P. Benzie, M. Magnor, and J. Watson, “Computer generated holograms from three dimensional meshes using an analytic light transport model,” Appl. Opt. 47(10), 1567–1574 (2008).
    [Crossref] [PubMed]
  8. H. Kim, J. Hahn, and B. Lee, “Mathematical modeling of triangle-mesh-modeled three-dimensional surface objects for digital holography,” Appl. Opt. 47(19), D117–D127 (2008).
    [Crossref] [PubMed]
  9. T. M. Kreis, M. Adams, and W. P. O. Jüptner, “Methods of digital holography: A comparison,” SPIE Proc. 3098, 224–233 (1997).
  10. 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(4), 857–867 (1998).
    [Crossref]
  11. 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(9), 1755–1762 (2003).
    [Crossref]
  12. S. J. Jeong and C. K. Hong, “Pixel-size-maintained image reconstruction of digital holograms on arbitrarily tilted planes by the angular spectrum method,” Appl. Opt. 47(16), 3064–3071 (2008).
    [Crossref] [PubMed]
  13. S. De Nicola, A. Finizio, G. Pierattini, P. Ferraro, and D. Alfieri, “Angular spectrum method with correction of anamorphism for numerical reconstruction of digital holograms on tilted planes,” Opt. Express 13(24), 9935–9940 (2005).
    [Crossref] [PubMed]
  14. F. Zhang, I. Yamaguchi, and L. P. Yaroslavsky, “Algorithm for reconstruction of digital holograms with adjustable magnification,” Opt. Lett. 29(14), 1668–1670 (2004).
    [Crossref] [PubMed]
  15. L. Yu and M. K. Kim, “Pixel resolution control in numerical reconstruction of digital holography,” Opt. Lett. 31(7), 897–899 (2006).
    [Crossref] [PubMed]
  16. D. Wang, J. Zhao, F. Zhang, G. Pedrini, and W. Osten, “High-fidelity numerical realization of multiple-step Fresnel propagation for the reconstruction of digital holograms,” Appl. Opt. 47(19), D12–D20 (2008).
    [Crossref] [PubMed]
  17. 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]
  18. Y. M. Engelberg and S. Ruschin, “Fast method for physical optics propagation of high-numerical-aperture beams,” J. Opt. Soc. Am. A 21(11), 2135–2145 (2004).
    [Crossref]
  19. M. Sypek, “Light propagation in the Fresnel region. New numerical approach,” Opt. Commun. 116(1-3), 43–48 (1995).
    [Crossref]
  20. E. A. Sziklas and A. E. Siegman, “Mode calculations in unstable resonators with flowing saturable gain. 2: Fast Fourier transform method,” Appl. Opt. 14(8), 1874–1889 (1975).
    [Crossref] [PubMed]
  21. F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula,” Appl. Opt. 45(6), 1102–1110 (2006).
    [Crossref] [PubMed]
  22. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), chap. 2.2.

2008 (6)

2007 (1)

2006 (2)

2005 (2)

2004 (2)

2003 (1)

1998 (1)

1997 (2)

I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22(16), 1268–1270 (1997).
[Crossref] [PubMed]

T. M. Kreis, M. Adams, and W. P. O. Jüptner, “Methods of digital holography: A comparison,” SPIE Proc. 3098, 224–233 (1997).

1995 (1)

M. Sypek, “Light propagation in the Fresnel region. New numerical approach,” Opt. Commun. 116(1-3), 43–48 (1995).
[Crossref]

1994 (1)

1975 (1)

1967 (1)

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11(3), 77–79 (1967).
[Crossref]

Adams, M.

T. M. Kreis, M. Adams, and W. P. O. Jüptner, “Methods of digital holography: A comparison,” SPIE Proc. 3098, 224–233 (1997).

Ahrenberg, L.

Alfieri, D.

Benzie, P.

De Nicola, S.

Delen, N.

Engelberg, Y. M.

Ferraro, P.

Finizio, A.

Goodman, J. W.

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11(3), 77–79 (1967).
[Crossref]

Hahn, J.

Hong, C. K.

Hooker, B.

