Abstract

Using the rigorous wave-front formulation for scalar wave diffraction of Kraus [ J. Opt. Soc. Am. A 6, 1196 ( 1989); J. Opt. Soc. Am. A 9, 1132 ( 1992)], it is shown that the two-dimensional integral used to calculate the diffraction of spherical waves by a circular aperture may be reduced to a one-dimensional integral by choosing an appropriate coordinate frame. Both the two-dimensional integral and the one-dimensional integral must be evaluated numerically, but because each dimension must be sampled at approximately N locations to calculate accurately the integral (where N is the number of wavelengths across the aperture) the two-dimensional integration will require of the order of N2 evaluations of the integrand, whereas the one-dimensional integration will require of the order of only N evaluations, a substantial decrease in computing time for apertures that are large compared with optical wavelengths.

© 1994 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), Chap. 8.
  2. E. Wolf, E. W. Marchand, “Comparison of the Kirchhoff and Rayleigh–Sommerfeld theories of diffraction at an aperture,”J. Opt. Soc. Am. 54, 587–594 (1964).
    [CrossRef]
  3. W. H. Southwell, “Validity of the Fresnel approximation in the near field,”J. Opt. Soc. Am. 71, 7–14 (1981).
    [CrossRef]
  4. A. M. Steane, H. N. Rutt, “Diffraction calculations in the near field and validity of the Fresnel approximation,” J. Opt. Soc. Am. A 6, 1809–1814 (1989).
    [CrossRef]
  5. C. J. R. Sheppard, M. Hrynevych, “Diffraction by a circular aperture: a generalization of Fresnel diffraction theory,” J. Opt. Soc. Am. A 9, 274–281 (1992).
    [CrossRef]
  6. J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluations of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
    [CrossRef]
  7. H. G. Kraus, “Finite element area and line integral transforms for generalization of aperture function and geometry in Kirchhoff scalar diffraction theory,” Opt. Eng. 32, 368–383 (1993).
    [CrossRef]
  8. J. J. Stamnes, “Hybrid integration technique for efficient and accurate computation of diffraction integrals,” J. Opt. Soc. Am. A 6, 1330–1342 (1989).
    [CrossRef]
  9. H. G. Kraus, “Huygens–Fresnel–Kirchhoff wave-front diffraction formulation: spherical waves,” J. Opt. Soc. Am. A 6, 1196–1205 (1989).
    [CrossRef]
  10. H. G. Kraus, “Huygens–Fresnel–Kirchhoff wave-front diffraction formulations for spherical waves and Gaussian laser beams: discussion and errata,” J. Opt. Soc. Am. A 9, 1132–1134 (1992).
    [CrossRef]
  11. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986), Chap. 4.
  12. E. Isaacson, H. B. Keller, Analysis of Numerical Methods (Wiley, New York, 1966).
  13. C. de Jong, “Precise mathematical description of the diffraction pattern beyond a square aperture in an absorbing medium,” Opt. Eng. 30, 641–645 (1991).
    [CrossRef]

1993 (1)

H. G. Kraus, “Finite element area and line integral transforms for generalization of aperture function and geometry in Kirchhoff scalar diffraction theory,” Opt. Eng. 32, 368–383 (1993).
[CrossRef]

1992 (2)

1991 (1)

C. de Jong, “Precise mathematical description of the diffraction pattern beyond a square aperture in an absorbing medium,” Opt. Eng. 30, 641–645 (1991).
[CrossRef]

1989 (3)

1983 (1)

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluations of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

1981 (1)

1964 (1)

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 8.

de Jong, C.

C. de Jong, “Precise mathematical description of the diffraction pattern beyond a square aperture in an absorbing medium,” Opt. Eng. 30, 641–645 (1991).
[CrossRef]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986), Chap. 4.

Hrynevych, M.

Isaacson, E.

E. Isaacson, H. B. Keller, Analysis of Numerical Methods (Wiley, New York, 1966).

Keller, H. B.

E. Isaacson, H. B. Keller, Analysis of Numerical Methods (Wiley, New York, 1966).

Kraus, H. G.

Marchand, E. W.

Pedersen, H. M.

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluations of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986), Chap. 4.

Rutt, H. N.

Sheppard, C. J. R.

Southwell, W. H.

Spjelkavik, B.

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluations of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

Stamnes, J. J.

J. J. Stamnes, “Hybrid integration technique for efficient and accurate computation of diffraction integrals,” J. Opt. Soc. Am. A 6, 1330–1342 (1989).
[CrossRef]

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluations of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

Steane, A. M.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986), Chap. 4.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986), Chap. 4.

Wolf, E.

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (5)

Opt. Acta (1)

J. J. Stamnes, B. Spjelkavik, H. M. Pedersen, “Evaluations of diffraction integrals using local phase and amplitude approximations,” Opt. Acta 30, 207–222 (1983).
[CrossRef]

Opt. Eng. (2)

H. G. Kraus, “Finite element area and line integral transforms for generalization of aperture function and geometry in Kirchhoff scalar diffraction theory,” Opt. Eng. 32, 368–383 (1993).
[CrossRef]

C. de Jong, “Precise mathematical description of the diffraction pattern beyond a square aperture in an absorbing medium,” Opt. Eng. 30, 641–645 (1991).
[CrossRef]

Other (3)

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), Chap. 8.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986), Chap. 4.

