Abstract

The second Rayleigh–Sommerfeld (RS) diffraction integral, wherein the normal derivative is specified, is evaluated in simple closed form for all axial points when a divergent or convergent spherical wave is incident upon a circular aperture or disk. These evaluations (solutions) are compared with known corresponding solutions of the first RS diffraction integral. These sets of solutions are intercompared with their mean value, i.e., the derived solutions of the Kirchhoff diffraction integral. The three diffraction formulations are shown to be in agreement for incident divergent spherical waves when the source and observation points are equally distant from the aperture or disk. Conversely, for convergent spherical waves, the three formulations are never in exact agreement for focal and observation points located at finite distances from the aperture, though at optical frequencies the relative error at the geometric focal point is vanishingly small. The second RS formulation predicts, in the limit of plane waves incident on a disk, that the axial irradiance is everywhere equal to the incident irradiance, whereas the first RS formulation predicts that the irradiance goes to zero at the back of the disk.

© 1973 Optical Society of America

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References

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  1. B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens’ Principle, 2nd ed. (Clarendon Press, Oxford, England, 1950).
  2. Rayleigh, Philos. Mag. 43, 259 (1897).
  3. A. Sommerfeld, Optics — Lectures on Theoretical Physics, Vol. IV (Academic, New York, 1954), p. 199.
  4. H. Osterberg and L. W. Smith, J. Opt. Soc. Am. 51, 1050 (1961).
    [Crossref]
  5. N. Mukunda, J. Opt. Soc. Am. 52, 336 (1962).
    [Crossref]
  6. E. Wolf and E. W. Marchand, J. Opt. Soc. Am. 54, 587 (1964).
    [Crossref]
  7. Reference 4, Eq. (6).
  8. Reference 4, Eq. (16).
  9. G. W. Farnell, J. Opt. Soc. Am. 48, 643 (1958).
    [Crossref]
  10. Reference 4, Figs. 3 and 4.
  11. A. B. Schafer, J. Opt. Soc. Am. 57, 638 (1967).
  12. E. Wolf, in Proceedings of the Symposium on Modern Optics, edited by Jerome Fox (Polytechnic Press, Brooklyn, New York, 1967).
  13. E. Marchand and E. Wolf, J. Opt. Soc. Am. 56, 1712 (1966).
    [Crossref]

1967 (1)

A. B. Schafer, J. Opt. Soc. Am. 57, 638 (1967).

1966 (1)

1964 (1)

1962 (1)

1961 (1)

1958 (1)

1897 (1)

Rayleigh, Philos. Mag. 43, 259 (1897).

Baker, B. B.

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

Copson, E. T.

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

Farnell, G. W.

Marchand, E.

Marchand, E. W.

Mukunda, N.

Osterberg, H.

Rayleigh,

Rayleigh, Philos. Mag. 43, 259 (1897).

Schafer, A. B.

A. B. Schafer, J. Opt. Soc. Am. 57, 638 (1967).

Smith, L. W.

Sommerfeld, A.

A. Sommerfeld, Optics — Lectures on Theoretical Physics, Vol. IV (Academic, New York, 1954), p. 199.

Wolf, E.

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

E. Wolf and E. W. Marchand, J. Opt. Soc. Am. 54, 587 (1964).
[Crossref]

E. Wolf, in Proceedings of the Symposium on Modern Optics, edited by Jerome Fox (Polytechnic Press, Brooklyn, New York, 1967).

J. Opt. Soc. Am. (6)

Philos. Mag. (1)

Rayleigh, Philos. Mag. 43, 259 (1897).

Other (6)

A. Sommerfeld, Optics — Lectures on Theoretical Physics, Vol. IV (Academic, New York, 1954), p. 199.

Reference 4, Figs. 3 and 4.

Reference 4, Eq. (6).

Reference 4, Eq. (16).

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

E. Wolf, in Proceedings of the Symposium on Modern Optics, edited by Jerome Fox (Polytechnic Press, Brooklyn, New York, 1967).

