Abstract

Kirchhoff’s diffraction theory, which is often criticized because of an apparent internal inconsistency, is shown to be a rigorous solution to a certain boundary-value problem that has a clear physical meaning. This new interpretation of Kirchhoff’s theory is a direct consequence of the Rubinowicz theory of the boundary diffraction wave.

We argue that these true boundary conditions of Kirchhoff’s theory are physically reasonable for diffraction at an aperture in a black screen whose linear dimensions are large compared with the wavelength. The boundary values which Kirchhoff’s solution takes in the plane of the aperture and in the near zone on the axis for the case of a normally incident plane wave diffracted at a circular aperture are compared with previously published results of experiments with microwaves, and reasonable agreement is found. (Strict agreement cannot be expected since the screens used in the microwave experiments were not black.) Moreover, Kirchhoff’s solution is found to be in closer agreement with the experimental results than the “manifestly consistent” Rayleigh–Sommerfeld theory.

We also suggest an extension of the Kirchhoff theory, which might provide a physically reasonable approximation to the solution of the problems of diffraction at an aperture in a black screen whose linear dimensions are of the order of magnitude of or smaller than the wavelength of the light.

© 1966 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1965), 3rd ed.
  2. B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens’ Principle (Clarendon Press, Oxford, England, 1950), 2nd ed.
  3. H. Poincaré, Théorie Mathématique de la Lumière (Georges Carré, Paris, 1892), Pt. II, pp. 187–188.
  4. G. Toraldo di Francia, Atti Fond. Giorgio Ronchi 11, 503 (1956).
  5. M. Boron, Optik (Julius Springer-Verlag, Berlin, 1933: re printed 1965), p. 152.
  6. W. Franz, Z. Physik 125, 563 (1949).
    [CrossRef]
  7. S. A. Schelkunoff, Commun. Pure Appl. Math. 4, 43 (1951).
    [CrossRef]
  8. C. J. Bouwkamp, Rept. Progr. Phys. 17, 35 (1954).
    [CrossRef]
  9. Some of the modified theories are reviewed in Ref. 8, Sec. 4.
  10. (a)F. Kottler, Ann. Physik (Leipzig) 70, 405 (1923). (b) in Progress in Optics IV, E. Wolf, Ed. (North-Holland Publ. Co. Amsterdam and John Wiley & Sons Inc., New York, 1965), p. 281.
    [CrossRef]
  11. A. Rubinowicz, Ann. Physik (Leipzig) 53, 257 (1917). For other references and an account of the theory, see Ref. 1, Sec. 8.9.
    [CrossRef]
  12. The experimental evidence is tentative in the sense that it uses data obtained from diffraction experiments with screens that are not black.
  13. When the incident wave is a plane wave, propagated in a direction specified by the unit vector p, (2.6) must be replaced byK(ξ,η;x,y)=14πp×sˆ1−p·sˆ, where ŝ= Q′P/|Q′P|. A similar change must then be made in the expression for W(i), given by Eq. (3.1).In (2.6), and in several other places, an adjacent pair of capital letters indicates a vector. Also, ŝ represents a vector, even though not printed in boldface type.
  14. A. Rubinowicz, Acta Phys. Polon. 12, 225 (1953).
  15. (a)K. Miyamoto and E. Wolf, J. Opt. Soc Am. 52, 615 (1962); (b)J. Opt. Soc Am. 52, 626 (1962).
    [CrossRef]
  16. J. B. Keller, R. M. Lewis, and B. D. Seckler, J. Appl. Phys. 28, 570 (1957).
    [CrossRef]
  17. M. J. Ehrlich, S. Silver, and G. Held, J. Appl. Phys. 26, 336 (1955). Similar experimental curves were published by J. Buchsbaum, A. R. Milne, D. C. Hogg, G. Bekefi, and G. A. Woonton, J. Appl. Phys. 26, 706 (1955); C. L. Andrews, Phys. Rev. 71, 777 (1947); J. Appl. Phys. 21, 761 (1950). In the last-quoted paper, Andrews pointed out that his experimental results are consistent with Thomas Young’s ideas about the origin of diffraction. Since the Rubinowicz theory of the boundary diffraction wave may be regarded as a mathematical refinement of Young’s ideas, Andrews’s views are essentially in qualitative agreement with those of the present paper.
    [CrossRef]
  18. Some aspects of the Rayleigh–Sommerfeld theory are discussed in the papers by E. W. Marchand and E. Wolf, J. Opt. Soc. Am. 52, 761 (1962); J. Opt. Soc. Am. 54, 587 (1964). In the first of these two papers, the theory was associated with the names of Rayleigh and Kirchhoff, but it seems more appropriate to associate it with the names of Rayleigh and Sommerfeld.
    [CrossRef]
  19. N. Mukunda, J. Opt. Soc. Am. 52, 336 (1962).
    [CrossRef]
  20. We have shown elsewhere [J. Opt. Soc. Am. 54, 587 (1964)] that, if the linear dimensions of the aperture are large compared with the wavelength, the difference in the behavior of the Rayleigh–Sommerfeld solutions and of the Kirchhoff solution in the plane of the aperture does not lead to significantly different results for the predicted field in the far zone, in the neighborhood of the forward direction.
    [CrossRef]
  21. See also R. W. Dunham, J. Opt. Soc. Am. 54, 1102 (1964).
    [CrossRef]
  22. R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley and Los Angeles, Calif., 1964).
  23. It is clear from an examination of the left-hand side of (B4) that, if Q′P′≠ 0, the expression could only vanish if cos(SQ′,Q′P′) = 1, i.e., if the directions SQ′and Q′P′were parallel. This is impossible, since S is situated off the plane of the aperture, whereas both Q′and P′are in this plane.
  24. O. D. Kellogg, Foundations of Potential Theory (Dover Publications, Inc., New York, 1953), p. 144.