E. Isaacson, H. B. Keller, Analysis of Numerical Methods (Wiley, New York, 1966).

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Figures (6)

Fig. 1
Fig. 1

Geometry of expanding waves (see Kraus9).

Fig. 2
Fig. 2

Construction of z″ axis for expanding waves.

Fig. 3
Fig. 3

Geometry of converging waves.

Fig. 4
Fig. 4

Construction of z″ axis for converging waves.

Fig. 5
Fig. 5

Geometry of plane waves.

Fig. 6
Fig. 6

Construction of z″ axis for plane waves.

Equations (20)

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E P ( r P , ϕ , z ) = C 1 θ = 0 θ 1 ϕ = 0 2 π [ ( i k - 1 R ) cos θ + ( i k - 1 ρ ) ] exp ( i k R ) R sin θ d ϕ d θ ,
R = [ r P 2 + 2 ρ 2 ( 1 - cos θ ) + z + 2 z ρ ( 1 - cos θ ) - 2 r P ρ sin θ cos ( ϕ - ϕ ) ] 1 / 2 , cos θ = r P sin θ cos ( ϕ - ϕ ) - ρ + ( z + ρ ) cos θ R , sin θ 1 = a / ρ , C 1 = - E 0 ρ exp ( i k ρ ) 4 π ,
z = z + ρ ( 1 - cos α ) cos α = x P - ρ sin α sin α .
ϕ L ( θ , α ) = 0 , 0 θ θ 1 - α , ϕ L ( θ , α ) = cos - 1 ( cos θ cos α - cos θ 1 sin θ sin α ) , θ 1 - α θ θ 1 + α .
ψ ( θ , α ) = 2 π , 0 θ θ 1 - α , ψ ( θ , α ) = 2 cos - 1 ( cos θ 1 - cos θ cos α sin θ sin α ) , θ 1 - α θ θ 1 + α .
E P ( x P , z ) = E P ( 0 , z ) = C 1 θ = θ 0 α + θ 1 ψ ( θ , α ) [ ( i k - 1 R ) cos θ + ( i k - 1 ρ ) ] × exp ( i k R ) R sin θ d θ ,
R = [ 2 ρ 2 ( 1 - cos θ ) + z 2 + 2 z ρ ( 1 - cos θ ) ] 1 / 2 , cos θ = ( z + ρ ) cos θ - ρ R , θ 0 = { 0 α < θ 1 α - θ 1 α θ 1 ,
E P ( r P , ϕ , z ) = C 2 θ = 0 θ 1 ϕ = 0 2 π [ ( i k - 1 R ) cos θ + ( i k + 1 ρ ) ] × exp ( i k R ) R sin θ d ϕ d θ ,
R = [ r P 2 + ρ 2 + z 2 + 2 z ρ cos θ - 2 r P ρ sin θ cos ( ϕ - ϕ ) ] 1 / 2 , cos θ = z cos θ + ρ - r P sin θ cos ( ϕ - ϕ ) R , sin θ 1 = a / ρ , C 2 = - E 0 ρ exp ( - i k ρ ) 4 π .
z = x P sin α = z cos α .
E P ( x P , z ) = C 2 θ = θ 0 α + θ 1 ψ ( θ , α ) [ ( i k - 1 R ) cos θ + ( i k + 1 ρ ) ] × exp ( i k R ) R sin θ d θ ,
R = ( ρ 2 + z 2 + 2 z ρ cos θ ) 1 / 2 , cos θ = z cos θ + ρ R , θ 0 = { 0 α < θ 1 α - θ 1 α θ 1 .
E P ( r P , ϕ , z ) = C 3 ϕ = 0 2 π r = 0 a [ ( i k - 1 R ) cos θ + i k ] × exp ( i k R ) R r d r d ϕ ,
R = [ r P 2 + r 2 + z 2 + 2 r P r cos ( ϕ - ϕ ) ] 1 / 2 , cos θ = z R , C 3 = - E 0 exp ( i k z ) 4 π .
ϕ L ( r , x P ) = 0 , 0 r a - x P , ϕ L ( r , x P ) = cos - 1 ( a 2 - r 2 - x P 2 2 x P r ) , a - x P < r a + x P .
ψ ( r , x P ) = 2 π , 0 r a - x P , ψ ( r , x P ) = 2 cos - 1 ( r 2 + x P 2 - a 2 2 x P r ) , a - x P < r a + x P .
E P ( x P , z ) = C 3 r = r 0 a + x P ψ ( r , x P ) [ ( i k - 1 R ) cos θ + i k ] × exp ( i k R ) R r d r ,
R = ( r 2 + z 2 ) 1 / 2 , cos θ = z R , r 0 = { 0 x P < a x P - a x P a .
ρ sin θ cos ϕ L - ρ sin ( θ 1 - α ) = - ρ sin ( θ 1 + α ) - ρ sin ( θ 1 - α ) ρ cos ( θ 1 + α ) - ρ cos ( θ 1 - α ) × [ ρ cos θ - ρ cos ( θ 1 - α ) ] .
x P 2 + 2 x P r cos ϕ L + r 2 cos 2 ϕ L + r 2 sin 2 ϕ L = a 2 ,

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