Abstract

Starting with the most elementary multiple scattering problem, a symmetrically excited pair of monopoles, we use both “successive-scattering” and “self-consistent” procedures to obtain the closed-form multiply scattered amplitude in terms of the singly scattered value. Then we represent the field for an arbitrary configuration of arbitrary scatterers in terms of the individual, multiply scattered amplitudes, and specify these in terms of singly scattered amplitudes by a system of self-consistent integral equations. Applications are made to fixed configurations (two scatterers, and the diffraction grating), and to random distributions (models for rough surfaces, and for random media). Simple explicit results in terms of isolated scatterer functions are derived, and some of their essential features are given in graphs: the coupling effects for two dielectric rods (intensity and phase) are graphed versus spacing for both polarizations; for the diffraction grating, when a surface wave is “near grazing,” the spectral orders show essential features of the anomalies measured by Wood, Strong, and Palmer; for a lossless rough surface, the graphs for both polarizations for the coherently reflected intensity and phase are analogous to those obtained for reflection from a lossy interface (the corresponding results for the incoherent scattering accounting for the loss from the “main beam”); and for a slab region containing a random distribution of nearly transparent large scatterers, we graph the macroscopic parameters, and the coherent and incoherent, forward-scattered intensities versus the fractional volume.

© 1962 Optical Society of America

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References

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  1. Additional introductory material on scattering problems and reviews of recent developments will be found in the writer’s “Electromagnetic waves,” Physics Today 13, (7) 30 (1960), and “On multiple scattering of waves,” J. Research Natl. Bureau Standards 64D, 715 (1960).
  2. All numerical computations (three significant figures, rounded off) for the graphs of this paper were done by S. E. Bergstrom with a desk calculator.
  3. V. Twersky, J. Appl. Phys. 23, 407 (1952).
    [Crossref]
  4. Such series are discussed in some detail by the writer in J. Acoust. Soc. Am. 24, 42 (1952).
  5. F. Zaviska, Ann. Physik 40, 1023 (1913).
    [Crossref]
  6. K. A. Breuckner, Phys. Rev. 89, 834 (1953); Phys. Rev. 97, 1353 (1955).
    [Crossref]
  7. K. M. Watson, Phys. Rev. 87, 575 (1953).
    [Crossref]
  8. R. F. Millar, Can. J. Phys. 38, 272 (1960).
    [Crossref]
  9. See literature survey by V. Twersky, in J. Research Natl. Bur. Standards64D, 715 (1960); part of the Report of the United States of America National Committee to the XIIIth General Assembly of the International Scientific Radio Union, London, England.
  10. The two-dimensional case is treated by V. Twersky, in Report EDL-E60, Sylvania Electronic Defense Laboratories, 1961; to be published in “Proceedings of Symposium on Electromagnetic Waves,” University of Wisconsin (April, 1961). The three-dimensional case is treated by V. Twersky, in Report EDL-E61, Sylvania Electronic Electronic Defense Laboratories, 1961. [J. Math. Phys. (to be published)].
  11. S. N. Karp and N. Zitron, Research Report EM-126, Institute of Mathematical Sciences, New York University (1959).
  12. S. N. Karp, Proceedings of the Symposium on Microwave Optics, 1953, p. 198 [Electronic Research Directorate, AFCRC (1959)].
  13. V. Twersky, IRE Trans. AP-4, 330 (1956), Proceedings of the URSI-Michigan International Symposium on Electromagnetic Theory; , Sylvania Electronic Defense Laboratories (1957); Report EDL-E28, Sylvania Electronic Defense Laboratories (1958).
  14. Rayleigh, Proc. Roy. Soc. London A79, 399 (1907); see also Rayleigh’s earlier paper in Phil. Mag. 14, 60 (1907).
  15. W. v. Ignatowsky, Ann. Physik 44, 369 (1914).
    [Crossref]
  16. F. M. Schwerd, Die Beugungsercheinungen aus den Fundamentalgesetzen der Undulations Theorie Analytisch Entwickelt (Schwan und Goetz’schen Hosbuchhandlung, Mannheim, Germany, 1835).
  17. R. W. Wood, Phys. Rev. 48, 928 (1935); Phil. Mag. 4, 396 (1902); ibid. 23, 310 (1912).
    [Crossref]
  18. L. R. Ingersoll, J. Astrophys. 51, 129 (1920); Phys. Rev. 17, 493 (1921).
    [Crossref]
  19. J. Strong, Phys. Rev. 49, 291 (1936).
    [Crossref]
  20. H. C. Palmer, J. Opt. Soc. Am. 42, 269 (1952); ibid. 46, 50 (1956).
    [Crossref]
  21. U. Fano, Ann. Physik 32, 393 (1938).
    [Crossref]
  22. K. Artmann, Z. Physik 119, 529 (1942).
    [Crossref]
  23. V. Twersky, J. Appl. Phys. 25, 859 (1954).
    [Crossref]
  24. J. E. Burke and V. Twersky, Report EDL-M266, Sylvania Electronic Defense Laboratories (1960).
  25. J. E. Burke and V. Twersky, Report EDL-E44, Sylvania Electronic Defense Laboratories (1960).
  26. S. N. Karp and J. Radlow, IRE Trans AP-4, 654 (1956).
  27. B. A. Lippmann and A. Oppenheim, “Towards a Theory of Wood’s Anomalies,” Technical Research Group, New York (1954).
  28. R. F. Millar, Can. J. Phys. 39, 81 (1961).
    [Crossref]
  29. A. Hessel, “Guiding and scattering by sinusoidally-modulated reactance surfaces,” , Polytechnic Institute of Brooklyn, 1960; A. Hessel and A. A. Oliner, “On the theory of Wood’s anomalies” in , Polytechnic Institute of Brooklyn, 1961.
  30. V. Twersky, J. Appl. Phys. 23, 1099 (1952).
    [Crossref]
  31. L. L. Foldy, Phys. Rev. 67, 107 (1945).
    [Crossref]
  32. M. Lax, Revs. Modern Phys. 23(1951); Phys. Rev. 80, 621 (1952).
    [Crossref]
  33. V. Twersky, J. Acoust. Soc. Am. 29, 209 (1957); IRE Trans. AP-7, S5307 (1959), Proceedings of the URSI-Toronto International Symposium on Electromagnetics; IRE Trans. AP-5, 81 (1957).
    [Crossref]
  34. Rayleigh, Phil. Mag. 14, 350 (1907).
  35. V. Twersky, J. Acoust. Soc. Am. 22, 539 (1950); ibid. 23, 336 (1951); J. Appl. Phys.22, 825 (1951), , Nuclear Development Associates; J. Appl. Phys. 24, 659 (1953).
    [Crossref]
  36. V. Twersky, Research Report EM-58, Institute of Mathematical Sciences, New York University (1953).
  37. M. A. Biot, J. Acoust. Soc. Am. 29, 1193 (1957); J. Appl. Phys. 28, 1455 (1957); J. Appl. Phys. 29, 998 (1958).
    [Crossref]
  38. J. R. Wait, IRE Trans. AP-7, Part I, S154; Part II, S163 (1959).
  39. V. Twersky, Report EDL-E26, Sylvania Electronic Defense Laboratories, 1958.
  40. V. Twersky, Report EDL-E36, Sylvania Electronic Defense Laboratories, 1959.
  41. V. Twersky, Research Report EM-59, Institute of Mathematical Sciences, New York University (1953).
  42. R. J. Urick and W. S. Ament, J. Acoust. Soc. Am. 21, 115 (1949).
    [Crossref]
  43. F. Reiche, Ann. Physik 50, 1, 121 (1916).
  44. Rayleigh, Phil. Mag. 47, 375 (1899).
  45. C. I. Beard and V. Twersky, Report EDL-E46; Report EDL-E47, Sylvania Electronic Defense Laboratories, 1960.