1964 (2)

1962 (3)

1957 (1)

J. B. Keller, R. M. Lewis, and B. D. Seckler, J. Appl. Phys. 28, 570 (1957).
[CrossRef]

1956 (1)

G. Toraldo di Francia, Atti Fond. Giorgio Ronchi 11, 503 (1956).

1955 (1)

M. J. Ehrlich, S. Silver, and G. Held, J. Appl. Phys. 26, 336 (1955). Similar experimental curves were published by J. Buchsbaum, A. R. Milne, D. C. Hogg, G. Bekefi, and G. A. Woonton, J. Appl. Phys. 26, 706 (1955); C. L. Andrews, Phys. Rev. 71, 777 (1947); J. Appl. Phys. 21, 761 (1950). In the last-quoted paper, Andrews pointed out that his experimental results are consistent with Thomas Young’s ideas about the origin of diffraction. Since the Rubinowicz theory of the boundary diffraction wave may be regarded as a mathematical refinement of Young’s ideas, Andrews’s views are essentially in qualitative agreement with those of the present paper.
[CrossRef]

1954 (1)

C. J. Bouwkamp, Rept. Progr. Phys. 17, 35 (1954).
[CrossRef]

1953 (1)

A. Rubinowicz, Acta Phys. Polon. 12, 225 (1953).

1951 (1)

S. A. Schelkunoff, Commun. Pure Appl. Math. 4, 43 (1951).
[CrossRef]

1949 (1)

W. Franz, Z. Physik 125, 563 (1949).
[CrossRef]

1923 (1)

(a)F. Kottler, Ann. Physik (Leipzig) 70, 405 (1923). (b) in Progress in Optics IV, E. Wolf, Ed. (North-Holland Publ. Co. Amsterdam and John Wiley & Sons Inc., New York, 1965), p. 281.
[CrossRef]

1917 (1)

A. Rubinowicz, Ann. Physik (Leipzig) 53, 257 (1917). For other references and an account of the theory, see Ref. 1, Sec. 8.9.
[CrossRef]

Baker, B. B.

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens’ Principle (Clarendon Press, Oxford, England, 1950), 2nd ed.

Born, M.

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

Boron, M.

M. Boron, Optik (Julius Springer-Verlag, Berlin, 1933: re printed 1965), p. 152.

Bouwkamp, C. J.

C. J. Bouwkamp, Rept. Progr. Phys. 17, 35 (1954).
[CrossRef]

Copson, E. T.

