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.

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  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).
  2. E. W. Marchand and E. Wolf, J. Opt. Soc. Am. 56, 1712 (1966).
  3. J. B. Keller, (a) J. Appl. Phvs. 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 byF. J. Kottler, in Progress in Optics, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, and J. Wiley & Sons, New York) 4, 281 (1965); 6, 331 (1967), and the preface to Vol. 4.
  5. A. Rubinowicz (a) 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)2 COS2(ψ/2).

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).

Keller, J. B.

J. B. Keller, (a) J. Appl. Phvs. 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 byF. J. Kottler, in Progress in Optics, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, and J. Wiley & Sons, New York) 4, 281 (1965); 6, 331 (1967), and the preface to Vol. 4.

Marchand, E. W.

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

Rubinowicz, A.

A. Rubinowicz (a) 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).

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

Other (8)

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).

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

J. B. Keller, (a) J. Appl. Phvs. 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).

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

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

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)2 COS2(ψ/2).

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