Abstract

The subject of virtual rays and its application is introduced, for the first time in this country to our knowledge, as a new method in ray optics. The method of virtual rays uses a representation of the electromagnetic field in terms of a continuum of a special class of rays that do not obey the usual geometrical optics along the entire path of propagation. It yields, in the asymptotic limit as λ → 0, the correct field representation at caustics, shadow, and transition regions, i.e., in which geometrical optics and its extensions, such as the geometrical theory of diffraction, fail. The method of virtual rays is used to solve for the high-frequency diffraction from a perfectly or imperfectly conducting semi-infinite wedge.

© 1994 Optical Society of America

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References

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  1. Yu. I. Orlov, Ph.D. dissertation (Moscow University, Moscow, 1969).
  2. L. A. Vainshtein, E. A. Tishchenko, “Wave-tracing and shortwave diagnostics of a cylindrical plasma,” Sov. Phys. Tech. Phys. 21, 1338–1343 (1976).
  3. L. A. Vainshtein, E. S. Birger, N. B. Konyukhova, E. L. Kosarev, G. P. Prudkovskii, “Shortwave plasma diagnostics,” Sov. J. Plasma Phys. 2, 362–369 (1976).
  4. A. Vainshtein, P. Ya. Ufimtsev, “Virtual rays in the problem of diffraction from a wedge,” Radiotekh. Elektron. 2, 625–633 (1982).
  5. D. R. Jackson, N. G. Alexopoulos, “Scattering by the meothod of virtual rays,” Final Rep. (Phraxos Research and Development, Inc., Santa Monica, Calif., 1992).
  6. N. G. Alexopoulos, “The method of virtual rays in scattering theory,” in Digest of the URSI Symposium (Union Radio-Scientifique Internationale, Chicago, Ill., 1992), p. 479.
  7. G. Fanceschetti, A. Brancaccio, N. G. Alexopoulos, “Virtual rays—the half-plane,” in Digest of the URSI Symposium (Union Radio-Scientifique Internationale, Chicago, Ill., 1992), p. 480.
  8. N. G. Alexopoulos, G. Franceschetti, P. Ya. Ufimtsev, “Virtual rays—surface ray excitation,” in Digest of the URSI Symposium (Union Radio-Scientifique Internationale, Chicago, Ill., 1992), p. 481.
  9. D. R. Jackson, N. G. Alexopoulos, “Scattering from coated and impedance wedges using the method of virtual rays,” in Digest of the URSI Symposium (Union Radio-Scientifique Internationale, Chicago, Ill., 1992), p. 482.
  10. P. Ya. Ufimtsev, N. G. Alexopoulos, “Virtual rays and a cone diffraction problem,” in Digest of the URSI Symposium (Union Radio-Scientifique Internationale, Chicago, Ill., 1992), p. 483.
  11. R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961).
  12. S. W. Lee, “Path integrals for solving some electromagnetic edge diffraction problems,” J. Math. Phys. 19, 1414–1422 (1978).
    [CrossRef]
  13. R. W. Ziolkowski, “A path integral Riemann space approach to the electromagnetic wedge diffraction problem,” J. Math.Phys. 27, 2271–2281 (1986).
    [CrossRef]
  14. M.R. Spiegel, Mathematical Handbook, Schaum’s Outline Series in Mathematics (McGraw-Hill, New York, 1968), p. 109.
  15. N. Bleistein, R. A. Handelsman, Asymptotic Expansion of Integrals (Holt, Rinehart & Winston, New York, 1975).
  16. There is a typographical error in this formula: the term ain front of the cosine should be ka.
  17. S. Ramo, J. R. Whinnery, T. Van Duzer, Fields and Waves in Communication Electronics (Wiley, New York, 1965), Chap. 6, pp. 356–365.
  18. R. Tiberio, G. Pelosi, G. Manara, “A uniform GTD formulation for the diffraction by a wedge with impedance faces,” IEEE Trans. Antennas Propag. AP-33, 867–873 (1985).
    [CrossRef]
  19. G. D. Maliuzhinets, “Excitation, reflection, and emission of surface waves from a wedge with given face impedances,” Sov. Phys. Dokl. 3, 752–755 (1958).There is evidently a typographical error in Figs. 1, 2, and 4 of Ref. 18: the TM and the TE solutions are reversed.
  20. M. I. Herman, J. L. Volakis, “High frequency scattering from a double impedance wedge,” IEEE Trans. Antennas Propag. 36, 664–678 (1988).
    [CrossRef]
  21. P. Ya. Ufimtsev, “Transverse diffusion for diffraction by a wedge,” Radiotekh. Elektron. 10, 1013–1022 (1965).
  22. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed. (Academic, Orlando, Fla., 1980), Chap. 1, p. 36.

