Abstract

The dispersion relations and the fields for surface waves in interfaces formed by an isotropic metal and a uniaxial crystal are obtained for the cases in which the plane of incidence is a symmetry plane. These results are compared with those obtained for attenuated total reflection in Kretschmann’s configuration. An adequate coincidence, similar to that for the isotropic case, is obtained.

© 1996 Optical Society of America

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References

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  1. J. Zenneck, “Über die Fortpflanzung Ebener Elektromagnetischer Wellen längst einer Ebenen Leiterfläche und ihre Bezichung zur Drahtlosen Telegraphie,” Ann. Phys. (Leipzig) 23, 846–866 (1907).
  2. A. Sommerfeld, “Über die Ausbreitung der Wellen in der drahtlosen Telegraphie,” Ann. Phys. (Leipzig) 28, 665–736 (1909).
  3. A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,”Z. Phys. 216, 398–410 (1968).
    [CrossRef]
  4. T. Tamir, “Inhomogeneous wave types at planar interfaces. II. Surface waves,” Optik (Stuttgart) 37(2), 204–228 (1973).
  5. T. Tamir, “Inhomogeneous wave types at planar interfaces. III. Leaky waves,” Optik (Stuttgart) 37(3), 269–299 (1973).
  6. A. Hartstein, E. Burstein, J. J. Brion, R. F. Wallis, “Surface polaritons on semi-infinite anisotropic media,” Surf. Sci. 34, 81–89 (1973).
    [CrossRef]
  7. V. Presa, J. M. Simon, “Reflection and transmission coefficients of multilayer on a uniaxial substrate,” Optik (Stuttgart) 98(4) 181–185 (1995).
  8. E. Kretschmann, “Die Bestimmung Optischer Konstanten von Metallen durch Anregung von Oberflächenplasma–schwingungen,”Z. Phys. 241, 313–324 (1971).
    [CrossRef]
  9. A. Lakthakia, “Would Brewster recognize today’s Brewster angle?” Opt. News 15(6), 14–18 (1994).
    [CrossRef]
  10. M. C. Simon, L. I. Perez, “Reflection and transmission coefficients in uniaxial crystals,” J. Mod. Opt. 38, 503–518 (1991).
    [CrossRef]
  11. R. M. Echarri, M. T. Garea, “Behaviour of the Poynting vector in uniaxial absorbent media,” Pure Appl. Opt. 3, 931–941 (1994).
    [CrossRef]
  12. L. I. Perez, “Coeficientes de reflexión en medios uniaxiales en la reflexión total,” An. Asoc. Física Argen. 5, 224–228 (1993).
  13. D. Rogovin, T. P. Shen, “Excitation of surface polaritons via optical nonlinearities: threshold behavior,” Phys. Rev. B 37, 1121–1135 (1988).
    [CrossRef]
  14. J. M. Simon, V. A. Presa, “Surface electromagnetic waves at the interface with anisotropic media,” J. Mod. Opt. 42, 2201–2211 (1995).
    [CrossRef]
  15. J. M. Simon, V. A. Presa, “Behaviour of the phases in the observation of surface electromagnetic waves,” J. Mod. Opt. 36, 649–657 (1989).
    [CrossRef]

1995 (2)

V. Presa, J. M. Simon, “Reflection and transmission coefficients of multilayer on a uniaxial substrate,” Optik (Stuttgart) 98(4) 181–185 (1995).

J. M. Simon, V. A. Presa, “Surface electromagnetic waves at the interface with anisotropic media,” J. Mod. Opt. 42, 2201–2211 (1995).
[CrossRef]

1994 (2)

R. M. Echarri, M. T. Garea, “Behaviour of the Poynting vector in uniaxial absorbent media,” Pure Appl. Opt. 3, 931–941 (1994).
[CrossRef]

A. Lakthakia, “Would Brewster recognize today’s Brewster angle?” Opt. News 15(6), 14–18 (1994).
[CrossRef]

1993 (1)

L. I. Perez, “Coeficientes de reflexión en medios uniaxiales en la reflexión total,” An. Asoc. Física Argen. 5, 224–228 (1993).

1991 (1)

M. C. Simon, L. I. Perez, “Reflection and transmission coefficients in uniaxial crystals,” J. Mod. Opt. 38, 503–518 (1991).
[CrossRef]

1989 (1)

J. M. Simon, V. A. Presa, “Behaviour of the phases in the observation of surface electromagnetic waves,” J. Mod. Opt. 36, 649–657 (1989).
[CrossRef]

1988 (1)

D. Rogovin, T. P. Shen, “Excitation of surface polaritons via optical nonlinearities: threshold behavior,” Phys. Rev. B 37, 1121–1135 (1988).
[CrossRef]

1973 (3)

T. Tamir, “Inhomogeneous wave types at planar interfaces. II. Surface waves,” Optik (Stuttgart) 37(2), 204–228 (1973).

T. Tamir, “Inhomogeneous wave types at planar interfaces. III. Leaky waves,” Optik (Stuttgart) 37(3), 269–299 (1973).

A. Hartstein, E. Burstein, J. J. Brion, R. F. Wallis, “Surface polaritons on semi-infinite anisotropic media,” Surf. Sci. 34, 81–89 (1973).
[CrossRef]

1971 (1)

E. Kretschmann, “Die Bestimmung Optischer Konstanten von Metallen durch Anregung von Oberflächenplasma–schwingungen,”Z. Phys. 241, 313–324 (1971).
[CrossRef]

1968 (1)

A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,”Z. Phys. 216, 398–410 (1968).
[CrossRef]

1909 (1)

A. Sommerfeld, “Über die Ausbreitung der Wellen in der drahtlosen Telegraphie,” Ann. Phys. (Leipzig) 28, 665–736 (1909).

1907 (1)

J. Zenneck, “Über die Fortpflanzung Ebener Elektromagnetischer Wellen längst einer Ebenen Leiterfläche und ihre Bezichung zur Drahtlosen Telegraphie,” Ann. Phys. (Leipzig) 23, 846–866 (1907).