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Figures (3)

Fig. 1
Fig. 1

Plot of the relative axial-irradiance distributions for a divergent spherical wave incident on a circular aperture. The source point is located 100 wavelengths from the aperture, which is 10 wavelengths in radius. The abcissa is in units of wavelengths, with the origin at the aperture plane.

Fig. 2
Fig. 2

Plot of the relative axial-irradiance distributions for a divergent spherical wave incident on a circular disk. The source point is located 200 wavelengths from the disk, which is 20 wavelengths in radius. The abcissa is in units of wavelengths, with the origin at the center of the disk.

Fig. 3
Fig. 3

Plot of the relative axial irradiance |UII|2 in the vicinity of the focal point of a converging spherical wave, against distance measured in units of wavelengths from the geometrical focus. The focal distance is 100 wavelengths and the radius of the aperture is 10 and 20 wavelengths. All three diffraction formulations produce, on this scale, the same curves.

Equations (57)

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U I ( x , y , z ) = 1 2 π U i ( ξ , η , 0 ) z ( exp ( i k r ) r ) d ξ d η .
r = [ ( x ξ ) 2 + ( y η ) 2 + z 2 ] 1 2 ,
U II ( x , y , z ) = 1 2 π U i ( ξ , η , z ) z | z = 0 exp ( i k r ) r d ξ d η .
U i ( ξ , η , z ) z | z = 0
U k ( x , y , z ) = 1 2 [ U I ( x , y , z ) + U II ( x , y , z ) ] .
δ ( x , y , z ) = 1 2 [ U I ( x , y , z ) U II ( x , y , z ) ] .
ξ = ρ sin ϕ , η = ρ cos ϕ .
U i ( ξ , η , 0 ) = [ exp ( i k r s ) ] / r s ,
r s = ( ρ 2 + d 2 ) 1 2
U i ( ξ , η , z ) z | z = 0 = d ρ ρ ( exp ( i k r s ) r s ) .
U II ( 0 , 0 , z ) = d R 1 R 2 ρ ( exp ( i k r s ) r s ) exp ( i k r 0 ) r 0 d ρ ,
r 0 = ( ρ 2 + z 2 ) 1 2
U II ( 0 , 0 , z ) = d exp [ i k ( r 0 + r s ) ] r 0 r s | R 1 R 2 + d R 1 R 2 exp ( i k r 0 ) r 0 ρ ( exp ( i k r s ) r s ) d ρ .
ρ ( exp ( i k r 0 ) r 0 ) = ρ z z ( exp ( i k r 0 ) r 0 ) .
U II ( 0 , 0 , z ) = d exp [ i k ( r 0 + r s ) ] r 0 r s | R 1 R 2 d z U I ( 0 , 0 , z ) .
U I ( 0 , 0 , z ) = ( z / r 0 ) H ( r 0 + r s ) ,
U II ( 0 , 0 , z ) = ( d / r s ) H ( r 0 + r s ) ,
U k ( 0 , 0 , z ) = 1 2 ( z / r 0 + d / r s ) H ( r 0 + r s ) ,
δ ( 0 , 0 , z ) = 1 2 ( z / r 0 + d / r s ) G ( r 0 + r s ) .
H ( r 0 + r s ) = { exp [ i k ( r 0 + r s ) ] } / ( r 0 + r s )
