Abstract

The Huygens–Fresnel diffraction integral has been formulated for incident spherical waves with use of the Kirchhoff obliquity factor and the wave front as the surface of integration instead of the aperture plane. Accurate numerical integration calculations were used to investigate very-near-field (a few aperture diameters or less) diffraction for the well-established case of a circular aperture. It is shown that the classical aperture-plane formulation degenerates when the wave front, as truncated at the aperture, has any degree of curvature to it, whereas the wave-front formulation produces accurate results from ∞ up to one aperture diameter behind the aperture plane. It is also shown that the Huygens–Fresnel–Kirchhoff incident-plane-wave-aperture-plane-integration and incident-spherical-wave-wave-front-integration formulations produce equally accurate results for apertures with exit f-numbers as small as 1.

© 1989 Optical Society of America

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References

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  1. B. B. Baker, E. T. Copson, The Mathematical Theory of Huygens’s Principle, 2nd ed. (Oxford U. Press, Oxford, 1969).
  2. E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Pa., 1987).
  3. A. Sommerfeld, Optics, Vol. 4 of Lectures on Theoretical Physics (Academic, New York, 1964).
  4. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  5. C. S. Williams, O. A. Beckland, Optics: A Short Course for Scientists and Engineers (Wiley-Interscience, New York, 1972).
  6. F. A. Jenkins, H. E. White, Fundamental Optics, 4th ed. (McGraw-Hill, New York, 1976).
  7. A. Nussbaum, R. A. Phillips, Contemporary Optics for Scientists and Engineers (Prentice-Hall, Englewood Cliffs, N.J., 1976).
  8. J. A. Hudson, “Fresnel–Kirchhoff diffraction in optical systems: an approximate computational algorithm,” Appl. Opt. 23, 2292–2295 (1984).
    [CrossRef] [PubMed]
  9. Y. P. Kathuria, “Fresnel and far-field diffraction due to an elliptical aperture,” J. Opt. Soc. Am. A 2, 852–857 (1985).
    [CrossRef]
  10. Y. P. Kathuria, “Computer modelling of three-dimensional Fresnel-diffraction pattern at circular, rectangular and square apertures,” Opt. Appl. 14, 509–514 (1984).
  11. D. S. Burch, “Fresnel diffraction by a circular aperture,” Am. J. Phys. 53, 255–260 (1985).
    [CrossRef]
  12. E. Wolf, E. W. Marchand, “Comparison of the Kirchhoff and the Rayleigh–Sommerfeld theories of diffraction at an aperture,”J. Opt. Soc. Am. 54, 587–594 (1964).
    [CrossRef]
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    [CrossRef]
  14. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  20. A. J. Campillo, J. E. Pearson, S. L. Shapiro, N. J. Terrell, “Fresnel diffraction effects in the design of high-power laser systems,” Appl. Phys. Lett. 23, 85–87 (1973).
    [CrossRef]
  21. J. T. Wesley, A. F. Behof, “Optical diffraction pattern measurements using a self-scanning photodiode array interfaced to a microcomputer,” Am. J. Phys. 55, 835–844 (1987).
    [CrossRef]
  22. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).
  23. B. T. Landesman, H. H. Barrett, “Gaussian amplitude functions that are exact solutions to the scalar Helmholtz equation,” J. Opt. Soc. Am. A 5, 1610–1619 (1988).
    [CrossRef]
  24. H. G. Kraus, “Huygens–Fresnel–Kirchhoff wave-front diffraction formulation: paraxial and exact Gaussian laser beams,” submitted to J. Opt. Soc. Am. A.

1988 (1)

1987 (1)

J. T. Wesley, A. F. Behof, “Optical diffraction pattern measurements using a self-scanning photodiode array interfaced to a microcomputer,” Am. J. Phys. 55, 835–844 (1987).
[CrossRef]

1985 (2)

Y. P. Kathuria, “Fresnel and far-field diffraction due to an elliptical aperture,” J. Opt. Soc. Am. A 2, 852–857 (1985).
[CrossRef]

D. S. Burch, “Fresnel diffraction by a circular aperture,” Am. J. Phys. 53, 255–260 (1985).
[CrossRef]

1984 (2)

Y. P. Kathuria, “Computer modelling of three-dimensional Fresnel-diffraction pattern at circular, rectangular and square apertures,” Opt. Appl. 14, 509–514 (1984).

