Abstract

Lommel’s problem, the diffraction of spherical waves by a circular aperture and opaque disk, is extended to the case of a concentric array of ring-shaped apertures with arbitrary distribution of circle radii. The evaluation of the diffracted amplitude resulting from the entire array by the obvious process of numerical summation, applied to amplitude expressions pertaining to the sequence of circles involved, is found unsuitable and, instead, series expansions are investigated in which the location of the off-axis observation point and the interference effects of the ring openings are formally separated. The already available solutions of Lommel’s problem are examined from this viewpoint, and two new expansions in complex power series are derived. By means of two novel multiplication theorems for Lommel’s functions of two variables, convenient expansions in series of these functions are obtained for the amplitude diffracted by the annular array, thus affording the basis of a general theory of diffraction by ring systems, including the case of phase-reversal construction. Furthermore, equivalent general expressions are directly obtained from the power series formulas mentioned previously. A first application is made to the special case of the Soret zone plate. Finally Hopkins’ assumption of radially non-uniform amplitude is briefly dealt with in connection with the circular aperture and the annular systems.

© 1952 Optical Society of America

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References

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  1. Rayleigh, Scientific Papers, Vol. III, p. 87.
  2. C. T. Lane, Can. J. Research 2, 26 (1930).
    [Crossref]
  3. E. Lommel, Abhandl. k. bayer. Akad. Wiss. XV, 229 (1884).
  4. B. R. A. Nijboer, Physica 13, 605 (1947).
    [Crossref]
  5. F. Zernike and B. R. A. Nijboer, La Théorie des Images Optiques (Editions de la Revue d’Optique, Paris, 1949), p. 227.
  6. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University Press, London, 1922), second edition, p. 133.
  7. R. W. Wood, Phil. Mag. 45, 511 (1898).
    [Crossref]
  8. G. F. Hull, Am. J. Phys. 17, 559 (1949).
    [Crossref]
  9. I. Maddaus, (June3, 1948).
  10. G. Toraldo di Francia, Atti fondazione “Giorgio Ronchi” 6, 3 (1951).
  11. D. Gabor, Proc. Roy. Soc. (London),  197A, 454 (1949).
    [Crossref]
  12. G. L. Rogers, Nature 166, 237 (1950).
    [Crossref]
  13. G. L. Rogers, Nature 166, 1027 (1950).
    [Crossref]
  14. H. H. Hopkins, Proc. Phys. Soc. (London) 62B, 22 (1949).
    [Crossref]

1951 (1)

G. Toraldo di Francia, Atti fondazione “Giorgio Ronchi” 6, 3 (1951).

1950 (2)

G. L. Rogers, Nature 166, 237 (1950).
[Crossref]

G. L. Rogers, Nature 166, 1027 (1950).
[Crossref]

1949 (3)

H. H. Hopkins, Proc. Phys. Soc. (London) 62B, 22 (1949).
[Crossref]

D. Gabor, Proc. Roy. Soc. (London),  197A, 454 (1949).
[Crossref]

G. F. Hull, Am. J. Phys. 17, 559 (1949).
[Crossref]

1947 (1)

B. R. A. Nijboer, Physica 13, 605 (1947).
[Crossref]

1930 (1)

C. T. Lane, Can. J. Research 2, 26 (1930).
[Crossref]

1898 (1)

R. W. Wood, Phil. Mag. 45, 511 (1898).
[Crossref]

1884 (1)

E. Lommel, Abhandl. k. bayer. Akad. Wiss. XV, 229 (1884).

Gabor, D.

D. Gabor, Proc. Roy. Soc. (London),  197A, 454 (1949).
[Crossref]

Hopkins, H. H.

H. H. Hopkins, Proc. Phys. Soc. (London) 62B, 22 (1949).
[Crossref]

Hull, G. F.

G. F. Hull, Am. J. Phys. 17, 559 (1949).
[Crossref]

Lane, C. T.

C. T. Lane, Can. J. Research 2, 26 (1930).
[Crossref]

Lommel, E.

E. Lommel, Abhandl. k. bayer. Akad. Wiss. XV, 229 (1884).

Maddaus, I.

I. Maddaus, (June3, 1948).

Nijboer, B. R. A.