1988 (1)

M. I. Herman, J. L. Volakis, “High frequency scattering from a double impedance wedge,” IEEE Trans. Antennas Propag. 36, 664–678 (1988).
[CrossRef]

1986 (1)

R. W. Ziolkowski, “A path integral Riemann space approach to the electromagnetic wedge diffraction problem,” J. Math.Phys. 27, 2271–2281 (1986).
[CrossRef]

1985 (1)

R. Tiberio, G. Pelosi, G. Manara, “A uniform GTD formulation for the diffraction by a wedge with impedance faces,” IEEE Trans. Antennas Propag. AP-33, 867–873 (1985).
[CrossRef]

1982 (1)

A. Vainshtein, P. Ya. Ufimtsev, “Virtual rays in the problem of diffraction from a wedge,” Radiotekh. Elektron. 2, 625–633 (1982).

1978 (1)

S. W. Lee, “Path integrals for solving some electromagnetic edge diffraction problems,” J. Math. Phys. 19, 1414–1422 (1978).
[CrossRef]

1976 (2)

L. A. Vainshtein, E. A. Tishchenko, “Wave-tracing and shortwave diagnostics of a cylindrical plasma,” Sov. Phys. Tech. Phys. 21, 1338–1343 (1976).

L. A. Vainshtein, E. S. Birger, N. B. Konyukhova, E. L. Kosarev, G. P. Prudkovskii, “Shortwave plasma diagnostics,” Sov. J. Plasma Phys. 2, 362–369 (1976).

1965 (1)

P. Ya. Ufimtsev, “Transverse diffusion for diffraction by a wedge,” Radiotekh. Elektron. 10, 1013–1022 (1965).

1958 (1)

G. D. Maliuzhinets, “Excitation, reflection, and emission of surface waves from a wedge with given face impedances,” Sov. Phys. Dokl. 3, 752–755 (1958).There is evidently a typographical error in Figs. 1, 2, and 4 of Ref. 18: the TM and the TE solutions are reversed.

Alexopoulos, N. G.

D. R. Jackson, N. G. Alexopoulos, “Scattering by the meothod of virtual rays,” Final Rep. (Phraxos Research and Development, Inc., Santa Monica, Calif., 1992).

N. G. Alexopoulos, “The method of virtual rays in scattering theory,” in Digest of the URSI Symposium (Union Radio-Scientifique Internationale, Chicago, Ill., 1992), p. 479.

G. Fanceschetti, A. Brancaccio, N. G. Alexopoulos, “Virtual rays—the half-plane,” in Digest of the URSI Symposium (Union Radio-Scientifique Internationale, Chicago, Ill., 1992), p. 480.

N. G. Alexopoulos, G. Franceschetti, P. Ya. Ufimtsev, “Virtual rays—surface ray excitation,” in Digest of the URSI Symposium (Union Radio-Scientifique Internationale, Chicago, Ill., 1992), p. 481.

D. R. Jackson, N. G. Alexopoulos, “Scattering from coated and impedance wedges using the method of virtual rays,” in Digest of the URSI Symposium (Union Radio-Scientifique Internationale, Chicago, Ill., 1992), p. 482.

P. Ya. Ufimtsev, N. G. Alexopoulos, “Virtual rays and a cone diffraction problem,” in Digest of the URSI Symposium (Union Radio-Scientifique Internationale, Chicago, Ill., 1992), p. 483.

Birger, E. S.

L. A. Vainshtein, E. S. Birger, N. B. Konyukhova, E. L. Kosarev, G. P. Prudkovskii, “Shortwave plasma diagnostics,” Sov. J. Plasma Phys. 2, 362–369 (1976).

Bleistein, N.

N. Bleistein, R. A. Handelsman, Asymptotic Expansion of Integrals (Holt, Rinehart & Winston, New York, 1975).

Brancaccio, A.

G. Fanceschetti, A. Brancaccio, N. G. Alexopoulos, “Virtual rays—the half-plane,” in Digest of the URSI Symposium (Union Radio-Scientifique Internationale, Chicago, Ill., 1992), p. 480.

Fanceschetti, G.

G. Fanceschetti, A. Brancaccio, N. G. Alexopoulos, “Virtual rays—the half-plane,” in Digest of the URSI Symposium (Union Radio-Scientifique Internationale, Chicago, Ill., 1992), p. 480.

Franceschetti, G.

N. G. Alexopoulos, G. Franceschetti, P. Ya. Ufimtsev, “Virtual rays—surface ray excitation,” in Digest of the URSI Symposium (Union Radio-Scientifique Internationale, Chicago, Ill., 1992), p. 481.

Gradshteyn, S.

S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed. (Academic, Orlando, Fla., 1980), Chap. 1, p. 36.

Handelsman, R. A.

N. Bleistein, R. A. Handelsman, Asymptotic Expansion of Integrals (Holt, Rinehart & Winston, New York, 1975).

Harrington, R. F.

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961).

Herman, M. I.

M. I. Herman, J. L. Volakis, “High frequency scattering from a double impedance wedge,” IEEE Trans. Antennas Propag. 36, 664–678 (1988).
[CrossRef]

Jackson, D. R.

D. R. Jackson, N. G. Alexopoulos, “Scattering from coated and impedance wedges using the method of virtual rays,” in Digest of the URSI Symposium (Union Radio-Scientifique Internationale, Chicago, Ill., 1992), p. 482.

