Abstract

A novel method for simulating field propagation is presented. The method, based on the angular spectrum of plane waves and coordinate rotation in the Fourier domain, removes geometric limitations posed by conventional propagation calculation and enables us to calculate complex amplitudes of diffracted waves on a plane not parallel to the aperture. This method can be implemented by using the fast Fourier transformation twice and a spectrum interpolation. It features computation time that is comparable with that of standard calculation methods for diffraction or propagation between parallel planes. To demonstrate the method, numerical results as well as a general formulation are reported for a single-axis rotation.

© 2003 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Chap. 3.10.
  2. G. Harburn, J. K. Ranniko, R. P. Williams, “An aspect of phase in Fraunhofer diffraction patterns,” Optik (Stuttgart) 48, 321–328 (1977).
  3. S. Ganci, “Fourier diffraction through a tilted slit,” Eur. J. Phys. 2, 158–160 (1981).
    [CrossRef]
  4. K. Patorski, “Fraunhofer diffraction patterns of tilted planar objects,” Opt. Acta 30, 673–679 (1983).
    [CrossRef]
  5. H. J. Rabal, N. Bolognini, E. E. Sicre, “Diffraction by a tilted aperture: coherent and partially coherent cases,” Opt. Acta 32, 1309–1311 (1985).
    [CrossRef]
  6. D. Leseberg, C. Frère, “Computer-generated holograms of 3-D objects composed of tilted planar segments,” Appl. Opt. 27, 3020–3024 (1988).
    [CrossRef] [PubMed]
  7. D. Leseberg, “Computer-generated three-dimensional image holograms,” Appl. Opt. 31, 223–229 (1992).
    [CrossRef] [PubMed]
  8. Y. Takaki, H. Ohzu, “Hybrid holographic microscopy: visualization of three-dimensional object information by use of viewing angles,” Appl. Opt. 39, 5302–5308 (2000).
    [CrossRef]
  9. K. Matsushima, M. Takai, “Recurrence formulas for fast creation of synthetic three-dimensional holograms,” Appl. Opt. 39, 6587–6594 (2000).
    [CrossRef]
  10. K. Matsushima, H. Schimmel, F. Wyrowski, “New creation algorithm for digitally synthesized holograms in surface model by diffraction from tilted planes,” in Practical Holography XVI and Holographic Materials VIII, S. A. Benton, S. H. Stevenson, T. J. Trout, eds., Proc. SPIE4659, 53–60 (2002).
    [CrossRef]
  11. T. M. Lehmann, C. Gönner, K. Spitzer, “Large-sized local interpolators,” in Proceedings of The International Association of Science and Technology for Development (IASTED) Conference on Computer Graphics and Imaging (ACTA Press, Calgary, Alberta, Canada, 1999), pp. 156–161.

2000 (2)

1992 (1)

1988 (1)

1985 (1)

H. J. Rabal, N. Bolognini, E. E. Sicre, “Diffraction by a tilted aperture: coherent and partially coherent cases,” Opt. Acta 32, 1309–1311 (1985).
[CrossRef]

1983 (1)

K. Patorski, “Fraunhofer diffraction patterns of tilted planar objects,” Opt. Acta 30, 673–679 (1983).
[CrossRef]

1981 (1)

S. Ganci, “Fourier diffraction through a tilted slit,” Eur. J. Phys. 2, 158–160 (1981).
[CrossRef]

1977 (1)

G. Harburn, J. K. Ranniko, R. P. Williams, “An aspect of phase in Fraunhofer diffraction patterns,” Optik (Stuttgart) 48, 321–328 (1977).

Bolognini, N.

H. J. Rabal, N. Bolognini, E. E. Sicre, “Diffraction by a tilted aperture: coherent and partially coherent cases,” Opt. Acta 32, 1309–1311 (1985).
[CrossRef]

Frère, C.

Ganci, S.

S. Ganci, “Fourier diffraction through a tilted slit,” Eur. J. Phys. 2, 158–160 (1981).
[CrossRef]

Gönner, C.

T. M. Lehmann, C. Gönner, K. Spitzer, “Large-sized local interpolators,” in Proceedings of The International Association of Science and Technology for Development (IASTED) Conference on Computer Graphics and Imaging (ACTA Press, Calgary, Alberta, Canada, 1999), pp. 156–161.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Chap. 3.10.

Harburn, G.

G. Harburn, J. K. Ranniko, R. P. Williams, “An aspect of phase in Fraunhofer diffraction patterns,” Optik (Stuttgart) 48, 321–328 (1977).

Lehmann, T. M.

T. M. Lehmann, C. Gönner, K. Spitzer, “Large-sized local interpolators,” in Proceedings of The International Association of Science and Technology for Development (IASTED) Conference on Computer Graphics and Imaging (ACTA Press, Calgary, Alberta, Canada, 1999), pp. 156–161.

Leseberg, D.

Matsushima, K.

