Abstract

With a view to elucidating the effect of a well-known mathematical inconsistency in Kirchhoff’s diffraction theory, a comparison is made of the predictions relating to the field diffracted at an aperture, based on Kirchhoff’s theory (UK) and on formulas due to Rayleigh and Sommerfeld (UR). It is shown that, when the incident wave is plane or spherical, the difference δ = UKUR represents a boundary wave, i.e., a wave which may be thought of as originating at each point of the edge of the aperture. It is shown further that, when the linear dimensions of the aperture are large compared with the wavelength, the boundary values of δ in the plane of the aperture change very rapidly and almost periodically from point to point, with the mean period close to the wavelength of the incident radiation. This result is shown to imply that if the linear dimensions of the aperture are large compared with the wavelength, the two theories predict essentially the same behavior for the diffracted field in the far zone, at moderate angles of diffraction.

© 1964 Optical Society of America

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References

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  1. H. Poincaré, Theorie Mathematique de la Lumière (George Carré, Paris, II, 1892), pp. 187–188.
  2. G. Toraldo di Francia, Atti Fond. Giorgio Ronchi, Publisher, Ist. Nazl. Ottica 11, §6 (1956).
  3. B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens’ Principle (Clarendon Press, Oxford, England, 1950), 2nd ed.
  4. A. Sommerfeld, Optics (Academic Press Inc., New York, 1954), p. 199;see also A. Rubinowicz, Die Beugungswelle in der Kirchhoffschen Theorie der Beugung (Polska Akademi Nauk, Warszawa, 1957), pp. 77–83.
  5. See, for example, Lord Rayleigh, Phil. Mag. 43, 259 (1897);see also his Scientific Papers4 (Cambridge University Press, 1903), p. 283.
  6. C. J. Bouwkamp, Rept. Progr. Phys.17, 41–42 (1954).
    [Crossref]
  7. H. Osterberg and L. W. Smith, J. Opt. Soc. Am. 51, 1050 (1961).
    [Crossref]
  8. H. M. Nussenzweig, Solution of a Diffraction Problem (Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro, 1957);Notas de Fisica, Suppl. AO, Vol. 3.
  9. S. Silver, J. Opt. Soc. Am. 52, 137 (1962).
    [Crossref]
  10. M. Born and E. Wolf, Principles of Optics (Pergamon Press, Ltd., London, 1959).
  11. N. Mukunda, J. Opt. Soc. Am. 52, 336 (1962).
    [Crossref]
  12. E. W. Marchand and E. Wolf, J. Opt. Soc. Am. 52, 761 (1962).
    [Crossref]
  13. This is so, since UK and UI obey this condition and therefore so does the difference δ= UK−UI.
  14. E. Wolf, Proc. Phys. Soc. (London) 74, 280 (1959), Appendix.
  15. The reader should not confuse the symbols δ and δD. The first represents the “error wave” defined by (2.6), and the latter symbol is used in the present section only to denote the Dirac delta function.
  16. K. Miyamoto and E. Wolf, J. Opt. Soc. Am. 52, 624 (1962).
  17. Since the integrand in (A1) is periodic in φ with period 2π, we may take the limits of integration to be φ0 and φ0+2π rather than 0 and 2π[as in (3.12)], where φ0 is any convenient constant. The reason for this formal change is to avoid having a stationary point at the end of the interval of integration [cf. (A11) below].

1962 (4)

S. Silver, J. Opt. Soc. Am. 52, 137 (1962).
[Crossref]

N. Mukunda, J. Opt. Soc. Am. 52, 336 (1962).
[Crossref]

E. W. Marchand and E. Wolf, J. Opt. Soc. Am. 52, 761 (1962).
[Crossref]

K. Miyamoto and E. Wolf, J. Opt. Soc. Am. 52, 624 (1962).

1961 (1)

1959 (1)

E. Wolf, Proc. Phys. Soc. (London) 74, 280 (1959), Appendix.

1956 (1)

G. Toraldo di Francia, Atti Fond. Giorgio Ronchi, Publisher, Ist. Nazl. Ottica 11, §6 (1956).

1897 (1)

See, for example, Lord Rayleigh, Phil. Mag. 43, 259 (1897);see also his Scientific Papers4 (Cambridge University Press, 1903), p. 283.

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, Ltd., London, 1959).

Bouwkamp, C. J.

C. J. Bouwkamp, Rept. Progr. Phys.17, 41–42 (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.

Marchand, E. W.

Miyamoto, K.

K. Miyamoto and E. Wolf, J. Opt. Soc. Am. 52, 624 (1962).

Mukunda, N.

