Abstract

A planar structure consisting of layers of different media is considered. The material of each layer is linear and homogeneous but anisotropic and, in general, exhibits biaxial, gyroelectric, and gyromagnetic properties. Thus most materials used in practice fall into the general category considered here. The structure has free space on one side and an isotropic medium (e.g., a dielectric or a metal) on the other side. A plane electromagnetic wave with arbitrary direction and polarization is incident upon the structure. The propagation constants and the components of the electric and magnetic fields are determined in each layer. A chain-matrix approach is implemented to obtain reflection and transmission coefficients for the structure.

© 1985 Optical Society of America

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References

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  1. M. H. Engineer, B. R. Nag, “Propagation of electromagnetic waves in rectangular guides filled with a semiconductor in the presence of a transverse magnetic field,” IEEE Trans. Microwave Theory Tech. MTT-13, 641–646 (1965).
    [CrossRef]
  2. A. N. Tikhonov, “The propagation of a continuous electromagnetic wave in a laminarly anisotropic medium,” Sov. Phys. Dokl. 4, 566–570 (1959).
  3. D. N. Chetaev, “On the field of a low-frequency electric dipole situated on the surface of a uniform anisotropic conducting halfspace,” Sov. Phys. Tech. Phys. 7, 991–995 (1963).
  4. J. R. Wait, “Electromagnetic fields of a dipole over an anisotropic half-space,” Can. J. Phys. 44, 2387–2401 (1966).
    [CrossRef]
  5. J. G. Negi, P. D. Saraf, “Effects of anisotropy on quasi-static fields of a vertical electric dipole buried in a layered anisotropic earth,” Radio Sci. 8, 155–165 (1973).
    [CrossRef]
  6. V. G. Daniele, R. S. Zich, “Radiation by arbitrary sources in anisotropic stratified media,” Radio Sci. 8, 63–70 (1973).
    [CrossRef]
  7. A. D. Bresler, N. Marcuvitz, “Operator methods in electromagnetic field theory,” (Polytechnic Institute of Brooklyn, New York, 1956), p. 88.
  8. A. D. Bresler, N. Marcuvitz, “Operator methods in electromagnetic field theory,” (Polytechnic Institute of Brooklyn, New York, 1957), p. 118.
  9. J. A. Kong, “Electromagnetic fields due to dipole antennas over stratified anisotropic media,” Geophysics 37, 985–996 (1972).
    [CrossRef]
  10. C. M. Tang, “Electromagnetic fields due to dipole antennas embedded in stratified anisotropic media,”IEEE Trans. Antennas Propag. AP-27, 665–670 (1979).
    [CrossRef]
  11. S. M. Ali, S. F. Mahmoud, “Electromagnetic fields of buried sources in stratified anisotropic media,”IEEE Trans. Antennas Propag. AP-27, 671–678 (1979).
    [CrossRef]
  12. D. K. Paul, R. K. Shevgaonkar, “Multimode propagation in anisotropic optical waveguides,” Radio Sci. 16, 525–533 (1981).
    [CrossRef]
  13. J. A. Kong, Theory of Electromagnetic Waves (Wiley, New York, 1975).
  14. B. Lax, K. J. Button, Microwave Ferrites and Ferrimagnetics (McGraw-Hill, New York, 1962).
  15. P. Hlawiczka, Gyrotropic Waveguides (Academic, New York, 1981).
  16. S. Visnovsky, R. Krishnan, “Complex Faraday effect in multilayer structures,”J. Opt. Soc. Am. 71, 315–320 (1981).
    [CrossRef]
  17. T. A. Martynova, “Propagation of electromagnetic waves in waveguide with local layered gyrotropic filling,” Radio Eng. Electron. Phys. (USSR) 18, 1320–1323 (1973).
  18. A. I. Semenenko, F. S. Mironov, “Matrix method in the optics of anisotropic layered media,” Opt. Spectrosc. 41, 262–265 (1976).
  19. N. J. Damaskos, A. L. Maffett, P. L. E. Uslenghi, “Dispersion relation for general anisotropic media,”IEEE Trans. Antennas Propag. AP-30, 991–993 (1982).
    [CrossRef]
  20. R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960).
  21. C. H. Krueger, “A computer program for determining the reflection and transmission properties of multilayer plane impedance boundaries,” (Wright-Patterson Air Force Base, Ohio, 1967), p. 27.
  22. N. G. Alexopoulos, P. L. E. Uslenghi, “Reflection and transmission for materials with arbitrarily graded parameters,”J. Opt. Soc. Am. 71, 1508–1512 (1981).
    [CrossRef]