1961 (1)

R. F. Millar, Can. J. Phys. 39, 81 (1961).
[Crossref]

1960 (2)

Additional introductory material on scattering problems and reviews of recent developments will be found in the writer’s “Electromagnetic waves,” Physics Today 13, (7) 30 (1960), and “On multiple scattering of waves,” J. Research Natl. Bureau Standards 64D, 715 (1960).

R. F. Millar, Can. J. Phys. 38, 272 (1960).
[Crossref]

1959 (1)

J. R. Wait, IRE Trans. AP-7, Part I, S154; Part II, S163 (1959).

1957 (2)

V. Twersky, J. Acoust. Soc. Am. 29, 209 (1957); IRE Trans. AP-7, S5307 (1959), Proceedings of the URSI-Toronto International Symposium on Electromagnetics; IRE Trans. AP-5, 81 (1957).
[Crossref]

M. A. Biot, J. Acoust. Soc. Am. 29, 1193 (1957); J. Appl. Phys. 28, 1455 (1957); J. Appl. Phys. 29, 998 (1958).
[Crossref]

1956 (2)

S. N. Karp and J. Radlow, IRE Trans AP-4, 654 (1956).

V. Twersky, IRE Trans. AP-4, 330 (1956), Proceedings of the URSI-Michigan International Symposium on Electromagnetic Theory; , Sylvania Electronic Defense Laboratories (1957); Report EDL-E28, Sylvania Electronic Defense Laboratories (1958).

1954 (1)

V. Twersky, J. Appl. Phys. 25, 859 (1954).
[Crossref]

1953 (2)

K. A. Breuckner, Phys. Rev. 89, 834 (1953); Phys. Rev. 97, 1353 (1955).
[Crossref]

K. M. Watson, Phys. Rev. 87, 575 (1953).
[Crossref]

1952 (4)

V. Twersky, J. Appl. Phys. 23, 407 (1952).
[Crossref]

Such series are discussed in some detail by the writer in J. Acoust. Soc. Am. 24, 42 (1952).

H. C. Palmer, J. Opt. Soc. Am. 42, 269 (1952); ibid. 46, 50 (1956).
[Crossref]

V. Twersky, J. Appl. Phys. 23, 1099 (1952).
[Crossref]

1951 (1)

M. Lax, Revs. Modern Phys. 23(1951); Phys. Rev. 80, 621 (1952).
[Crossref]

1950 (1)

V. Twersky, J. Acoust. Soc. Am. 22, 539 (1950); ibid. 23, 336 (1951); J. Appl. Phys.22, 825 (1951), , Nuclear Development Associates; J. Appl. Phys. 24, 659 (1953).
[Crossref]

1949 (1)

R. J. Urick and W. S. Ament, J. Acoust. Soc. Am. 21, 115 (1949).
[Crossref]

1945 (1)

L. L. Foldy, Phys. Rev. 67, 107 (1945).
[Crossref]

1942 (1)

K. Artmann, Z. Physik 119, 529 (1942).
[Crossref]

1938 (1)

U. Fano, Ann. Physik 32, 393 (1938).
[Crossref]

1936 (1)

J. Strong, Phys. Rev. 49, 291 (1936).
[Crossref]

1935 (1)

R. W. Wood, Phys. Rev. 48, 928 (1935); Phil. Mag. 4, 396 (1902); ibid. 23, 310 (1912).
[Crossref]

1920 (1)

L. R. Ingersoll, J. Astrophys. 51, 129 (1920); Phys. Rev. 17, 493 (1921).
[Crossref]

1916 (1)

F. Reiche, Ann. Physik 50, 1, 121 (1916).

1914 (1)

W. v. Ignatowsky, Ann. Physik 44, 369 (1914).
[Crossref]

1913 (1)

F. Zaviska, Ann. Physik 40, 1023 (1913).
[Crossref]

1907 (2)

Rayleigh, Proc. Roy. Soc. London A79, 399 (1907); see also Rayleigh’s earlier paper in Phil. Mag. 14, 60 (1907).

Rayleigh, Phil. Mag. 14, 350 (1907).

1899 (1)

Rayleigh, Phil. Mag. 47, 375 (1899).

Ament, W. S.

R. J. Urick and W. S. Ament, J. Acoust. Soc. Am. 21, 115 (1949).
[Crossref]

Artmann, K.

K. Artmann, Z. Physik 119, 529 (1942).
[Crossref]

Beard, C. I.

C. I. Beard and V. Twersky, Report EDL-E46; Report EDL-E47, Sylvania Electronic Defense Laboratories, 1960.

Biot, M. A.

M. A. Biot, J. Acoust. Soc. Am. 29, 1193 (1957); J. Appl. Phys. 28, 1455 (1957); J. Appl. Phys. 29, 998 (1958).
[Crossref]

Breuckner, K. A.

K. A. Breuckner, Phys. Rev. 89, 834 (1953); Phys. Rev. 97, 1353 (1955).
[Crossref]

Burke, J. E.

J. E. Burke and V. Twersky, Report EDL-M266, Sylvania Electronic Defense Laboratories (1960).

J. E. Burke and V. Twersky, Report EDL-E44, Sylvania Electronic Defense Laboratories (1960).

Fano, U.

U. Fano, Ann. Physik 32, 393 (1938).
[Crossref]

Foldy, L. L.

L. L. Foldy, Phys. Rev. 67, 107 (1945).
[Crossref]

Hessel, A.

A. Hessel, “Guiding and scattering by sinusoidally-modulated reactance surfaces,” , Polytechnic Institute of Brooklyn, 1960; A. Hessel and A. A. Oliner, “On the theory of Wood’s anomalies” in , Polytechnic Institute of Brooklyn, 1961.

Ignatowsky, W. v.

W. v. Ignatowsky, Ann. Physik 44, 369 (1914).
[Crossref]

Ingersoll, L. R.

L. R. Ingersoll, J. Astrophys. 51, 129 (1920); Phys. Rev. 17, 493 (1921).
[Crossref]

Karp, S. N.

S. N. Karp and J. Radlow, IRE Trans AP-4, 654 (1956).

S. N. Karp, Proceedings of the Symposium on Microwave Optics, 1953, p. 198 [Electronic Research Directorate, AFCRC (1959)].

S. N. Karp and N. Zitron, Research Report EM-126, Institute of Mathematical Sciences, New York University (1959).

Lax, M.

M. Lax, Revs. Modern Phys. 23(1951); Phys. Rev. 80, 621 (1952).
[Crossref]

Lippmann, B. A.