B. R. A. Nijboer, Physica 13, 605 (1947).
[Crossref]

F. Zernike and B. R. A. Nijboer, La Théorie des Images Optiques (Editions de la Revue d’Optique, Paris, 1949), p. 227.

Rayleigh,

Rayleigh, Scientific Papers, Vol. III, p. 87.

Rogers, G. L.

G. L. Rogers, Nature 166, 237 (1950).
[Crossref]

G. L. Rogers, Nature 166, 1027 (1950).
[Crossref]

Toraldo di Francia, G.

G. Toraldo di Francia, Atti fondazione “Giorgio Ronchi” 6, 3 (1951).

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University Press, London, 1922), second edition, p. 133.

Wood, R. W.

R. W. Wood, Phil. Mag. 45, 511 (1898).
[Crossref]

Zernike, F.

F. Zernike and B. R. A. Nijboer, La Théorie des Images Optiques (Editions de la Revue d’Optique, Paris, 1949), p. 227.

Abhandl. k. bayer. Akad. Wiss. (1)

E. Lommel, Abhandl. k. bayer. Akad. Wiss. XV, 229 (1884).

Am. J. Phys. (1)

G. F. Hull, Am. J. Phys. 17, 559 (1949).
[Crossref]

Atti fondazione “Giorgio Ronchi” (1)

G. Toraldo di Francia, Atti fondazione “Giorgio Ronchi” 6, 3 (1951).

Can. J. Research (1)

C. T. Lane, Can. J. Research 2, 26 (1930).
[Crossref]

Nature (2)

G. L. Rogers, Nature 166, 237 (1950).
[Crossref]

G. L. Rogers, Nature 166, 1027 (1950).
[Crossref]

Phil. Mag. (1)

R. W. Wood, Phil. Mag. 45, 511 (1898).
[Crossref]

Physica (1)

B. R. A. Nijboer, Physica 13, 605 (1947).
[Crossref]

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

H. H. Hopkins, Proc. Phys. Soc. (London) 62B, 22 (1949).
[Crossref]

Proc. Roy. Soc. (London) (1)

D. Gabor, Proc. Roy. Soc. (London),  197A, 454 (1949).
[Crossref]

Other (4)

F. Zernike and B. R. A. Nijboer, La Théorie des Images Optiques (Editions de la Revue d’Optique, Paris, 1949), p. 227.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University Press, London, 1922), second edition, p. 133.

Rayleigh, Scientific Papers, Vol. III, p. 87.

I. Maddaus, (June3, 1948).

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Equations (30)

