Abstract

In an earlier paper dealing with a consistent formulation of Kirchhoff’s theory, an extension of the theory was proposed which seems to be applicable to treatment of diffraction at very small apertures in black screens. The extension consists of the addition to Kirchhoff’s solution of the effect of waves arising from multiple diffraction at the edge of the aperture. In the present paper, calculations based on this modified theory are presented. Corrections to Kirchhoff’s theory are obtained for the case of a plane wave incident normally on a small circular aperture in a black screen. Appreciable departures from Kirchhoff’s theory are found only when the diameter of the aperture is small compared with the wavelength of the incident wave or when the angles of diffraction are very large. It is also shown that in the asymptotic limit ka→∞ (k is the wave number, a the radius of the aperture) our results are consistent with those obtained on the basis of Keller’s geometrical theory of diffraction.

© 1969 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. There are a number of studies relating to diffraction at small apertures in conducting screens. For references and review see C. J. Bouwkamp, Rept. Prog. Phys. 18, 35 (1954).
    [CrossRef]
  2. E. W. Marchand and E. Wolf, J. Opt. Soc. Am. 56, 1712 (1966).
    [CrossRef]
  3. (a)J. B. Keller, J. Appl. Phys. 28, 426 (1957); (b)J. Appl. Phys. 28, 570 (1957) (with R. M. Lewis and D. Seckler); (c) in Calculus of Variations, L. M. Graves, Ed. (McGraw–Hill Book Co., New York, 1958) p. 27; (d)J. Opt. Soc. Am. 52, 116 (1962).
  4. In this connection, see also articles by F. J. Kottler, in Progress in Optics, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, and J. Wiley & Sons, New York) 4, 281 (1965); Progress in Optics 6, 331 (1967), and the preface to Vol. 4.
    [CrossRef]
  5. (a)A. Rubinowicz, Ann. Phys. (Leipzig) 53, 257 (1917); (b)Die Beugungswelle in der Kirchhoffschen Theorie der Beugung, Zweite Aufl. (Springer-Verlag, Berlin, 1966).
  6. M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon Press, London and New York, 1965).
  7. Explicit expressions for the multiply scattered waves are given in § 5 of Ref. 2.
  8. The choice of our normalization of A2 and AK is based on the behavior of these quantities for small values of ka. It may be shown that, for sufficiently small values of ka, A2(ψ; ka) ≈ (ka/4π), AK(ψ; ka) ≈ −(i/2)(ka)2cos2(ψ/2).

1966 (1)

1957 (1)

(a)J. B. Keller, J. Appl. Phys. 28, 426 (1957); (b)J. Appl. Phys. 28, 570 (1957) (with R. M. Lewis and D. Seckler); (c) in Calculus of Variations, L. M. Graves, Ed. (McGraw–Hill Book Co., New York, 1958) p. 27; (d)J. Opt. Soc. Am. 52, 116 (1962).

1954 (1)

There are a number of studies relating to diffraction at small apertures in conducting screens. For references and review see C. J. Bouwkamp, Rept. Prog. Phys. 18, 35 (1954).
[CrossRef]

1917 (1)

(a)A. Rubinowicz, Ann. Phys. (Leipzig) 53, 257 (1917); (b)Die Beugungswelle in der Kirchhoffschen Theorie der Beugung, Zweite Aufl. (Springer-Verlag, Berlin, 1966).

Born, M.

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon Press, London and New York, 1965).

Bouwkamp, C. J.

There are a number of studies relating to diffraction at small apertures in conducting screens. For references and review see C. J. Bouwkamp, Rept. Prog. Phys. 18, 35 (1954).
[CrossRef]

Keller, J. B.

(a)J. B. Keller, J. Appl. Phys. 28, 426 (1957); (b)J. Appl. Phys. 28, 570 (1957) (with R. M. Lewis and D. Seckler); (c) in Calculus of Variations, L. M. Graves, Ed. (McGraw–Hill Book Co., New York, 1958) p. 27; (d)J. Opt. Soc. Am. 52, 116 (1962).