B. A. Lippmann and A. Oppenheim, “Towards a Theory of Wood’s Anomalies,” Technical Research Group, New York (1954).

Millar, R. F.

R. F. Millar, Can. J. Phys. 39, 81 (1961).
[Crossref]

R. F. Millar, Can. J. Phys. 38, 272 (1960).
[Crossref]

Oppenheim, A.

B. A. Lippmann and A. Oppenheim, “Towards a Theory of Wood’s Anomalies,” Technical Research Group, New York (1954).

Palmer, H. C.

Radlow, J.

S. N. Karp and J. Radlow, IRE Trans AP-4, 654 (1956).

Rayleigh,

Rayleigh, Phil. Mag. 14, 350 (1907).

Rayleigh, Proc. Roy. Soc. London A79, 399 (1907); see also Rayleigh’s earlier paper in Phil. Mag. 14, 60 (1907).

Rayleigh, Phil. Mag. 47, 375 (1899).

Reiche, F.

F. Reiche, Ann. Physik 50, 1, 121 (1916).

Schwerd, F. M.

F. M. Schwerd, Die Beugungsercheinungen aus den Fundamentalgesetzen der Undulations Theorie Analytisch Entwickelt (Schwan und Goetz’schen Hosbuchhandlung, Mannheim, Germany, 1835).

Strong, J.

J. Strong, Phys. Rev. 49, 291 (1936).
[Crossref]

Twersky, V.

V. Twersky, J. Acoust. Soc. Am. 29, 209 (1957); IRE Trans. AP-7, S5307 (1959), Proceedings of the URSI-Toronto International Symposium on Electromagnetics; IRE Trans. AP-5, 81 (1957).
[Crossref]

V. Twersky, IRE Trans. AP-4, 330 (1956), Proceedings of the URSI-Michigan International Symposium on Electromagnetic Theory; , Sylvania Electronic Defense Laboratories (1957); Report EDL-E28, Sylvania Electronic Defense Laboratories (1958).

V. Twersky, J. Appl. Phys. 25, 859 (1954).
[Crossref]

V. Twersky, J. Appl. Phys. 23, 1099 (1952).
[Crossref]

V. Twersky, J. Appl. Phys. 23, 407 (1952).
[Crossref]

V. Twersky, J. Acoust. Soc. Am. 22, 539 (1950); ibid. 23, 336 (1951); J. Appl. Phys.22, 825 (1951), , Nuclear Development Associates; J. Appl. Phys. 24, 659 (1953).
[Crossref]

V. Twersky, Research Report EM-58, Institute of Mathematical Sciences, New York University (1953).

J. E. Burke and V. Twersky, Report EDL-M266, Sylvania Electronic Defense Laboratories (1960).

J. E. Burke and V. Twersky, Report EDL-E44, Sylvania Electronic Defense Laboratories (1960).

See literature survey by V. Twersky, in J. Research Natl. Bur. Standards64D, 715 (1960); part of the Report of the United States of America National Committee to the XIIIth General Assembly of the International Scientific Radio Union, London, England.

The two-dimensional case is treated by V. Twersky, in Report EDL-E60, Sylvania Electronic Defense Laboratories, 1961; to be published in “Proceedings of Symposium on Electromagnetic Waves,” University of Wisconsin (April, 1961). The three-dimensional case is treated by V. Twersky, in Report EDL-E61, Sylvania Electronic Electronic Defense Laboratories, 1961. [J. Math. Phys. (to be published)].

V. Twersky, Report EDL-E26, Sylvania Electronic Defense Laboratories, 1958.

V. Twersky, Report EDL-E36, Sylvania Electronic Defense Laboratories, 1959.

V. Twersky, Research Report EM-59, Institute of Mathematical Sciences, New York University (1953).

C. I. Beard and V. Twersky, Report EDL-E46; Report EDL-E47, Sylvania Electronic Defense Laboratories, 1960.

Urick, R. J.

R. J. Urick and W. S. Ament, J. Acoust. Soc. Am. 21, 115 (1949).
[Crossref]

Wait, J. R.

J. R. Wait, IRE Trans. AP-7, Part I, S154; Part II, S163 (1959).

Watson, K. M.

K. M. Watson, Phys. Rev. 87, 575 (1953).
[Crossref]

Wood, R. W.

R. W. Wood, Phys. Rev. 48, 928 (1935); Phil. Mag. 4, 396 (1902); ibid. 23, 310 (1912).
[Crossref]

Zaviska, F.

F. Zaviska, Ann. Physik 40, 1023 (1913).
[Crossref]

Zitron, N.

S. N. Karp and N. Zitron, Research Report EM-126, Institute of Mathematical Sciences, New York University (1959).

Ann. Physik (4)

F. Zaviska, Ann. Physik 40, 1023 (1913).
[Crossref]

W. v. Ignatowsky, Ann. Physik 44, 369 (1914).
[Crossref]

U. Fano, Ann. Physik 32, 393 (1938).
[Crossref]

F. Reiche, Ann. Physik 50, 1, 121 (1916).

Can. J. Phys. (2)

R. F. Millar, Can. J. Phys. 39, 81 (1961).
[Crossref]

R. F. Millar, Can. J. Phys. 38, 272 (1960).
[Crossref]

IRE Trans (1)

S. N. Karp and J. Radlow, IRE Trans AP-4, 654 (1956).

IRE Trans. (2)

V. Twersky, IRE Trans. AP-4, 330 (1956), Proceedings of the URSI-Michigan International Symposium on Electromagnetic Theory; , Sylvania Electronic Defense Laboratories (1957); Report EDL-E28, Sylvania Electronic Defense Laboratories (1958).

J. R. Wait, IRE Trans. AP-7, Part I, S154; Part II, S163 (1959).

J. Acoust. Soc. Am. (5)

M. A. Biot, J. Acoust. Soc. Am. 29, 1193 (1957); J. Appl. Phys. 28, 1455 (1957); J. Appl. Phys. 29, 998 (1958).
[Crossref]

R. J. Urick and W. S. Ament, J. Acoust. Soc. Am. 21, 115 (1949).
[Crossref]

Such series are discussed in some detail by the writer in J. Acoust. Soc. Am. 24, 42 (1952).

V. Twersky, J. Acoust. Soc. Am. 29, 209 (1957); IRE Trans. AP-7, S5307 (1959), Proceedings of the URSI-Toronto International Symposium on Electromagnetics; IRE Trans. AP-5, 81 (1957).
[Crossref]

V. Twersky, J. Acoust. Soc. Am. 22, 539 (1950); ibid. 23, 336 (1951); J. Appl. Phys.22, 825 (1951), , Nuclear Development Associates; J. Appl. Phys. 24, 659 (1953).
[Crossref]

J. Appl. Phys. (3)

V. Twersky, J. Appl. Phys. 23, 1099 (1952).
[Crossref]

V. Twersky, J. Appl. Phys. 25, 859 (1954).
[Crossref]

V. Twersky, J. Appl. Phys. 23, 407 (1952).
[Crossref]

J. Astrophys. (1)

L. R. Ingersoll, J. Astrophys. 51, 129 (1920); Phys. Rev. 17, 493 (1921).
[Crossref]

J. Opt. Soc. Am. (1)

Phil. Mag. (2)

Rayleigh, Phil. Mag. 14, 350 (1907).

