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.

© 1962 Optical Society of America

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References

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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 Symposia 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); Ann. Physik 73, 339 (1924).
    [Crossref]
  5. N. G. van Kampen, Physica 14, 575 (1949).
    [Crossref]
  6. J. B. Keller, R. M. Lewis, and B. D. Seckler, J. Appl. Phys. 28, 570 (1957).
    [Crossref]
  7. W. Braunbek, Z. Physik 127, 381 (1950); Z. Physik 127, 405 (1950).
    [Crossref]
  8. J. Coulson and G. G. Becknell, Phys. Rev. 20, 594 (1922); Phys. Rev. 20, 607 (1922).
    [Crossref]
  9. K. Nienhuis, Thesis, Groningen, 1948.
  10. J. B. Keller, J. Appl. Phys. 28, 426 (1957).
    [Crossref]
  11. S. N. Karp and A. Russek, J. Appl. Phys. 27, 886 (1956).
    [Crossref]
  12. S. N. Karp and J. B. Keller, Optica Acta (Paris) 8, 61 (1961).
    [Crossref]
  13. C. J. Bouwkamp, Repts. Progr. in Phys. 17, 35 (1954).
    [Crossref]
  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).
    [Crossref]
  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).
    [Crossref]
  20. J. B. Keller, IRE Trans. Antennas and Prop. AP-9, 411 (1961).
    [Crossref]
  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).
    [Crossref]
  24. L. Kraus and L. Levine, Commun. Pure Appl. Math. 14, 49 (1961).
    [Crossref]
  25. L. B. Felsen, J. Appl. Phys. 26, 138 (1955).
    [Crossref]
  26. K. M. Siegel, J. W. Crispin, and C. E. Schensted, J. Appl. Phys. 26, 309 (1955).
    [Crossref]
  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).
    [Crossref]
  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).
    [Crossref]
  31. D. Magiros and J. B. Keller, Commun. Pure. Appl. Math. 14, 457 (1961).
    [Crossref]
  32. J. B. Keller, J. Appl. Phys. 30, 1452 (1952).
    [Crossref]
  33. W. Franz and K. Depperman, Ann. Physik 10, 361 (1952).
    [Crossref]
  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).
    [Crossref]
  37. B. D. Seckler and J. B. Keller, J. Acoust. Soc. Am. 31, 192 (1958).
    [Crossref]

1961 (4)

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

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

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

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

1960 (4)

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

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

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

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

1959 (2)

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

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

1958 (1)

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

1957 (3)

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

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

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

1956 (2)

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

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

1955 (3)

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

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

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

1954 (1)

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

1952 (2)

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

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

1950 (1)

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

1949 (1)

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

1922 (1)

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

1917 (1)

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

Becknell, G. G.

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

Bouwkamp, C. J.

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

Braunbek, W.

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

Buchal, R. N.

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

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); Phys. Rev. 20, 607 (1922).
[Crossref]

Crispin, J. W.

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

Depperman, K.

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

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).
[Crossref]

Franz, W.

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

Friedrichs, K. O.

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

Karp, S. N.

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

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

Keller, J. B.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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 Symposia 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.

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

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

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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

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).
[Crossref]

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).
[Crossref]

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); Ann. Physik 73, 339 (1924).
[Crossref]

Russek, A.

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

Schensted, C. E.

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

Seckler, B. D.

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

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

Siegel, K. M.

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

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).
[Crossref]

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).

Ann. Physik (2)

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

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

Can. J. Phys. (1)

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

Commun. Pure Appl. Math. (3)

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

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

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

Commun. Pure. Appl. Math. (1)

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

IRE Trans. Antennas and Prop. (2)

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

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

IRE Trans. Antennas and Propagation (2)

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

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

J. Acoust. Soc. Am. (2)

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

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

J. Appl. Phys. (7)

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

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

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

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

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

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

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

J. Math. and Mech. (1)

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

Optica Acta (Paris) (1)

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

Phys. Rev. (1)

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

Physica (1)

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

Repts. Progr. in Phys. (1)

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

Z. Physik (1)

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

Other (11)

K. Nienhuis, Thesis, Groningen, 1948.

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 Symposia 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).

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).

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).

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).

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

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Figures (21)

Fig. 1
Fig. 1

(a) The cone of diffracted rays produced by an incident ray which hits the edge of a thin screen obliquely. (b) The plane of diffracted rays produced by a ray normally incident on the edge of a thin screen.