1982

N. J. Damaskos, A. L. Maffett, P. L. E. Uslenghi, “Dispersion relation for general anisotropic media,”IEEE Trans. Antennas Propag. AP-30, 991–993 (1982).
[CrossRef]

1981

1979

C. M. Tang, “Electromagnetic fields due to dipole antennas embedded in stratified anisotropic media,”IEEE Trans. Antennas Propag. AP-27, 665–670 (1979).
[CrossRef]

S. M. Ali, S. F. Mahmoud, “Electromagnetic fields of buried sources in stratified anisotropic media,”IEEE Trans. Antennas Propag. AP-27, 671–678 (1979).
[CrossRef]

1976

A. I. Semenenko, F. S. Mironov, “Matrix method in the optics of anisotropic layered media,” Opt. Spectrosc. 41, 262–265 (1976).

1973

T. A. Martynova, “Propagation of electromagnetic waves in waveguide with local layered gyrotropic filling,” Radio Eng. Electron. Phys. (USSR) 18, 1320–1323 (1973).

J. G. Negi, P. D. Saraf, “Effects of anisotropy on quasi-static fields of a vertical electric dipole buried in a layered anisotropic earth,” Radio Sci. 8, 155–165 (1973).
[CrossRef]

V. G. Daniele, R. S. Zich, “Radiation by arbitrary sources in anisotropic stratified media,” Radio Sci. 8, 63–70 (1973).
[CrossRef]

1972

J. A. Kong, “Electromagnetic fields due to dipole antennas over stratified anisotropic media,” Geophysics 37, 985–996 (1972).
[CrossRef]

1966

J. R. Wait, “Electromagnetic fields of a dipole over an anisotropic half-space,” Can. J. Phys. 44, 2387–2401 (1966).
[CrossRef]

1965

M. H. Engineer, B. R. Nag, “Propagation of electromagnetic waves in rectangular guides filled with a semiconductor in the presence of a transverse magnetic field,” IEEE Trans. Microwave Theory Tech. MTT-13, 641–646 (1965).
[CrossRef]

1963

D. N. Chetaev, “On the field of a low-frequency electric dipole situated on the surface of a uniform anisotropic conducting halfspace,” Sov. Phys. Tech. Phys. 7, 991–995 (1963).

1959

A. N. Tikhonov, “The propagation of a continuous electromagnetic wave in a laminarly anisotropic medium,” Sov. Phys. Dokl. 4, 566–570 (1959).

Alexopoulos, N. G.

Ali, S. M.

S. M. Ali, S. F. Mahmoud, “Electromagnetic fields of buried sources in stratified anisotropic media,”IEEE Trans. Antennas Propag. AP-27, 671–678 (1979).
[CrossRef]

Bresler, A. D.

A. D. Bresler, N. Marcuvitz, “Operator methods in electromagnetic field theory,” (Polytechnic Institute of Brooklyn, New York, 1956), p. 88.

A. D. Bresler, N. Marcuvitz, “Operator methods in electromagnetic field theory,” (Polytechnic Institute of Brooklyn, New York, 1957), p. 118.

Button, K. J.

B. Lax, K. J. Button, Microwave Ferrites and Ferrimagnetics (McGraw-Hill, New York, 1962).

Chetaev, D. N.

D. N. Chetaev, “On the field of a low-frequency electric dipole situated on the surface of a uniform anisotropic conducting halfspace,” Sov. Phys. Tech. Phys. 7, 991–995 (1963).

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960).

Damaskos, N. J.

N. J. Damaskos, A. L. Maffett, P. L. E. Uslenghi, “Dispersion relation for general anisotropic media,”IEEE Trans. Antennas Propag. AP-30, 991–993 (1982).
[CrossRef]

Daniele, V. G.

V. G. Daniele, R. S. Zich, “Radiation by arbitrary sources in anisotropic stratified media,” Radio Sci. 8, 63–70 (1973).
[CrossRef]

Engineer, M. H.