Rayleigh, Phil. Mag. 47, 375 (1899).

Phys. Rev. (5)

L. L. Foldy, Phys. Rev. 67, 107 (1945).
[Crossref]

R. W. Wood, Phys. Rev. 48, 928 (1935); Phil. Mag. 4, 396 (1902); ibid. 23, 310 (1912).
[Crossref]

J. Strong, Phys. Rev. 49, 291 (1936).
[Crossref]

K. A. Breuckner, Phys. Rev. 89, 834 (1953); Phys. Rev. 97, 1353 (1955).
[Crossref]

K. M. Watson, Phys. Rev. 87, 575 (1953).
[Crossref]

Physics Today (1)

Additional introductory material on scattering problems and reviews of recent developments will be found in the writer’s “Electromagnetic waves,” Physics Today 13, (7) 30 (1960), and “On multiple scattering of waves,” J. Research Natl. Bureau Standards 64D, 715 (1960).

Proc. Roy. Soc. London (1)

Rayleigh, Proc. Roy. Soc. London A79, 399 (1907); see also Rayleigh’s earlier paper in Phil. Mag. 14, 60 (1907).

Revs. Modern Phys. (1)

M. Lax, Revs. Modern Phys. 23(1951); Phys. Rev. 80, 621 (1952).
[Crossref]

Z. Physik (1)

K. Artmann, Z. Physik 119, 529 (1942).
[Crossref]

Other (15)

J. E. Burke and V. Twersky, Report EDL-M266, Sylvania Electronic Defense Laboratories (1960).

J. E. Burke and V. Twersky, Report EDL-E44, Sylvania Electronic Defense Laboratories (1960).

B. A. Lippmann and A. Oppenheim, “Towards a Theory of Wood’s Anomalies,” Technical Research Group, New York (1954).

A. Hessel, “Guiding and scattering by sinusoidally-modulated reactance surfaces,” , Polytechnic Institute of Brooklyn, 1960; A. Hessel and A. A. Oliner, “On the theory of Wood’s anomalies” in , Polytechnic Institute of Brooklyn, 1961.

V. Twersky, Research Report EM-58, Institute of Mathematical Sciences, New York University (1953).

F. M. Schwerd, Die Beugungsercheinungen aus den Fundamentalgesetzen der Undulations Theorie Analytisch Entwickelt (Schwan und Goetz’schen Hosbuchhandlung, Mannheim, Germany, 1835).

All numerical computations (three significant figures, rounded off) for the graphs of this paper were done by S. E. Bergstrom with a desk calculator.

See literature survey by V. Twersky, in J. Research Natl. Bur. Standards64D, 715 (1960); part of the Report of the United States of America National Committee to the XIIIth General Assembly of the International Scientific Radio Union, London, England.

The two-dimensional case is treated by V. Twersky, in Report EDL-E60, Sylvania Electronic Defense Laboratories, 1961; to be published in “Proceedings of Symposium on Electromagnetic Waves,” University of Wisconsin (April, 1961). The three-dimensional case is treated by V. Twersky, in Report EDL-E61, Sylvania Electronic Electronic Defense Laboratories, 1961. [J. Math. Phys. (to be published)].

S. N. Karp and N. Zitron, Research Report EM-126, Institute of Mathematical Sciences, New York University (1959).

S. N. Karp, Proceedings of the Symposium on Microwave Optics, 1953, p. 198 [Electronic Research Directorate, AFCRC (1959)].

C. I. Beard and V. Twersky, Report EDL-E46; Report EDL-E47, Sylvania Electronic Defense Laboratories, 1960.

V. Twersky, Report EDL-E26, Sylvania Electronic Defense Laboratories, 1958.

V. Twersky, Report EDL-E36, Sylvania Electronic Defense Laboratories, 1959.

V. Twersky, Research Report EM-59, Institute of Mathematical Sciences, New York University (1953).

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Figures (29)

Fig. 1
Fig. 1

Plane wave ϕ traveling in direction i is incident on arbitrary object whose surface is S. Scattered wave u is observed in direction o at distance r from origin.

Fig. 2
Fig. 2

The simplest multiple scattering problem: Plane wave “normally incident” on pair of identical monopoles.

Fig. 3
Fig. 3

Schematic for successive scattering procedure. Figures 3(a), (b), and (c) correspond, respectively, to the first, second, and third “orders of scattering” of the “top” element. The first order is excited by the primary field, and the second and third orders are excited, respectively, by the first and second orders arriving from the “bottom” scatterer. Summing the above plus the remaining terms of the infinite series of orders of scattering gives A[1+AH+(AH)2+⋯]=A/(1−AH)=B.

Fig. 4
Fig. 4

Scattered intensity (relative to singly scattered value) for two parallel fine dielectric cylinders for normal incidence (normal to axis of pair) and E parallel (polarization parallel to rod length) as function of path difference between origins; see Eq. (12) for pair of monopoles. Solid curve is I0=|B0/A0|2=1/|1−A0H0(1)(2kb)|2 vs. x=2kb=4πb/λ for A0=iC0C02 with C 0 = π ρ 2 ( η 2 - 1 ) / 4 = 1 4. Here η is the index of refraction of the rods, and ρ2=(ka)2 for circular rods of radius a, or ρ2=k2aa′ for elliptic rods of semiaxis a and a′. Dashed curve is obtained on using H 0 ( x ) ~ ( 2 / i π x ) 1 2 e i x. Plots for two perfectly conducting wires and E parallel [for which case C0π/2 ln (0.89ρ)] would show the same oscillating structure (shifted by π) but the departures from single-scattering theory for the same value of ρ would be much more pronounced. Faint lines indicate breakdown of (12) because of higher order multipoles induced by closeness of monopoles; for such cases, more complete forms may be gotten from (38) and (35).

Fig. 5
Fig. 5

Scattered phase (relative to singly scattered value) corresponding to intensity curve shown in Fig. 4. Solid curve is graph of p0=tan−1 [Im (B0/A0)/Re (B0/A0)] in degrees versus x=2kb.

Fig. 6
Fig. 6

Schematic for self-consistent procedure. The “top” scatterer is excited by the primary field (1) plus the net initially unknown field arriving from the “bottom” scatterer [BH(2b)]. The response of the top scatterer is given by [1+BH(2b)]A=B=A/(1−AH).

Fig. 7
Fig. 7

Schematic for scattering of a wave φ traveling in the direction i by a configuration of many objects.

Fig. 8
Fig. 8

Coordinates for plane wave incident on two arbitrary scatterers whose separation is 2b.

Fig. 9
Fig. 9

Scattered intensity (relative to single scattered value) for two parallel fine dielectric cylinders for normal incidence and E perpendicular as function of path difference between origins: see discussion after Eq. (37) for pair of dipoles. Solid curve is I1=|B1/A1|2=1/|1−A1H1(1)(2kb)/kb|2 versus x=2kb for A1=iC1C12 with C1=πρ2(η2−1)/4 (η2+1)=1/20=C0/5; dashed curve is obtained on using H 1 ( x ) ~ - i ( 2 / i π x ) 1 2 e i x. This figure supplements that of Fig. 4 (whose legend should be consulted for ρ and η); for example, for identical pairs of rods and η=2, Figs. 4 and 9 correspond to polarization parallel and perpendicular to the rods, respectively.