Kottler, F. J.

In this connection, see also articles by F. J. Kottler, in Progress in Optics, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, and J. Wiley & Sons, New York) 4, 281 (1965); Progress in Optics 6, 331 (1967), and the preface to Vol. 4.
[CrossRef]

Marchand, E. W.

Rubinowicz, A.

(a)A. Rubinowicz, Ann. Phys. (Leipzig) 53, 257 (1917); (b)Die Beugungswelle in der Kirchhoffschen Theorie der Beugung, Zweite Aufl. (Springer-Verlag, Berlin, 1966).

Wolf, E.

E. W. Marchand and E. Wolf, J. Opt. Soc. Am. 56, 1712 (1966).
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon Press, London and New York, 1965).

Ann. Phys. (Leipzig) (1)

(a)A. Rubinowicz, Ann. Phys. (Leipzig) 53, 257 (1917); (b)Die Beugungswelle in der Kirchhoffschen Theorie der Beugung, Zweite Aufl. (Springer-Verlag, Berlin, 1966).

J. Appl. Phys. (1)

(a)J. B. Keller, J. Appl. Phys. 28, 426 (1957); (b)J. Appl. Phys. 28, 570 (1957) (with R. M. Lewis and D. Seckler); (c) in Calculus of Variations, L. M. Graves, Ed. (McGraw–Hill Book Co., New York, 1958) p. 27; (d)J. Opt. Soc. Am. 52, 116 (1962).

J. Opt. Soc. Am. (1)

Rept. Prog. Phys. (1)

There are a number of studies relating to diffraction at small apertures in conducting screens. For references and review see C. J. Bouwkamp, Rept. Prog. Phys. 18, 35 (1954).
[CrossRef]

Other (4)

In this connection, see also articles by F. J. Kottler, in Progress in Optics, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, and J. Wiley & Sons, New York) 4, 281 (1965); Progress in Optics 6, 331 (1967), and the preface to Vol. 4.
[CrossRef]

M. Born and E. Wolf, Principles of Optics, 3rd ed. (Pergamon Press, London and New York, 1965).

Explicit expressions for the multiply scattered waves are given in § 5 of Ref. 2.

The choice of our normalization of A2 and AK is based on the behavior of these quantities for small values of ka. It may be shown that, for sufficiently small values of ka, A2(ψ; ka) ≈ (ka/4π), AK(ψ; ka) ≈ −(i/2)(ka)2cos2(ψ/2).

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 (17)

Fig. 1
Fig. 1

Illustration of the notation relating to Eqs. (1.1). a denotes the direct geometrical beam, the shadow region.

Fig. 2
Fig. 2

The structure of the singly (U1(s)), doubly (U2(s)) and triply (U3(s)) scattered boundary waves.

Fig. 3
Fig. 3

Notation relating to the scattered boundary waves U1(s)(P) and U2(s)(P).

Fig. 4
Fig. 4

The amplitude |U2(s)| of the doubly scattered boundary wave in the near field, as a function of the normalized off-axis distance r/a, in plane at distance z=λ/2, from the plane of the aperture. The incident field is a monochromatic plane wave of wavelength λ falling normally on the plane of the aperture. The aperture is circular, of radius a.

Fig. 5
Fig. 5

Same as Fig. 4, in plane at the distance z=λ/10 from the plane of the aperture.

Fig. 6
Fig. 6

Same as Fig. 4, in plane at the distance z=λ/100 from the plane of the aperture.

Fig. 7
Fig. 7

The amplitude |U2(s)| of the doubly scattered boundary wave on the axis, as a function of the normalized distance kz from the aperture (z=actual distance, k=2π/λ, λ=wavelength). The incident field is a monochromatic plane wave of wavelength λ, falling normally on the plane of the aperture. The aperture is circular, of radius a.

