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Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 52, Iss. 7 — Jul. 1, 1962
  • pp: 761–763

Boundary Diffraction Wave in the Domain of the Rayleigh-Kirchhoff Diffraction Theory

E. W. MARCHAND and E. WOLF  »View Author Affiliations

JOSA, Vol. 52, Issue 7, pp. 761-763 (1962)

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Maggi and Rubinowicz formulated a boundary wave theory of diffraction at an aperture, which may be regarded as developments of early ideas of Young about the nature of diffraction. These investigations were based on the Kirchhoff diffraction theory. In the present paper it is shown that a theory of the boundary diffraction wave may also be formulated by taking as starting point the representation of the field in terms of the Rayleigh diffraction integrals, which are free from a mathematical inconsistency inherent in the Kirchhoff theory. It is shown that the Rayleigh integrals (with physically approximate but mathematically consistent boundary conditions of the Kirchhoff type) may be transformed into the sum of two terms. One term represents the effect of a disturbance which originates in each point of the boundary of the aperture (boundary wave); the other represents the combined effect of disturbances originating in certain special points situated within the aperture. In the special cases when the field incident upon the aperture is plane or spherical, the second term represents precisely the disturbances predicted by geometrical optics, in strict analogy with the classical results of Maggi and Rubinowicz. In these special cases the boundary wave of the present theory differs from the boundary wave of the earlier theories only in the form of an inclination factor.

E. W. MARCHAND and E. WOLF, "Boundary Diffraction Wave in the Domain of the Rayleigh-Kirchhoff Diffraction Theory," J. Opt. Soc. Am. 52, 761-763 (1962)

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  1. G. A. Maggi, Ann. di Mat. IIa 16, 21 (1888).
  2. A. Rubinowicz, Ann. Physik 53, 257 (1917).
  3. A. Rubinowicz, Acta Phys. Polonica 12, 225 (1953).
  4. A comprehensive account of previous researches on the boundary wave will be found in A. Rubinowicz, Die Beugungswelle in der Kirchhoffschen Theorie der Beugung (Polska Akademia Nauk, Warszawa, 1957). See also M. Born and E. Wolf, Principles of Optics (Pergamon Press, London and New York, 1959), Sec. 8.9.
  5. K. Miyamoto and E. Wolf, J. Opt. Soc. Am. 52, 615 (1962).
  6. K. Miyamoto and E. Wolf, J. Opt. Soc. Am. 52, 626 (1962).
  7. A. Rubinowicz, J. Opt. Soc. Am. 52, 717 (1962).
  8. A. Rubinowicz, J. Opt. Soc. Am. 52, 717 (1962).
  9. A. Rubinowicz, Acta Phys. Polon. 21, 61 (1962).
  10. See, for example, Lord Rayleigh, Theory of Sound (Dover Publications, Inc., New York, 1945), 2nd ed., Vol. II, p. 278.
  11. N. Mukunda, J. Opt. Soc. Am. 52, 336 (1962).
  12. In this connection it is of interest to note a result of a recent investigation of H. Osterberg and L. W. Smith [J. Opt. Soc. Am. 51, 1050 (1961)]. These authors considered, on the basis of the Rayleigh-Kirchhoff theory, the diffraction of a spherical wave by a circular aperture, the center of the spherical wave being assumed to lie on the normal through the center of the aperture. They found that the field at a point P on this normal, in the half-space into which the wave is propagated, may be regarded as the combined effect of a wave deflected to P at the rim of the aperture and a wave reaching P directly through the aperture. This result illustrates, for this special case, the require decomposition of the Rayleigh-Kirchhoff integral.
  13. In reference 9 Rubinowicz showed [his Eq. (8.3)] that the expression (3.12) may also be expressed in the form WK(P,Q)=curlQM(P,Q)+WK(P,A), where [Equation].

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