M. H. Engineer, B. R. Nag, “Propagation of electromagnetic waves in rectangular guides filled with a semiconductor in the presence of a transverse magnetic field,” IEEE Trans. Microwave Theory Tech. MTT-13, 641–646 (1965).
[CrossRef]

Hlawiczka, P.

P. Hlawiczka, Gyrotropic Waveguides (Academic, New York, 1981).

Kong, J. A.

J. A. Kong, “Electromagnetic fields due to dipole antennas over stratified anisotropic media,” Geophysics 37, 985–996 (1972).
[CrossRef]

J. A. Kong, Theory of Electromagnetic Waves (Wiley, New York, 1975).

Krishnan, R.

Krueger, C. H.

C. H. Krueger, “A computer program for determining the reflection and transmission properties of multilayer plane impedance boundaries,” (Wright-Patterson Air Force Base, Ohio, 1967), p. 27.

Lax, B.

B. Lax, K. J. Button, Microwave Ferrites and Ferrimagnetics (McGraw-Hill, New York, 1962).

Maffett, A. L.

N. J. Damaskos, A. L. Maffett, P. L. E. Uslenghi, “Dispersion relation for general anisotropic media,”IEEE Trans. Antennas Propag. AP-30, 991–993 (1982).
[CrossRef]

Mahmoud, S. F.

S. M. Ali, S. F. Mahmoud, “Electromagnetic fields of buried sources in stratified anisotropic media,”IEEE Trans. Antennas Propag. AP-27, 671–678 (1979).
[CrossRef]

Marcuvitz, N.

A. D. Bresler, N. Marcuvitz, “Operator methods in electromagnetic field theory,” (Polytechnic Institute of Brooklyn, New York, 1957), p. 118.

A. D. Bresler, N. Marcuvitz, “Operator methods in electromagnetic field theory,” (Polytechnic Institute of Brooklyn, New York, 1956), p. 88.

Martynova, T. A.

T. A. Martynova, “Propagation of electromagnetic waves in waveguide with local layered gyrotropic filling,” Radio Eng. Electron. Phys. (USSR) 18, 1320–1323 (1973).

Mironov, F. S.

A. I. Semenenko, F. S. Mironov, “Matrix method in the optics of anisotropic layered media,” Opt. Spectrosc. 41, 262–265 (1976).

Nag, B. R.

M. H. Engineer, B. R. Nag, “Propagation of electromagnetic waves in rectangular guides filled with a semiconductor in the presence of a transverse magnetic field,” IEEE Trans. Microwave Theory Tech. MTT-13, 641–646 (1965).
[CrossRef]

Negi, J. G.

J. G. Negi, P. D. Saraf, “Effects of anisotropy on quasi-static fields of a vertical electric dipole buried in a layered anisotropic earth,” Radio Sci. 8, 155–165 (1973).
[CrossRef]

Paul, D. K.

D. K. Paul, R. K. Shevgaonkar, “Multimode propagation in anisotropic optical waveguides,” Radio Sci. 16, 525–533 (1981).
[CrossRef]

Saraf, P. D.

J. G. Negi, P. D. Saraf, “Effects of anisotropy on quasi-static fields of a vertical electric dipole buried in a layered anisotropic earth,” Radio Sci. 8, 155–165 (1973).
[CrossRef]

Semenenko, A. I.

A. I. Semenenko, F. S. Mironov, “Matrix method in the optics of anisotropic layered media,” Opt. Spectrosc. 41, 262–265 (1976).

Shevgaonkar, R. K.

D. K. Paul, R. K. Shevgaonkar, “Multimode propagation in anisotropic optical waveguides,” Radio Sci. 16, 525–533 (1981).
[CrossRef]

Tang, C. M.

C. M. Tang, “Electromagnetic fields due to dipole antennas embedded in stratified anisotropic media,”IEEE Trans. Antennas Propag. AP-27, 665–670 (1979).
[CrossRef]

Tikhonov, A. N.

A. N. Tikhonov, “The propagation of a continuous electromagnetic wave in a laminarly anisotropic medium,” Sov. Phys. Dokl. 4, 566–570 (1959).

Uslenghi, P. L. E.

N. J. Damaskos, A. L. Maffett, P. L. E. Uslenghi, “Dispersion relation for general anisotropic media,”IEEE Trans. Antennas Propag. AP-30, 991–993 (1982).
[CrossRef]

N. G. Alexopoulos, P. L. E. Uslenghi, “Reflection and transmission for materials with arbitrarily graded parameters,”J. Opt. Soc. Am. 71, 1508–1512 (1981).
[CrossRef]

Visnovsky, S.