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens’ Principle (Clarendon Press, Oxford, England, 1950), 2nd ed.

Dunham, R. W.

Ehrlich, M. J.

M. J. Ehrlich, S. Silver, and G. Held, J. Appl. Phys. 26, 336 (1955). Similar experimental curves were published by J. Buchsbaum, A. R. Milne, D. C. Hogg, G. Bekefi, and G. A. Woonton, J. Appl. Phys. 26, 706 (1955); C. L. Andrews, Phys. Rev. 71, 777 (1947); J. Appl. Phys. 21, 761 (1950). In the last-quoted paper, Andrews pointed out that his experimental results are consistent with Thomas Young’s ideas about the origin of diffraction. Since the Rubinowicz theory of the boundary diffraction wave may be regarded as a mathematical refinement of Young’s ideas, Andrews’s views are essentially in qualitative agreement with those of the present paper.
[CrossRef]

Franz, W.

W. Franz, Z. Physik 125, 563 (1949).
[CrossRef]

Held, G.

M. J. Ehrlich, S. Silver, and G. Held, J. Appl. Phys. 26, 336 (1955). Similar experimental curves were published by J. Buchsbaum, A. R. Milne, D. C. Hogg, G. Bekefi, and G. A. Woonton, J. Appl. Phys. 26, 706 (1955); C. L. Andrews, Phys. Rev. 71, 777 (1947); J. Appl. Phys. 21, 761 (1950). In the last-quoted paper, Andrews pointed out that his experimental results are consistent with Thomas Young’s ideas about the origin of diffraction. Since the Rubinowicz theory of the boundary diffraction wave may be regarded as a mathematical refinement of Young’s ideas, Andrews’s views are essentially in qualitative agreement with those of the present paper.
[CrossRef]

Keller, J. B.

J. B. Keller, R. M. Lewis, and B. D. Seckler, J. Appl. Phys. 28, 570 (1957).
[CrossRef]

Kellogg, O. D.

O. D. Kellogg, Foundations of Potential Theory (Dover Publications, Inc., New York, 1953), p. 144.

Kottler, F.

(a)F. Kottler, Ann. Physik (Leipzig) 70, 405 (1923). (b) in Progress in Optics IV, E. Wolf, Ed. (North-Holland Publ. Co. Amsterdam and John Wiley & Sons Inc., New York, 1965), p. 281.
[CrossRef]

Lewis, R. M.

J. B. Keller, R. M. Lewis, and B. D. Seckler, J. Appl. Phys. 28, 570 (1957).
[CrossRef]

Luneburg, R. K.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley and Los Angeles, Calif., 1964).

Marchand, E. W.

Miyamoto, K.

(a)K. Miyamoto and E. Wolf, J. Opt. Soc Am. 52, 615 (1962); (b)J. Opt. Soc Am. 52, 626 (1962).
[CrossRef]

Mukunda, N.

Poincaré, H.

H. Poincaré, Théorie Mathématique de la Lumière (Georges Carré, Paris, 1892), Pt. II, pp. 187–188.

Rubinowicz, A.

A. Rubinowicz, Acta Phys. Polon. 12, 225 (1953).

A. Rubinowicz, Ann. Physik (Leipzig) 53, 257 (1917). For other references and an account of the theory, see Ref. 1, Sec. 8.9.
[CrossRef]

Schelkunoff, S. A.

S. A. Schelkunoff, Commun. Pure Appl. Math. 4, 43 (1951).
[CrossRef]

Seckler, B. D.

J. B. Keller, R. M. Lewis, and B. D. Seckler, J. Appl. Phys. 28, 570 (1957).
[CrossRef]

Silver, S.

M. J. Ehrlich, S. Silver, and G. Held, J. Appl. Phys. 26, 336 (1955). Similar experimental curves were published by J. Buchsbaum, A. R. Milne, D. C. Hogg, G. Bekefi, and G. A. Woonton, J. Appl. Phys. 26, 706 (1955); C. L. Andrews, Phys. Rev. 71, 777 (1947); J. Appl. Phys. 21, 761 (1950). In the last-quoted paper, Andrews pointed out that his experimental results are consistent with Thomas Young’s ideas about the origin of diffraction. Since the Rubinowicz theory of the boundary diffraction wave may be regarded as a mathematical refinement of Young’s ideas, Andrews’s views are essentially in qualitative agreement with those of the present paper.
[CrossRef]

Toraldo di Francia, G.