K. Matsushima, M. Takai, “Recurrence formulas for fast creation of synthetic three-dimensional holograms,” Appl. Opt. 39, 6587–6594 (2000).
[CrossRef]

K. Matsushima, H. Schimmel, F. Wyrowski, “New creation algorithm for digitally synthesized holograms in surface model by diffraction from tilted planes,” in Practical Holography XVI and Holographic Materials VIII, S. A. Benton, S. H. Stevenson, T. J. Trout, eds., Proc. SPIE4659, 53–60 (2002).
[CrossRef]

Ohzu, H.

Patorski, K.

K. Patorski, “Fraunhofer diffraction patterns of tilted planar objects,” Opt. Acta 30, 673–679 (1983).
[CrossRef]

Rabal, H. J.

H. J. Rabal, N. Bolognini, E. E. Sicre, “Diffraction by a tilted aperture: coherent and partially coherent cases,” Opt. Acta 32, 1309–1311 (1985).
[CrossRef]

Ranniko, J. K.

G. Harburn, J. K. Ranniko, R. P. Williams, “An aspect of phase in Fraunhofer diffraction patterns,” Optik (Stuttgart) 48, 321–328 (1977).

Schimmel, H.

K. Matsushima, H. Schimmel, F. Wyrowski, “New creation algorithm for digitally synthesized holograms in surface model by diffraction from tilted planes,” in Practical Holography XVI and Holographic Materials VIII, S. A. Benton, S. H. Stevenson, T. J. Trout, eds., Proc. SPIE4659, 53–60 (2002).
[CrossRef]

Sicre, E. E.

H. J. Rabal, N. Bolognini, E. E. Sicre, “Diffraction by a tilted aperture: coherent and partially coherent cases,” Opt. Acta 32, 1309–1311 (1985).
[CrossRef]

Spitzer, K.

T. M. Lehmann, C. Gönner, K. Spitzer, “Large-sized local interpolators,” in Proceedings of The International Association of Science and Technology for Development (IASTED) Conference on Computer Graphics and Imaging (ACTA Press, Calgary, Alberta, Canada, 1999), pp. 156–161.

Takai, M.

Takaki, Y.

Williams, R. P.

G. Harburn, J. K. Ranniko, R. P. Williams, “An aspect of phase in Fraunhofer diffraction patterns,” Optik (Stuttgart) 48, 321–328 (1977).

Wyrowski, F.

K. Matsushima, H. Schimmel, F. Wyrowski, “New creation algorithm for digitally synthesized holograms in surface model by diffraction from tilted planes,” in Practical Holography XVI and Holographic Materials VIII, S. A. Benton, S. H. Stevenson, T. J. Trout, eds., Proc. SPIE4659, 53–60 (2002).
[CrossRef]

Appl. Opt. (4)

Eur. J. Phys. (1)

S. Ganci, “Fourier diffraction through a tilted slit,” Eur. J. Phys. 2, 158–160 (1981).
[CrossRef]

Opt. Acta (2)

K. Patorski, “Fraunhofer diffraction patterns of tilted planar objects,” Opt. Acta 30, 673–679 (1983).
[CrossRef]

H. J. Rabal, N. Bolognini, E. E. Sicre, “Diffraction by a tilted aperture: coherent and partially coherent cases,” Opt. Acta 32, 1309–1311 (1985).
[CrossRef]

Optik (Stuttgart) (1)

G. Harburn, J. K. Ranniko, R. P. Williams, “An aspect of phase in Fraunhofer diffraction patterns,” Optik (Stuttgart) 48, 321–328 (1977).

Other (3)

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996), Chap. 3.10.

K. Matsushima, H. Schimmel, F. Wyrowski, “New creation algorithm for digitally synthesized holograms in surface model by diffraction from tilted planes,” in Practical Holography XVI and Holographic Materials VIII, S. A. Benton, S. H. Stevenson, T. J. Trout, eds., Proc. SPIE4659, 53–60 (2002).
[CrossRef]

T. M. Lehmann, C. Gönner, K. Spitzer, “Large-sized local interpolators,” in Proceedings of The International Association of Science and Technology for Development (IASTED) Conference on Computer Graphics and Imaging (ACTA Press, Calgary, Alberta, Canada, 1999), pp. 156–161.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Definition of coordinate system and schemes for (a) a tilted screen and (b) a tilted aperture.

Fig. 2
Fig. 2

Geometry for (a) a tilted screen and (b) a tilted aperture.

Fig. 3
Fig. 3

Curves for transforming Fourier frequency u in source coordinates into uˆ in reference coordinates; v=0.

Fig. 4
Fig. 4

Schematic diagram of spectrum shift: (a) spectrum in source coordinates, (b) reference coordinates, and (c) shift of spectral origin.

Fig. 5
Fig. 5

Schematic setup for numerical simulation of single-axis rotation. One-dimensional rigorous amplitudes for comparison are calculated at sampling points (dots).

Fig. 6
Fig. 6

SNRs in one-dimensional amplitudes for several interpolation algorithms. Here nth and Cubic 8 stand for nth iterative linear interpolation and cubic interpolation at 8 points, respectively. The rotation angle is 30°.

Fig. 7
Fig. 7

SNR versus rotation angle. Cubic interpolation at 8 points is used in calculation.