Fig. 10
Fig. 10

Scattered phase (relative to singly scattered value) corresponding to intensity curve shown in Fig. 9. Solid curve is graph of p1=tan−1[Im(B1/A1)/(Re(B1/A1)] in degrees versus x=2kb. This graph supplements Fig. 5 in the sense that Fig. 9 supplements Fig. 4.

Fig. 11
Fig. 11

Plane wave traveling in direction θ0 incident on infinite grating of arbitrary identical elements spaced d apart.

Fig. 12
Fig. 12

The solution for a plane wave±ϕ(πθ0) exciting a grating of arbitrary identical protuberances on a ground plane [Fig. 12(b)] is obtained by superposing solutions for ϕ(θ0) and±ϕ(πθ0) exciting the analogous “full grating” [Fig. 12(a)].

Fig. 13
Fig. 13

Scattered intensities Jμ=|μ|2 of (47) versus

Fig. 14
Fig. 14

Plots of phases pν=tan−1 (Im ℛν/Re ℛν) corresponding to Fig. 13.

Fig. 15
Fig. 15

Figure 15(a) shows plots of I=|F10/ f ˆ 10|2 versus normalized range q2=±2(kd∊)2/π(ka)2 for sin θ 1 5 8, sinθμ=sinθ0+3μ/8, sin θ 0 = 1 4 ± 2 / 2. The curves marked 1, 2, 3 and 4 correspond, respectively, to π ( k a ) 2 / 2 = 1 4 , 1 8 , 1 16, and 1 32. The anomaly relative to |F|2=| f ˆ|2≈|f|2 (i.e., relative to I=1) becomes more pronounced as the scatterers decrease in size. Curve 1 is also related to Fig. 13 in that the only parameter that differs is kd: the present spacing is twice that of Fig. 13. Figure 15(b) shows the corresponding phases p=tan−1(Im F10/Re F10).

Fig. 16
Fig. 16

Schematic for scattering by models for “random screens” and “rough surfaces.”

Fig. 17
Fig. 17

Plots of coherent power-reflection coefficients

Fig. 18
Fig. 18

Plots of total phase change p in degrees for coherent reflection versus angle of incidence α in degrees for same situations as Fig. 17. Here and everywhere in the text our coefficients, etc., are for the E field for ⊥ (or−) case and for the H field for || (or+) case (i.e., we normalize all functions with respect to appropriate field component perpendicular to plane of incidence to facilitate using simple scalar notation). Corresponding to R=R|| at α=0, we now have pp||=180°, i.e., surface is “isotropically rough.” The fact that p||=0 precisely at α=45° is a consequence of neglecting quadrupole terms etc.; zero is also shifted for semi-ellipsoids, etc.

Fig. 19
Fig. 19

Plots of “forward” (o=i, specular direction) and backward (o=−i′, back toward source) differential scattering cross sections corresponding to Fig. 17 versus angle of incidence in degrees. Curves are based on σ=ρ|f(o,i)/k(1−Z)|2 of (90) and the explicit forms for hemispheres given in 1957 IRE paper.33 For perpendicular polarization [σ∝cos4α], one curve suffices for both forward and back scattering, but two are required for parallel polarization [σ||(f)∝(1−2 sin2α)2 and σ||(b)∝(1+sin2α)2]. The fact that σ||(f)=0 at α=45° is a consequence of neglecting quadrupole terms, etc., in the computations.

Fig. 20
Fig. 20

Analogous plots of coherent reflection coefficients of (87) versus angle of incidence α in degrees for model of “striated surface” consisting of fine perfectly conducting semicircles on ground plane. Symbols ⊥ and || stand for polarization perpendicular (−) and parallel (+), respectively, to plane of incidence (or parallel and perpendicular to axes of cylinders, respectively). Explicit forms for f required in Z are obtained from (58) and (60) for k a = k b = 1 2 and ρ / k = 1 2 (i.e., ρ a = 1 4, or average effective separation of centers equals two diameters as for Fig. 17). Note pair of R curves are similar to those of anisotropic lossy dielectric, i.e., values for the two polarizations are not equal for normal incidence. As for Fig. 17, the total scattering from unit area is given by (1−R) cosα=ρP.

Fig. 21
Fig. 21

Phase curves corresponding to Fig. 20. Curves differ essentially from those of Fig. 18 in that now at α=0, pp||≠180° (“anisotropic rough surface”). As for Fig. 18, the fact that p||=0 at α=45° is a consequence of neglect of quadrupole terms, etc., for semicircle. From (58) and (88), we see that for semi-ellipses we have tanp||∝Im f+α[b−(a+b) sin2α] (plus quadrupole terms,24 etc.); to present order, p||=0 when sin2α=b/(a+b). (Details for elliptic cylinders were given by Burke and Twersky, “On scattering and reflection of waves by elliptically striated surfaces,” Washington Meeting of URSI, May 1960).

Fig. 22
Fig. 22

Forward (θ=α) and backward (θ=−α) differential scattering cross sections σ=(2ρ/πk)|jα, πα)/(1−Z)|2 corresponding to Fig. 20. The present σ’s are proportional to the respective angular factors cited for Fig. 19 and thus show essentially the same gross features. As for Fig. 19 the zero of σ||(f) at α=45° is a consequence of neglecting quadrupole terms, etc., in computations. As for (21), location of zero for elliptic semicylinders is given by sin2α=b/(a+b).

Fig. 23
Fig. 23

Schematic for scattering of plane wave by a slab region of scatterers.

Fig. 24
Fig. 24

Plots of coherent coefficients of (130) for slab of thickness d. Right-hand scale gives phase change

Fig. 25
Fig. 25

Plots of the coherent scattered field C=e−2β, the forward scattered incoherent field I=q(1−e−2β) for q = 1 2, the total forward scattered field T=C+I, and the “optical noise” I/C for the parameters of Fig. 24.

Fig. 26
Fig. 26

Large-scale dynamical model of “compressible gas” whose molecules are 1.5-in. diam. styrofoam spheres; gas was designed to provide controlled “random distribution” for microwave measurements. (A five-minute movie for different concentrations of spheres was shown at the OSA meeting.)

Fig. 27
Fig. 27

Large-scale dynamical model of “gas” of 1 32 -in . Lucite sphere “molecules.” (Recessed cell on back face of gas-container near base holds polonium rod to ionize air and prevent build up of static charge by friction; model of Fig. 26 has similar, longer rod.)

Fig. 28
Fig. 28

Top view of large-scale dynamical model of “two-dimensional gas” described in text.

Fig. 29
Fig. 29

Large scale adjustable “one-dimensional gas” of “planar scatterers”; both spacings and orientation of grids are variable. The periodic limiting cases include anisotropic dielectrics and crystals, and “optically active” media.