G. Toraldo di Francia, Atti Fond. Giorgio Ronchi 11, 503 (1956).

Wolf, E.

Acta Phys. Polon. (1)

A. Rubinowicz, Acta Phys. Polon. 12, 225 (1953).

Ann. Physik (Leipzig) (2)

(a)F. Kottler, Ann. Physik (Leipzig) 70, 405 (1923). (b) in Progress in Optics IV, E. Wolf, Ed. (North-Holland Publ. Co. Amsterdam and John Wiley & Sons Inc., New York, 1965), p. 281.
[CrossRef]

A. Rubinowicz, Ann. Physik (Leipzig) 53, 257 (1917). For other references and an account of the theory, see Ref. 1, Sec. 8.9.
[CrossRef]

Atti Fond. Giorgio Ronchi (1)

G. Toraldo di Francia, Atti Fond. Giorgio Ronchi 11, 503 (1956).

Commun. Pure Appl. Math. (1)

S. A. Schelkunoff, Commun. Pure Appl. Math. 4, 43 (1951).
[CrossRef]

J. Appl. Phys. (2)

J. B. Keller, R. M. Lewis, and B. D. Seckler, J. Appl. Phys. 28, 570 (1957).
[CrossRef]

M. J. Ehrlich, S. Silver, and G. Held, J. Appl. Phys. 26, 336 (1955). Similar experimental curves were published by J. Buchsbaum, A. R. Milne, D. C. Hogg, G. Bekefi, and G. A. Woonton, J. Appl. Phys. 26, 706 (1955); C. L. Andrews, Phys. Rev. 71, 777 (1947); J. Appl. Phys. 21, 761 (1950). In the last-quoted paper, Andrews pointed out that his experimental results are consistent with Thomas Young’s ideas about the origin of diffraction. Since the Rubinowicz theory of the boundary diffraction wave may be regarded as a mathematical refinement of Young’s ideas, Andrews’s views are essentially in qualitative agreement with those of the present paper.
[CrossRef]

J. Opt. Soc Am. (1)

(a)K. Miyamoto and E. Wolf, J. Opt. Soc Am. 52, 615 (1962); (b)J. Opt. Soc Am. 52, 626 (1962).
[CrossRef]

J. Opt. Soc. Am. (4)

Rept. Progr. Phys. (1)

C. J. Bouwkamp, Rept. Progr. Phys. 17, 35 (1954).
[CrossRef]

Z. Physik (1)

W. Franz, Z. Physik 125, 563 (1949).
[CrossRef]

Other (10)

Some of the modified theories are reviewed in Ref. 8, Sec. 4.

R. K. Luneburg, Mathematical Theory of Optics (University of California Press, Berkeley and Los Angeles, Calif., 1964).

It is clear from an examination of the left-hand side of (B4) that, if Q′P′≠ 0, the expression could only vanish if cos(SQ′,Q′P′) = 1, i.e., if the directions SQ′and Q′P′were parallel. This is impossible, since S is situated off the plane of the aperture, whereas both Q′and P′are in this plane.

O. D. Kellogg, Foundations of Potential Theory (Dover Publications, Inc., New York, 1953), p. 144.

M. Boron, Optik (Julius Springer-Verlag, Berlin, 1933: re printed 1965), p. 152.

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

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens’ Principle (Clarendon Press, Oxford, England, 1950), 2nd ed.

H. Poincaré, Théorie Mathématique de la Lumière (Georges Carré, Paris, 1892), Pt. II, pp. 187–188.

The experimental evidence is tentative in the sense that it uses data obtained from diffraction experiments with screens that are not black.

When the incident wave is a plane wave, propagated in a direction specified by the unit vector p, (2.6) must be replaced byK(ξ,η;x,y)=14πp×sˆ1−p·sˆ, where ŝ= Q′P/|Q′P|. A similar change must then be made in the expression for W(i), given by Eq. (3.1).In (2.6), and in several other places, an adjacent pair of capital letters indicates a vector. Also, ŝ represents a vector, even though not printed in boldface type.