Nussenzweig, H. M.

H. M. Nussenzweig, Solution of a Diffraction Problem (Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro, 1957);Notas de Fisica, Suppl. AO, Vol. 3.

Osterberg, H.

Poincaré, H.

H. Poincaré, Theorie Mathematique de la Lumière (George Carré, Paris, II, 1892), pp. 187–188.

Rayleigh, Lord

See, for example, Lord Rayleigh, Phil. Mag. 43, 259 (1897);see also his Scientific Papers4 (Cambridge University Press, 1903), p. 283.

Silver, S.

S. Silver, J. Opt. Soc. Am. 52, 137 (1962).
[Crossref]

Smith, L. W.

Sommerfeld, A.

A. Sommerfeld, Optics (Academic Press Inc., New York, 1954), p. 199;see also A. Rubinowicz, Die Beugungswelle in der Kirchhoffschen Theorie der Beugung (Polska Akademi Nauk, Warszawa, 1957), pp. 77–83.

Toraldo di Francia, G.

G. Toraldo di Francia, Atti Fond. Giorgio Ronchi, Publisher, Ist. Nazl. Ottica 11, §6 (1956).

Wolf, E.

E. W. Marchand and E. Wolf, J. Opt. Soc. Am. 52, 761 (1962).
[Crossref]

K. Miyamoto and E. Wolf, J. Opt. Soc. Am. 52, 624 (1962).

E. Wolf, Proc. Phys. Soc. (London) 74, 280 (1959), Appendix.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Ltd., London, 1959).

Atti Fond. Giorgio Ronchi, Publisher, Ist. Nazl. Ottica (1)

G. Toraldo di Francia, Atti Fond. Giorgio Ronchi, Publisher, Ist. Nazl. Ottica 11, §6 (1956).

J. Opt. Soc. Am. (5)

Phil. Mag. (1)

See, for example, Lord Rayleigh, Phil. Mag. 43, 259 (1897);see also his Scientific Papers4 (Cambridge University Press, 1903), p. 283.

Proc. Phys. Soc. (London) (1)

E. Wolf, Proc. Phys. Soc. (London) 74, 280 (1959), Appendix.

Other (9)

The reader should not confuse the symbols δ and δD. The first represents the “error wave” defined by (2.6), and the latter symbol is used in the present section only to denote the Dirac delta function.

This is so, since UK and UI obey this condition and therefore so does the difference δ= UK−UI.

Since the integrand in (A1) is periodic in φ with period 2π, we may take the limits of integration to be φ0 and φ0+2π rather than 0 and 2π[as in (3.12)], where φ0 is any convenient constant. The reason for this formal change is to avoid having a stationary point at the end of the interval of integration [cf. (A11) below].

H. Poincaré, Theorie Mathematique de la Lumière (George Carré, Paris, II, 1892), pp. 187–188.

C. J. Bouwkamp, Rept. Progr. Phys.17, 41–42 (1954).
[Crossref]

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

A. Sommerfeld, Optics (Academic Press Inc., New York, 1954), p. 199;see also A. Rubinowicz, Die Beugungswelle in der Kirchhoffschen Theorie der Beugung (Polska Akademi Nauk, Warszawa, 1957), pp. 77–83.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, Ltd., London, 1959).

H. M. Nussenzweig, Solution of a Diffraction Problem (Centro Brasileiro de Pesquisas Fisicas, Rio de Janeiro, 1957);Notas de Fisica, Suppl. AO, Vol. 3.

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Figures (6)

F. 1
F. 1

Illustrating the notation.

F. 2
F. 2

Illustrating the notation.

F. 3
F. 3

Notation relating to determining the behavior of δ in the plane of the aperture.

F. 4
F. 4

The amplitude [(a), (b)] and the argument [(c)] of δ in the plane of the aperture. (Normally incident plane wave, circular aperture.)

F. 5
F. 5

The normalized amplitude distribution A of the angular spectrum of plane waves representing δ. u = ( k x 2 + k y 2 ) 1 2 / k. (Normally incident plane wave, circular aperture, ka = 100.)

F. 6
F. 6

The normalized amplitude distribution of the radiation patterns aδ of the error wave δ. C = −iπa2/λ. (Normally incident plane wave, circular aperture, ka = 100.)

Tables (1)

Tables Icon

Table I Comparison of the normalized amplitude distribution of the radiation patterns aδ and aK of the δ-field and of the Kirchhoff field, respectively. C = −(i/λ)πa2. (Normally incident plane wave, circular aperture, ka = 100.)