Fig. 8
Fig. 8

The amplitude |U1(s)| of the singly scattered boundary wave on the axis, as a function of the normalized distance kz from the aperture (z=actual distance, k=2π/λ). The incident field is a monochromatic plane wave of wavelength λ, falling normally on the plane of the aperture. The aperture is circular, of radius a.

Fig. 9
Fig. 9

The amplitude |UK| of the Kirchhoff field on the axis as a function of the normalized distance kz from the aperture (z=actual distance, k=2π/λ). The incident field is a monochromatic plane wave of wavelength λ, falling normally on the plane of the aperture. The aperture is circular, of radius a.

Fig. 10
Fig. 10

Illustration of the notation relating to the far field. The incident field is a monochromatic plane wave of wavelength λ, falling normally on the plane of the aperture.

Fig. 11
Fig. 11

The normalized amplitude of the radiation pattern of the doubly scattered boundary wave (U2(s)) as a function of the angle of scattering. The incident field is a monochromatic plane wave of wavelength λ, falling normally on the plane of the aperture. The aperture is circular, of radius a; k=2π/λ.

Fig. 12
Fig. 12

The normalized amplitude of the radiation pattern of the Kirchhoff field UK as a function of the angle ψ of scattering. The incident field is a monochromatic plane wave of wavelength λ, falling normally on the plane of the aperture. The aperture is circular, of radius a; k=2π/λ.

Fig. 13
Fig. 13

The ratio |A2/AK| of the amplitudes of the radiation patterns of the doubly scattered field and of the Kirchhoff field. The incident field is a monochromatic plane wave of wavelength λ, falling normally on the plane of the aperture. The aperture is circular, of radius a; k=2π/λ.

Fig. 14
Fig. 14

Notation relating to Formulae (4.8)–(4.14)

Fig. 15
Fig. 15

Notation relating to Keller’s geometrical theory of diffraction. The projections of the incident and the diffracted rays into a plane normal to the edge. The angles α and θ are the the angles between these projections and the negative z axis, measured as shown in the figure. The edge is perpendicular to the plane of the figure.

Fig. 16
Fig. 16

Notation relating to Keller’s theory of diffraction; doubly diffracted rays.

Fig. 17
Fig. 17

Notation relating to Keller’s theory of diffraction; singly diffracted rays.

Equations (104)