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Figures (6)

Fig. 1
Fig. 1

Illustrating the notation.

Fig. 2
Fig. 2

Regions A and B relating to the definition (3.3) of (x,y,z). Region A is represented by the unshaded area lying on the right of the plane z = 0.

Fig. 3
Fig. 3

(a) Measured principal component of the electric field in the plane of a circular aperture of diameter D in a conducting screen, as a function of the radial distance x (measured in wavelengths). The incident field is a plane wave of unit amplitude falling normally onto the plane of the aperture. [From M. J. Ehrlich, S. Silver, and G. Held, J. Appl. Phys. 26, 342 (1955)]. (b) The field amplitude |UK|, calculated from Eq. (3.4), in the plane of a circular aperture, of diameter D, as a function of the radial distance x (measured in wavelengths). The incident field is a plane wave falling normally onto the plane of the aperture.

Fig. 4
Fig. 4

(a) Measured axial distribution of the principal tangential component of the electric and magnetic fields E and H. for circular aperture of diameter D in a conducting screen, as a function of the distance z (measured in wavelengths) from the aperture. The incident field is a plane wave of unit amplitude falling normally on the plane of the aperture. [From M. J. Ehrlich, S. Silver, and G. Held, J. Appl. Phys. 26, 345 (1955)]. (b) The axial distribution of the field amplitude UK, calculated from (3.2), for circular aperture of diameter D, as a function of the distance z (measured in wavelengths) from the aperture. The incident field is a plane wave of unit amplitude, falling normally on the plane of the aperture.

Fig. 5
Fig. 5

The field amplitudes |UI| (denoted by I) and |UII| (denoted by II) given by the Rayleigh–Sommerfeld formulae of the first and the second kind, respectively, for the field in the plane of a circular aperture of diameter D, as a function of the radial distance x (measured in wavelengths). The incident field is a plane wave falling normally onto the plane of the aperture.

Fig. 6
Fig. 6

Illustrating the notation relating to the singly (U1(s)), doubly (U2(s)), and triply (U3(s)) scattered boundary waves.

Equations (68)

