Abstract

<p>As a first step towards a generalization of the Maggi-Rubinowicz theory of the boundary diffraction wave, a new vector potential W(<i>Q,P</i>) is associated with any monochromatic scalar wavefield <i>U</i>(<i>P</i>). This potential has the property that the normal component of its curl, taken with respect to the coordinates of any point <i>Q</i> on a closed surface <i>S</i> surrounding a field point <i>P</i>, is equal to the integrand of the Helmholtz-Kirchhoff integral; that is, [Equation], where <i>s</i> is the distance <i>QP</i> and ∂/∂<i>n</i> denotes the differentiation along the inward unit normal n to <i>S</i>.</p><p>Further it is shown that the vector potential always has singularities at some points <i>Q</i> on <i>S</i> and that the field at <i>P</i> may be rigorously expressed as the sum of disturbances propagated from these points alone.</p><p>A closed expression for the vector potential associated with any given monochromatic wavefield that obeys the Sommerfeld radiation condition at infinity is derived and it is shown that in the special case when <i>U</i> is a spherical wave, this expression reduces to that found by G. A. Maggi and A. Rubinowicz in their researches on the boundary diffraction wave.</p>

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  1. T. Young, Phil. Trans. Roy. Soc. 20, 26 (1802). Also Miscellaneous Works of the late Thomas Young (John Murray, London, 1855), Vol. 1, p. 151.
  2. Oeuvrés complètes d'Augustin Fresnel (Imprimerie Impériale, Paris, 1866), Vol. I, pp. 89, 129.
  3. A. Rubinowicz, Nature 180, 160 (1957).
  4. A. Rubinowicz, Die Beugungswelle in der Kirchhoffschen Theorie der Beugung (Polska Akademia Nauk, Warsaw, 1957).
  5. M. Born and E. Wolf, Principles of Optics (Pergamon Press, London and New York, 1959).
  6. G. A. Maggi, Ann. di Mat. IIa, 16, 21 (1888).
  7. A. Rubinowicz, Ann. Physik 53, 257 (1917).
  8. Since more complicated boundaries are in general encountered in the study of diffraction phenomena, the term "edge diffraction wave" would seem to be more appropriate but we conform here to the usual terminology.
  9. The equivalence of the formally different solutions of Maggi and Rubinowicz was first noted by F. Kottler [Ann. Physik 70, 405 (1923); 71, 457 (1923)] in his important papers dealing with the intricate concept of a black screen, the saltus interpretation of Kirchhoff theory and the generalization of the scalar Huygens-Fresnel Principle to electromagnetic waves. In this work the boundary wave also played an important role. Accounts of the researches of Maggi and Kottler are given in B. B. Baker and E. T. Copson, The Mathentatical Theory of Huygens' Principle (Oxford University Press, Oxford, 1950), 2nd ed.
  10. R. S. Ingarden [Acta Phys. Polonica 14, 77 (1955)] considered an extension of the Young-Maggi-Rubinowicz theory to waves which are of more complicated form, but which are restricted in that they have to obey the laws of geometrical optics. Even with this restriction no simple generalization was found by him.
  11. N. G. van Kampen, Physica 14, 575 (1949).
  12. A. Rubinowicz, Phys. Rev. 54, 931 (1938).
  13. A. Rubinowicz, Acta Phys. Polonica 12, 225 (1953).
  14. In some cases the singular points may not be discrete. In particular, in certain symmetrical situations one may expect that the singular points will form a continuous line. Such cases would need more refined analysis and will not be considered here.
  15. B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens' Principle (Oxford University Press, 1950), 2nd ed., p. 79.
  16. E. Wolf, Proc. Phys. Soc. (London) 74, 280 (1959).
  17. If there are singularities of the field in the region under consideration, A (px,py) may change on crossing a plane containing each singularity [cf., expression (6.8) below for the spherical wave].
  18. R. Courant and D. Hilbert, Methoden der Mathematischen Physik (Springer-Verlag, Berlin, 1937), Vol. II, p. 187.
  19. There are no contributions to W arising from Ai+ or Ai-, since one of these is neglected for physical reasons already explained, and the other obviously tends to zero as s0 → ∞.

Baker, B. B.

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens' Principle (Oxford University Press, 1950), 2nd ed., p. 79.

Born, M.

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

Copson, E. T.

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens' Principle (Oxford University Press, 1950), 2nd ed., p. 79.

Courant, R.

R. Courant and D. Hilbert, Methoden der Mathematischen Physik (Springer-Verlag, Berlin, 1937), Vol. II, p. 187.

Hilbert, D.