Fig. 2
Fig. 2

The projection of incident and diffracted rays into a plane normal to the edge of a screen. The angles α and θ are those between the projections and the normal to the screen, measured as shown in the figure. The edge is normal to the plane of the figure.

Fig. 3
Fig. 3

The diffracted rays produced by a plane wave normally incident on a slit in a thin screen. The two incident rays which hit the slit edges are shown, with some of the singly diffracted rays they produce. One diffracted ray from each edge is shown crossing the slit and hitting the opposite edge, producing doubly diffracted rays.

Fig. 4
Fig. 4

The far-field diffraction pattern of a slit of width 2a hit normally by a plane wave; ka=8. The solid curve based upon Eq. (6) results from single diffraction, and applies to a screen on which u=0 or ∂u/∂n=0. The dashed curve includes the effects of multiple diffraction for a screen on which u=0. The dots are based upon the exact solution of the reduced wave equation for a screen on which u=0.11 The ordinate is k|f(φ)| and the abscissa is φ in radians.

Fig. 5
Fig. 5

The far-field diffraction pattern of a slit of width 2a hit normally by a plane wave; ka=10π. The curve, based on Eq. (6), results from single diffraction for a screen on which u=0 or ∂u/∂n=0. The ordinate is k|fs(φ)| and the abscissa is φ in radians. The value at φ=0 is finite but too high to be shown.

Fig. 6
Fig. 6

The transmission cross section of a slit of width 2a as a function of ka, for normal incidence with u=0 on the screen. The solid curve, based on Eq. (10), results from single and double diffraction; the dashed curve includes single and all multiple diffraction. The dots are based upon the exact solution of the reduced wave equation with u=0 on the screen.11 The ordinate is σ/2a and the abscissa is ka.

Fig. 7
Fig. 7

The transmission cross section of a slit of width 2a as a function of ka, for normal incidence with ∂u/∂n=0 on the screen. The curve, based on Eq. (16), results from single and double diffraction. The dots are based upon the exact solution of the reduced wave equation.13 The ordinate is σ/2a and the abscissa is ka.

Fig. 8
Fig. 8

A pair of neighboring incident rays hitting a curved edge, and some of the resulting diffracted rays. The two cones of diffracted rays intersect at the caustic, which is at the distance ρ1 from the edge along the rays.

Fig. 9
Fig. 9

A tube of rays and two small portions of wave fronts normal to them a distance r apart. The neighboring rays intersect at the two centers of curvature of the wave fronts. They are at the distances ρ1 and ρ2 from one wave front and at ρ1+r and ρ2+r from the other. The ratio of the areas of the wavefront sections is seen to be ρ1ρ2/(ρ1+r)(ρ2+r).

Fig. 10
Fig. 10

The diffraction pattern of a circular hole of radius a hit normally by a plane wave; ka=3π. The solid curve, based on Eq. (25) results from single diffraction with either u=0 or ∂u/∂n=0 on the screen. The dashed curve, for the case u=0 on the screen, also includes multiple diffraction. The dotted curve near φ=0 includes the caustic and shadow-boundary corrections given by (30).

Fig. 11
Fig. 11

The transmission cross section σ of a circular aperture of radius a in a thin screen on which u=0. The wave is normally incident. The solid curve, based on Eq. (36), results from single and double diffraction; the dashed curve also includes all multiple diffractions. The dots and the broken curve up to ka=5 are based upon the exact solution of the reduced wave equation.13 The ordinate is σ/πa2 and the abscissa ka.

Fig. 12
Fig. 12

The transmission cross section σ for normal incidence on a circular hole in a screen on which ∂u/∂n=0. The ordinate is ∂/πa2 and the abscissa is ka. The solid curve is based upon Eq. (40) which results from single and double diffraction. The encircled points and the dashed curve are the exact values computed by Bouwkamp.13

Fig. 13
Fig. 13

Experimental and theoretical values of the electromagnetic back scattering cross section σEM of a finite circular metal cone for axial incidence. The ordinate is σEM/πa2 and the abscissa is ka where a is the radius of the base of the cone. The circles are measured values and the curves are computed using the singly and doubly diffracted fields from the edge. The half-angle γ of the cone is indicated on each curve.