Wait, J. R.

J. R. Wait, “Electromagnetic fields of a dipole over an anisotropic half-space,” Can. J. Phys. 44, 2387–2401 (1966).
[CrossRef]

Zich, R. S.

V. G. Daniele, R. S. Zich, “Radiation by arbitrary sources in anisotropic stratified media,” Radio Sci. 8, 63–70 (1973).
[CrossRef]

Can. J. Phys.

J. R. Wait, “Electromagnetic fields of a dipole over an anisotropic half-space,” Can. J. Phys. 44, 2387–2401 (1966).
[CrossRef]

Geophysics

J. A. Kong, “Electromagnetic fields due to dipole antennas over stratified anisotropic media,” Geophysics 37, 985–996 (1972).
[CrossRef]

IEEE Trans. Antennas Propag.

C. M. Tang, “Electromagnetic fields due to dipole antennas embedded in stratified anisotropic media,”IEEE Trans. Antennas Propag. AP-27, 665–670 (1979).
[CrossRef]

S. M. Ali, S. F. Mahmoud, “Electromagnetic fields of buried sources in stratified anisotropic media,”IEEE Trans. Antennas Propag. AP-27, 671–678 (1979).
[CrossRef]

N. J. Damaskos, A. L. Maffett, P. L. E. Uslenghi, “Dispersion relation for general anisotropic media,”IEEE Trans. Antennas Propag. AP-30, 991–993 (1982).
[CrossRef]

IEEE Trans. Microwave Theory Tech.

M. H. Engineer, B. R. Nag, “Propagation of electromagnetic waves in rectangular guides filled with a semiconductor in the presence of a transverse magnetic field,” IEEE Trans. Microwave Theory Tech. MTT-13, 641–646 (1965).
[CrossRef]

J. Opt. Soc. Am.

Opt. Spectrosc.

A. I. Semenenko, F. S. Mironov, “Matrix method in the optics of anisotropic layered media,” Opt. Spectrosc. 41, 262–265 (1976).

Radio Eng. Electron. Phys. (USSR)

T. A. Martynova, “Propagation of electromagnetic waves in waveguide with local layered gyrotropic filling,” Radio Eng. Electron. Phys. (USSR) 18, 1320–1323 (1973).

Radio Sci.

D. K. Paul, R. K. Shevgaonkar, “Multimode propagation in anisotropic optical waveguides,” Radio Sci. 16, 525–533 (1981).
[CrossRef]

J. G. Negi, P. D. Saraf, “Effects of anisotropy on quasi-static fields of a vertical electric dipole buried in a layered anisotropic earth,” Radio Sci. 8, 155–165 (1973).
[CrossRef]

V. G. Daniele, R. S. Zich, “Radiation by arbitrary sources in anisotropic stratified media,” Radio Sci. 8, 63–70 (1973).
[CrossRef]

Sov. Phys. Dokl.

A. N. Tikhonov, “The propagation of a continuous electromagnetic wave in a laminarly anisotropic medium,” Sov. Phys. Dokl. 4, 566–570 (1959).

Sov. Phys. Tech. Phys.

D. N. Chetaev, “On the field of a low-frequency electric dipole situated on the surface of a uniform anisotropic conducting halfspace,” Sov. Phys. Tech. Phys. 7, 991–995 (1963).

Other

R. E. Collin, Field Theory of Guided Waves (McGraw-Hill, New York, 1960).

C. H. Krueger, “A computer program for determining the reflection and transmission properties of multilayer plane impedance boundaries,” (Wright-Patterson Air Force Base, Ohio, 1967), p. 27.

A. D. Bresler, N. Marcuvitz, “Operator methods in electromagnetic field theory,” (Polytechnic Institute of Brooklyn, New York, 1956), p. 88.

A. D. Bresler, N. Marcuvitz, “Operator methods in electromagnetic field theory,” (Polytechnic Institute of Brooklyn, New York, 1957), p. 118.