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

U K ( P ) = U ( i ) ( P ) + U 1 ( s ) ( P ) when P is in a , = U 1 ( s ) ( P ) when P is in .
U 1 ( s ) ( P ) = Γ U ( i ) ( Q 1 ) exp ( i k s 1 P ) s 1 P × K ( SQ 1 , Q 1 P ) · d l 1 ,
K ( SQ , QP ) = 1 4 π SQ × QP SQ OP - SQ · OP
U ( P ) = U K ( P ) + n = 2 U n ( s ) ( P )
= U ( i ) ( P ) + n = 1 U n ( s ) ( P ) ,
= 1 if P a , ( direct beam ) , = 0 if P , ( shadow region ) .
K ( Q j Q j + 1 , Q j + 1 Q j + 2 ) · d l = 0.
U n ( s ) ( P ) = 0     if     n 3.
U ( P ) = U K ( P ) + U 2 ( s ) ( P )
= U ( i ) ( P ) + U 1 ( s ) ( P ) + U 2 ( s ) ( P ) .
U ( i ) ( r ) = exp ( i k p · r )
U 1 ( s ) ( P ) = 1 4 π Γ exp ( i k p · r ) ( e i k s 1 P s 1 P ) ( p × s 1 P s 1 P - p · s 1 P ) · d l 1 ,
U 2 ( s ) ( P ) = 1 ( 4 π ) 2 Γ Γ exp ( i k p · r ) e i k s 12 s 12 e i k s 2 P s 2 P × [ ( p × s 12 ) · d l 1 s 12 - p · s 12 ] [ ( s 12 × s 2 P ) · d l 2 s 12 s 2 P - s 12 · s 2 P ] .
s 12 = 2 a sin ( θ 2 - θ 1 ) / 2 ,
s 2 P = [ a 2 + r 2 + z 2 - 2 r a cos ( θ 2 - ϕ ) ] 1 2 ,
( p × s 12 ) · d l 1 = - ( 1 2 ) p 3 s 12 2 d θ 1 ,
s 12 - p · s 12 = s 12 [ 1 ± ρ sin ( ( θ 1 + θ 2 ) / 2 - χ ) ] ,
( s 12 × s 2 P ) · d l 2 = - ( 1 2 ) z s 12 2 d θ 2 ,
s 12 s 2 P - s 12 · s 2 P = s 12 [ s 2 P + s 12 / 2 ± r sin ( ( θ 1 + θ 2 ) / 2 - ϕ ) ] ,
U 2 ( s ) ( P ) = 1 ( 8 π ) 2 p 3 z 0 2 π 0 2 π × s 12 exp { i k [ a ρ cos ( θ 1 - χ ) + s 12 + s 2 P ] } s 2 P F G d θ 1 d θ 2 ,
F = 1 ± ρ sin ( ( θ 1 + θ 2 ) / 2 - χ ) ,
G = s 2 P + ( 1 2 ) s 12 ± r sin ( ( θ 1 + θ 2 ) / 2 - ϕ ) ,
I ( P ) = 0 2 π 0 2 π × s 12 2 exp { i k [ a ρ cos ( θ 1 - χ ) + s 12 + s 2 P ] } s 2 P ( s 12 - p · s 12 ) ( s 12 s 2 P - s 12 · s 2 P ) d θ 1 d θ 2 ,
U 2 ( s ) ( P ) 0 as z + 0.
p 1 = p 2 = 0 ,             p 3 = 1 ,
ϕ = π / 2 , x = 0 and r = y ,
U 2 ( s ) ( P ) = z ( 8 π ) 2 0 2 π 0 2 π × s 12 exp [ i k ( s 12 + s 2 P ) ] s 2 P [ s 2 P + ( s 12 + 2 ) y cos ( ( θ 1 + θ 2 ) / 2 ) ] d θ 1 d θ 2 .
U 2 ( s ) ( P ) = a z ( 4 π ) 2 0 2 π d α 0 π × sin α 0 exp [ i k ( 2 a sin α 0 + s 2 P ) ] s 2 P [ s 2 P + a sin α 0 - y cos ( α 0 - α ) ] d α 0 ,
s 2 P = ( a 2 + y 2 + z 2 - 2 a y sin α ) 1 2 .
U 2 ( s ) ( z ) = a z ( 4 π ) 2 0 2 π d α 0 π × sin α 0 [ exp i k ( 2 a sin α 0 + s 2 P ) ] s 2 P [ s 2 P + a sin α 0 ] d α 0 ,
