Abstract

Complex contour integration techniques have been used to derive an asymptotic solution of the Huygens–Fresnel integral in circular coordinates. The solution (which is valid everywhere, including across the shadow boundary) provides a computationally efficient means to propagate uniform plane waves with high Fresnel numbers or to permit the evaluation of high-Fresnel-number unstable resonator modes. The results are interpreted as consisting of a geometrical term plus diffraction terms originating from two points on the circular aperture.

© 1978 Optical Society of America

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References

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  1. P. Horwitz, J. Opt. Soc. Am. 63, 1529–1543 (1973).
    [Crossref]
  2. P. Horwitz, Appl. Opt. 15, 167–178 (1976).
    [Crossref] [PubMed]
  3. G. T. Moore, R. J. McCarthy, J. Opt. Soc. Am. 67, 228–241, (1977).
    [Crossref]
  4. M. Abramowitz, J. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), pp. 364 and 302.
  5. The idea of restoring the Hankel function to the solution occurred to the author when he became aware of a post-deadline paper given at the 1977 meeting of the Optical Society of America at Toronto by R. R. Butts and P. V. Avizonis. They derived a solution of Eq. (1) in terms of asymptotic expansions of two Bessel functions, which they replaced by the Bessel functions. However, their solution, although correct near the axis, contained a singularity at the shadow boundry.

1977 (1)

1976 (1)

1973 (1)

P. Horwitz, J. Opt. Soc. Am. 63, 1529–1543 (1973).
[Crossref]

Abramowitz, M.

M. Abramowitz, J. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), pp. 364 and 302.

Horwitz, P.

P. Horwitz, Appl. Opt. 15, 167–178 (1976).
[Crossref] [PubMed]

P. Horwitz, J. Opt. Soc. Am. 63, 1529–1543 (1973).
[Crossref]

McCarthy, R. J.

Moore, G. T.

Stegun, J. A.

M. Abramowitz, J. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), pp. 364 and 302.

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

Other (2)

M. Abramowitz, J. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965), pp. 364 and 302.

The idea of restoring the Hankel function to the solution occurred to the author when he became aware of a post-deadline paper given at the 1977 meeting of the Optical Society of America at Toronto by R. R. Butts and P. V. Avizonis. They derived a solution of Eq. (1) in terms of asymptotic expansions of two Bessel functions, which they replaced by the Bessel functions. However, their solution, although correct near the axis, contained a singularity at the shadow boundry.

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Figures (2)

Fig. 1
Fig. 1

Complex contour used for the evaluation of Eq. (10).

Fig. 2
Fig. 2

Comparison of the amplitude and phase calculated using direct numerical integration of the Huygens–Fresnel integral and the asymptotic solution for a uniform plane wave incident on a circular aperture with radius a = 3.464 cm, Fresnel number N = 25, and l = 0. The fields are plotted out to 1.15 A where the shadow boundary is at A. The two cases plotted are nearly exactly overlapping.

Equations (25)

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f l 1 ( r ) i l + 1 2 t 0 a f l ( ρ ) J l ( 2 t r ρ ) × exp [ - i t ( ρ 2 + r 2 ) ] ρ d ρ ,
J l ( χ ) = 1 2 [ H l ( 1 ) ( χ ) + H l ( 2 ) ( χ ) ] .
f l 1 ( r ) = i l + 1 t [ I ( 1 ) exp ( - i ψ l ) + I ( 2 ) exp ( i ψ l ) ] ,
I ( 1 ) = 0 a f l ( ρ ) G l ( 1 ) ( 2 t r ρ ) exp [ - i t ( ρ - r ) 2 ] ρ d ρ ,
I ( 2 ) = 0 a f l ( ρ ) G l ( 2 ) ( 2 t r ρ ) exp [ - i t ( ρ + r ) 2 ] ρ d ρ .
H l ( 1 ) ( Z ) = G l ( 1 ) ( Z ) exp ( i Z ) exp ( - i ψ l ) ,
H l ( 2 ) ( Z ) = G l ( 2 ) ( Z ) exp ( - i Z ) exp ( i ψ l ) ,
G l ( 1 ) ( Z ) = ( 2 / π Z ) 1 / 2 [ P l ( Z ) + i Q l ( Z ) ] ,
G l ( 2 ) ( Z ) = ( 2 / π Z ) 1 / 2 [ P l ( Z ) - i Q l ( Z ) ] .
I = f l ( Z ) G l ( 1 ) ( 2 t r Z ) exp [ - i t ( Z - r ) 2 ] Z d Z ,
I ( 1 ) = - I 1 - I 2 - I 3 .
I 1 = - i exp [ - i t ( a - r ) 2 ] 0 f l ( a - i y ) G l ( 1 ) [ 2 t r ( a - i y ) ] exp [ - 2 t ( a - r ) y ] exp ( i t y 2 ) ( a - i y ) d y .
I 1 = - i f l ( a ) G l ( 1 ) ( 2 t r a ) a ( π / 2 t ) 1 / 2 ϕ [ ( 2 t / π ) 1 / 2 ( a - r ) ] × exp [ - i t ( a - r ) 2 ] + .
I 2 = i - f l ( r + i y ) G l ( 1 ) [ 2 t r ( r + i y ) ] × exp ( i t y 2 ) ( r + i y ) d y .
I 2 = i f l ( r ) G l ( 1 ) ( 2 t r 2 ) ( i π / t ) 1 / 2 r + .
I 3 = exp ( - i t r 2 ) 0 f l ( i y ) H l ( 1 ) ( 2 t r i y ) × exp ( i t y 2 ) y d y .
I = f l ( z ) G l ( 2 ) ( 2 t r Z ) exp [ i t ( Z + r ) 2 ] Z d Z ,
I ( 2 ) = - I 4 - I 5 ,
I 4 = - i f l ( a ) G l ( 2 ) ( 2 t r a ) a ( π / 2 t ) 1 / 2 ϕ [ ( 2 t / π ) 1 / 2 ( a + r ) ] × exp ( - i t ( a + r ) 2 ] + ,
f l 1 ( r ) = f l ( r ) - i l f l ( a ) ( π t / 2 ) 1 / 2 a exp [ - i t ( a 2 + r 2 ) ] × { H l ( 1 ) ( 2 t r a ) ϕ [ ( 2 t / π ) 1 / 2 ( a - r ) ] + H l ( 2 ) ( 2 t r a ) ϕ [ ( 2 t / π ) 1 / 2 ( a + r ) ] } .
0 exp ( i t y 2 ) d y = 1 2 ( i π / t ) 1 / 2 ,
0 y exp ( i t y 2 ) d y = i / 2 t .
0 y 2 exp ( i t y 2 ) d y = ( i / 4 t ) ( i π / t ) 1 / 2 .
0 exp ( - c y ) exp ( i t y 2 ) d y = ( π / 2 t ) 1 / 2 ϕ [ ( c / 2 ) ( 2 / π t ) 1 / 2 ] ,
0 y exp ( - c y ) exp ( i t y 2 ) d y = ( i / 2 t ) × { 1 - ( π c / 2 ) ( 2 / π t ) 1 / 2 ϕ [ ( c / 2 ) ( 2 / π t ) 1 / 2 ] } .

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