J. A. Kong, Theory of Electromagnetic Waves (Wiley, New York, 1975).

B. Lax, K. J. Button, Microwave Ferrites and Ferrimagnetics (McGraw-Hill, New York, 1962).

P. Hlawiczka, Gyrotropic Waveguides (Academic, New York, 1981).

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Figures (4)

Fig. 1
Fig. 1

Cross-sectional view of layered structure.

Fig. 2
Fig. 2

Top view of layered structure.

Fig. 3
Fig. 3

Reflection coefficient R for the metal-backed two-layer structure of Section 8, at β = 45°.

Fig. 4
Fig. 4

Reflection (R) and transmission (T) coefficients for the two-layer structure of Section 8 backed by free space, at β = 45° and for (a) α = 0°, (b) α = 45°, and (c) α = 90°.

Equations (76)

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× H = j ω E ,             × E = - j ω μ H ,
E = ( E x E y E z ) ,             H = ( H x H y H z ) ,
= 0 ( 1 4 0 5 2 0 0 0 3 ) ,             μ = μ 0 ( μ 1 μ 4 0 μ 5 μ 2 0 0 0 μ 3 ) ,
k ^ 0 = x ^ sin γ cos β + y ^ sin γ sin β - z ^ cos γ ,
E 0 = [ x ^ ( sin α cos β cos γ - cos α sin β ) + y ^ ( sin α sin β cos γ + cos α cos β ) + z ^ sin α sin γ ] exp ( - j k 0 k ^ 0 · r ) ,
H 0 = Y 0 k ^ 0 × E 0 = Y 0 [ x ^ ( cos α cos β cos γ + sin α sin β ) + y ^ ( cos α sin β cos γ - sin α cos β ) + z ^ cos α sin γ ] exp ( - j k 0 k ^ 0 · r ) ,
r = x x ^ + y y ^ + z z ^
Z 0 = Y 0 - 1 = ( μ 0 / 0 ) 1 / 2 ,             k 0 = ω ( 0 μ 0 ) 1 / 2 = 2 π / λ 0
k l = k l · x ^ 0 x ^ 0 + k l z z ^ ,
k l = k x x ^ + k y y ^ + k l z z ^ ,
k x = k 0 x = k 0 sin γ cos β ,
k y = k 0 y = k 0 sin γ sin β
k 0 z = k 0 cos γ ,             for { incident reflected } field in z 0.
exp ( - j k l · r ) = exp [ - j ( k x x + k y y ) ] exp ( - j k l z z ) ,
k l = k 0 κ l ,
κ z 4 + A κ z 2 + B = 0 ,
A = ( h 3 + h μ μ 3 ) sin 2 γ + 4 μ 4 + 5 μ 5 - ( 1 μ 2 + 2 μ 1 ) ,
B = sin 4 γ 3 μ 3 { 1 μ 1 cos 4 β + 2 μ 2 sin 4 β + [ 1 μ 2 + 2 μ 1 - ( 4 + 5 ) ( μ 4 + μ 5 ) ] sin 2 β cos 2 β + [ h ( μ 4 + μ 5 ) + h μ ( 4 + 5 ) ] sin β cos β } - [ h μ 3 ( μ 1 μ 2 - μ 4 μ 5 ) + h μ 3 ( 1 2 - 4 5 ) ] sin 2 γ + ( 1 2 - 4 5 ) ( μ 1 μ 2 - μ 4 μ 5 ) ,
h = 1 cos 2 β + 2 sin 2 β + ( 4 + 5 ) sin β cos β ,
h μ = μ 1 cos 2 β + μ 2 sin 2 β + ( μ 4 + μ 5 ) sin β cos β .