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U K ( P ) = 1 4 π A { U ( i ) ( Q ) n ( e i k s s ) e i k s s U ( i ) ( Q ) n } d A .
U ( i ) ( x , y , z ) = e i k r / r
f 0 ( x , y ) = U ( i ) ( x , y , 0 ) + U ( s ) ( x , y , 0 ) when P A = U ( s ) ( x , y , 0 ) when P B ,
U ( s ) ( x , y , 0 ) = Γ U ( i ) ( ξ , η , 0 ) e i k s 0 s 0 K ( ξ , η ; x , y ) · dl ,
f 0 ( x , y ) = 0 ( x , y ) U ( i ) ( x , y , 0 ) + Γ U ( i ) ( ξ , η , 0 ) e i k s 0 s 0 K ( ξ , η ; x , y ) · dl ,
0 ( x , y ) = 1 if ( x , y , 0 ) A = 0 if ( x , y , 0 ) B .
K ( ξ , η ; x , y ) = 1 4 π S Q × Q P | S Q | | Q P | S Q · Q P .
2 U K + k 2 U K = 0 ,
U K ( x , y , z ) f 0 ( x , y )
W ( i ) ( Q , P ) W ( i ) ( ξ , η , ζ ; x , y , z ) ,
W ( i ) ( ξ , η , ζ ; x , y , z ) = U ( i ) ( ξ , η , ζ ) e i k s s [ 1 4 π S Q × Q P | S Q | | Q P | S Q · Q P ] , P = ( x , y , z ) , Q = ( ξ , η , ζ ) , s = Q P = [ ( x ξ ) 2 + ( y η ) 2 + ( z ζ ) 2 ] 1 2 .
U K ( x , y , z ) = ( x , y , z ) U ( i ) ( x , y , z ) + Γ W ( i ) ( ξ , η , 0 ; x , y , z ) · dl .
( x , y , z ) = 1 , if ( x , y , z ) A , = 0 , if ( x , y , z ) B , } ,
Lim z + 0 U K ( x , y , z ) = 0 ( x , y ) U ( i ) ( x , y , 0 ) + Γ W ( i ) ( ξ , η , 0 ; x , y , 0 ) · dl .
Lim z + 0 U K ( x , y , z ) = f 0 ( x , y ) ,
2 U K + k 2 U K = 0.
U I ( P ) = 1 2 π A U ( i ) ( Q ) n ( e i k s s ) d A ,
U II ( P ) = 1 2 π A U ( i ) ( Q ) n e i k s s d A .
Lim z + 0 U I ( x , y , z ) = 0 ( x , y ) U ( i ) ( x , y , 0 ) ,
U K ( P ) = 1 2 [ U I ( P ) + U II ( P ) ] ,
Lim z + 0 U II ( x , y , z ) = Lim z + 0 2 U K ( x , y , 0 ) Lim z + 0 U I ( x , y , 0 ) ,
Lim z + 0 U II ( x , y , z ) = 0 U ( i ) ( x , y , 0 ) + 2 Γ W ( i ) ( ξ , η , 0 ; x , y , 0 ) · dl .
U K ( P ) = ( P ) U ( i ) ( P ) + U 1 ( s ) ( P ) ,
U 1 ( s ) ( P ) = Γ U ( i ) ( Q 1 ) e i k s 1 P s 1 P K ( S Q 1 , Q 1 P ) · d l 1 .
U ( P ) = ( P ) U ( i ) ( P ) + n = 1 U n ( s ) ( P ) ,
U 2 ( s ) ( P ) = Γ U 1 ( s ) ( Q 2 ) e i k s 2 P s 2 P K ( Q 1 Q 2 , Q 2 P ) · d l 2 = Γ Γ U ( i ) ( Q 1 ) e i k ( s 12 + s 2 P ) s 12 · s 2 P K ( S Q 1 , Q 1 Q 2 ) · d l 1 × K ( Q 1 Q 2 , Q 2 P ) · d l 2 , U n ( s ) ( P ) = Γ U n 1 ( s ) ( Q n ) e i k s n P s n P K ( Q n 1 Q n , Q n P ) · d l n = Γ Γ U ( i ) ( Q 1 ) e i k ( s 12 + s 23 + s n P ) s 12 s 23 s n P × K ( S Q 1 , Q 1 Q 2 ) · d l 1 K ( Q 1 Q 2 , Q 2 Q 3 ) · d l 2 × K ( Q n 1 Q n , Q n P ) · d l n .
s 12 = Q 1 Q 2 , s 23 = Q 2 Q 3 , s n 1 , n = Q n 1 Q n , s n P = Q n P ,
K ( Q j Q j + 1 , Q j + 1 Q j + 2 ) = 0.
U 3 ( s ) ( P ) = U 4 ( s ) ( P ) = U n ( s ) ( P ) = = 0.