Fig. 14
Fig. 14

Comparison of the Kirchhoff edge-diffraction coefficients D1, D2, and Dk with the Braunbek coefficients DB1 and DB2 for normal incidence (α=0). The latter coincide with our D, given by Eq. (2) with the upper and lower signs, respectively. D1 and D2 result from the two usual modifications of the Kirchhoff theory and D k = 1 2 ( D 1 + D 2 ) comes from the usual form of it. The ordinate is - 2 ( 2 π k ) 1 2 e - i π / 4 sin θ D and the abscissa is θ. All the coefficients are singular on the shadow boundary θ=π but the singular factor sinθ has been removed. The dark side of the screen is at θ=3π/2 and the illuminated side at θ=−π/2.

Fig. 15
Fig. 15

The acoustic torque T per unit length on a strip of width 2a as a function of the angle of incidence α for ka=5. The vertical scale is the value of T a 2 P 2 / ρ 0 c 2 π 1 2 ( k a ) 3 2 - 1, where P is the amplitude of the incident pressure wave, ρ0 is the density of the medium and c is its sound speed. The values of α at which T=0 are equilibrium angles which are stable if Tα<0 and unstable if Tα>0. The curve does not apply near α=π/2, which corresponds to grazing incidence, at which T=0. T is an odd function of α.

Fig. 16
Fig. 16

The acoustic torque T per unit length on a strip of width 2a as a function of ka for α=π/4 (45°). The vertical scale is the same as in Fig. 15.

Fig. 17
Fig. 17

The acoustic torque T on a circular disk of radius a as a function of the angle of incidence α for ka=5. The vertical scale is the value of T|P2a3/ρ0c(ka)2|−1. P, ρ0, and c are defined in the caption of Fig. 15. The values of x at which T=0 are equilibrium values, stable if Tα<0 and unstable if Tα>0. The curve does not apply near α=0 (normal incidence) nor α=π/2 (grazing incidence).

Fig. 18
Fig. 18

Cross section of a screen of width 2b with a rounded end of radius b. A plane wave is normally incident upon it from the left. The dashed line is the shadow boundary. An incident ray which grazes the end of the screen is shown together with one of the surface diffracted rays which it produces in the shadow region. The angle between this ray and the shadow boundary is θ.

Fig. 19
Fig. 19

Cross section of an infinitely thin screen with a cylindrical tip of radius b. A plane wave is normally incident upon it from the left. The dotted line is the shadow boundary. An incident ray which grazes the tip is shown together with two of the surface diffracted rays which it produces. One of them is reflected from the screen, and both ultimately make the angle θ with the shadow boundary.

Fig. 20
Fig. 20

The far-field amplitude |f(π/4,kb)| in the shadow of the screen of Fig. 19 in the direction θ=π/4 as a function of kb. The upper curves and points apply to a screen on which ∂u/∂n=0 while the lower ones pertain to a screen on which u=0. The encircled points were obtained by laborious numerical computation of the series solution of the boundary-value problem; the curves were obtained from the formulas given by the geometrical theory of diffraction. The dashed curve was obtained from a formula in which an improved expression for α was used.

Fig. 21
Fig. 21

The far-field amplitude |f(π/4,kb)| in the shadow of the screen of Fig. 18 in the direction θ=π/4 as a function of kb. The curves are based upon the geometrical theory of diffraction while the points at kb=0 are determined from the solution for diffraction by a half-plane. The upper curve and point apply to a screen on which ∂u/∂n=0; the lower ones to a screen on which u=0. Probably only the decreasing portions of the curves are correct, and most likely the amplitude decreases monotonically for all values of kb.

Equations (41)