J. A. Hudson, “Fresnel–Kirchhoff diffraction in optical systems: an approximate computational algorithm,” Appl. Opt. 23, 2292–2295 (1984).
[CrossRef] [PubMed]

1981 (1)

1979 (1)

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am J. Phys. 47, 974–980 (1979).
[CrossRef]

1978 (1)

1974 (1)

1973 (1)

A. J. Campillo, J. E. Pearson, S. L. Shapiro, N. J. Terrell, “Fresnel diffraction effects in the design of high-power laser systems,” Appl. Phys. Lett. 23, 85–87 (1973).
[CrossRef]

1964 (2)

Baker, B. B.

B. B. Baker, E. T. Copson, The Mathematical Theory of Huygens’s Principle, 2nd ed. (Oxford U. Press, Oxford, 1969).

Barrett, H. H.

Beckland, O. A.

C. S. Williams, O. A. Beckland, Optics: A Short Course for Scientists and Engineers (Wiley-Interscience, New York, 1972).

Behof, A. F.

J. T. Wesley, A. F. Behof, “Optical diffraction pattern measurements using a self-scanning photodiode array interfaced to a microcomputer,” Am. J. Phys. 55, 835–844 (1987).
[CrossRef]

Born, M.

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

Burch, D. S.

D. S. Burch, “Fresnel diffraction by a circular aperture,” Am. J. Phys. 53, 255–260 (1985).
[CrossRef]

Campillo, A. J.

A. J. Campillo, J. E. Pearson, S. L. Shapiro, N. J. Terrell, “Fresnel diffraction effects in the design of high-power laser systems,” Appl. Phys. Lett. 23, 85–87 (1973).
[CrossRef]

Copson, E. T.

B. B. Baker, E. T. Copson, The Mathematical Theory of Huygens’s Principle, 2nd ed. (Oxford U. Press, Oxford, 1969).

Feiock, F. D.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

Goodman, J. W.

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

Harris, F. S.

Harvey, J. E.

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am J. Phys. 47, 974–980 (1979).
[CrossRef]

Hecht, E.

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Pa., 1987).

Hudson, J. A.

Jenkins, F. A.

F. A. Jenkins, H. E. White, Fundamental Optics, 4th ed. (McGraw-Hill, New York, 1976).

Kathuria, Y. P.

Y. P. Kathuria, “Fresnel and far-field diffraction due to an elliptical aperture,” J. Opt. Soc. Am. A 2, 852–857 (1985).
[CrossRef]

Y. P. Kathuria, “Computer modelling of three-dimensional Fresnel-diffraction pattern at circular, rectangular and square apertures,” Opt. Appl. 14, 509–514 (1984).

Kraus, H. G.

H. G. Kraus, “Huygens–Fresnel–Kirchhoff wave-front diffraction formulation: paraxial and exact Gaussian laser beams,” submitted to J. Opt. Soc. Am. A.

Landesman, B. T.

Marchand, E. W.

Marsh, J. S.

Nussbaum, A.

A. Nussbaum, R. A. Phillips, Contemporary Optics for Scientists and Engineers (Prentice-Hall, Englewood Cliffs, N.J., 1976).

Pearson, J. E.

A. J. Campillo, J. E. Pearson, S. L. Shapiro, N. J. Terrell, “Fresnel diffraction effects in the design of high-power laser systems,” Appl. Phys. Lett. 23, 85–87 (1973).
[CrossRef]

Phillips, R. A.

A. Nussbaum, R. A. Phillips, Contemporary Optics for Scientists and Engineers (Prentice-Hall, Englewood Cliffs, N.J., 1976).

Shapiro, S. L.

A. J. Campillo, J. E. Pearson, S. L. Shapiro, N. J. Terrell, “Fresnel diffraction effects in the design of high-power laser systems,” Appl. Phys. Lett. 23, 85–87 (1973).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).

Smith, R. C.

Sommerfeld, A.

A. Sommerfeld, Optics, Vol. 4 of Lectures on Theoretical Physics (Academic, New York, 1964).

Southwell, W. H.

Terrell, N. J.