R. Courant and D. Hilbert, Methoden der Mathematischen Physik (Springer-Verlag, Berlin, 1937), Vol. II, p. 187.

Ingarden, R. S.

R. S. Ingarden [Acta Phys. Polonica 14, 77 (1955)] considered an extension of the Young-Maggi-Rubinowicz theory to waves which are of more complicated form, but which are restricted in that they have to obey the laws of geometrical optics. Even with this restriction no simple generalization was found by him.

Kottler, F.

The equivalence of the formally different solutions of Maggi and Rubinowicz was first noted by F. Kottler [Ann. Physik 70, 405 (1923); 71, 457 (1923)] in his important papers dealing with the intricate concept of a black screen, the saltus interpretation of Kirchhoff theory and the generalization of the scalar Huygens-Fresnel Principle to electromagnetic waves. In this work the boundary wave also played an important role. Accounts of the researches of Maggi and Kottler are given in B. B. Baker and E. T. Copson, The Mathentatical Theory of Huygens' Principle (Oxford University Press, Oxford, 1950), 2nd ed.

Maggi, G. A.

G. A. Maggi, Ann. di Mat. IIa, 16, 21 (1888).

Rubinowicz, A.

A. Rubinowicz, Nature 180, 160 (1957).

A. Rubinowicz, Ann. Physik 53, 257 (1917).

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

A. Rubinowicz, Phys. Rev. 54, 931 (1938).

A. Rubinowicz, Die Beugungswelle in der Kirchhoffschen Theorie der Beugung (Polska Akademia Nauk, Warsaw, 1957).

van Kampen, N. G.

N. G. van Kampen, Physica 14, 575 (1949).

Wolf, E.

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

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

Young, T.

T. Young, Phil. Trans. Roy. Soc. 20, 26 (1802). Also Miscellaneous Works of the late Thomas Young (John Murray, London, 1855), Vol. 1, p. 151.

Other (19)

T. Young, Phil. Trans. Roy. Soc. 20, 26 (1802). Also Miscellaneous Works of the late Thomas Young (John Murray, London, 1855), Vol. 1, p. 151.

Oeuvrés complètes d'Augustin Fresnel (Imprimerie Impériale, Paris, 1866), Vol. I, pp. 89, 129.

A. Rubinowicz, Nature 180, 160 (1957).

A. Rubinowicz, Die Beugungswelle in der Kirchhoffschen Theorie der Beugung (Polska Akademia Nauk, Warsaw, 1957).

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

G. A. Maggi, Ann. di Mat. IIa, 16, 21 (1888).

A. Rubinowicz, Ann. Physik 53, 257 (1917).

Since more complicated boundaries are in general encountered in the study of diffraction phenomena, the term "edge diffraction wave" would seem to be more appropriate but we conform here to the usual terminology.

The equivalence of the formally different solutions of Maggi and Rubinowicz was first noted by F. Kottler [Ann. Physik 70, 405 (1923); 71, 457 (1923)] in his important papers dealing with the intricate concept of a black screen, the saltus interpretation of Kirchhoff theory and the generalization of the scalar Huygens-Fresnel Principle to electromagnetic waves. In this work the boundary wave also played an important role. Accounts of the researches of Maggi and Kottler are given in B. B. Baker and E. T. Copson, The Mathentatical Theory of Huygens' Principle (Oxford University Press, Oxford, 1950), 2nd ed.

R. S. Ingarden [Acta Phys. Polonica 14, 77 (1955)] considered an extension of the Young-Maggi-Rubinowicz theory to waves which are of more complicated form, but which are restricted in that they have to obey the laws of geometrical optics. Even with this restriction no simple generalization was found by him.

N. G. van Kampen, Physica 14, 575 (1949).

A. Rubinowicz, Phys. Rev. 54, 931 (1938).

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

In some cases the singular points may not be discrete. In particular, in certain symmetrical situations one may expect that the singular points will form a continuous line. Such cases would need more refined analysis and will not be considered here.

B. B. Baker and E. T. Copson, The Mathematical Theory of Huygens' Principle (Oxford University Press, 1950), 2nd ed., p. 79.

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

If there are singularities of the field in the region under consideration, A (px,py) may change on crossing a plane containing each singularity [cf., expression (6.8) below for the spherical wave].

R. Courant and D. Hilbert, Methoden der Mathematischen Physik (Springer-Verlag, Berlin, 1937), Vol. II, p. 187.

There are no contributions to W arising from Ai+ or Ai-, since one of these is neglected for physical reasons already explained, and the other obviously tends to zero as s0 → ∞.

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