Abstract

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.

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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.
  3. A. J. W. Sommerfeld, Optics (Academic Press, Inc., New York, 1954).
  4. A. Rubinowicz, Ann. Physik 53, 257 (1917); 73, 339 (1924).
  5. N. G. van Kampen, Physica 14, 575 (1949).
  6. J. B. Keller, R. M. Lewis, and B. D. Seckler, J. Appl. Phys. 28, 570 (1957).
  7. W. Braunbek, Z. Physik 127, 381 (1950); 127, 405 (1950).
  8. J. Coulson and G. G. Becknell, Phys. Rev. 20, 594 (1922); 20, 607 (1922).
  9. K. Nienhuis, Thesis, Groningen, 1948.
  10. J. B. Keller, J. Appl. Phys. 28, 426 (1957).
  11. S. N. Karp and A. Russek, J. Appl. Phys. 27, 886 (1956).
  12. S. N. Karp and J. B. Keller, Optica Acta (Paris) 8, 61 (1961).
  13. C. J. Bouwkamp, Repts. Progr. in Phys. 17, 35 (1954).
  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).
  16. R. N. Buchal and J. B. Keller, Commun. Pure Appl. Math. 8, 85 (1960).
  17. H. Levine and T. T. Wu, Tech. Rept. 71, Applied Mathematics and Statistics Laboratory, Stanford University, Stanford, California, (July, 1957).
  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).
  25. L. B. Felsen, J. Appl. Phys. 26, 138 (1955).
  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).
  28. B. R. Levy and J. B. Keller, Commun. Pure Appl. Math. 12, 159 (1959).
  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).
  31. D. Magiros and J. B. Keller, Commun. Pure. Appl. Math. 14, 457 (1961).
  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).

Becknell, G. G.

J. Coulson and G. G. Becknell, Phys. Rev. 20, 594 (1922); 20, 607 (1922).

Bouwkamp, C. J.

C. J. Bouwkamp, Repts. Progr. in Phys. 17, 35 (1954).

Braunbek, W.

W. Braunbek, Z. Physik 127, 381 (1950); 127, 405 (1950).

Buchal, R. N.

R. N. Buchal and J. B. Keller, Commun. Pure Appl. Math. 8, 85 (1960).

Burke, J. E.

J. E. Burke and J. B. Keller, Research Rept. EDL-E49, Electronic Defense Laboratories, Sylvania Electronic Systems, Mountain View, California (April, 1960).

J. E. Burke and J. B. Keller, Research Rept. EDL-E48, Electronic Defense Laboratories, Sylvania Electronic Systems, Mountain View, California (March, 1960).

Coulson, J.

J. Coulson and G. G. Becknell, Phys. Rev. 20, 594 (1922); 20, 607 (1922).

Crispin, J. W.

K. M. Siegel, J. W. Crispin, and C. E. Schensted, J. Appl. Phys. 26, 309 (1955).

Depperman, K.

W. Franz and K. Depperman, Ann. Physik 10, 361 (1952).

DeVore, R.

R. DeVore and R. Kouyoumjian, "The back scattering from a circular disk," URSI-IRE Spring meeting, Washington D. C. (May, 1961).

Felsen, L. B.

L. B. Felsen, J. Appl. Phys. 26, 138 (1955).

Franz, W.

W. Franz and K. Depperman, Ann. Physik 10, 361 (1952).

Friedrichs, K. O.

K. O. Friedrichs and J. B. Keller, J. Appl. Phys. 26, 961 (1955).

Karp, S. N.

S. N. Karp and J. B. Keller, Optica Acta (Paris) 8, 61 (1961).

S. N. Karp and A. Russek, J. Appl. Phys. 27, 886 (1956).

Keller, J. B.

J. E. Burke and J. B. Keller, Research Rept. EDL-E48, Electronic Defense Laboratories, Sylvania Electronic Systems, Mountain View, California (March, 1960).

J. B. Keller, R. M. Lewis, and B. D. Seckler, J. Appl. Phys. 28, 570 (1957).

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).

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.