D. R. Jackson, N. G. Alexopoulos, “Scattering by the meothod of virtual rays,” Final Rep. (Phraxos Research and Development, Inc., Santa Monica, Calif., 1992).

Konyukhova, N. B.

L. A. Vainshtein, E. S. Birger, N. B. Konyukhova, E. L. Kosarev, G. P. Prudkovskii, “Shortwave plasma diagnostics,” Sov. J. Plasma Phys. 2, 362–369 (1976).

Kosarev, E. L.

L. A. Vainshtein, E. S. Birger, N. B. Konyukhova, E. L. Kosarev, G. P. Prudkovskii, “Shortwave plasma diagnostics,” Sov. J. Plasma Phys. 2, 362–369 (1976).

Lee, S. W.

S. W. Lee, “Path integrals for solving some electromagnetic edge diffraction problems,” J. Math. Phys. 19, 1414–1422 (1978).
[CrossRef]

Maliuzhinets, G. D.

G. D. Maliuzhinets, “Excitation, reflection, and emission of surface waves from a wedge with given face impedances,” Sov. Phys. Dokl. 3, 752–755 (1958).There is evidently a typographical error in Figs. 1, 2, and 4 of Ref. 18: the TM and the TE solutions are reversed.

Manara, G.

R. Tiberio, G. Pelosi, G. Manara, “A uniform GTD formulation for the diffraction by a wedge with impedance faces,” IEEE Trans. Antennas Propag. AP-33, 867–873 (1985).
[CrossRef]

Orlov, Yu. I.

Yu. I. Orlov, Ph.D. dissertation (Moscow University, Moscow, 1969).

Pelosi, G.

R. Tiberio, G. Pelosi, G. Manara, “A uniform GTD formulation for the diffraction by a wedge with impedance faces,” IEEE Trans. Antennas Propag. AP-33, 867–873 (1985).
[CrossRef]

Prudkovskii, G. P.

L. A. Vainshtein, E. S. Birger, N. B. Konyukhova, E. L. Kosarev, G. P. Prudkovskii, “Shortwave plasma diagnostics,” Sov. J. Plasma Phys. 2, 362–369 (1976).

Ramo, S.

S. Ramo, J. R. Whinnery, T. Van Duzer, Fields and Waves in Communication Electronics (Wiley, New York, 1965), Chap. 6, pp. 356–365.

Ryzhik, I. M.

S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed. (Academic, Orlando, Fla., 1980), Chap. 1, p. 36.

Spiegel, M.R.

M.R. Spiegel, Mathematical Handbook, Schaum’s Outline Series in Mathematics (McGraw-Hill, New York, 1968), p. 109.

Tiberio, R.

R. Tiberio, G. Pelosi, G. Manara, “A uniform GTD formulation for the diffraction by a wedge with impedance faces,” IEEE Trans. Antennas Propag. AP-33, 867–873 (1985).
[CrossRef]

Tishchenko, E. A.

L. A. Vainshtein, E. A. Tishchenko, “Wave-tracing and shortwave diagnostics of a cylindrical plasma,” Sov. Phys. Tech. Phys. 21, 1338–1343 (1976).

Ufimtsev, P. Ya.

A. Vainshtein, P. Ya. Ufimtsev, “Virtual rays in the problem of diffraction from a wedge,” Radiotekh. Elektron. 2, 625–633 (1982).

P. Ya. Ufimtsev, “Transverse diffusion for diffraction by a wedge,” Radiotekh. Elektron. 10, 1013–1022 (1965).

N. G. Alexopoulos, G. Franceschetti, P. Ya. Ufimtsev, “Virtual rays—surface ray excitation,” in Digest of the URSI Symposium (Union Radio-Scientifique Internationale, Chicago, Ill., 1992), p. 481.

P. Ya. Ufimtsev, N. G. Alexopoulos, “Virtual rays and a cone diffraction problem,” in Digest of the URSI Symposium (Union Radio-Scientifique Internationale, Chicago, Ill., 1992), p. 483.

Vainshtein, A.

A. Vainshtein, P. Ya. Ufimtsev, “Virtual rays in the problem of diffraction from a wedge,” Radiotekh. Elektron. 2, 625–633 (1982).

Vainshtein, L. A.

L. A. Vainshtein, E. S. Birger, N. B. Konyukhova, E. L. Kosarev, G. P. Prudkovskii, “Shortwave plasma diagnostics,” Sov. J. Plasma Phys. 2, 362–369 (1976).

L. A. Vainshtein, E. A. Tishchenko, “Wave-tracing and shortwave diagnostics of a cylindrical plasma,” Sov. Phys. Tech. Phys. 21, 1338–1343 (1976).

Van Duzer, T.

S. Ramo, J. R. Whinnery, T. Van Duzer, Fields and Waves in Communication Electronics (Wiley, New York, 1965), Chap. 6, pp. 356–365.

Volakis, J. L.