S 0 = z + d .
S 1 = ( z 2 + a 2 ) 1 2 + ( d 2 + a 2 ) 1 2 .
cos θ s = d / ( d 2 + a 2 ) 1 2 and cos θ 0 = z / ( z 2 + a 2 ) 1 2 .
U I a ( 0 , 0 , z ) = [ exp ( i k S 0 ) ] / S 0 [ cos θ 0 ] × [ exp ( i k S 1 ) ] / S 1 ,
U II a ( 0 , 0 , z ) = [ exp ( i k S 0 ) ] / S 0 [ cos θ s ] × [ exp ( i k S 1 ) ] / S 1 ,
U k a ( 0 , 0 , z ) = [ exp ( i k S 0 ) ] / S 0 [ cos θ 0 + cos θ s ] × [ exp ( i k S 1 ) ] / 2 S 1 ,
δ a ( 0 , 0 , z ) = [ cos θ 0 cos θ s ] [ exp ( i k S 1 ) ] / 2 S 1 .
I 0 = d 2 .
| U I a ( 0 , 0 , 0 ) | 2 = I 0 ,
| U II a ( 0 , 0 , 0 ) | 2 = I 0 4 sin 2 ( k a / 2 ) ,
| U k a ( 0 , 0 , 0 ) | 2 = I 0 [ 5 4 cos ( k a ) ] / 4 .
U I d ( 0 , 0 , z ) = [ cos θ 0 ] [ exp ( i k S 1 ) ] / S 1 ,
U II d ( 0 , 0 , z ) = [ cos θ s ] [ exp ( i k S 1 ) ] / S 1 ,
U k d ( 0 , 0 , z ) = [ cos θ 0 + cos θ s ] [ exp ( i k S 1 ) ] / 2 S 1 ,
δ d ( 0 , 0 , z ) = [ cos θ 0 cos θ s ] [ exp ( i k S 1 ) ] / 2 S 1 .
U I a + U I d = U II a + U II d = U k a + U k d = { exp [ i k ( z + d ) ] } / ( z + d ) , z 0
| U I d ( 0 , 0 , z ) | 2 = I 0 cos 2 θ 0 ,
| U II d ( 0 , 0 , z ) | 2 = I 0 ,
| U k d ( 0 , 0 , z ) | 2 = I 0 cos 4 ( θ 0 / 2 ) .
U i ( ξ , η , 0 ) = [ exp ( i k r f ) ] / r f ,
r f = ( f 2 + ρ 2 ) 1 2
U i ( ξ , η , z ) z | z = 0 = f ρ ρ ( exp ( i k r f ) r f ) .
U II ( 0 , 0 , z ) = f exp [ i k ( r 0 r f ) ] r 0 r f | R 1 R 2 f z U I ( 0 , 0 , z ) ,
U I ( 0 , 0 , z ) = ( z / r 0 ) H ( r 0 r f ) ,
U II ( 0 , 0 , z ) = ( f / r j ) H ( r 0 r f ) ,
U k ( 0 , 0 , z ) = 1 2 ( z / r 0 + f / r f ) H ( r 0 r f ) ,
δ ( 0 , 0 , z ) = 1 2 ( z / r 0 + f / r f ) H ( r 0 r f ) .
H ( r 0 r f ) = { exp [ i k ( r 0 r f ) ] } / ( r 0 r f )
D 1 = ( z 2 + a 2 ) 1 2 ( f 2 + a 2 ) 1 2 .
D 0 = z f ,
cos θ f = f / ( f 2 + a 2 ) 1 2 .
U I ( 0 , 0 , z ) = [ cos θ 0 ] [ exp ( i k D 1 ) ] / D 1 [ exp ( i k D 0 ) ] / D 0 ,
U II ( 0 , 0 , z ) = [ cos θ f ] [ exp ( i k D 1 ) ] / D 1 [ exp ( i k D 0 ) ] / D 0 ,
U k ( 0 , 0 , z ) = [ cos θ 0 + cos θ f ] [ exp ( i k D 1 ) ] / 2 D 1 [ exp ( i k D 0 ) ] / D 0 ,
δ ( 0 , 0 , z ) = [ cos θ 0 cos θ f ] [ exp ( i k D 1 ) ] / 2 D 1 .
| δ ( 0 , 0 , f ) | = ( 1 / 2 f ) sin 2 θ f .
U I ( 0 , 0 , f ) = i k ( 1 cos θ f ) ( 1 / 2 f ) sin 2 θ f , U II ( 0 , 0 , f ) = i k ( 1 cos θ f ) + ( 1 / 2 f ) sin 2 θ f , U k ( 0 , 0 , f ) = i k ( 1 cos θ f ) .