E l z = sin γ ( C l ( 1 ) - 1 { α l 1 exp [ - j k 0 κ l z ( 1 ) z ] + b l 1 exp [ + j k 0 κ l z ( 1 ) z ] } + a l 2 exp [ - j k 0 κ l z ( 2 ) z ] + b 12 exp [ + j k 0 κ l z ( 2 ) z ] ) ,
Z 0 H l z = sin γ ( a l 1 exp [ - j k 0 κ l z ( 1 ) z ] - b l 1 exp [ + j k 0 κ l z ( 1 ) z ] + C l ( 2 ) { a l 2 exp [ - j k 0 κ l z ( 2 ) z ] - b l 2 exp [ + j k 0 κ l z ( 2 ) z ] } ) ,
E l x y = i = 1 , 2 P l x y ( i ) { a l i exp [ - j k 0 κ l z ( i ) z ] - b l i exp [ + j k 0 κ l z ( i ) z ] } ,
Z 0 H l x y = i = 1 , 2 Q l x y ( i ) { a l i exp [ - j k 0 κ l z ( i ) z ] + b l i exp [ + j k 0 κ l z ( i ) z ] } ,
P l x ( i ) = - 1 h l [ l 3 cos β κ l z ( i ) Φ l ( i ) + μ l 3 ( l 2 sin β + l 4 cos β ) Ψ l ( i ) ] ,
P l y ( i ) = - 1 h l [ l 3 sin β κ l z ( i ) Φ l ( i ) - μ l 3 ( l 5 sin β + l 1 cos β ) Ψ l ( i ) ] ,
Q l x ( i ) = - 1 h l μ [ μ l 3 cos β κ l x ( i ) Ψ l ( i ) - l 3 ( μ l 2 sin β + μ l 4 cos β ) Φ l ( i ) ] ,
Q l y ( i ) = - 1 h l μ [ μ l 3 sin β κ l z ( i ) Ψ l ( i ) + l 3 ( μ l 5 sin β + μ l 1 cos β ) Φ l ( i ) ] ,
Φ l ( 1 ) = C l ( 1 ) - 1 ,             Φ l ( 2 ) = Ψ l ( 1 ) = 1 , Ψ l ( 2 ) = C l ( 2 ) ,
C l ( i ) = l 3 κ l z ( i ) ( l 1 μ l 4 - l 5 μ l 1 ) cos 2 β + ( l 4 μ l 2 - l 2 μ l 5 ) sin 2 β + K l sin β cos β h l h l μ sin 2 γ + μ l 3 κ l z ( i ) 2 h l + μ l 3 ( l 4 l 5 - l 1 l 2 ) h l μ ,
K l = l 1 μ l 2 -     l 2 μ l 1 + l 4 μ l 4 - l 5 μ l 5 .
C l ( 1 ) - 1 = 0 ,             C l ( 2 ) = 0.
D l = k 0 d l ,
A l i = a l i exp [ j κ l z ( i ) D l ] ,             B l i = b l i exp [ - j κ l z ( i ) D l ] ,
θ l + 1 ( i ) = κ l + 1 , z ( i ) ( D l + 1 - D l ) ;
α l ( A l 1 A l 2 B l 1 B l 2 ) = β l ( A l + 1 , 1 A l + 1 , 2 B l + 1 , 1 B l + 1 , 2 ) ,
α l = [ P l x ( 1 ) P l x ( 2 ) - P l x ( 1 ) - P l x ( 2 ) P l y ( 1 ) P l y ( 2 ) - P l y ( 1 ) - P l y ( 2 ) Q l x ( 1 ) Q l x ( 2 ) Q l x ( 1 ) Q l x ( 2 ) Q l y ( 1 ) Q l y ( 2 ) Q l y ( 1 ) Q l y ( 2 ) ] ,
β l = { P l + 1 , x ( 1 ) exp [ - j θ l + 1 ( 1 ) ] P l + 1 , x ( 2 ) exp [ - j θ l + 1 ( 2 ) ] - P l + 1 , x ( 1 ) exp [ j θ l + 1 ( 1 ) ] - P l + 1 , x ( 2 ) exp [ j θ l + 1 ( 2 ) ] P l + 1 , y ( 1 ) exp [ - j θ l + 1 ( 1 ) ] P l + 1 , y ( 2 ) exp [ - j θ l + 1 ( 2 ) ] - P l + 1 , y ( 1 ) exp [ j θ l + 1 ( 1 ) ] - P l + 1 , y ( 2 ) exp [ j θ l + 1 ( 2 ) ] Q l + 1 , x ( 1 ) exp [ - j θ l + 1 ( 1 ) ] Q l + 1 , x ( 2 ) exp [ - j θ l + 1 ( 2 ) ] Q l + 1 , x ( 1 ) exp [ j θ l + 1 ( 1 ) ] Q l + 1 , x ( 2 ) exp [ j θ l + 1 ( 2 ) ] Q l + 1 , y ( 1 ) exp [ - j θ l + 1 ( 1 ) ] Q l + 1 , y ( 2 ) exp [ - j θ l + 1 ( 2 ) ] Q l + 1 , y ( 1 ) exp [ j θ l + 1 ( 1 ) ] Q l + 1 , y ( 2 ) exp [ j θ l + 1 ( 2 ) ] } .
