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Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 52, Iss. 2 — Feb. 1, 1962
  • pp: 116–130

Geometrical Theory of Diffraction

JOSEPH B. KELLER  »View Author Affiliations

JOSA, Vol. 52, Issue 2, pp. 116-130 (1962)

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The geometrical theory of diffraction is an extension of geometrical optics which accounts for diffraction. It introduces diffracted rays in addition to the usual rays of geometrical optics. These rays are produced by incident rays which hit edges, corners, or vertices of boundary surfaces, or which graze such surfaces. Various laws of diffraction, analogous to the laws of reflection and refraction, are employed to characterize the diffracted rays. A modified form of Fermat’s principle, equivalent to these laws, can also be used. Diffracted wave fronts are defined, which can be found by a Huygens wavelet construction. There is an associated phase or eikonal function which satisfies the eikonal equation. In addition complex or imaginary rays are introduced. A field is associated with each ray and the total field at a point is the sum of the fields on all rays through the point. The phase of the field on a ray is proportional to the optical length of the ray from some reference point. The amplitude varies in accordance with the principle of conservation of energy in a narrow tube of rays. The initial value of the field on a diffracted ray is determined from the incident field with the aid of an appropriate diffraction coefficient. These diffraction coefficients are determined from certain canonical problems. They all vanish as the wavelength tends to zero. The theory is applied to diffraction by an aperture in a thin screen diffraction by a disk, etc., to illustrate it. Agreement is shown between the predictions of the theory and various other theoretical analyses of some of these problems. Experimental confirmation of the theory is also presented. The mathematical justification of the theory on the basis of electromagnetic theory is described. Finally, the applicability of this theory, or a modification of it, to other branches of physics is explained.

JOSEPH B. KELLER, "Geometrical Theory of Diffraction," J. Opt. Soc. Am. 52, 116-130 (1962)

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  1. J. B. Keller, "The geometrical theory of diffraction," Proceedings of the Symposium on Microwave Optics, Eaton Electronics Research Laboratory, McGill University, Montreal, Canada (June, 1953).
  2. See J. B. Keller, in Calculus of Variations and its Applications, Proceedings of Svmnposia in Applied Math, edited by L. M. Graves (McGraw-Hill Book Company, Inc., New York and American Mathematical Society, Providence, Rhode Island, 1958), Vol. 8.
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  7. W. Braunbek, Z. Physik 127, 381 (1950); 127, 405 (1950).
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  14. J. E. Burke and J. B. Keller, Research Rept. EDL-E48, Electronic Defense Laboratories, Sylvania Electronic Systems, Mountain View, California (March, 1960).
  15. H. Levine, Institute of Mathematical Sciences, New York University, New York, Research Rept. EM-84 (1955).
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  18. R. DeVore and R. Kouyoumjian, "The back scattering from a circular disk," URSI-IRE Spring meeting, Washington D. C. (May, 1961).
  19. J. B. Keller, IRE Trans. Antennas and Prop. AP-8, 175 (1960).
  20. J. B. Keller, IRE Trans. Antennas and Prop. AP-9, 411 (1961).
  21. J. E. Keys and R. I. Primich, Defense Research Telecommunications Establishment, Ottawa, Canada, Rept. 1010 (May, 1959).
  22. J. E. Burke and J. B. Keller, Research Rept. EDL-E49, Electronic Defense Laboratories, Sylvania Electronic Systems, Mountain View, California (April, 1960).
  23. J. B. Keller, J. Acoust. Soc. Am. 29, 1085 (1957).
  24. L. Kraus and L. Levine, Commun. Pure Appl. Math. 14, 49 (1961).
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  26. K. M. Siegel, J. W. Crispin, and C. E. Schensted, J. Appl. Phys. 26, 309 (1955).
  27. J. B. Keller, IRE Trans. Antennas and Propagation AP-4, 243 (1956).
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  29. B. R. Levy, J. Math. and Mech. 9, 147 (1960).
  30. B. R. Levy and J. B. Keller, Can. J. Phys. 38, 128 (1960).
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  32. J. B. Keller, J. Appl. Phys. 30, 1452 (1952).
  33. W. Franz and K. Depperman, Ann. Physik 10, 361 (1952).
  34. B. R. Levy and J. B. Keller, IRE Trans. Antennas and Propagation, AP-7, 552 (1959).
  35. N. A. Logan and K. S. Yee, Symposium on Electromagnetic Theory, U. S. Army Mathematics Research Center, University of Wisconsin, Madison, Wisconsin (April, 1961).
  36. K. O. Friedrichs and J. B. Keller, J. Appl. Phys. 26, 961 (1955).
  37. B. D. Seckler and J. B. Keller, J. Acoust. Soc. Am. 31, 192 (1958).

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