U ( P ) = U K ( P ) + U 2 ( s ) ( P ) ,
2 U + k 2 U = 0
Lim z + 0 U ( x , y , z ) = f ( x , y )
U ( x , y , z ) = 1 2 π f ( x , y ) × { z ( e i k s s ) } z = 0 d x d y ,
s = [ ( x x ) 2 + ( y y ) 2 + ( z z ) 2 ] 1 2 .
| f ( x , y ) | < B R , | f ( x , y ) x | < B R , | f ( x , y ) y | < B R ,
| U ( x , y , z ) | < C R , | U ( x , y , z ) R | < C R ,
| U / R i k U | < D / R 2 .
f 0 ( s ) ( P ) = Γ U ( i ) ( Q ) e i k s 0 s 0 × [ 1 4 π S Q × Q P | S Q | | Q P | S Q · Q P ] · dl .
ξ = ξ ( l ) , η = η ( l ) ,
ξ ( l ) d l , η ( l ) d l , 0.
S Q = ( ξ x 0 , η y 0 , z 0 ) , Q P = ( x ξ , y η , 0 ) .
( S Q × Q P ) · dl = | ξ x 0 η y 0 z 0 x ξ y η 0 ξ ( l ) η ( l ) 0 | d l .
| S Q | | Q P | S Q · Q P = [ ( ξ x 0 ) 2 + ( η y 0 ) 2 + z 0 2 ] 1 2 [ ( x ξ ) 2 + ( y η ) 2 ] 1 2 [ ( ξ x 0 ) ( x ξ ) + ( η y 0 ) ( y η ) ] .
f 0 ( x , y ) , f 0 ( x , y ) x , f 0 ( x , y ) y ,
f 0 ( s ) ( P ) = 1 4 π Γ e i k s 0 s 0 p × s ˆ 1 p · s ˆ · dl ,
f 0 ( s ) ( P ) = 1 4 π 0 2 π exp ( i k a σ ) σ 2 [ 1 ρ cos ψ ] d ψ ,
σ = [ 1 2 ρ cos ψ + ρ 2 ] 1 2 , ρ = d / a .
f 0 ( s ) ( P ) = 1 4 π [ 0 π ( 1 ρ 2 ) exp ( i k a σ ) 1 2 σ cos ψ + ρ 2 d ψ + 0 π exp ( i k a σ ) d ψ ] .
Lim ρ 1 0 π exp ( i k a σ ) d ψ = 0 π exp [ 2 i k a sin ψ ] d ψ .
A ( ρ ) = 0 π ( 1 ρ 2 ) exp ( i k a σ ) 1 2 ρ cos ψ + ρ 2 d ψ ,
A ( ρ ) = A 1 ( ρ ) + A 2 ( ρ ) ,
A 1 ( ρ ) = δ π ( 1 ρ 2 ) exp ( i k a σ ) 1 2 ρ cos ψ + ρ 2 d ψ ,
A 2 ( ρ ) = 0 δ ( 1 ρ 2 ) exp ( i k a σ ) 1 2 ρ cos ψ + ρ 2 d ψ ,
lim ρ 1 A 1 ( ρ ) = 0 ,
σ ~ ( 1 2 ρ + ρ 2 ) 1 2 = | 1 ρ | ,
exp ( i k α σ ) ~ exp ( i k | 1 ρ | ) .
A 2 ( ρ ) ~ ( 1 ρ 2 ) exp ( i k a | 1 ρ | ) 0 δ d ψ 1 + ρ 2 2 ρ cos ψ = 2 signum ( 1 ρ 2 ) exp ( i k a | 1 ρ | ) × tan 1 | 1 + ρ 1 ρ tan δ 2 | .
lim ρ 1 + A 2 ( ρ ) = π , lim ρ 1 A 2 ( ρ ) = + π .
Lim ρ 1 + f 0 ( s ) ( P ) = 1 4 1 4 π 0 π exp [ 2 i k a sin ψ ] d ψ , Lim ρ 1 f 0 ( s ) ( P ) = 1 4 1 4 π 0 π exp [ 2 i k a sin ψ ] d ψ .
lim ρ 1 + f 0 ( x , y ) = 1 4 1 4 π 0 π exp [ 2 i k a sin ψ ] d ψ , lim ρ 1 f 0 ( x , y ) = 3 4 1 4 π 0 π exp [ 2 i k a sin ψ ] d ψ .
Δ f 0 = lim ρ 1 + f 0 ( x , y ) lim ρ 1 f 0 ( x , y ) = 1 2 ,
f 0 ( x , y ) = 1 4 π Γ U ( i ) ( ξ , η , 0 ) e i k s 0 s 0 ( r ˆ × s ˆ ) · l 1 r ˆ · s ˆ d l ,
r ˆ = S Q | S Q | , s ˆ = Q P | Q P |
| f 0 ( x , y ) | 1 4 π Max { | U ( i ) ( ξ , η , 0 ) | | ( r ˆ × s ˆ ) · l 1 r ˆ · s ˆ | } Γ 1 s d l ,
| f 0 ( x , y ) | M Γ 1 s d l ,
| R f 0 | < B , | R 2 f 0 x | < B , | R 2 f 0 y | < B ,
| f 0 | < B R , | f 0 x | < B R , | f 0 y | < B R ,
K(ξ,η;x,y)=14πp×sˆ1p·sˆ,