γ l = α l - 1 β l ,
( A 11 A 12 B 11 B 12 ) = Γ ( A n 1 A n 2 B n 1 B n 2 ) ,
Γ = { Γ h m } = l = 1 n - 1 γ l             ( h , m = 1 , 2 , 3 , 4 ) .
Γ = α 1 - 1 ( l = 2 n - 1 δ l ) β n - 1 ,
δ l = β l - 1 α l - 1 ,             2 l n - 1
E r = [ x ^ ( - E sin β - E cos β cos γ ) + y ^ ( E cos β - E sin β cos γ ) + z ^ E sin γ ] exp ( - j k 0 k ^ r · r ) ,
Z 0 H r = [ x ^ ( E sin β - E cos β cos γ ) + y ^ ( - E cos β - E sin β cos γ ) + z ^ E sin γ ] exp ( - j k 0 k ^ r · r ) ,
k ^ r = x ^ sin γ cos β + y ^ sin γ sin β + z ^ cos β
A n 1 = B n 1 ,             A n 2 = B n 2 .
E cos β cos γ + E sin β + A n 1 ( u 1 x + u 3 x ) + A n 2 ( u 2 x + u 4 x ) = sin α cos β cos γ - cos α sin β ,
E sin β cos γ - E cos β + A n 1 ( u 1 y + u 3 y ) + A n 2 ( u 2 y + u 4 y ) = sin α sin β cos γ + cos α cos β ,
- E sin β + E cos β cos γ + A n 1 ( v 1 x + v 3 x ) + A n 2 ( v 2 x + v 4 x ) = cos α cos β cos γ + sin α sin β ,
E cos β + E sin β cos γ + A n 1 ( v 1 y + v 3 y ) + A n 2 ( v 2 y + v 4 y ) = cos α sin β cos γ - sin α cos β ,
u i x y = P 1 x y ( 1 ) ξ 1 i - + P 1 x y ( 2 ) ξ 2 i - ,
v i x y = Q 1 x y ( 1 ) ξ 1 i + + Q 1 x y ( 2 ) ξ 2 i + ,
ξ 1 i = Γ 1 i exp [ - j κ 1 z ( 1 ) D 1 ] Γ 3 i exp [ j κ 1 z ( 1 ) D 1 ] ,
ξ 2 i = Γ 2 i exp [ - j κ 1 z ( 2 ) D 1 ] Γ 4 i exp [ j κ 1 z ( 2 ) D 1 ] ,
R = E 2 + E 2 .
Z t = ( μ t / t ) ,             N t = [ t μ t / ( 0 μ 0 ) ] .
k t = N t k 0 k ^ t ,
k ^ t = x ^ cos β sin γ t + y ^ sin β sin γ t - z ^ cos γ t ,
γ t = arcsin ( sin γ N t )
E t = Z t H t × k ^ t = [ x ^ ( - E t sin β + E t cos β cos γ t ) + y ^ ( E t cos β + E t sin β cos γ t ) + z ^ E t sin γ t ] exp ( j k 0 Δ t ) ,
Δ t = - x sin γ cos β - y sin γ sin β + N t ( z + d n ) cos γ t .
E cos β cos γ + E sin β + A n 1 u 1 x + A n 2 u 2 x + B n 1 u 3 x + B n 2 u 4 x = sin α cos β cos γ - cos α sin β ,
E sin β cos γ - E cos β + A n 1 u 1 y + A n 2 u 2 y + B n 1 u 3 y + B n 2 u 4 y = sin α sin β cos γ + cos α cos β ,
- E sin β + E cos β cos γ + A n 1 v 1 x + A n 2 v 2 x + B n 1 v 3 x + B n 2 v 4 x = cos α cos β cos γ + sin α sin β ,