Equations (152)

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( 2 + k 2 ) ψ e - i ω t = 0 ,
k = 2 π / λ = ω / c
φ = e i k · r ,             k · r = ( k i ) · ( r o ) ,
u ~ g ( o , i ) H ( r ) ,             r ,
ψ = φ + u ,
u = A H ( r ) ,
Ψ = e i k x + U
U = U + + U - ,             U ± = G ± H ( r ± ) ,             G + = G - = B .
U ± ~ B H ( r ) e i k b sin θ .
U = U + + U - ~ [ B H ( r ) ] 2 cos ( k b sin θ ) = G H ( r ) ,
A H ( r ± ) [ 1 + A H ( 2 b ) + ( A H ) 2 + ] = B H ( r ± ) .
B = A / [ 1 - A H ( 2 b ) ] ,
B = A + A · B H ( 2 b ) .
B s = A s [ 1 + t B t H ( b s - b t ) ] ,
B s = A s + A s t A t H ( b s - b t ) + A s t A t H ( b s - b t ) × p A p H ( b t - b p ) + ,
u ( r , i ) = 1 4 i [ H 0 ( 1 ) ( k r - r ) n u ( r , i ) - u ( r , i ) n H 0 ( 1 ) ( k r - r ) ] d S ( r ) ;
- e i k r - r / 4 π r - r = h 0 ( 1 ) ( k r - r ) k / 4 π i .
u ( r , i ) = { 0 ( k r - r ) , u ( r , i ) } ,
g ( o , i ) = { e - i k o · r , u ( r , i ) } .
- Re g ( i , i ) = M g ( o , i ) 2 ,
- Re g ( i , i ) = M g ( o , i ) 2 + M p a = M ( p s + p a ) = M p ,
u ( r , i ) = 1 π c e i k p · r g ( p , i ) d τ = 1 π c e i k r cos ( θ - τ ) g ( τ , θ t ) d τ
u ( r , i ) = 1 2 π c e i k p · r g ( p , i ) d Ω
u ( r , i ) = c e i k p · r g ( p , i ) .
Ψ ( r ) = φ + U ,             U = s U s ( r - b s ) e i k · b s
U s = { 0 ( k r s - r s ) , U s ( r s ) } s ,
G s ( o , i ) = { e - i k o · r s , U s ( r s ) } s .
U s = c e i k p · r s G s ( p , i ) .
U t = u t ( i ) + s c e i k ( b t - b s ) · ( p - i ) u t ( p ) G s ( p , i ) ,
G t ( o , i ) = g t ( o , i ) + s c e i k ( b t - b s ) · ( p - i ) g t ( o , p ) G s ( p , i ) .
Ψ = φ + s e i k · b s c ( s ) e i k p · ( r - b s ) G s ( p , i )
g s ( θ , α ) = n = - A n s ( α ) e i n θ = n m A n m s e i n θ + i m α , G s ( θ , α ) = n = - B n s ( α ) e i n θ ,
B n t = A n t ( α ) + s m p A n m t B p s E ( s , t ; p , m ) , E ( s , t ; p , m ) = e - i k b t s cos ( β t s - α ) H p + m ( 1 ) ( k b t s ) × e i ( p + m ) ( β t s + π / 2 ) ,
g s ( θ , α ) = A n s e i n ( θ - α ) ,
B n t = A n t [ e - i n α + s p B p s E ( s , t ; p , - n ) ] ;
- Re B = A / ( 1 - A H ) 2 ( 1 + J ) = B 2 ( 1 + J )
U ( r ) = e i δ U + ( r + ) + e - i δ U - ( r - ) ,             δ = k · b = k i · b .
G ( o , i ) = e i ( δ - Δ ) G + ( o , i ) + e - i ( δ - Δ ) G - ( o , i ) ,             Δ = k o · b
D n ± = d n ± [ e - i n γ + p D p H ± ( p - n ) ( 1 ) ( 2 k b ) ] .
D 0 ± = d 0 ± ( 1 + D 0 H 0 ) = [ d 0 ± ( 1 + d 0 H 0 ) ] / [ 1 - d 0 + d 0 - H 0 2 ] .
D 1 + ± D - 1 + = d 1 + [ e - i γ ± e i γ + ( D 1 - ± D - 1 - ) ( H 1 ± H 2 ) ] = d 1 + [ e - i γ ± e i γ ] [ 1 + d 1 - ( H 0 ± H 2 ) ] 1 - d 1 + d 1 - ( H 0 ± H 2 ) 2
D 0 ± C = d 0 ± [ ( 1 + d 0 H 0 ) ( 1 - 4 d 1 + d 1 - H 1 2 ) - 2 d 0 d 1 ± H 1 2 ( 1 + 2 d 1 H 1 ) ] + i 2 d 0 ± H 1 sin γ { d 1 ( 1 + 2 d 1 ± H 1 ) - d 0 d 1 ± [ H 0 ( 1 + 2 d 1 H 1 ) + 2 d 1 H 1 2 ] } , ( D 1 ± - D - 1 ± ) C = - i 2 d 1 ± sin γ [ ( 1 + 2 d 1 H 1 ) × ( 1 - 2 d 0 + d 0 - H 0 2 ) - 2 d 0 ± d 1 H 1 2 ( 1 + d 0 H 0 ) ] ± 2 d 1 ± H 1 { d 0 ( 1 + d 0 ± H 0 ) - 2 d 0 ± d 1 [ H 1 ( 1 + d 0 H 0 ) + d 0 H 1 2 ] } . C = 1 - d 0 + d 0 - H 0 2 - 4 d 1 + d 1 - H 1 2 - 2 ( d 0 - d 1 + + d 0 + d 1 - ) H 1 2 + 4 d 0 + d 0 - d 1 + d 1 - ( H 1 4 + H 0 2 H 1 2 + 2 H 0 H 1 2 H 1 ) .
F ± = ( e i 2 k b i 2 k b ) e - i 2 k b i · b ± × [ 1 + ( i 4 k b ) D ± 1 ! + ( i 4 k b ) 2 D ± ( D ± - 1.2 ) 2 ! + ] D = - 1 sin 2 τ [ 2 β 2 + sin τ τ sin τ τ ] ,
G ± ( o , i ) = g ± ( o , i ) + F ± g ± ( o , b ± ) × [ g ( b ± , i ) + F g ( b ± , b ) g ± ( b , i ) 1 - F g ( b ± , b ) F ± g ± ( b , b ± ) ] .
Ψ > = φ + U > = φ + 2 μ = - G ( θ μ , θ 0 ) k d cos θ μ e i k y sin θ μ + i k x cos θ μ ,
sin θ μ = sin θ 0 + 2 μ π k d = sin θ 0 + μ λ d ,             μ = 0 , ± 1 , ± 2 , .
Ψ < = U < = 2 C μ G ( π - θ μ , θ 0 ) e i k y sin θ μ - i k x cos θ μ , C μ = 1 / k d cos θ μ .
μ + = [ m + ] = [ d ( 1 - sin θ 0 ) / λ ] ,
μ - = [ m - ] = [ d ( 1 + sin θ 0 ) / λ ] .
- Re G ( θ 0 , θ 0 ) = μ - μ + C μ [ G ( θ μ , θ 0 ) 2 + G ( π - θ μ , θ 0 ) 2 ] .