Fig. 8
Fig. 8

Two-dimensional distribution of amplitudes and phase calculated on tilted reference planes. The phase factor due to the carrier frequency appearing on the tilted plane is eliminated in depicted phase images.

Fig. 9
Fig. 9

Computation time for rotation and translational propagation. Measurements were made by using a Pentium III processor with 1 GHz frequency.

Fig. 10
Fig. 10

Schematic depiction of the spectral fold. Each side with respect to the maximum or minimum value in frequency transformation curves corresponds to the sign of the k^z element of wave vectors in the reference coordinates.

Tables (1)

Tables Icon

Table 1 Summary of Parameters Used for Numerical Simulations

Equations (43)

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

G(u, v)=F{g(x, y)}=-g(x, y)exp[-i2π(ux+vy)]dxdy
g(x, y)=F-1{G(u, v)}=-G(u, v)exp[i2π(ux+vy)]dudv.
u(x, y, z=0; u, v)=G(u, v)exp[i2π(ux+vy)].
u(x, y, z=0; k)=A expi 2πλ (kxx+kyy),
k=2πλ [kxkykz].
k=2π[uvw(u, v)],
F(uˆ, vˆ)=F{f(uˆ, vˆ)}=-f(xˆ, yˆ)exp[-i2π(uˆxˆ+vˆyˆ)]dxˆdyˆ.
kˆ=2π[uˆvˆwˆ(uˆ, vˆ)],
kˆ=Tk,k=T-1kˆ.
T-1=a1a2a3a4a5a6a7a8a9,
u=α(uˆ, vˆ)=a1uˆ+a2vˆ+a3wˆ(uˆ, vˆ),
v=β(uˆ, vˆ)=a4uˆ+a5vˆ+a6wˆ(uˆ, vˆ).
F(uˆ, vˆ)=G(α(uˆ, vˆ), β(uˆ, vˆ)).
Etotal-|G(u, v)|2dudv.
|G(α(uˆ, vˆ), β(uˆ, vˆ))|2|J(uˆ, vˆ)|duˆdvˆ,
J(uˆ, vˆ)=αuˆβvˆ-αvˆβuˆ=(a2a6-a3a5) uˆwˆ(uˆ, vˆ)+(a3a4-a1a6) vˆwˆ(uˆ, vˆ)+(a1a5-a2a4),
Etotal-|G(α(uˆ, vˆ), β(uˆ, vˆ))|2|J(uˆ, vˆ)|duˆdvˆ.
f(xˆ, yˆ)=-F(uˆ, vˆ)exp[i2π(uˆxˆ+vˆyˆ)]×|J(uˆ, vˆ)|duˆdvˆ.
f(xˆ, yˆ)=F-1{F(uˆ, vˆ)|J(uˆ, vˆ)|}.
J(uˆ, vˆ)(a1a5-a2a4)
f(xˆ, yˆ)F-1{F(uˆ, vˆ)}.
Gd(u, v)=G(u, v)exp[i2πd(λ-2-u2-v2)1/2].
T-1=cos φ0sin φ010-sin φ0cos φ.
F(uˆ, vˆ)=Gd(uˆ cos φ+wˆ(uˆ, vˆ)sin φ, vˆ),
J(uˆ, vˆ)=cos φ-uˆwˆ(uˆ, vˆ)sin φ.
f(xˆ, yˆ)=F-1Gd(uˆ cos φ+wˆ(uˆ, vˆ)sin φ, vˆ)×cos φ-uˆwˆ(uˆ, vˆ)sin φ.
f(xˆ, yˆ)F-1{Gd(uˆ cos φ+wˆ(uˆ, vˆ)sin φ, vˆ)}.
f(xˆ, yˆ)=F-1{F(u, vˆ)|J(u, vˆ)|}exp[i2πu^0x].
SNR= |f(xˆ, yˆ)|2dxˆdyˆ |f(xˆ, yˆ)-αfrig(xˆ, yˆ)|2dxˆdyˆ,
α= f(xˆ, yˆ)frig*(xˆ, yˆ)dxˆdyˆ |frig(xˆ, yˆ)|2dxˆdyˆ.
uˆ=u cos φ-w(u, v)sin φ,
wˆ=u sin φ+w(u, v)cos φ.
u sin φ+w(u, v)cos φ=0,
k^z(u, v)=2π[u sin φ+w(u, v)cos φ].
T=1|T-1|A1A2A3A4A5A6A7A8A9.
A3=(-1)(1+3)a2a3a5a6=a2a6-a3a5,
A6=(-1)(3+2)a1a3a4a6=a3a4-a1a6.
A9=(-1)(3+3)a1a2a4a5=a1a5-a2a4.
J(uˆ, vˆ)=A3uˆwˆ+A6vˆwˆ+A9.
A3uˆwˆ=A3A1u+A2v+A3wA7u+A8v+A9w.
A3uˆwˆ=A3A1u/w+A2v/w+A3A7u/w+A8v/w+A9A32A9,
A6vˆwˆA62A9.
J(uˆ, vˆ)A32+A62+A92A9.

Metrics