s 2 P = ( a 2 + z 2 ) 1 2 .
U 2 ( s ) ( z ) = a z 4 π exp ( i k s 2 P ) s 2 P × 0 π / 2 sin α 0 exp { i k ( 2 a sin α 0 ) } s 2 P + a sin α 0 d α 0 .
y = R sin ψ ,             z = R cos ψ .
s 2 P = ( a 2 + R 2 - 2 a R sin ψ sin α ) 1 2 = R [ 1 - 2 ( a / R ) sin ψ sin α + ( a / R ) 2 ] 1 2 .
s 2 P ~ R [ 1 - ( a / R ) sin ψ sin α ] .
U 2 ( s ) ( P ) = a cos ψ ( 4 π ) 2 0 2 π d α 0 π × sin α 0 exp [ i k ( 2 a sin α 0 + R - a sin ψ sin α ) ] R + a sin α 0 - R sin ψ cos ( α 0 - α ) d α 0 .
U 2 ( s ) ( P ) = A 2 ( ψ ; k a ) [ exp ( i k R ) / k R ] ,
A 2 ( ψ ; k a ) = k a cos ψ ( 4 π ) 2 0 2 π d α 0 π × sin α 0 exp [ i k a ( 2 sin α 0 - sin ψ sin α ) ] 1 - sin ψ cos ( α 0 - α ) d α 0 .
A 2 ( 0 ; k a ) = k a 4 π 0 π / 2 sin α 0 exp ( 2 i k a sin α 0 ) d α 0
= k a 4 π 0 π / 2 sin α 0 cos ( 2 k a sin α 0 ) d α 0 + i k a 8 J 1 ( 2 k a ) ,
H 0 ( z ) = 2 π 0 π / 2 sin ( z sin α 0 ) d α 0 ,
U 2 ( s ) = 0 2 π d α 0 π g ( α 0 , α ) exp [ i k a f ( α 0 , α ) ] d α 0 ,
g ( α 0 , α ) = z ( 4 π ) 2 sin α 0 s 2 P [ s 2 P + sin α 0 - y cos ( α 0 - α ) ] ,
f ( α 0 , α ) = 2 sin α 0 + s 2 P ,
y = y / a ,             z = z / a ,
s 2 P = s 2 P / a = ( 1 + y 2 + z 2 - 2 y sin α ) 1 2 .
U 2 ( s ) ( P ) ~ z 8 π k a exp ( 2 i k a ) ( 2 y ) 1 2 [ exp ( i k a l 1 ) l 1 1 2 ( l 1 + 1 - y ) - i exp ( i k a l 2 ) l 2 1 2 ( l 2 + 1 + y ) ] ,
l 1 = ( 1 - 2 y + y 2 + z 2 ) 1 2 , l 2 = ( 1 + 2 y + y 2 + z 2 ) 1 2 .
r 1 = a l 1 ,             r 2 = a l 2
U 2 ( s ) ( P ) ~ ( 2 / y ) 1 2 z exp ( 2 i k a ) 16 π k [ exp ( i k r 1 ) r 1 1 2 ( a - y + r 1 ) - i exp ( i k r 2 ) r 2 1 2 ( a + y + r 2 ) ] .
r 1 = [ ( a - y ) 2 + z 2 ] 1 2 ,             r 2 = [ ( a + y ) 2 + z 2 ] 1 2 .
a - y = r 1 cos δ 1 , z = r 1 sin δ 1 , a + y = r 2 cos δ 2 , z = r 2 sin δ 2 .
z y 1 2 1 a - y + r 1 = sin δ 1 a 1 2 ( 1 + cos δ 1 ) ( 1 - r 1 cos δ 1 / a ) 1 2 ,
z y 1 2 1 a + y + r 2 = - i sin δ 2 a 1 2 ( 1 + cos δ 12 ) ( 1 - r 2 cos δ 2 / a ) 1 2 ,
U 2 ( s ) ~ exp ( 2 i k a ) 8 π k ( 2 a ) 1 2 [ sin δ 1 1 + cos δ 1 exp ( i k r 1 ) [ r 1 ( 1 - r 1 cos δ 1 / a ) ] 1 2 + sin δ 2 1 + cos δ 2 exp ( i k r 2 ) [ r 2 ( 1 - r 2 cos δ 2 / a ) ] 1 2 ] .
U ( P ) = D U ( i ) [ r ( 1 + r / ρ ) ] - 1 2 exp ( i k r ) .
ρ = - ρ sin 2 β ( ρ β ˙ sin β + cos δ ) - 1 .
