Abstract

A fast and accurate numerical method for free-space beam propagation between arbitrarily oriented planes is developed. The only approximation made in the development of the method was that the vector nature of light was ignored. The method is based on evaluating the Rayleigh–Sommerfeld diffraction integral by use of the fast Fourier transform with a special transformation to handle tilts and offsets of planes. The fundamental aspects of a software package based on the developed method are presented. A numerical example realized with the software package is presented to establish the validity of the method.

© 1998 Optical Society of America

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References

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  1. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 382.
  2. R. E. Wagner, W. J. Tomlinson, “Coupling efficiency of optics in single-mode fiber components,” Appl. Opt. 21, 2671–2688 (1982).
    [CrossRef] [PubMed]
  3. H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
    [CrossRef]
  4. S. Ganci, “Fourier diffraction through a tilted slit,” Eur. J. Phys. 2, 158–160 (1981).
    [CrossRef]
  5. K. Patorski, “Fraunhofer diffraction patterns of tilted planar objects,” Opt. Acta 30, 673–679 (1983).
    [CrossRef]
  6. 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]
  7. D. Leseberg, C. Frère, “Computer generated holograms of 3-D objects composed of tilted planar segments,” Appl. Opt. 27, 3020–3024 (1988).
    [CrossRef] [PubMed]
  8. C. Frère, D. Leseberg, “Large objects reconstructed from computer-generated holograms,” Appl. Opt. 28, 2422–2425 (1989).
    [CrossRef] [PubMed]
  9. T. Tommasi, B. Bianco, “Frequency analysis of light diffraction between rotated planes,” Opt. Lett. 17, 556–558 (1992).
    [CrossRef] [PubMed]
  10. T. Tommasi, B. Bianco, “Computer-generated holograms of tilted planes by a spatial frequency approach,” J. Opt. Soc. Am. A 10, 299–305 (1993).
    [CrossRef]
  11. B. Bianco, T. Tommasi, “Space-variant optical interconnection through the use of computer-generated holograms,” Appl. Opt. 34, 7573–7580 (1995).
    [CrossRef] [PubMed]
  12. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 1.
  13. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 17.
  14. A. Sommerfeld, Optics: Lectures on Theoretical Physics (Academic, New York, 1964), p. 197.
  15. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 3.
  16. E. Lalor, “Conditions for the validity of the angular spectrum of plane waves,” J. Opt. Soc. Am. 58, 1235–1237 (1968).
    [CrossRef]
  17. E. A. Sziklas, A. E. Siegman, “Mode calculations in unstable resonators with flowing saturable gain,” Appl. Opt. 14, 1874–1889 (1975).
    [CrossRef] [PubMed]
  18. W. H. Southwell, “Validity of the Fresnel approximations in the near field,” J. Opt. Soc. Am. 71, 7–14 (1981).
    [CrossRef]
  19. G. Thomas, R. Finney, Calculus and Analytic Geometry, 7th ed. (Addison-Wesley, New York, 1990), p. 780.
  20. G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), p. 195.
  21. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 195.
  22. A. Louri, M. Major, “Generalized methodology for modeling and simulating optical interconnection networks using diffraction analysis,” Appl. Opt. 34, 4052–4064 (1995).
    [CrossRef] [PubMed]
  23. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 2.
  24. W. Press, S. Teukolsky, W. Vetterling, B. Flannery, Numerical Recipes in fortran, 2nd ed. (Cambridge U. Press, New York, 1992), p. 497.

1995 (2)

1993 (1)

1992 (1)

1989 (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]

1982 (1)

1981 (2)

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

W. H. Southwell, “Validity of the Fresnel approximations in the near field,” J. Opt. Soc. Am. 71, 7–14 (1981).
[CrossRef]

1975 (1)

1968 (1)

1966 (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Arfken, G.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), p. 195.

Bianco, B.

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]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 195.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 17.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 1.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 382.

Finney, R.

G. Thomas, R. Finney, Calculus and Analytic Geometry, 7th ed. (Addison-Wesley, New York, 1990), p. 780.

Flannery, B.

W. Press, S. Teukolsky, W. Vetterling, B. Flannery, Numerical Recipes in fortran, 2nd ed. (Cambridge U. Press, New York, 1992), p. 497.

Frère, C.

Ganci, S.

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

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 3.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 2.

Kogelnik, H.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Lalor, E.

Leseberg, D.

Li, T.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Louri, A.

Major, M.

Patorski, K.

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

Press, W.

W. Press, S. Teukolsky, W. Vetterling, B. Flannery, Numerical Recipes in fortran, 2nd ed. (Cambridge U. Press, New York, 1992), p. 497.

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]

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]

Siegman, A. E.

Sommerfeld, A.

A. Sommerfeld, Optics: Lectures on Theoretical Physics (Academic, New York, 1964), p. 197.

Southwell, W. H.

Sziklas, E. A.

Teukolsky, S.

W. Press, S. Teukolsky, W. Vetterling, B. Flannery, Numerical Recipes in fortran, 2nd ed. (Cambridge U. Press, New York, 1992), p. 497.

Thomas, G.