M. I. Herman, J. L. Volakis, “High frequency scattering from a double impedance wedge,” IEEE Trans. Antennas Propag. 36, 664–678 (1988).
[CrossRef]

Whinnery, J. R.

S. Ramo, J. R. Whinnery, T. Van Duzer, Fields and Waves in Communication Electronics (Wiley, New York, 1965), Chap. 6, pp. 356–365.

Ziolkowski, R. W.

R. W. Ziolkowski, “A path integral Riemann space approach to the electromagnetic wedge diffraction problem,” J. Math.Phys. 27, 2271–2281 (1986).
[CrossRef]

IEEE Trans. Antennas Propag. (2)

R. Tiberio, G. Pelosi, G. Manara, “A uniform GTD formulation for the diffraction by a wedge with impedance faces,” IEEE Trans. Antennas Propag. AP-33, 867–873 (1985).
[CrossRef]

M. I. Herman, J. L. Volakis, “High frequency scattering from a double impedance wedge,” IEEE Trans. Antennas Propag. 36, 664–678 (1988).
[CrossRef]

J. Math. Phys. (1)

S. W. Lee, “Path integrals for solving some electromagnetic edge diffraction problems,” J. Math. Phys. 19, 1414–1422 (1978).
[CrossRef]

J. Math.Phys. (1)

R. W. Ziolkowski, “A path integral Riemann space approach to the electromagnetic wedge diffraction problem,” J. Math.Phys. 27, 2271–2281 (1986).
[CrossRef]

Radiotekh. Elektron. (2)

A. Vainshtein, P. Ya. Ufimtsev, “Virtual rays in the problem of diffraction from a wedge,” Radiotekh. Elektron. 2, 625–633 (1982).

P. Ya. Ufimtsev, “Transverse diffusion for diffraction by a wedge,” Radiotekh. Elektron. 10, 1013–1022 (1965).

Sov. J. Plasma Phys. (1)

L. A. Vainshtein, E. S. Birger, N. B. Konyukhova, E. L. Kosarev, G. P. Prudkovskii, “Shortwave plasma diagnostics,” Sov. J. Plasma Phys. 2, 362–369 (1976).

Sov. Phys. Dokl. (1)

G. D. Maliuzhinets, “Excitation, reflection, and emission of surface waves from a wedge with given face impedances,” Sov. Phys. Dokl. 3, 752–755 (1958).There is evidently a typographical error in Figs. 1, 2, and 4 of Ref. 18: the TM and the TE solutions are reversed.

Sov. Phys. Tech. Phys. (1)

L. A. Vainshtein, E. A. Tishchenko, “Wave-tracing and shortwave diagnostics of a cylindrical plasma,” Sov. Phys. Tech. Phys. 21, 1338–1343 (1976).

Other (13)

Yu. I. Orlov, Ph.D. dissertation (Moscow University, Moscow, 1969).

M.R. Spiegel, Mathematical Handbook, Schaum’s Outline Series in Mathematics (McGraw-Hill, New York, 1968), p. 109.

N. Bleistein, R. A. Handelsman, Asymptotic Expansion of Integrals (Holt, Rinehart & Winston, New York, 1975).

There is a typographical error in this formula: the term ain front of the cosine should be ka.

S. Ramo, J. R. Whinnery, T. Van Duzer, Fields and Waves in Communication Electronics (Wiley, New York, 1965), Chap. 6, pp. 356–365.

D. R. Jackson, N. G. Alexopoulos, “Scattering by the meothod of virtual rays,” Final Rep. (Phraxos Research and Development, Inc., Santa Monica, Calif., 1992).

N. G. Alexopoulos, “The method of virtual rays in scattering theory,” in Digest of the URSI Symposium (Union Radio-Scientifique Internationale, Chicago, Ill., 1992), p. 479.

G. Fanceschetti, A. Brancaccio, N. G. Alexopoulos, “Virtual rays—the half-plane,” in Digest of the URSI Symposium (Union Radio-Scientifique Internationale, Chicago, Ill., 1992), p. 480.

N. G. Alexopoulos, G. Franceschetti, P. Ya. Ufimtsev, “Virtual rays—surface ray excitation,” in Digest of the URSI Symposium (Union Radio-Scientifique Internationale, Chicago, Ill., 1992), p. 481.

D. R. Jackson, N. G. Alexopoulos, “Scattering from coated and impedance wedges using the method of virtual rays,” in Digest of the URSI Symposium (Union Radio-Scientifique Internationale, Chicago, Ill., 1992), p. 482.

P. Ya. Ufimtsev, N. G. Alexopoulos, “Virtual rays and a cone diffraction problem,” in Digest of the URSI Symposium (Union Radio-Scientifique Internationale, Chicago, Ill., 1992), p. 483.

R. F. Harrington, Time-Harmonic Electromagnetic Fields (McGraw-Hill, New York, 1961).

S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, 4th ed. (Academic, Orlando, Fla., 1980), Chap. 1, p. 36.

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Figures (16)

Fig. 1
Fig. 1

Rays that propagate from the source to the plane x = x0 and then to the observation point O.