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k = 2 π ( R + R ) / λ R R ,             l = 2 π σ / λ R ,
u P = 2 i ω 0 a exp ( - i k ρ 2 / 2 ) ρ J 0 ( l ρ ) d ρ ,
2 i ω ϕ ( a , k , l ) ,
y = k a 2 ,             z = l a ,
ϕ ( a , y , z ) = a 2 i y { e i z 2 / 2 y - e - i y / 2 [ V 0 ( y , z ) + i V 1 ( y , z ) ] } ϕ ( a , y , z ) = a 2 i y e - i y / 2 [ - U 2 ( y , z ) + i U 1 ( y , z ) ] } .
U n ( y , z ) = p = 0 ( - 1 ) p ( y z ) n + 2 p J n + 2 p ( z ) V n ( y , z ) = p = 0 ( - 1 ) p ( z y ) n + 2 p J n + 2 p ( z ) } .
ϕ * ( a , y , z ) = e i y / 4 ( 2 π ) 1 2 y - 1 2 n = 0 ( - i ) n ( 2 n + 1 ) × J n + 1 2 ( y 4 ) J 2 n + 1 ( z ) / z .
ϕ ( a , k , l ) = 1 i k { e i l 2 / 2 k - r = 0 1 r ! ( i l 2 2 k ) r F r ( i k a 2 / 2 ) } ,
F r ( x ) e - x p = 0 r x p / p ! .
z z μ + 1 C ν ( z ) d z = - ( μ 2 - ν 2 ) z z μ - 1 C ν ( z ) d z + [ z μ + 1 C ν + 1 ( z ) + ( μ - ν ) z μ C ν ( z ) ] .
ϕ ( a , k , l ) = 1 l 2 r = 0 ( - 1 ) r r ! ( 2 i k l 2 ) r · p = 0 r ( - 1 ) p ( l a ) 2 p ( 2 p p ! ) 2 · { ( l a ) J 1 ( l a ) + 2 p J 0 ( l a ) } .
Δ u P = 2 i ω [ ϕ ( b n , k , l ) - ϕ ( a n , k , l ) ] , 2 i ω ϒ ( a n , b n , k , l ) ,
u P = 2 i ω n = 1 N ϒ ( a n , b n , k , l ) .
u P = 2 i ω { - n = 1 N ϒ ( a n , b n , k , l ) + ϕ ( b N , k , l ) } .
u P = 2 i ω { 2 n = 1 N ϒ ( a n , b n , k , l ) - ϕ ( b N , k , l ) } ,
u P = 2 i ω { - 2 n = 1 N ϒ ( a n , b n , k , l ) + 2 ϕ ( b N , k , l ) - ϕ ( a N , k , l ) } ,
V n ( α 2 y , α z ) = m = 0 ( - 1 ) m ( α 2 - 1 ) m y m 2 m m ! V n + m ( y , z ) U n ( α 2 y , α z ) = m = 0 ( α 2 - 1 ) m y m 2 m m ! U n - m ( y , z ) } ,
k a n 2 = ( a n / ) 2 y l a n = ( a n / ) z } ,             k b n 2 = ( b n / ) 2 y l b n = ( b n / ) z } .
u P ( y , z , N ) = 2 ω 2 y m = 0 ( - 1 ) m y m 2 m m ! × [ V m ( y , z ) + i V m + 1 ( y , z ) ] · ( f m - i g m ) ,
u P ( y , z , N ) = 2 ω 2 y m = 0 y m 2 m m ! × [ U 2 - m ( y , z ) - i U 1 - m ( y , z ) ] · ( f m - i g m ) ,
f m - i g m n = 1 N { [ ( a n ) 2 - 1 ] m exp [ - i ( a n ) 2 y 2 ] - [ ( b n ) 2 - 1 ] m exp [ - i ( b n ) 2 y 2 ] } ,
U n ( y , z ) + U n + 2 ( y , z ) = ( y z ) n J n ( z ) V n ( y , z ) + V n + 2 ( y , z ) = ( z y ) n J n ( z ) } .
u P ( y , z , N ) = 2 ω 2 y r = 0 1 r ! ( i z 2 2 y ) r · n = 1 N { F r ( i ( a n ) 2 y 2 ) - F r ( i ( b n ) 2 y 2 ) } ,
u P ( y , z , N ) = 2 ω 2 y r = 0 ( - 1 ) r r ! ( 2 i y z 2 ) r · p = 0 r ( - 1 ) p z 2 p ( 2 p p ! ) 2 · n = 1 N { ( b n ) 2 p · [ ( b n z ) J 1 ( b n z ) + 2 p J 0 ( b n z ) ] - ( a n ) 2 p · [ ( a n z ) J 1 ( a n z ) + 2 p J 0 ( a n z ) ] }
u p ( y , z , N ) = 2 ω a 1 2 y m = 0 ( - 1 ) m y m 2 m m ! × [ V m ( y , z ) + i V m + 1 ( y , z ) ] · s = 1 2 N ( - 1 ) s - 1 e - i s y / 2 ( s - 1 ) m ,
u P ( y , z , N ) = 2 ω a 1 2 y m = 0 y m 2 m m ! [ U 2 - m ( y , z ) - i U 1 - m ( y , z ) ] · s = 1 2 N ( - 1 ) s - 1 e - i s y / 2 ( s - 1 ) m ,
u P ( y , 0 , N ) = - 2 ω a 1 2 y s = 1 2 N ( - 1 ) s e - i s y / 2 .
u P u P * = 4 ω 2 ( a 1 2 y ) 2 · [ sin ( N y / 2 ) cos ( y / 4 ) ] 2 .
u P ( 0 , z , N ) = 2 π λ a 1 2 z m = 0 ( - 1 ) m z m 2 m m ! J m - 1 ( z ) · s = 1 2 N ( - 1 ) s - 1 ( s - 1 ) m ,
Y n ( α 2 y , α z ) = m = 0 ( - 1 ) m ( α 2 - 1 ) m y m 2 m m ! × { Y m + n ( y , z ) - m V m + n ( y , z ) } X n ( α 2 y , α z ) = m = 0 ( α 2 - 1 ) m y m 2 m m ! × { X n - m ( y , z ) - m U n - m ( y , z ) } } ,