S. N. Karp and J. B. Keller, Optica Acta (Paris) 8, 61 (1961).

R. N. Buchal and J. B. Keller, Commun. Pure Appl. Math. 8, 85 (1960).

J. E. Burke and J. B. Keller, Research Rept. EDL-E49, Electronic Defense Laboratories, Sylvania Electronic Systems, Mountain View, California (April, 1960).

J. B. Keller, J. Acoust. Soc. Am. 29, 1085 (1957).

J. B. Keller, IRE Trans. Antennas and Prop. AP-8, 175 (1960).

J. B. Keller, IRE Trans. Antennas and Prop. AP-9, 411 (1961).

K. O. Friedrichs and J. B. Keller, J. Appl. Phys. 26, 961 (1955).

B. R. Levy and J. B. Keller, IRE Trans. Antennas and Propagation, AP-7, 552 (1959).

B. D. Seckler and J. B. Keller, J. Acoust. Soc. Am. 31, 192 (1958).

J. B. Keller, J. Appl. Phys. 28, 426 (1957).

B. R. Levy and J. B. Keller, Can. J. Phys. 38, 128 (1960).

D. Magiros and J. B. Keller, Commun. Pure. Appl. Math. 14, 457 (1961).

J. B. Keller, J. Appl. Phys. 30, 1452 (1952).

J. B. Keller, IRE Trans. Antennas and Propagation AP-4, 243 (1956).

B. R. Levy and J. B. Keller, Commun. Pure Appl. Math. 12, 159 (1959).

Keys, J. E.

J. E. Keys and R. I. Primich, Defense Research Telecommunications Establishment, Ottawa, Canada, Rept. 1010 (May, 1959).

Kouyoumjian, R.

R. DeVore and R. Kouyoumjian, "The back scattering from a circular disk," URSI-IRE Spring meeting, Washington D. C. (May, 1961).

Kraus, L.

L. Kraus and L. Levine, Commun. Pure Appl. Math. 14, 49 (1961).

Levine, H.

H. Levine and T. T. Wu, Tech. Rept. 71, Applied Mathematics and Statistics Laboratory, Stanford University, Stanford, California, (July, 1957).

H. Levine, Institute of Mathematical Sciences, New York University, New York, Research Rept. EM-84 (1955).

Levine, L.

L. Kraus and L. Levine, Commun. Pure Appl. Math. 14, 49 (1961).

Levy, B. R.

B. R. Levy, J. Math. and Mech. 9, 147 (1960).

B. R. Levy and J. B. Keller, Can. J. Phys. 38, 128 (1960).

B. R. Levy and J. B. Keller, Commun. Pure Appl. Math. 12, 159 (1959).

B. R. Levy and J. B. Keller, IRE Trans. Antennas and Propagation, AP-7, 552 (1959).

Lewis, R. M.

J. B. Keller, R. M. Lewis, and B. D. Seckler, J. Appl. Phys. 28, 570 (1957).

Logan, N. A.

N. A. Logan and K. S. Yee, Symposium on Electromagnetic Theory, U. S. Army Mathematics Research Center, University of Wisconsin, Madison, Wisconsin (April, 1961).

Magiros, D.

D. Magiros and J. B. Keller, Commun. Pure. Appl. Math. 14, 457 (1961).

Nienhuis, K.

K. Nienhuis, Thesis, Groningen, 1948.

Primich, R. I.

J. E. Keys and R. I. Primich, Defense Research Telecommunications Establishment, Ottawa, Canada, Rept. 1010 (May, 1959).

Rubinowicz, A.

A. Rubinowicz, Ann. Physik 53, 257 (1917); 73, 339 (1924).

Russek, A.

S. N. Karp and A. Russek, J. Appl. Phys. 27, 886 (1956).

Schensted, C. E.

K. M. Siegel, J. W. Crispin, and C. E. Schensted, J. Appl. Phys. 26, 309 (1955).

Seckler, B. D.

B. D. Seckler and J. B. Keller, J. Acoust. Soc. Am. 31, 192 (1958).

J. B. Keller, R. M. Lewis, and B. D. Seckler, J. Appl. Phys. 28, 570 (1957).

Siegel, K. M.

K. M. Siegel, J. W. Crispin, and C. E. Schensted, J. Appl. Phys. 26, 309 (1955).

Sommerfeld, A. J. W.