A. J. Campillo, J. E. Pearson, S. L. Shapiro, N. J. Terrell, “Fresnel diffraction effects in the design of high-power laser systems,” Appl. Phys. Lett. 23, 85–87 (1973).
[CrossRef]

Wesley, J. T.

J. T. Wesley, A. F. Behof, “Optical diffraction pattern measurements using a self-scanning photodiode array interfaced to a microcomputer,” Am. J. Phys. 55, 835–844 (1987).
[CrossRef]

White, H. E.

F. A. Jenkins, H. E. White, Fundamental Optics, 4th ed. (McGraw-Hill, New York, 1976).

Williams, C. S.

C. S. Williams, O. A. Beckland, Optics: A Short Course for Scientists and Engineers (Wiley-Interscience, New York, 1972).

Wolf, E.

Am J. Phys. (1)

J. E. Harvey, “Fourier treatment of near-field scalar diffraction theory,” Am J. Phys. 47, 974–980 (1979).
[CrossRef]

Am. J. Phys. (2)

J. T. Wesley, A. F. Behof, “Optical diffraction pattern measurements using a self-scanning photodiode array interfaced to a microcomputer,” Am. J. Phys. 55, 835–844 (1987).
[CrossRef]

D. S. Burch, “Fresnel diffraction by a circular aperture,” Am. J. Phys. 53, 255–260 (1985).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

A. J. Campillo, J. E. Pearson, S. L. Shapiro, N. J. Terrell, “Fresnel diffraction effects in the design of high-power laser systems,” Appl. Phys. Lett. 23, 85–87 (1973).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (2)

Opt. Appl. (1)

Y. P. Kathuria, “Computer modelling of three-dimensional Fresnel-diffraction pattern at circular, rectangular and square apertures,” Opt. Appl. 14, 509–514 (1984).

Other (11)

B. B. Baker, E. T. Copson, The Mathematical Theory of Huygens’s Principle, 2nd ed. (Oxford U. Press, Oxford, 1969).

E. Hecht, Optics, 2nd ed. (Addison-Wesley, Reading, Pa., 1987).

A. Sommerfeld, Optics, Vol. 4 of Lectures on Theoretical Physics (Academic, New York, 1964).

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

C. S. Williams, O. A. Beckland, Optics: A Short Course for Scientists and Engineers (Wiley-Interscience, New York, 1972).

F. A. Jenkins, H. E. White, Fundamental Optics, 4th ed. (McGraw-Hill, New York, 1976).

A. Nussbaum, R. A. Phillips, Contemporary Optics for Scientists and Engineers (Prentice-Hall, Englewood Cliffs, N.J., 1976).

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

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

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986).

H. G. Kraus, “Huygens–Fresnel–Kirchhoff wave-front diffraction formulation: paraxial and exact Gaussian laser beams,” submitted to J. Opt. Soc. Am. A.

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Figures (9)

Fig. 1
Fig. 1

Geometry for Huygens–Fresnel–Kirchhoff aperture-plane diffraction formulation for spherical waves.

Fig. 2
Fig. 2

Geometry for Huygens–Fresnel–Kirchhoff wave-front diffraction formulation for spherical waves.

Fig. 3
Fig. 3

Normalized intensity on optic axis in observation plane, In(r′ = 0), versus f′-number (exiting f-number) for even Fresnel numbers 2 through 10.

Fig. 4
Fig. 4

Normalized intensity, In, versus observation-plane radius, r′, for Fr = 2: (a) Table 4, case 4 (Table 2, case 4); (b) Table 3, case 4.

Fig. 5
Fig. 5

Normalized intensity, In, versus observation-plane radius, r′, for Fr = 4: (a) Table 4, case 8 (Table 2, case 8); (b) Table 3, case 8.

Fig. 6
Fig. 6

Normalized intensity, In, versus observation-plane radius, r′, for Fr = 6: (a) Table 4, case 12 (Table 2, case 12); (b) Table 3, case 12.

Fig. 7
Fig. 7

Normalized intensity, In, versus observation-plane radius, r′, for Fr = 8: (a) Table 4, case 16 (Table 2, case 16); (b) Table 3, case 16.

Fig. 8
Fig. 8

Normalized intensity, In, versus observation plane radius, r′, for Fr = 10: (a) Table 4, case 20 (Table 2, case 20); (b) Table 3, case 20.