Equations (77)

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U K ( P ) = 1 4 π a { U ( i ) ( Q ) z ( exp i k s s ) exp i k s s U ( i ) ( Q ) z } d S .
U = U ( i ) ( Q ) when Q is in a = 0 when Q is on B + } ,
U / z = U ( i ) ( Q ) / z when Q is in a = 0 when Q is on B + } ,
U K ( P ) = 1 2 [ U I ( P ) + U II ( P ) ] ,
U I ( P ) = 1 2 π a U ( i ) ( Q ) z ( exp i k s s ) d S ,
U I I ( P ) = 1 2 π a U ( i ) ( Q ) z exp i k s s d S .
δ I ( P ) = U K ( P ) U I ( P ) ,
δ I I ( P ) = U K ( P ) U II ( P ) .
δ ( P ) = δ I ( P ) = δ II ( P ) = 1 2 [ U II ( P ) U I ( P ) ] .
2 δ ( P ) + k 2 δ ( P ) = 0 .
U I ( P ) = U I ( B ) ( P ) + U I ( S ) ( P ) ,
U II ( P ) = U II ( B ) ( P ) + U II ( S ) ( P ) .
δ ( P ) = 1 2 [ U II ( B ) ( P ) U I ( B ) ( P ) ] + 1 2 [ U II ( S ) ( P ) U I ( S ) ( P ) ] .
δ ( P ) = 1 2 [ U II ( B ) ( P ) U I ( B ) ( P ) ] .
U ( i ) ( P ) = exp ( i k p · r ) ,
U α ( B ) ( P ) = 1 4 π Γ exp ( i k p · r ) exp i k s s × [ ŝ × p 1 + ŝ · p ŝ * × p 1 + ŝ * · p ] · l d l .
U I ( B ) ( P ¯ ) = 0 ,
U II ( B ) ( P ¯ ) = 1 2 π Γ exp ( i k p · r ) exp i k s s × [ ŝ × p 1 + ŝ · p ] · l d l ,
δ ( P ¯ ) = 1 2 U II ( B ) ( P ¯ ) ,
U K ( P ¯ ) = U K ( B ) ( P ¯ ) + U ( G ) ( P ¯ ) = δ ( P ¯ ) + U II ( P ¯ ) = δ ( P ¯ ) + U II ( B ) ( P ¯ ) + U ( G ) ( P ¯ ) ,
δ ( P ¯ ) = U K ( B ) ( P ¯ )
= 1 2 U II ( B ) ( P ¯ ) .
δ ( P ¯ ) = 1 2 U II ( B ) ( P ¯ ) = 1 4 π Γ exp ( i k p · r ) exp i k s s s × p s + s · p · l d l .
x = a cos φ , y = a sin φ , x ¯ = r cos φ ¯ , y ¯ = r sin φ ¯ .
r : x , y , 0 , p : 0 , 0 , 1 , s : x x ¯ , y y ¯ , 0 , l : sin φ , cos φ , 0 . }
s = [ a 2 + r 2 2 a r cos ( φ φ ¯ ) ] 1 2 p · r = p · s = 0 ( s × p ) · l = a + r cos ( φ φ ¯ ) . }
δ ( P ¯ ) = a 4 π 0 2 π exp ( i k s ) s 2 [ r cos ( φ φ ¯ ) a ] d φ ,
ρ = ( r / a ) , σ = ( s / a ) ,
δ ( 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 .
k a 1 .
δ ( P ¯ ) exp ( i k a ) 2 ( 2 π k a ρ ) 1 2 { exp [ i ( k a ρ π / 4 ) ] ( 1 + ρ ) 1 2 + exp [ i ( k a ρ π / 4 ) ] ( 1 ρ ) 1 2 } if 0 < ρ < 1 ,
exp ( i k a ρ ) 2 ( 2 π k a ρ ) 1 2 { exp [ i ( k a π / 4 ) ] ( ρ + 1 ) 1 2 exp [ i ( k a π / 4 ) ] ( ρ 1 ) 1 2 } if ρ > 1 .
δ ( 0 ) = 1 2 exp ( i k a ) .
Δ ρ = 2 π / k a = λ / a ,
Δ r = λ .
δ ( P ¯ ) exp ( i k a ) ( 2 π k a ρ ) 1 2 cos ( k a ρ π / 4 ) 1 2 exp ( i k a ) J 0 ( k a ρ ) , ( 0 < ρ < 1 ) ,