- Re g ( θ 0 , θ 0 ) = 1 2 π 0 2 π g ( θ , θ 0 ) 2 d θ = 1 2 π - π / 2 π / 2 [ g ( θ , θ 0 ) 2 + g ( π - θ , θ 0 ) 2 ] d θ = m - m + C μ [ g ( θ μ , θ 0 ) 2 + g ( π - θ μ , θ 0 ) 2 ] d μ ,
G ( θ , θ 0 ) = g ( θ , θ 0 ) + S C μ [ g ( θ , θ μ ) G ( θ μ , θ 0 ) + g ( θ , π - θ μ ) G ( π - θ μ , θ 0 ) ] ,
S = [ μ = - - - d μ ] e i k x cos θ μ | x 0 ,
G ( θ , θ 0 ) g ( θ , θ 0 )
Ψ = φ + 2 C ν F ν 0 ± e i k r cos ( θ - θ ν ) ν ± e i k r cos ( θ - θ ν ) , F ν 0 ± = G ± ( θ ν , θ 0 ) ± G ± ( θ ν , π - θ 0 ) ,
f ν 0 ± = f ± ( θ ν , π - θ 0 ) = g ± ( θ ν , θ 0 ) ± g ± ( θ ν , π - θ 0 ) ,
F ν 0 = f ν 0 + S f ν μ C μ F μ 0 .
- Re F 00 = μ - μ + C μ F μ 0 2 ;             cos θ 0 = μ - μ + μ 2 cos θ μ ,             μ = δ μ 0 + 2 C μ F μ 0 ,
- Re f 00 = 1 π - π / 2 π / 2 f μ 0 2 d θ μ = m - m + C μ f μ 0 2 d μ .
F μ 0 f ν 0 + f ν m C m F m 0 ;
F m 0 = f m 0 + f m m C m F m 0 = f m 0 / ( 1 - C m f m m ) ,
F ν 0 f ν 0 + f ν m C m f m 0 1 - C m f m m = f ν 0 + f ν m f m 0 k d cos θ m - f m m
f ν m [ 1 + C m f m m + ( C m f m m ) 2 + ] C m f m 0
a s θ m π 2 : f + i s n o n v a n i s h i n g ; f ν m - f m 0 - cos θ m 0 ,             f m m - cos 2 θ m 0.
F ν 0 + f ν 0 + - f ν m + f m 0 + / f m m + ,             θ m π / 2 ,
F ν 0 + f ν 0 + - f ν m + f m 0 + / Re f m m + .
F ν 0 + j ν 0 - j ν m j m 0 / ρ m m ,             j ν μ = i Im f ν μ + ,             ρ ν μ = Re F ν μ + .
F ν 0 + ρ ν 0 + ( j ν 0 / j ν m j m 0 ) × [ j ν 0 ρ m m - j ν m ρ m 0 - j m 0 ρ ν m ] ,
Im f + ( θ , π - α ) - ( π k 2 / 2 ) [ a b - a ( a + b ) sin θ sin α ] , Im f - - [ π k 2 a ( a + b ) / 2 ] cos θ cos α .
- f 0 ν - f ν 0 * = 2 π - π / 2 π / 2 f μ 0 * f μ ν d θ μ = 2 m - m + C μ f μ 0 * f μ ν d μ
- Re f ν 0 1 π - π / 2 π / 2 Im f μ 0 Im f μ ν d θ μ ,
Re f + - π 2 k 4 16 [ 2 a 2 b 2 + a 2 ( a + b ) 2 sin θ sin α ] , Re f - - π 2 k 4 a 2 ( a + b ) 2 cos θ cos α 16 .
f ˆ ν 0 = f ν 0 + ( μ - μ + - m - m + d μ ) f ν μ C μ f ˆ ν μ .
f ν 0 = f ν 0 - m - m + f ν μ C μ f μ 0 d μ ,
f ˆ ν 0 = f ν 0 + μ - μ + f ¯ ν μ C μ f ˆ μ 0
- Re f ˆ 00 = μ - μ + C μ f ˆ μ 0 2 ;
f ν 0 i Im f ν 0 , f ˆ ν 0 i Im f ν 0 - μ - μ + Im f ν μ C μ Im f μ 0 , f ν 0 i Im f ν 0 - m - m + Im f ν μ C μ Im f μ 0 d μ ,
F ν 0 f ˆ ν 0 + f ˆ ν m C m f ˆ m 0 / ( 1 - C m f ˆ m m ) ,
U s ( r - b s ) = { 0 ( k R s ) , U s ( r s ) } s = { 0 ( k R s ) , Ψ ( r s ) } s ,
U = s U s = s U s ( r - b s ) s w ( b s ) d b s , U s s = { 0 ( k R s ) , Ψ } s s = { 0 , Ψ s } s ,
Ψ s = φ + U s = φ + U s s + t U t s t w s t ( b s ; b t ) d b t , U t s t = { 0 ( k R t ) , Ψ ( b t + r t ) s t } t ,
w s ( b s ) = ρ / N ,
Ψ = φ + U ( r ) = φ + ρ U ( r - b s ; b s ) s d b s , U s = { H 0 ( k R s ) , Ψ ( b s + r s ) s } ,
s w s t = ρ ( N - 1 ) / N ρ ,
Ψ ( b t + r ) s t Ψ t .
Ψ s Ψ + U s
Ψ = φ + s d b s w ( b s ) C e i k p · ( r - b s ) G s ( b s ; p , i ) s
G t ( b t ; o , i ) t = g t ( o , i ) e i k · b t + s C d b s w t s ( b t ; b s ) × C e i k p · ( b t - b s ) g t ( o , p ) G s ( b s ; p , i ) t s .
Ψ = φ + U = φ + ρ d b s C e i k p · ( r - b s ) G ( b s ; p ) ,
G ( b t ; o ) g ( o , i ) e i k · b t + ρ d b s C e i k p · ( b t - b s ) × g ( o , p ) G ( b s ; p ) ,
Ψ ( b s + r ) s = Ψ ( ζ + r ) s e i k · τ ξ ,             ζ = ζ n = ( b s · n ) n ,
G ( b s ; o ) = G ( ζ ; o ) e i k · τ ξ = { e - i k o · r , Ψ ( ζ + r ) s } e i k · τ ξ .
Ψ T = φ [ 1 + 2 C G ( i , i ) ] ,             C 2 = ρ 2 / k · n ,             C 3 = ρ 3 π / k k · n ,             φ = e i k i · r ,             k = k i ,
Ψ R = φ 2 C G ( i , i ) ,             φ = e i k i · r ,
G ( o , i ) = g ( o , i ) + g ( o , i ) C G ( i , i ) + g ( o , i ) C G ( i , i ) ,
Ψ ± = φ [ 1 + 2 C F ± ( i , i ) ] , F ± ( i , i ) = G ± ( i , i ) ± G ± ( i , i ) = G ± ( i , i ) ± G ± ( i , i ) = F ± ( i , i ) ,
F ± ( o , i ) = f ± ( o , i ) [ 1 - C F ± ( i , i ) ] , f ± ( o , i ) = g ± ( o , i ) ± g ± ( o , i ) = f ± ( o , i ) ,
F ( o , i ) = f ( o , i ) 1 - C f ( i , i ) f ( o , i ) 1 - Z .