D = exp ( i π / 4 ) ( cos θ - cos α ) 2 ( 2 π k ) 1 2 sin β ( sin θ + sin α ) .
U ( i ) = 1 ,             r = r j , ρ = a ,             β = π / 2 ,             β ˙ = 0 , δ 1 = PA 1 A 2 ,             δ 2 = PA 2 A 1 , ρ j = - a / cos δ j ,             ( j = 1 , 2 ) , θ j = π / 2 + δ j ,             α j = 0 ,             ( j = 1 , 2 ) .
U j ( P ) = D j [ r j ( 1 - r j cos δ j / a ) ] - 1 2 exp ( i k r j ) ,
D j = - exp ( i π / 4 ) ( 1 + sin δ j ) / 2 ( 2 π k ) 1 2 cos δ j .
U 1 ( A 2 ) = - exp [ i ( 2 k a - π / 4 ) ] / 4 ( π k a ) 1 2 .
U 1 = U 1 ( A 2 ) [ Eq . ( 5.7 ) above ] ,             r = r 2 , ρ = a ,             β = ( π / 2 ) ,             β ˙ = 0 , δ = δ 2 ,             ρ = - a / cos δ 2 , θ = π / 2 + δ 2 ,             α = π / 2.
D 2 = - sin δ 2 exp ( i π / 4 ) 2 ( 2 π k ) 1 2 ( 1 + cos δ 2 ) .
U 12 ( P ) = sin δ 2 1 + cos δ 2 exp ( i 2 k a ) 8 π k ( 2 a ) 1 2 exp ( i k r 2 ) [ r 2 ( 1 - r 2 cos δ 2 / a ) ] 1 2 .
U 2 ( A 1 ) = exp [ i ( 2 k a + π / 4 ) ] / 4 ( π k a ) 1 2 .
U 2 = U 2 ( A 1 ) , [ Eq . ( 5.11 ) above ] , r = r 1 , ρ = a ,             β = ( π / 2 ) ,             β ˙ = 0 , δ = + δ 1 ,             ρ = - a / cos δ 1 , θ = π / 2 + δ 1 ,             α = π / 2.
D 1 = - sin δ 1 exp ( i π / 4 ) 2 ( 2 π k ) 1 2 ( 1 + cos δ 1 ) .
U 21 ( P ) = sin δ 1 1 + cos δ 1 exp ( i 2 k a ) 8 π k ( 2 a ) 1 2 exp ( i k r 1 ) [ r 1 ( 1 - r 1 cos δ 1 / a ) ] 1 2 .
U ( P ) = exp ( i 2 k a ) 8 π k ( 2 a ) 1 2 [ sin δ 1 1 + cos δ 1 exp ( i k r 1 ) [ r 1 ( 1 - r 1 cos δ 1 / a ) ] 1 2 + sin δ 2 1 + cos δ 2 exp ( i k r 2 ) [ r 2 ( 1 - r 2 cos δ 2 / a ) ] 1 2 ] .
U 2 ( s ) ( P ) = z ( 8 π ) 2 0 2 π 0 2 π × s 12 exp [ i k ( s 12 + s 2 P ) ] s 2 P [ s 2 P + 1 2 s 12 y cos ( ( θ 1 + θ 2 ) / 2 ) ] d θ 1 d θ 2 ,
α 1 = θ 1 / 2 ,             α 2 = θ 2 / 2.
s 12 = 2 a sin α 2 - α 1 ,
s 2 P = [ a 2 + y 2 + z 2 - 4 a y sin α 2 cos α 2 ] 1 2 .
U 2 ( s ) ( P ) = [ 2 a z / ( 4 π ) 2 ] ( A + B ) ,
A = 0 π d α 1 0 α 1 sin ( α 1 - α 2 ) exp { i k [ 2 a sin ( α 1 - α 2 ) + s 2 P ] } s 2 P [ s 2 P + a sin ( α 1 - α 2 ) + y cos ( α 1 + α 2 ) ] d α 2 ,
B = 0 π d α 2 0 α 2 sin ( α 2 - α 1 ) exp { i k [ 2 a sin ( α 2 - α 1 ) + s 2 P ] } s 2 P [ s 2 P + a sin ( α 2 - α 1 ) - y cos ( α 1 + α 2 ) ] d α 1 .