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u e = D A i r - 1 2 e i ( k r + ψ i ) = D u i r - 1 2 e i k r .
D = e i π / 4 2 ( 2 π k ) 1 2 sin β [ sec 1 2 ( θ - α ) ± csc 1 2 ( θ + α ) ] .
D = e i π / 4 sin π n n ( 2 π k ) 1 2 sin β [ ( cos π n - cos θ - α n ) - 1 ( cos π n - cos θ + α + π n ) - 1 ] .
u el ( P ) = - e i k ( r 1 - a sin α ) + i π / 4 2 ( 2 π k r 1 ) 1 2 [ sec 1 2 ( θ 1 + α ) ± csc 1 2 ( θ 1 - α ) ] - e i k ( r 2 + a sin α ) + i π / 4 2 ( 2 π k r 2 ) 1 2 + [ sec 1 2 ( θ 2 - α ) ± csc 1 2 ( θ 2 + α ) ] .
u el ( P ) = - ( k / 2 π r ) 1 2 e i k r + i π / 4 f s ( φ ) .
f s ( φ ) = i sin [ k a ( sin φ + sin α ) ] k sin 1 2 ( φ + α ) ± cos [ k a ( sin φ + sin α ) ] k cos 1 2 ( φ - α ) .
u el = - e i k a ( 2 + sin α ) + i π / 4 2 ( π k a ) 1 2 sec 1 2 ( π 2 - α ) .
f d ( - α ) = - 1 k ( π k a ) 1 2 [ e i 2 k a ( 1 + sin α ) + i π / 4 1 + sin α + e i 2 k a ( 1 - sin α ) + i π / 4 1 - sin α ] .
σ = 2 a cos α - 1 k ( π k a ) 1 2 [ cos [ 2 k a ( 1 + sin α ) - π / 4 1 + sin α + cos [ 2 k a ( 1 - sin α ) - π / 4 1 - sin α ] .
σ 2 a = 1 - cos ( 2 k a - π / 4 ) π 1 2 ( k a ) 3 2 .
u e = D u i n · r - 1 2 e i k r .
D ( θ ) = - 1 i k α D ( θ , π / 2 ) = - e - π / 4 2 ( 2 π ) 1 2 k 3 2 sin 2 β sin ( π 4 - θ 2 ) cos 2 ( π 4 - θ 2 ) .
u el n = - e i k a ( 2 + sin α ) + i π / 4 8 ( π k a 3 ) 1 2 · sin ( π 4 - α 2 ) cos 2 ( π 4 - α 2 ) .
u e 2 = - e i k a ( 2 + sin α ) + i k r 1 16 π ( k a ) 3 2 ( 2 k r 1 ) 1 2 · sin ( π 4 - α 2 ) cos 2 ( π 4 - α 2 ) · sin ( π 4 - θ 1 2 ) cos 2 ( π 4 - θ 1 2 ) , - e i k a ( 2 - sin α ) + i k r 2 16 π ( k a ) 3 2 ( 2 k r 2 ) 1 2 · sin ( π 4 + α 2 ) cos 2 ( π 4 + α 2 ) · sin ( π 4 - θ 2 2 ) cos 2 ( π 4 - θ 2 2 ) .
σ = 2 a cos α - 2 a 32 π 1 2 ( k a ) 5 2 × { sin 2 ( π 4 - α 2 ) cos 4 ( π 4 - α 2 ) sin [ 2 k a ( 1 + sin α ) - π / 4 ] + sin 2 ( π 4 + α 2 ) cos 4 ( π 4 + α 2 ) sin [ 2 k a ( 1 - sin α ) - π / 4 ] } .
σ 2 a = 1 - sin ( 2 k a - π / 4 ) 8 π 1 2 ( k a ) 5 2 .
u el = - t = - N t = N ( k 2 π r t ) 1 2 e i k ( r t - t b sin α ) + i π / 4 f s ( φ ) .
u el = - ( k 2 π r ) 1 2 e i k r + i π / 4 f s ( φ ) t = - N t = N e - i k t b ( sin φ + sin α ) = - ( k 2 π r ) 1 2 e i k r + i π / 4 f s ( φ ) × sin [ ( N + 1 2 ) k b ( sin φ + sin α ) ] sin [ 1 2 k b ( sin φ + sin α ) ] .
D ( θ , n ) = 1 i k α D ( - π / 2 , θ , n ) = 2 e - i π / 4 sin ( π / n ) n 2 k 3 2 ( 2 π ) 1 2 × sin [ ( θ + π 2 ) / n ] { cos π n - cos [ ( θ + π 2 ) / n ] } .