Ψ ± = φ ( 1 + 2 Z ± 1 - Z ± ) = φ ( 1 + Z ± 1 - Z ± ) .
A ± = ± 1 + Z ± 1 - Z ± ,             R ± = | 1 + Z ± 1 - Z ± | 2 , Z ± = C f ± ( i , i ) = C [ g ± ( i , i ) ± g ± ( i , i ) ] ,
ν ± = tan - 1 | 2 Im Z ± 1 - Z ± 2 | .
V = Ψ 2 - Ψ 2
I = Re Ψ * Ψ / i k - Re [ Ψ * Ψ / i k ] .
σ ( o , i ) = D ρ | f ( o , i ) 1 - Z | 2 ,             D 2 = 2 π k ,             D 3 = 1 k 2 ,
J = R i + I R i + σ ( s , i ) r - b s m - 1 s d b s ,             s = r - b s r - b s
1 = R + ρ P i · n ,             P = P a + P s ,             ρ P s = 1 2 σ ( s , i ) d Ω s
- Re f ( i , i ) = M p = M ( p a + p s ) ,             M 2 = k / 4 ,             M 3 = k 2 / 4 π ;             p s = D 1 2 f ( s , i ) 2 d Ω s
Z - β 0 ,             Z + 1 / β ;
± ( 1 + Z ± ) / ( 1 - Z ± )
R ± - 1 = 4 Re Z ± 1 - Z ± 2 β 0 ;
σ - β 4 ,             σ + β 2 .
R R ± 1 = 1 + 4 Re Z ± = 1 - ρ p ± / i · n , ρ p s = 1 2 σ 1 d Ω s = ρ D 1 2 f ( s , i ) 2 d Ω s .
Z = m Z m ,             Z = Z ( μ ) d μ ,
σ = m σ m ;             σ = σ ( μ ) d μ ,
Ψ = φ [ 1 + C 0 z e - i γ ζ G ( ζ ; k ) d ζ ] + φ C z d e i γ ζ G ( ζ ; k ) d ζ ,
Ψ e i k · r A ( 0 , z ) + e i k · r A ( z , d ) ,
G ( ζ ; k o ) = A ( 0 , ζ ) e i k · z g ( k o , k i ) + A ( ζ , d ) e i k · z g ( k o , k i ) ,
ψ = A 1 e i K · r + A 2 e i K · r ,
K ± 2 = k 2 sin 2 α + ( Δ ± Γ ) 2 ,             Δ = i ( S - - S + ) / 2 , Γ 2 = [ γ - i ( S + + S - ) / 2 ] 2 + R + R - ,             γ = k cos α ; S + = C g ( i , i ) ,             R + = C g ( i , i ) ,             S - = C g ( i , i ) , R - = C g ( i , i ) .
K 2 = k 2 - i 2 γ S - S 2 + R 2 = k 2 μ , ( K · n ) 2 = Γ 2 = [ γ - i ( S - R ) ] [ γ - i ( S + R ) ] ,
μ } = 1 B = 1 + S - R i γ             if             y 0 φ = { H i E i ;             B = z Ψ outside z Ψ inside | z = 0 , d .
Ψ t = T e i k · r ,             T = ( 1 - Q 2 ) e i d ( Γ - γ ) / D , Ψ r = r i k · r ,             = - Q ( 1 - e i 2 d Γ ) / D ,             D = 1 - Q 2 e i 2 Γ d ; Q = S + i ( γ - Γ ) R = R S + i ( γ + Γ ) = Z - 1 Z + 1 , Z = [ i γ + S + R i γ + S - R ] 1 2 ,
K 2 = k 2 + 4 π ρ f + ( 2 π ρ k cos α ) 2 ( f 2 - f 2 ) ;             f = f ( k , k ) ,             f = f ( k , k ) ,
K 2 = k 2 + 4 π ρ f + ( 2 π ρ k ) 2 ( f 2 - f 2 ) = [ k + 2 π ρ k ( f + f ) ] [ k + 2 π ρ k ( f - f ) ]
K 2 = k 2 + 4 π ρ f ,
K k + 2 π ρ f / k ,
G ( ζ ; k o ) = A e i K · z g ( k o , K ) + A e i K · z g ( k o , K )
K ± 2 = k 2 sin 2 α + Γ ± 2 , 1 = S i ( Γ + - γ ) - R i ( Γ + + γ ) = R i ( Γ - - γ ) - S i ( Γ - + γ ) , S = C g ( k , K ) ,             S = C g ( k , K ) ,             R = C g ( k , K ) ,             R = C g ( k , K ) .
K 2 = k 2 + i R ( Γ - γ ) - i S ( Γ + γ ) = k 2 μ ,
B = 1 - S - R i Γ ,             1 B = μ }             if             y 0 φ = { H i E i .
Q = ( Γ - γ ) g ( k , K ) ( Γ + γ ) g ( k , K ) = Z - 1 Z + 1 ,             i Z = S Γ - γ + r Γ + γ .
G ( ζ ; o ) = [ G ( o ) e i Γ ζ + Q G ( o ) e i Γ ( 2 d - ζ ) ] / D ,
G ( o ) = g ( k o , k ) - Q g ( k o , k )
G ( 0 ) = ( 1 - Q ) g ( k o , K )
J = C + I = Re [ Ψ * Ψ / i k ] + I ,
I ρ D G ( ζ ; k s ) 2 r - b s m - 1 s d b s = ρ D 0 d d ζ G ( ζ ; k s ) 2 s · n s d Ω , s = r - b s r - b s .
Γ = γ - i C G ( i ) ,
2 Im Γ = - 2 C Re G ( i ) P ρ sec α ,
2 + T 2 + ρ sec α D 0 d [ G ( ζ ; k o ) 2 d Ω ] d ζ = 1 ,
2 Im Γ = D ρ G ( o ) 2 d Ω cos α ( 1 - Q 2 ) = 2 C M G ( o ) 2 1 - Q 2
- Re G ( i ) = M G ( o ) 2 1 - Q 2 = M P ,
g ( k , K ) ( i k 2 v / 2 π ) ( K - K ) [ 1 + i ( K - k ) b ] ,             g ( k , K ) g ( k , K ) ,
K = k η k + ( 2 π ρ / i k 2 ) g ( k , K ) ,             η = = μ .
Re ( K - k ) = [ w / ( 1 + w ) ] ( K - k ) = W ( K - k ) , Im K = [ b w / ( 1 + w ) 2 ] ( K - k ) 2 = b W ( 1 - W ) ( K - k ) 2 ,
C = T 2 = e - 2 Im K d e - 2 β .
G 2 = g ( k o , K ) 2 e - 2 Im K ζ ,
I = ( 1 - e - 2 β ) q ( δ ) ,
J · n = C + I = e - 2 β + ( 1 - e - 2 β ) q ( δ ) .
I / C = q ( e 2 β - 1 ) .
x 2 = ± ( k d ) 2 = ± ( 8 π / 3 ) 2             for             sin θ μ = sin θ 0 + 3 μ / 4
R = ( 1 + Z ) / ( 1 - Z ) 2
p = Re ( K - k ) d = k d ( η - 1 ) W = 5.82 W