Jeong, S. J.

Jüptner, W.

Jüptner, W. P. O.

T. M. Kreis, M. Adams, and W. P. O. Jüptner, “Methods of digital holography: A comparison,” SPIE Proc. 3098, 224–233 (1997).

Kim, H.

Kim, M. K.

Kreis, T. M.

T. M. Kreis, M. Adams, and W. P. O. Jüptner, “Methods of digital holography: A comparison,” SPIE Proc. 3098, 224–233 (1997).

Lawrence, R. W.

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11(3), 77–79 (1967).
[Crossref]

Lee, B.

Magnor, M.

Matsushima, K.

Muffoletto, R. P.

Nakatsuji, T.

Osten, W.

Pedrini, G.

Pierattini, G.

Ruschin, S.

Schimmel, H.

Schnars, U.

Shen, F.

Siegman, A. E.

Sypek, M.

M. Sypek, “Light propagation in the Fresnel region. New numerical approach,” Opt. Commun. 116(1-3), 43–48 (1995).
[Crossref]

Sziklas, E. A.

Tohline, J. E.

Tyler, J. M.

Wang, A.

Wang, D.

Watson, J.

Wyrowski, F.

Yamaguchi, I.

Yaroslavsky, L. P.

Yu, L.

Zhang, F.

Zhang, T.

Zhao, J.

Appl. Opt. (10)

T. Nakatsuji and K. Matsushima, “Free-viewpoint images captured using phase-shifting synthetic aperture digital holography,” Appl. Opt. 47(19), D136–D143 (2008).
[Crossref] [PubMed]

K. Matsushima, “Formulation of the rotational transformation of wave fields and their application to digital holography,” Appl. Opt. 47(19), D110–D116 (2008).
[Crossref] [PubMed]

K. Matsushima, “Computer-generated holograms for three-dimensional surface objects with shade and texture,” Appl. Opt. 44(22), 4607–4614 (2005).
[Crossref] [PubMed]

L. Ahrenberg, P. Benzie, M. Magnor, and J. Watson, “Computer generated holograms from three dimensional meshes using an analytic light transport model,” Appl. Opt. 47(10), 1567–1574 (2008).
[Crossref] [PubMed]

H. Kim, J. Hahn, and B. Lee, “Mathematical modeling of triangle-mesh-modeled three-dimensional surface objects for digital holography,” Appl. Opt. 47(19), D117–D127 (2008).
[Crossref] [PubMed]

U. Schnars and W. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33(2), 179–181 (1994).
[Crossref] [PubMed]

S. J. Jeong and C. K. Hong, “Pixel-size-maintained image reconstruction of digital holograms on arbitrarily tilted planes by the angular spectrum method,” Appl. Opt. 47(16), 3064–3071 (2008).
[Crossref] [PubMed]

D. Wang, J. Zhao, F. Zhang, G. Pedrini, and W. Osten, “High-fidelity numerical realization of multiple-step Fresnel propagation for the reconstruction of digital holograms,” Appl. Opt. 47(19), D12–D20 (2008).
[Crossref] [PubMed]

E. A. Sziklas and A. E. Siegman, “Mode calculations in unstable resonators with flowing saturable gain. 2: Fast Fourier transform method,” Appl. Opt. 14(8), 1874–1889 (1975).
[Crossref] [PubMed]

F. Shen and A. Wang, “Fast-Fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula,” Appl. Opt. 45(6), 1102–1110 (2006).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

J. W. Goodman and R. W. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11(3), 77–79 (1967).
[Crossref]

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

Opt. Commun. (1)

M. Sypek, “Light propagation in the Fresnel region. New numerical approach,” Opt. Commun. 116(1-3), 43–48 (1995).
[Crossref]

Opt. Express (2)

Opt. Lett. (3)

SPIE Proc. (1)

T. M. Kreis, M. Adams, and W. P. O. Jüptner, “Methods of digital holography: A comparison,” SPIE Proc. 3098, 224–233 (1997).

Other (1)

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), chap. 2.2.

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Figures (11)

Fig. 1
Fig. 1 Definition of the coordinate system and the geometry of the model.