Fig. 2
Fig. 2

Virtual rays that propagate from a source at point S to the observation point O, conforming to a family of concentric circles.

Fig. 3
Fig. 3

Virtual rays that propagate from the line source at point S to the observation point O. (a) The origin is arbitrarily located. (b) The origin is located on the straight-line path SO.

Fig. 4
Fig. 4

Virtual rays that have a negative phase change along the circular path: (a) a ray with m = − 1, (b) a ray with m = 0.

Fig. 5
Fig. 5

Geometry used in the calculation of the path length: (a) ν > 0, (b) ν < 0.

Fig. 6
Fig. 6

Geometry of a wedge having reflection coefficients R0 and Rα on the top and the bottom faces, respectively. The interior wedge angle is α, the line source S is at (r1,ϕ1), and the observation point O is at (r2,ϕ).

Fig. 7
Fig. 7

Type-1 virtual rays used in the scattering from a reflecting wedge.

Fig. 8
Fig. 8

Type-2 virtual rays used in the scattering from a reflecting wedge: (a) ray picture for ϕ2π/2, (b) ray picture for ϕ2π/2.

Fig. 9
Fig. 9

Type-3 virtual rays used in the scattering from a reflecting wedge: (a) ray picture for ϕ1απ/2, (b) ray picture for ϕ1απ/2.

Fig. 10
Fig. 10

Type-4 virtual rays used in the scattering from a reflecting wedge.

Fig. 11
Fig. 11

Normalized total-field magnitude versus ϕ2 for plane-wave incidence upon a perfectly conducting wedge. ϕ0 = 360° − α = 90°, ϕ1 = 30°, r20 = 10.0/(2π). The results of the virtual-ray method (curve) are compared with those of the Maliuzhinets solution (dots): (a) TMz solution, (b) TEz solution.

Fig. 12
Fig. 12

Normalized total-field magnitude versus ϕ2 for plane-wave incidence upon an impedance wedge. ϕ0 = 360° − α = 90°, ϕ1 = 30°, r20 = 10.0/(2π), Zs0 = Z = η0/4. The results of the virtual-ray method (curve) are compared with those of the Maliuzhinets solution (dots): (a) TMz solution, (b) TEz solution.

Fig. 13
Fig. 13

Normalized total-field magnitude versus ϕ2 for plane-wave incidence upon a perfectly conducting wedge (solid curve) and upon the same wedge coated with a thin dielectric layer on both faces (dashed curve). The layer has the parameters r = 2.2 and μr = 1.0 and has a thickness of 0.02λ0. ϕ0 = 360° − α = 90°, ϕ1 = 30°, r20 = 10.0/(2π): (a) TMz solution, (b) TEz solution.

Fig. 14
Fig. 14

Normalized total-field magnitude versus ϕ2 for plane-wave incidence upon a perfectly conducting wedge (solid curve) and upon the same wedge coated with a thin magnetic layer on both faces (dashed curve). The layer has the parameters r = 1.0 and μr = 4.0 and has a thickness of 0.02λ0. ϕ0 = 360° − α = 90°, ϕ1 = 30°, r20 = 10.0/(2π): (a) TMz solution, (b) TEz solution.

Fig. 15
Fig. 15

Normalized diffracted-field magnitude versus ϕ2 for plane-wave incidence upon an impedance wedge, with ϕ1 = ϕ2 (monostatic radar cross section). ϕ0 = 360° − α = 90°, Zs0 = Z = η0/4. The results of the virtual-ray method (curve) are compared with those obtained from a code based on the Maliuzhinets solution (dots): (a) TMz solution, (b) TEz solution.

Fig. 16
Fig. 16

Geometry for the ray corresponding to the stationary phase point ν = ν0 = krst.

Equations (101)