A = 0 π d α 1 π π + α 1 sin ( α ¯ 2 - α 1 ) exp { i k [ 2 a sin ( α ¯ 2 - α 1 ) + s 2 P ] } s 2 P [ s 2 P + a sin ( α ¯ 2 - α 1 ) - y cos ( α 1 + α ¯ 2 ) ] d α ¯ 2 .
B = 0 π d α 1 α 1 π sin ( α 2 - α 1 ) exp { i k [ 2 a sin ( α 2 - α 1 ) + s 2 P ] } s 2 P [ s 2 P + a sin ( α 2 - α 1 ) - y cos ( α 1 + α 2 ) ] d α 2 .
A + B = 0 π d α 1 α 1 α 1 + π sin ( α 2 - α 1 ) exp { i k [ 2 a sin ( α 2 - α 1 ) + s 2 P ] } s 2 P [ s 2 P + a sin ( α 2 - α 1 ) - y cos ( α 1 + α 2 ) ] d α 2 .
A + B = 0 π d α 1 0 π sin α 0 exp { i k [ 2 a sin α 0 + s 2 P ] } s 2 P [ s 2 P + a sin α 0 - y cos ( α 0 + 2 α 1 ) ] d α 0 ,
s 2 P = [ a 2 + y 2 + z 2 - 2 a y sin 2 ( α 0 + α 1 ) ] 1 2 .
U 2 ( s ) ( P ) = a z ( 4 π ) 2 0 2 π d α 0 π sin α 0 exp { i k [ 2 a sin α 0 + s 2 P ] } s 2 P [ s 2 P + a sin α 0 - y cos ( α 0 - α ) ] d α 0 ,
s 2 P = ( a 2 + y 2 + z 2 - 2 a y sin α ) 1 2 .
U 2 ( s ) ( P ) = 0 2 π d α 0 π g ( α 0 , α ) exp { i k a f ( α 0 , α ) } d α 0
f / α 0 = f / α = 0.
U 2 ( s ) ( P ) = 2 π i k a j { σ ( A B - C 2 ) 1 2 g ( α 0 , α ) × exp [ i k a f ( α 0 , α ) ] } j ,
A = 2 f / α 0 2 ,             B = 2 f / α 2 ,             C = 2 f / α 0 α ,
σ = { + 1 if A B > C 2 , A > 0 , - 1 if A B > C 2 , A < 0 , - i if A B < C 2 . }
f ( α 0 , α ) = 2 sin α 0 + s 2 P ,
g ( α 0 , α ) = z ( 4 π ) 2 sin α 0 s 2 P [ s 2 P + sin α 0 - y cos ( α 0 - α ) ] ,
s 2 P = ( 1 + y 2 + z 2 - 2 y sin α ) 1 2 .
f / α 0 = 2 cos α 0 ,             f / α = ( - y / s 2 P ) cos α ,
A = - 2 sin α 0 ,             B = ( y / s 2 P ) sin α - [ y 2 / ( s 2 P ) 3 ] cos 2 α , C = 0.
α 0 = π / 2 ,             α = π / 2     and     α 0 = π / 2 ,             α = 3 π / 2.
A = - 2 ,             B = y / l 1 ,             C = 0 ,             σ = - i ,
l 1 = ( 1 - 2 y + y 2 + z 2 ) 1 2
f = 2 + l 1 , g = [ z / ( 4 π ) 2 ] / l 1 ( l 1 + 1 - y ) .
z 8 π k a exp ( 2 i k a ) ( 2 y ) 1 2 [ exp ( i k a l 1 ) l 1 1 2 ( l 1 + 1 - y ) ] .
A = - 2 ,             B = - y / l 2 ,             C = 0 ,             σ = - 1 ,
l 2 = ( 1 + 2 y + y 2 + z 2 ) 1 2
f = 2 + l 2 , g = [ z / ( 4 π ) 2 ] / l 2 ( l 2 + 1 + y ) .
z 8 π k a exp ( 2 i k a ) ( 2 y ) 1 2 [ - i exp ( i k a l 2 ) l 2 1 2 ( l 2 + 1 + y ) ] .
U 2 ( s ) ( P ) ~ z 8 π k a exp ( 2 i k a ) ( 2 y ) 1 2 [ exp ( i k a l 1 ) l 1 1 2 ( l 1 + 1 - y ) - i exp ( i k a l 2 ) l 2 1 2 ( l 2 + 1 + y ) ] .