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Fig. 2
Fig. 2 Conversion of a circular convolution into a linear convolution in cases where the origin of the coordinates system is placed at the center (a) and the left lower corner (b) of the sampling array.

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Fig. 3
Fig. 3 Numerical simulation for verifying accuracy of the AS.

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Fig. 4
Fig. 4 Comparison of the accuracy of the AS and Shift-FR. (a) One-dimensional amplitude distribution in the destination plane. (b) SNR of the AS and Shift-FR.

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Fig. 5
Fig. 5 Example of a sampled transfer function of the AS. Only the real part of the transfer function H(u;z) is depicted in the sampling interval Δu=(2Sx)−1=(2NxΔx)−1 , where Nx=1024 , Δx=2λ , and z=50Sx .

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Fig. 6
Fig. 6 Accuracy of the band-limited AS proposed in this work. (a) One-dimensional amplitude distribution in the destination plane. (b) SNR of the AS and Shift-FR. Parameters used in the calculation are the same as in Fig. 4.

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Fig. 7
Fig. 7 Regions in the (u,v) space to avoid aliasing errors of the sampled transfer function. The sampling intervals in the Fourier space are Δu=Δv=(2Sx)−1=(2Sy)−1 for the sampling windows Sx=NxΔx and Sy=NyΔy in the real space, where Nx=Ny=1024 and Δx=Δy=2λ .

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Fig. 8
Fig. 8 A schematic illustration of the approximated rectangular region.

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Fig. 9
Fig. 9 Amplitude images calculated by the original and band-limited AS. (a) Diffraction by a square aperture with dimensions Sx/2×Sy/2 and z = 100 Sx . (b) Diffraction by a circular aperture with diameter Sx /2 and z = 200 Sx .

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Fig. 10
Fig. 10 Model for the minimum bandwidth required for exact calculation of field propagation.

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Fig. 11
Fig. 11 SNR as a function of bandwidth in the AS in cases of the normal sampling (black solid line) and over sampling (red broken line). Here Wx=Sx/2 and z=50Sx . The other parameters are the same as in Fig. 4.

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Equations (24)

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

g(x,y,z)=∬g(x',y',0)exp(i2πr'λ−1)r'zr'(12πr'+1iλ)dx'dy',
g(x,y,z)=g(x,y,0)∗h(x,y,z),
h(x,y,z)=exp(i2πrλ−1)rzr(12πr+1iλ),
G(u,v;z)=G(u,v;0)H(u,v;z).
G(u,v;0)=∬g(x,y,0)exp[−i2π(ux+vy)]dxdy=F{g(x,y,0)},
H(u,v;z)=F{h(x,y,z)}=exp[i2πwz],
w=w(u,v)={(λ−2−u2−v2)1/2⋯u2+v2≤λ−20⋯otherwise
g(x,y,z)=F−1{G(u,v;0)exp[i2πw(u,v)z]}.
hFR(x,y;z)=1iλzexp[i2πλ−1(z+x2+y22z)],HFR(u,v;z)=exp{iπz[2λ−λ(u2+v2)]}.
H(u;z)=exp[iϕ(u)],ϕ(u)=2π(λ−2−u2)1/2z,
fu=12π∂ϕ∂u=uz[λ−2−u2]1/2.
Δu−1≥2|fu|.
|u|≤1[(2Δuz)2+1]1/2λ≡ulimit.
H′(u;z)=H(u;z)rect(u2ulimit),
fu=12π∂ϕ∂u=uz[λ−2−u2−v2]1/2,
fv=12π∂ϕ∂v=vz[λ−2−u2−v2]1/2,
Δu−1≥2|fu|andΔv−1≥2|fv|,
u2ulimit2+v2λ−2≤1,
u2λ−2+v2vlimit2≤1,
vlimit=[(2Δvz)2+1]−1/2λ−1.
|u|≤ulimitifSx≪2z,
|v|≤vlimitifSy≪2z,
H′(u,v;z)=H(u,v;z)rect(u2ulimit)rect(v2vlimit).
uneed≡2umax=2[(2zWx+Sx)2+1]−1/2λ−1.

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