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G ( r , r ) = H 0 ( 2 ) ( k r 12 ) ,
H 0 ( 2 ) ( k r 12 ) = n = H n ( 2 ) ( k r 1 ) J n ( k r 2 ) exp [ j n ( ϕ 2 ϕ 1 ) ] .
J n ( x ) = 1 / 2 [ H n ( 1 ) ( x ) + H n ( 2 ) ( x ) ]
G ( r , r ) = G 1 ( r , r ) + G 2 ( r , r ) ,
G 1 ( r , r ) = 1 2 n = H n ( 2 ) ( k r 1 ) H n ( 2 ) ( k r 2 ) exp [ j n ( ϕ 2 ϕ 1 ) ] ,
G 2 ( r , r ) = 1 2 n = H n ( 2 ) ( k r 1 ) H n ( 1 ) ( k r 2 ) exp [ j n ( ϕ 2 ϕ 1 ) ] .
n = f ( n ) = n = f ( 2 π n ) ,
f ( α ) = f ( x ) exp ( j α x ) d x .
f ( n ) = g ( n ) exp ( jnx ) ,
f ( α ) = [ g ( ν ) exp ( j ν x ) ] exp ( j α ν ) d v = g ( α + x ) .
n = g ( n ) exp ( jnx ) = n = g ( 2 π n + x ) ,
g ( n ) = 1 / 2 H n ( 2 ) ( k r 1 ) H n ( 2 ) ( k r 2 ) ,
x = ϕ 2 ϕ 1 .
G 1 ( r , r ) = n = 1 2 H ν ( 2 ) ( k r 1 ) H ν ( 2 ) ( k r 2 ) × exp [ j ν ( ϕ 2 ϕ 1 + 2 π n ) ] d ν .
G 2 ( r , r ) = n = 1 2 H ν ( 2 ) ( k r 1 ) H ν ( 1 ) ( k r 2 ) × exp [ j ν ( ϕ 2 ϕ 1 + 2 π n ) ] d ν .
H ν ( 2 ) ( k r ) ( 2 π k ) 1 / 2 { 1 [ r 2 ( ν k ) 2 ] 1 / 4 } × exp ( j { k [ r 2 ( ν k ) 2 ] 1 / 2 ν cos 1 ( ν k r ) π 4 } ) .
G 1 ( r , r ) = n = exp ( j { [ ( k r 1 ) 2 ν 2 ] 1 / 2 + [ ( k r 2 ) 2 ν 2 ] 1 / 2 ν cos 1 ( ν k r 1 ) ν cos 1 ( ν k r 2 ) π 2 + ν ( ϕ 2 + ϕ 2 2 π n ) } ) π [ ( k r 1 ) 2 ν 2 ] 1 / 4 [ ( k r 2 ) 2 ν 2 ] 1 / 4 d ν .
cos 1 ( ν k r 1 ) = π 2 sin 1 ( ν k r 1 )
ψ = ϕ 1 ϕ 2 π ,
S ( ν , ψ ) = [ ( k r 1 ) 2 ν 2 ] 1 / 2 + [ ( k r 2 ) 2 ν 2 ] 1 / 2 + ν sin 1 ( ν k r 1 ) + ν sin 1 ( ν k r 2 ) + ν ψ ,
G 1 ( r , r ) = n = j π exp [ j S ( ν , ψ + 2 π n ) ] π [ ( k r 1 ) 2 ν 2 ] 1 / 4 [ ( k r 2 ) 2 ν 2 ] 1 / 4 d ν .
G 2 ( r , r ) = n = j π exp [ j S ¯ ( ν , ψ + 2 π n ) ] π [ ( k r 1 ) 2 ν 2 ] 1 / 4 [ ( k r 2 ) 2 ν 2 ] 1 / 4 d ν ,
S ¯ ( ν , ψ ) = [ ( k r 1 ) 2 ν 2 ] 1 / 2 [ ( k r 2 ) 2 ν 2 ] 1 / 2 + ν sin 1 ( ν k r 1 ) ν sin 1 ( ν k r 2 ) + ν ψ + ν π .
g ( ψ ) = Γ ( ν ) exp [ j S ( ν , ψ ) ] d ν [ ( k r 1 ) 2 ν 2 ] 1 / 4 [ ( k r 2 ) 2 ν 2 ] 1 / 4 ,
g ¯ ( ψ ) = Γ ( ν ) exp [ j S ¯ ( ν , ψ ) ] d ν [ ( k r 1 ) 2 ν 2 ] 1 / 4 [ ( k r 2 ) 2 ν 2 ] 1 / 4 .
G 1 ( r , r ) = m = g ( ψ + 2 π m ) ,
G 2 ( r , r ) = m = g ¯ ( ψ + 2 π m ) .
G ( r , r ) = n = H ν ( 2 ) ( k r 1 ) J ν ( k r 2 ) × exp [ j ν ( ϕ 2 ϕ 1 + 2 π n ) ] d ν .
J ν ( λ ) exp [ ν cosh 1 ( ν / λ ) + ν 2 λ 2 ] 2 π ( ν 2 λ 2 ) 1 / 4 ,
r 12 = [ r 1 2 + r 2 2 2 r 1 r 2 cos ( ϕ 1 ϕ 2 ) ] 1 / 2 .
| ν | = k r ,
S ( ν , r 1 , ϕ 1 , r 2 , ϕ 2 ) = k ( l 1 + l c + l 2 ) .