A. J. W. Sommerfeld, Optics (Academic Press, Inc., New York, 1954).

van Kampen, N. G.

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

Wu, T. T.

H. Levine and T. T. Wu, Tech. Rept. 71, Applied Mathematics and Statistics Laboratory, Stanford University, Stanford, California, (July, 1957).

Yee, K. S.

N. A. Logan and K. S. Yee, Symposium on Electromagnetic Theory, U. S. Army Mathematics Research Center, University of Wisconsin, Madison, Wisconsin (April, 1961).

Other (37)

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).

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.

A. J. W. Sommerfeld, Optics (Academic Press, Inc., New York, 1954).

A. Rubinowicz, Ann. Physik 53, 257 (1917); 73, 339 (1924).

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

J. B. Keller, R. M. Lewis, and B. D. Seckler, J. Appl. Phys. 28, 570 (1957).

W. Braunbek, Z. Physik 127, 381 (1950); 127, 405 (1950).

J. Coulson and G. G. Becknell, Phys. Rev. 20, 594 (1922); 20, 607 (1922).

K. Nienhuis, Thesis, Groningen, 1948.

J. B. Keller, J. Appl. Phys. 28, 426 (1957).

S. N. Karp and A. Russek, J. Appl. Phys. 27, 886 (1956).

S. N. Karp and J. B. Keller, Optica Acta (Paris) 8, 61 (1961).

C. J. Bouwkamp, Repts. Progr. in Phys. 17, 35 (1954).

J. E. Burke and J. B. Keller, Research Rept. EDL-E48, Electronic Defense Laboratories, Sylvania Electronic Systems, Mountain View, California (March, 1960).

H. Levine, Institute of Mathematical Sciences, New York University, New York, Research Rept. EM-84 (1955).

R. N. Buchal and J. B. Keller, Commun. Pure Appl. Math. 8, 85 (1960).

H. Levine and T. T. Wu, Tech. Rept. 71, Applied Mathematics and Statistics Laboratory, Stanford University, Stanford, California, (July, 1957).

R. DeVore and R. Kouyoumjian, "The back scattering from a circular disk," URSI-IRE Spring meeting, Washington D. C. (May, 1961).

J. B. Keller, IRE Trans. Antennas and Prop. AP-8, 175 (1960).

J. B. Keller, IRE Trans. Antennas and Prop. AP-9, 411 (1961).

J. E. Keys and R. I. Primich, Defense Research Telecommunications Establishment, Ottawa, Canada, Rept. 1010 (May, 1959).

J. E. Burke and J. B. Keller, Research Rept. EDL-E49, Electronic Defense Laboratories, Sylvania Electronic Systems, Mountain View, California (April, 1960).

J. B. Keller, J. Acoust. Soc. Am. 29, 1085 (1957).

L. Kraus and L. Levine, Commun. Pure Appl. Math. 14, 49 (1961).

L. B. Felsen, J. Appl. Phys. 26, 138 (1955).

K. M. Siegel, J. W. Crispin, and C. E. Schensted, J. Appl. Phys. 26, 309 (1955).

J. B. Keller, IRE Trans. Antennas and Propagation AP-4, 243 (1956).

B. R. Levy and J. B. Keller, Commun. Pure Appl. Math. 12, 159 (1959).

B. R. Levy, J. Math. and Mech. 9, 147 (1960).

B. R. Levy and J. B. Keller, Can. J. Phys. 38, 128 (1960).

D. Magiros and J. B. Keller, Commun. Pure. Appl. Math. 14, 457 (1961).

J. B. Keller, J. Appl. Phys. 30, 1452 (1952).

W. Franz and K. Depperman, Ann. Physik 10, 361 (1952).

B. R. Levy and J. B. Keller, IRE Trans. Antennas and Propagation, AP-7, 552 (1959).

N. A. Logan and K. S. Yee, Symposium on Electromagnetic Theory, U. S. Army Mathematics Research Center, University of Wisconsin, Madison, Wisconsin (April, 1961).

K. O. Friedrichs and J. B. Keller, J. Appl. Phys. 26, 961 (1955).

B. D. Seckler and J. B. Keller, J. Acoust. Soc. Am. 31, 192 (1958).

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