E cos β + E sin β cos γ + A n 1 v 1 y + A n 2 v 2 y + B n 1 v 3 y + B n 2 v 4 y = cos α sin β cos γ - sin α cos β ,
E t cos β cos γ t - E t sin β = i = 1 , 2 P n x ( i ) ( A n i - B n i ) ,
E t sin β cos γ t + E t cos β = i = 1 , 2 P n y ( i ) ( A n i - B n i ) ,
E t sin β + E t cos β cos γ t = Z t Y 0 i = 1 , 2 Q n x ( i ) ( A n i + B n i ) ,
- E t cos β + E t sin β cos γ t = Z t Y 0 i = 1 , 2 Q n y ( i ) ( A n i + B n i ) .
T = E t 2 + E t 2 .
α - 1 = { α ˜ h m ; h , m = 1 , 2 , 3 , 4 } ,
α ˜ 11 = - α ˜ 31 = ½ P P y ( 2 ) , α ˜ 41 = - α ˜ 21 = ½ P P y ( 1 ) , α ˜ 32 = - α ˜ 12 = ½ P P x ( 2 ) , α ˜ 22 = - α ˜ 42 = ½ P P x ( 1 ) , α ˜ 13 = α ˜ 33 = ½ Q Q y ( 2 ) , α ˜ 23 = α ˜ 43 = - ½ Q Q y ( 1 ) , α ˜ 14 = α ˜ 34 = - ½ Q Q x ( 2 ) , α ˜ 24 = α ˜ 44 = ½ Q Q x ( 1 ) ,
P = [ P x ( 1 ) P y ( 2 ) - P x ( 2 ) P y ( 1 ) ] - 1 , Q = [ Q x ( 1 ) Q y ( 2 ) - Q x ( 2 ) Q y ( 1 ) ] - 1 .
δ = { δ h m ; h , m = 1 , 2 , 3 , 4 } ;
δ 11 = P [ P x ( 1 ) P y ( 2 ) cos θ ( 1 ) - P x ( 2 ) P y ( 1 ) cos θ ( 2 ) ] , δ 12 = - P P x ( 1 ) P x ( 2 ) [ cos θ ( 1 ) - cos θ ( 2 ) ] , δ 13 = - j Q [ P x ( 1 ) Q y ( 2 ) sin θ ( 1 ) - P x ( 2 ) Q y ( 1 ) sin θ ( 2 ) ] , δ 14 = j Q [ P x ( 1 ) Q x ( 2 ) sin θ ( 1 ) - P x ( 2 ) Q x ( 1 ) sin θ ( 2 ) ] , δ 21 = P P y ( 1 ) P y ( 2 ) [ cos θ ( 1 ) - cos θ ( 2 ) ] , δ 22 = - P [ P x ( 2 ) P y ( 1 ) cos θ ( 1 ) - P x ( 1 ) P y ( 2 ) cos θ ( 2 ) ] , δ 23 = - j Q [ P y ( 1 ) Q y ( 2 ) sin θ ( 1 ) - P y ( 2 ) Q y ( 1 ) sin θ ( 2 ) ] , δ 24 = j Q [ P y ( 1 ) Q x ( 2 ) sin θ ( 1 ) - P y ( 2 ) Q x ( 1 ) sin θ ( 2 ) ] , δ 31 = - j P [ P y ( 2 ) Q x ( 1 ) sin θ ( 1 ) - P y ( 1 ) Q x ( 2 ) sin θ ( 2 ) ] , δ 32 = j P [ P x ( 2 ) Q x ( 1 ) sin θ ( 1 ) - P x ( 1 ) Q x ( 2 ) sin θ ( 2 ) ] , δ 33 = Q [ Q x ( 1 ) Q y ( 2 ) cos θ ( 1 ) - Q x ( 2 ) Q y ( 1 ) cos θ ( 2 ) ] , δ 34 = - Q Q x ( 1 ) Q x ( 2 ) [ cos θ ( 1 ) - cos θ ( 2 ) ] , δ 41 = - j P [ P y ( 2 ) Q y ( 1 ) sin θ ( 1 ) - P y ( 1 ) Q y ( 2 ) sin θ ( 2 ) ] , δ 42 = j P [ P x ( 2 ) Q y ( 1 ) sin θ ( 1 ) - P x ( 1 ) Q y ( 2 ) sin θ ( 2 ) ] , δ 43 = Q Q y ( 1 ) Q y ( 2 ) [ cos θ ( 1 ) - cos θ ( 2 ) ] , δ 44 = - Q [ Q x ( 2 ) Q y ( 1 ) cos θ ( 1 ) - Q x ( 1 ) Q y ( 2 ) cos θ ( 2 ) ] .

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