l 1 = [ r 1 2 ( ν k ) 2 ] 1 / 2 ,
l 2 = [ r 2 2 ( ν k ) 2 ] 1 / 2 .
Δ ϕ = ϕ a ϕ b ( because ϕ 1 > ϕ 2 ) = ϕ 1 ϕ 2 β 1 β 2 = ϕ 1 ϕ 2 ( π / 2 α 1 ) ( π / 2 α 2 ) = π + ( ϕ 1 ϕ 2 ) + α 1 + α 2 = ψ + α 1 + α 2 = ψ + sin 1 ( ν k r 1 ) + sin 1 ( ν k r 2 ) ,
S ( ν , r 1 , ϕ 1 , r 2 , ϕ 2 ) = k l 1 + k l 2 + ν Δ ϕ .
S ( ν , ψ ) = [ ( k r 1 ) 2 ν 2 ] 1 / 2 + [ ( k r 2 ) 2 ν 2 ] 1 / 2 + ν [ ψ + sin 1 ( ν k r 1 ) + sin 1 ( ν k r 2 ) ] .
Δ ϕ = π γ 1 γ 2 = π [ β 1 ( π ϕ 1 ) ] [ β 2 ϕ 2 ] = π [ ( π / 2 α 1 ) ( π ϕ 1 ) ] [ ( π / 2 α 2 ) ϕ 2 ] = π + ( ϕ 2 ϕ 1 ) + sin 1 | ν k r 1 | + sin 1 | ν k r 2 | .
S ( ν , ψ ) = k l 1 + k l 2 ν Δ ϕ ,
g ( ψ ) = m = Γ ( ν ) exp [ j S ( ν , ψ + 2 π m ) ] d ν [ ( k r 1 ) 2 ν 2 ] 1 / 4 [ ( k r 2 ) 2 ν 2 ] 1 / 4 .
S ν = 0 ,
ν = ν 0 = k r 1 r 2 sin ( ψ ) r 12 .
S ( ν 0 , ψ ) = k r 12 .
g ( ψ ) Γ ( ν 0 ) exp ( j π / 4 ) exp ( j k r 12 ) ( 2 π k r 12 ) 1 / 2 .
E z = I 0 4 ω μ 0 H 0 ( 2 ) ( k r 12 ) ,
E z I 0 4 ω μ 0 ( 2 π k r 12 ) 1 / 2 exp ( j π / 4 ) exp ( j k r 12 ) .
Γ 0 = j ω μ 0 I 0 4 π .
S ( ν , ψ ) k ( r 1 + r 2 ) + ψ ν + ( r 1 + r 2 2 k r 1 r 2 ) ν 2 .
g ( ψ ) Γ 0 k r 1 r 2 exp [ j k ( r 1 + r 2 ) ] × exp { j [ ψ ν + ( r 1 + r 2 2 k r 1 r 2 ) ν 2 ] } d ν .
g 0 ( ψ ) = 0 exp [ j S ( ν , ψ ) ] d ψ [ ( k r 1 ) 2 ν 2 ] 1 / 4 [ ( k r 2 ) 2 ν 2 ] 1 / 4 .
G 1 = m = ( R 0 R α ) | m | g 0 ( ϕ 1 ϕ 2 π + 2 m α ) .
G 2 = R 0 m = ( R 0 R α ) | m | g 0 ( ϕ 1 + ϕ 2 π + 2 m α ) .
G 3 = R α m = ( R 0 R α ) | m | g 0 ( 2 α ϕ 1 ϕ 2 π + 2 m α ) .
G 4 = R 0 R α m = ( R 0 R α ) | m | g 0 ( 2 α ϕ 1 + ϕ 2 π + 2 m α ) .
g 0 ( ψ ) = Γ 0 [ 2 π j k ( r 1 + r 2 ) ] 1 / 2 exp [ j k ( r 1 + r 2 ) ] V ( s ) ,
V ( s ) = exp ( j s 2 / 2 ) 2 j π s sgn ( s ) exp ( j t 2 / 2 ) d t ,
s = ψ ( k r 1 r 2 r 1 + r 2 ) 1 / 2 .
g 0 ( ψ ) = ( 0 ) Γ ( ν ) exp [ j S ( ν , ψ ) ] d ν [ ( k r 1 ) 2 ν 2 ] 1 / 4 [ ( k r 2 ) 2 ν 2 ] 1 / 4 .
g 0 ( ψ ) = Γ 0 { ( 2 j π k R 12 ) 1 / 2 exp ( j k R 12 ) + [ 2 j π k ( r 1 + r 2 ) ] 1 / 2 × exp [ j k ( r 1 + r 2 ) ] V ( s ) } .
0 f ( ν ) exp [ j Ω g ( ν ) ] d ν j Ω f ( 0 ) g ( 0 ) exp [ j Ω g ( 0 ) ]
g 0 ( ψ ) ( 1 ψ ) j Γ 0 k r 1 r 2 exp [ j k ( r 1 + r 2 ) ] .
G = g 0 ( ψ 1 ) + R 0 g 0 ( ψ 2 ) + R α g 0 ( ψ 3 ) + R 0 R α g 0 ( ψ 4 ) + j Γ 0 k r 1 r 2 exp [ j k ( r 1 + r 2 ) ] [ S ( ψ 1 , p ) + R 0 S ( ψ 2 , p ) + R α S ( ψ 3 , p ) + R 0 R α S ( ψ 4 , p ) ] ,
S ( ψ , p ) = m = 1 p m ( 1 ψ + 2 m α + 1 ψ 2 m α ) .
S ( ψ , p ) = 1 2 α S ¯ ( ψ 2 α , p ) ,