u e = D u i [ r ( 1 + ρ 1 - 1 r ) ] - 1 2 e i k r .
1 ρ 1 = - β ˙ sin β - cos δ ρ sin 2 β .
u e = - e i k ( R + r ) + i π / 4 2 sin β ( 2 π k ) 1 2 [ r R ( r + R ) ] 1 2 × [ sec 1 2 ( θ - α ) ± csc 1 2 ( θ + α ) ] .
u el ( P ) = - e i k r + i π / 4 2 ( 2 π k ) 1 2 [ sec θ 1 2 ± csc θ 1 2 ] [ r 1 ( 1 - a - 1 r 1 sin θ 1 ) ] - 1 2 - e i k r 2 + i π / 4 2 ( 2 π k ) 1 2 [ sec θ 2 2 ± csc θ 2 2 ] × [ r 2 ( 1 - a - 1 r 2 sin θ 2 ) ] - 1 2 .
u el = - k e i k r 2 π r f s ( φ ) .
f s ( φ ) = k - 3 2 ( 2 π a / sin φ ) 1 2 { i sin [ k a sin φ - ( π / 4 ) ] sin ( φ / 2 ) ± cos [ k a sin φ - ( π / 4 ) ] cos ( φ / 2 ) } .
u el = - a 1 2 e i k ( x sin δ + a cos δ ) ( 2 π k ρ ) 1 2 ( x 2 + a 2 ) 1 4 [ sec ( δ 2 + π 4 ) ± csc ( δ 2 + π 4 ) ] × cos ( k ρ cos δ - π 4 ) .
e i k x sin δ J 0 ( k ρ cos δ ) ~ 2 e i k x sin δ ( 2 π k ρ cos δ ) 1 2 cos ( k ρ cos δ - π / 4 ) .
F = 1 2 ( 2 π k ρ cos δ ) 1 2 sec ( k ρ cos δ - π / 4 ) J 0 ( k ρ cos δ ) .
u el = - ( a cos δ ) 1 2 2 ( x 2 + a 2 ) 1 4 [ sec ( δ 2 + π 4 ) ± csc ( δ 2 + π 4 ) ] × J 0 ( k ρ cos δ ) exp [ i k ( x 2 + a 2 ) 1 2 ] .
u g + u el e i k x - a 2 x e i k x ( 1 + i k a 2 2 x ) 2 x a J 0 ( k a sin φ ) = - k e i k x 2 π x i π a 2 J 0 ( k a sin φ ) .
f s ( φ ) = π a k [ i J 1 ( k a sin φ ) sin ( φ / 2 ) ± J 0 ( k a sin φ ) cos ( φ / 2 ) ] .
u el = - e 2 i k a + i π / 4 ( 2 π k a ) 1 2 .
u e 2 = e i k ( r 1 + 2 a ) 2 π k a 1 2 [ r 1 ( 1 - a - 1 r 1 cos δ 1 ) ] - 1 2 sec δ 1 2 + e i k ( r 2 + 2 a ) 2 π k a 1 2 [ r 2 ( 1 - a - 1 r 2 cos δ 2 ) ] - 1 2 sec δ 2 2 .
u e 2 = - 2 e 2 i k a + π / 4 ( 2 π k a ) 1 2 sec δ 2 [ sec ( δ 2 + π 4 ) + csc ( δ 2 + π 4 ) ] - 1 u el .
u e 2 = - k e i k x 2 π x · [ - 2 k ( π a k ) 1 2 e 2 i k a - i π / 4 J 0 ( k a sin φ ) ] .
σ π a 2 = 1 - 2 sin [ 2 k a - π / 4 ] π 1 2 ( k a ) 3 2 .
u el n = - e 2 i k a - i π / 4 4 ( 2 π k a 3 ) 1 2 .
u e 2 = - i e i k ( r 1 + 2 a ) 16 π ( k 4 a 3 ) 1 2 [ r 1 ( 1 - a - 1 r 1 cos δ 1 ) ] - 1 2 sin ( δ 1 / 2 ) cos 2 ( δ 1 / 2 ) . - i e i k ( r 2 + 2 a ) 16 π ( k 4 a 3 ) 1 2 [ r 2 ( 1 - a - 1 r 2 cos δ 2 ) ] - 1 2 sin ( δ 2 / 2 ) cos 2 ( δ 2 / 2 ) .
u e 2 = - k e i k x 2 π x [ π 1 2 J 0 ( k a sin φ ) 4 k 5 2 a 1 2 e 2 i k a + i π / 4 ] .
σ π a 2 = 1 - 1 4 ( k a ) 2 + sin ( 2 k a + π / 4 ) 4 π 1 2 ( k a ) 5 2 .
u = C u i ( e i k r / r ) .