G. Thomas, R. Finney, Calculus and Analytic Geometry, 7th ed. (Addison-Wesley, New York, 1990), p. 780.

Tomlinson, W. J.

Tommasi, T.

Vetterling, W.

W. Press, S. Teukolsky, W. Vetterling, B. Flannery, Numerical Recipes in fortran, 2nd ed. (Cambridge U. Press, New York, 1992), p. 497.

Wagner, R. E.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 195.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 17.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 1.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 382.

Appl. Opt. (6)

Eur. J. Phys. (1)

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

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (1)

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]

Opt. Lett. (1)

Proc. IEEE (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Other (10)

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 382.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 1.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 17.

A. Sommerfeld, Optics: Lectures on Theoretical Physics (Academic, New York, 1964), p. 197.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 3.

G. Thomas, R. Finney, Calculus and Analytic Geometry, 7th ed. (Addison-Wesley, New York, 1990), p. 780.

G. Arfken, Mathematical Methods for Physicists, 3rd ed. (Academic, New York, 1985), p. 195.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 195.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 2.

W. Press, S. Teukolsky, W. Vetterling, B. Flannery, Numerical Recipes in fortran, 2nd ed. (Cambridge U. Press, New York, 1992), p. 497.

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Figures (14)

Fig. 1
Fig. 1

Coordinate systems involved in the diffraction integral, Eq. (3).

Fig. 2
Fig. 2

Relationship between k-vector direction and spatial frequencies for a particular coordinate system.

Fig. 3
Fig. 3

Change of plane-wave propagation direction when the coordinate system is rotated around the y axis.

Fig. 4
Fig. 4

Calculating phase accumulation that is due to propagation between two planes.

Fig. 5
Fig. 5

Perception of spatial-frequency changes depending on the orientation of the coordinate system.

Fig. 6
Fig. 6

Writing the field distribution with respect to an offset coordinate system.

Fig. 7
Fig. 7

Shifted prime coordinate system on the next plane.

Fig. 8
Fig. 8

Nucleus of the developed software.

Fig. 9
Fig. 9

Details of the implementing plane-wave propagation step.

Fig. 10
Fig. 10

Optical system analyzed in Section 6.

Fig. 11
Fig. 11

Intensity contours for 1%, 10%, 30%, 70%, and 90% on plane 3 calculated from plane 1 in one propagation step.

Fig. 12
Fig. 12

Intensity contours for 1%, 10%, 30%, 70%, and 90% on the intermediate plane making a 45° angle with the optical axis of plane 1.

Fig. 13
Fig. 13

Intensity contours for 1%, 10%, 30%, 70%, and 90% on the intermediate plane parallel to plane 1.

Fig. 14
Fig. 14

Intensity contours for 1%, 10%, 30%, 70%, and 90% on plane 3 calculated from the intermediate plane 2.

Equations (25)

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(2+k2)U=0,k=2πλ.
U(x, y, z)=-12πAU(x, y, 0)×zexp(-ikr)rdxdy,
U(x, y, z)=12πAU(x, y, 0)×exp(-ikr)rzrik+1rdxdy,
U(x, y, z)=F-1{F{U(x, y, z)}}=F-1F{U(x, y, 0)}×F12πzexp(-ikr)r.
F{U(x, y, 0)}=A0(νx, νy, 0)=AU(x, y, 0)×exp[-i2π(νxx+νyy)]dxdy.
F12πzexp(-ikr)r=exp-i2πz1λ2-νx2-νy2.
U(x, y, z)=A0(νx, νy, 0)exp[i2π(νxx+νyy)]×exp-i2πz1λ2-νx2-νy2dνxdνy.
α=sin θx,β=sin θy,
νx=sin θxλ,νy=sin θyλ.
f(xp, 0)=A exp-i 2πλxp sin θp,
f(xp, z0)=A exp-i 2πλz0 cos θp×exp-i 2πλxp sin θp,
f(xn, 0)=A exp-i 2πλz0 cos θp×exp-i 2πλxn sin θn.
OpA¯=OpOn¯·k|k|=z0ez·kxex+kyey+kzez|k|=z0kz|k|.
kz=2πλ1-(λνx)2-(λνy)2,
ϕ(OpA¯)=2πz01λ2-νx2-νy2.
(x, y, z)T=R(x, y, z)T,
(νx, νy, νz)=R(νx, νy, νz)T,
R=cos ψ0-sin ψ010sin ψ0cos ψ.
Im[νz]=0,νz>0
foffset(xn, 0)=A exp-i 2πλz0 cos θp×expi 2πλxn0 sin θn×exp-i 2πλxn sin θn.
x=x-x0,y=y-y0,z=z,
U(x, y, z)=A0(νx, νy, 0)exp[i2π(νxx+νyy)]×exp-i2πz1λ2-νx2-νy2dνxdνy,
A0(νx, νy, 0)=A0(νx, νy, 0)exp[i2π(νxx0+νyy0)].
νxx=iNj=i(λ/2)Njλ2,i, j=-N2, , N2-1,
νx=i(λ/2)N=sin θxλi=-N2, , N2-1

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