S ¯ ( c , p ) = m = 1 p m ( 1 c + m + 1 c m ) .
S ¯ ( c , p ) = 2 c m = 1 S m ,
S m = p m ( m + c ) ( m c ) .
S ¯ ( c , p ) = 2 c m = 1 ( S m S m a 1 S m a 2 S m a 3 ) 2 c p S 1 2 c p 2 S 2 2 c p 3 ( c 2 + 2 ) S 3 ,
S m a 1 = p m ( m ) ( m + 1 ) ,
S m a 2 = p m ( m ) ( m + 1 ) ( m + 2 ) ,
S m a 3 = ( c 2 + 2 ) p m ( m ) ( m + 1 ) ( m + 2 ) ( m + 3 ) ,
S 1 = S 1 ( p ) = m = 1 p m + 1 m ( m + 1 ) ,
S 2 = S 2 ( p ) = m = 1 p m + 2 m ( m + 1 ) ( m + 2 ) ,
S 3 = S 3 ( p ) = m = 1 p m + 3 m ( m + 1 ) ( m + 2 ) ( m + 3 ) .
S 3 ( p ) = S 2 ( p ) = S 1 ( p ) = m = 1 p m 1 = 1 1 p .
S 1 ( p ) = p + ( 1 p ) ln ( 1 p ) ,
S 2 ( p ) = 1 4 + p 2 2 + 1 4 ( 1 p ) 2 1 2 ( 1 p ) 2 ln ( 1 p ) ,
S 3 = 5 36 + p 4 + p 3 6 5 36 ( 1 p ) 3 + 1 6 ( 1 p ) 3 ln ( 1 p ) ,
R 0 ( θ ) = ± Z s 0 Z 0 Z s 0 + Z 0 ,
Z 0 TM = η 0 sec ( θ ) ,
Z 0 TE = η 0 cos ( θ ) ,
θ 0 max = max ( 0 , π / 2 ϕ 2 ) .
θ α max = max ( 0 , π / 2 + ϕ 1 α ) .
G = g 0 ( ψ 1 ) + R 0 ( θ 0 max ) g 0 ( ψ 2 ) + R α ( θ α max ) g 0 ( ψ 3 ) + R 0 ( θ 0 max ) R α ( θ α max ) g 0 ( ψ 4 ) + j Γ 0 k r 1 r 2 exp [ j k ( r 1 + r 2 ) ] × [ S ( ψ 1 , p ) + R 0 ( θ 0 max ) S ( ψ 2 , p ) + R α ( θ α max ) S ( ψ 3 , p ) + R 0 ( θ 0 max ) R α ( θ α max ) S ( ψ 4 , p ) ] .
tan ( θ 0 s ) = r 2 cos ( ϕ 2 ) r 1 cos ( ϕ 1 ) r 1 sin ( ϕ 1 ) r 2 sin ( ϕ 2 ) ,
tan ( θ α s ) = r 2 cos ( ϕ 2 α + π ) r 1 cos ( ϕ 1 α + π ) r 1 sin ( ϕ 1 α + π ) + r 2 sin ( ϕ 2 α + π ) ,
S ν = 0 .
S ν = ψ + sin 1 ( ν k r 1 ) + sin 1 ( ν k r 2 ) = ψ + α 1 + α 2 ,
α 1 + α 2 = ψ = π ( ϕ 1 ϕ 2 ) ,
ϕ 1 ϕ 2 = β 1 + β 2 .
r 12 sin ( ϕ 1 ϕ 2 ) = r 2 sin ( α 1 ) = r 1 r 2 r st ,
r st = r 1 r 2 sin ( ϕ 1 ϕ 2 ) r 12 ,
ν 0 = k r st = k r 1 r 2 sin ( ψ ) r 12 ,
G = [ 2 j π k ( r 1 + r 2 ) ] 1 / 2 W ( k ρ , ϕ 1 , ϕ 2 ) exp [ j k ( r 1 + r 2 ) ] ,
ρ = r 1 r 2 r 1 + r 2 ,
W ( k ρ , ϕ 1 , ϕ 2 ) = w ( k ρ , ϕ 1 ϕ 2 ) ± w ( k ρ , ϕ 1 + ϕ 2 ) .
w ( k ρ , x ) = V [ ( k ρ 2 ) 1 / 2 ( x π ) ] + [ S 1 ( x ) 1 x π ] exp ( j π / 4 ) ( 2 π k ρ ) 1 / 2 ,
S 1 ( x ) = 1 x π + m = 1 ( 1 x π + 2 m α + 1 x π 2 m α ) 1 x + π m = 1 ( 1 x + π + 2 m α + 1 x + π 2 m α ) .
S 1 ( x ) = π 2 α { cot [ π ( x π ) 2 α ] cot [ π ( x + π ) 2 α ] }
S 1 ( x ) = π α sin ( π 2 α ) cos ( π 2 α ) cos ( π x α ) .
w ( k ρ , x ) = V [ ( k ρ 2 ) 1 / 2 ( x π ) ] + [ π α sin ( π 2 α ) cos ( π 2 α ) cos ( π x α ) 1 x π ] exp ( j π / 4 ) ( 2 π k ρ ) 1 / 2 .

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