δ ( P ¯ ) i sin ( k a π / 4 ) ( 2 π k a ) 1 2 exp ( i k a ρ ) ρ , ( ρ 1 ) .
δ ( x , y , z ) = A ( k x , k y ) exp [ i ( k x x + k y y + k z z ) ] d k x d k y ,
k z = ( k 2 k x 2 k y 2 ) 1 2 when k x 2 + k y 2 k 2 ,
= i ( k x 2 + k y 2 k 2 ) 1 2 when k x 2 + k y 2 k 2 .
δ ( x , y , z ) = k x 2 + k y 2 k 2 A ( k x , k y ) exp [ i ( k x x + k y y + k z z ) ] d k x d k y + k x 2 + k y 2 > k 2 A ( k x , k y ) exp ( | k z | z ) × exp [ i ( k x x + k y y ) ] d k x d k y .
δ ( x , y , 0 ) = A ( k x , k y ) exp [ i ( k x x + k y y ) ] d k x d k y ,
A ( k x , k y ) = 1 4 π 2 δ ( x , y , 0 ) exp [ i ( k x x + k y y ) ] d x d y .
k x / 2 π = 1 / Δ x , k y / 2 π = 1 / Δ y .
δ ( x , y ) = exp { i [ k x 0 x + k y 0 y ] } ,
k x 0 = 2 π / ( Δ x ) 0 , k y 0 = 2 π / ( Δ y ) 0 ,
A ( k x , k y ) = δ D ( k x k x ( 0 ) ) δ D ( k y k y ( 0 ) ) ,
δ ( r ) = 2 π 0 A ( υ ) J 0 ( υ r ) υ d υ ,
A ( υ ) = ( 1 / 2 π ) 0 δ ( r ) J 0 ( υ r ) r d r ,
r = ( x 2 + y 2 ) 1 2 , υ = ( k x 2 + k y 2 ) 1 2 ,
δ ( r ) = J 0 ( υ ( 0 ) r ) ,
A ( υ ) = 1 2 π [ δ D ( υ υ ( 0 ) ) / υ ] .
υ ( 0 ) = k = 2 π / λ ,
k z = ( k 2 k x 2 k y 2 ) 1 2 = ( k 2 υ 0 2 ) 1 2 = 0 .
δ ( θ , φ ) a δ ( θ , φ ) ( e i k R / R ) ,
a δ ( θ , φ ) = 2 π i k cos θ A ( k sin θ cos φ , k sin θ sin φ ) .
δ ( θ ) = a δ ( θ ) ( e i k R / R ) ,
a δ ( θ ) = 2 π i k cos θ A ( k sin θ ) .
a K ( θ ) = C 2 J 1 ( k a sin θ ) k a sin θ ,
C = ( i / λ ) π a 2 .
| a δ ( 0 ) a K ( 0 ) | < 0.013 ,
δ ( P ¯ ) = 1 4 π φ 0 φ 0 + 2 π exp ( i k a σ ) σ 2 ( 1 ρ cos φ ) d φ ,
σ = ( 1 2 ρ cos φ + ρ 2 ) 1 2 .
δ ( x ) = α β f ( φ ) exp [ i x g ( φ ) ] d φ ,
f ( φ ) = 1 4 π 1 ρ cos φ σ 2 ,
g ( φ ) = σ ,
x = k a > 0 ,
α = φ 0 , β = φ 0 + 2 π .
δ ( x ) ( 2 π x ) 1 2 j j [ | g ( φ j ) | ] 1 2 f ( φ j ) exp [ i x g ( φ j ) ] ,
j = e ± i π / 4 according as g ( φ j ) 0 .
d g / d φ = 0 at φ = φ j .
φ 1 = π , φ 2 = 2 π .
g ( φ 1 ) = ρ / ( 1 + ρ ) , g ( φ 2 ) = ± ρ / ( 1 ρ ) according as ρ 1 ,
1 = exp ( i π / 4 ) , 2 = exp ( i π / 4 ) ,
g ( φ 1 ) = 1 + ρ , g ( φ 2 ) = ± ( 1 ρ ) according as ρ 1 , f ( φ 1 ) = ( 1 / 4 π ) [ 1 / ( 1 + ρ ) ] , f ( φ 2 ) = ( 1 / 4 π ) [ 1 / ( 1 ρ ) ] .
δ exp ( i k a ) 2 ( 2 π k a ρ ) 1 2 { exp [ i ( k a ρ π / 4 ) ] ( 1 + ρ ) 1 2 + exp [ i ( k a ρ π / 4 ) ] ( 1 ρ ) 1 2 } if 0 < ρ < 1 , exp ( i k a ρ ) 2 ( 2 π k a ρ ) 1 2 { exp [ i ( k a π / 4 ) ] ( ρ + 1 ) 1 2 exp [ i ( k a π / 4 ) ] ( ρ 1 ) 1 2 } if ρ > 1 .