Fig. 9
Fig. 9

Percent error in Fresnel number calculated by using z′ values from Table 4 in Eq. (10) versus aperture radius for even Fresnel numbers from 2 to 10 with ρ = 0.01 m, λ = 0.6328 m.

Tables (4)

Tables Icon

Table 1 Characteristic Circular-Aperture Fresnel-Number Equations of the Form Am2 + Bm + C for Plane-Wave and Spherical-Wave Huygens–Fresnel–Kirchhoff Diffraction Formulationsa

Tables Icon

Table 2 Huygens–Fresnel–Kirchhoff Aperture-Plane Diffraction Formulation: Plane-Wave Casesa

Tables Icon

Table 3 Huygens–Fresnel–Kirchhoff Aperture-Plane Diffraction Formulation: Spherical-Wave Casesa

Tables Icon

Table 4 Huygens–Fresnel–Kirchhoff Wave-Front Diffraction Formulation: Spherical-Wave Casesa

Equations (15)

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E P ( r , ϕ ) = C 0 2 π 0 a ( 1 + cos θ ) exp [ i k ( R + R ) ] R R r d r d ϕ ,
C = - ( E 0 / 2 λ ) i , k = 2 π / λ , R = ( r 2 + z 2 ) 1 / 2 , R = [ r 2 + r 2 - 2 r r cos ( ϕ - ϕ P ) + z 2 ] 1 / 2 , cos θ = ( - a ¯ 2 + b ¯ 2 + c ¯ 2 ) / 2 b c , a ¯ = [ r P 2 + r Q 2 - 2 r P r Q cos ( ϕ Q - ϕ P ) ] 1 / 2 , b ¯ = [ r Q 2 + r 2 - 2 r r Q cos ( ϕ - ϕ Q ) + z 2 ] 1 / 2 , c ¯ = R , ϕ Q = ϕ , r Q = r ( z + z ) / z .
Fr = ( 2 / λ ) [ ( z 2 + z 2 ) 1 / 2 + ( z 2 + a 2 ) 1 / 2 - ( z + z ) ] .
E p ( r , ϕ ) = C θ = 0 θ 1 ϕ = 0 2 π ( 1 + cos θ ) exp ( i k R ) sin θ R d θ d ϕ ,
C = - [ E 0 ρ exp ( i k ρ ) / 2 λ ] i , θ 1 = sin - 1 ( a / ρ ) , R = [ ρ 2 sin 2 θ + r P 2 - 2 ρ r P sin θ cos ( ϕ - ϕ P ) + ( z + h ) 2 ] 1 / 2 , h = ρ ( 1 - cos θ ) , cos θ = ( - a 2 + b 2 + c 2 ) / 2 b c , a = [ r P 2 + r Q 2 - 2 r P r Q cos ( ϕ P - ϕ Q ) ] 1 / 2 , b = [ ρ 2 sin 2 θ + r Q 2 - 2 r Q ρ sin θ cos ( ϕ - ϕ Q ) + ( z + h ) 2 ] 1 / 2 , c = R , ϕ Q = ϕ , r Q = ( ρ + z ) tan θ ,
A l = π ρ λ ( ρ + z ) [ z + λ ( 2 l - 1 ) 4 ] .
A t = 2 π ρ h ,
h = ρ [ 1 - ( 1 - a 2 ρ 2 ) 1 / 2 ] = ρ ( 1 - cos θ ) .
A t = l = 1 m A l ,
2 π ρ 2 [ 1 - ( 1 - a 2 ρ 2 ) 1 / 2 ] = π ρ λ ( ρ + z ) [ m z + λ 4 l = 1 m ( 2 l - 1 ) ] .
λ 2 4 ( ρ + z ) m 2 + λ z ( ρ + z ) m = 2 ρ [ 1 - ( 1 - a 2 ρ 2 ) 1 / 2 ] ,             m = Fr .
Fr = ( ρ + z ) a 2 λ ρ z = a 2 λ ( 1 ρ + 1 z ) .
Fr = a 2 λ z .
I n ( r , ϕ ) = ( 1 / 2 ) E P E P * / I max ,
S f = ( z s + h a ) / z p ,

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