Brion, J. J.

A. Hartstein, E. Burstein, J. J. Brion, R. F. Wallis, “Surface polaritons on semi-infinite anisotropic media,” Surf. Sci. 34, 81–89 (1973).
[CrossRef]

Burstein, E.

A. Hartstein, E. Burstein, J. J. Brion, R. F. Wallis, “Surface polaritons on semi-infinite anisotropic media,” Surf. Sci. 34, 81–89 (1973).
[CrossRef]

Echarri, R. M.

R. M. Echarri, M. T. Garea, “Behaviour of the Poynting vector in uniaxial absorbent media,” Pure Appl. Opt. 3, 931–941 (1994).
[CrossRef]

Garea, M. T.

R. M. Echarri, M. T. Garea, “Behaviour of the Poynting vector in uniaxial absorbent media,” Pure Appl. Opt. 3, 931–941 (1994).
[CrossRef]

Hartstein, A.

A. Hartstein, E. Burstein, J. J. Brion, R. F. Wallis, “Surface polaritons on semi-infinite anisotropic media,” Surf. Sci. 34, 81–89 (1973).
[CrossRef]

Kretschmann, E.

E. Kretschmann, “Die Bestimmung Optischer Konstanten von Metallen durch Anregung von Oberflächenplasma–schwingungen,”Z. Phys. 241, 313–324 (1971).
[CrossRef]

Lakthakia, A.

A. Lakthakia, “Would Brewster recognize today’s Brewster angle?” Opt. News 15(6), 14–18 (1994).
[CrossRef]

Otto, A.

A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,”Z. Phys. 216, 398–410 (1968).
[CrossRef]

Perez, L. I.

L. I. Perez, “Coeficientes de reflexión en medios uniaxiales en la reflexión total,” An. Asoc. Física Argen. 5, 224–228 (1993).

M. C. Simon, L. I. Perez, “Reflection and transmission coefficients in uniaxial crystals,” J. Mod. Opt. 38, 503–518 (1991).
[CrossRef]

Presa, V.

V. Presa, J. M. Simon, “Reflection and transmission coefficients of multilayer on a uniaxial substrate,” Optik (Stuttgart) 98(4) 181–185 (1995).

Presa, V. A.

J. M. Simon, V. A. Presa, “Surface electromagnetic waves at the interface with anisotropic media,” J. Mod. Opt. 42, 2201–2211 (1995).
[CrossRef]

J. M. Simon, V. A. Presa, “Behaviour of the phases in the observation of surface electromagnetic waves,” J. Mod. Opt. 36, 649–657 (1989).
[CrossRef]

Rogovin, D.

D. Rogovin, T. P. Shen, “Excitation of surface polaritons via optical nonlinearities: threshold behavior,” Phys. Rev. B 37, 1121–1135 (1988).
[CrossRef]

Shen, T. P.

D. Rogovin, T. P. Shen, “Excitation of surface polaritons via optical nonlinearities: threshold behavior,” Phys. Rev. B 37, 1121–1135 (1988).
[CrossRef]

Simon, J. M.

J. M. Simon, V. A. Presa, “Surface electromagnetic waves at the interface with anisotropic media,” J. Mod. Opt. 42, 2201–2211 (1995).
[CrossRef]

V. Presa, J. M. Simon, “Reflection and transmission coefficients of multilayer on a uniaxial substrate,” Optik (Stuttgart) 98(4) 181–185 (1995).

J. M. Simon, V. A. Presa, “Behaviour of the phases in the observation of surface electromagnetic waves,” J. Mod. Opt. 36, 649–657 (1989).
[CrossRef]

Simon, M. C.

M. C. Simon, L. I. Perez, “Reflection and transmission coefficients in uniaxial crystals,” J. Mod. Opt. 38, 503–518 (1991).
[CrossRef]

Sommerfeld, A.

A. Sommerfeld, “Über die Ausbreitung der Wellen in der drahtlosen Telegraphie,” Ann. Phys. (Leipzig) 28, 665–736 (1909).

Tamir, T.

T. Tamir, “Inhomogeneous wave types at planar interfaces. II. Surface waves,” Optik (Stuttgart) 37(2), 204–228 (1973).

T. Tamir, “Inhomogeneous wave types at planar interfaces. III. Leaky waves,” Optik (Stuttgart) 37(3), 269–299 (1973).

Wallis, R. F.

A. Hartstein, E. Burstein, J. J. Brion, R. F. Wallis, “Surface polaritons on semi-infinite anisotropic media,” Surf. Sci. 34, 81–89 (1973).
[CrossRef]

Zenneck, J.

J. Zenneck, “Über die Fortpflanzung Ebener Elektromagnetischer Wellen längst einer Ebenen Leiterfläche und ihre Bezichung zur Drahtlosen Telegraphie,” Ann. Phys. (Leipzig) 23, 846–866 (1907).

An. Asoc. Física Argen. (1)

L. I. Perez, “Coeficientes de reflexión en medios uniaxiales en la reflexión total,” An. Asoc. Física Argen. 5, 224–228 (1993).

Ann. Phys. (Leipzig) (2)

J. Zenneck, “Über die Fortpflanzung Ebener Elektromagnetischer Wellen längst einer Ebenen Leiterfläche und ihre Bezichung zur Drahtlosen Telegraphie,” Ann. Phys. (Leipzig) 23, 846–866 (1907).

A. Sommerfeld, “Über die Ausbreitung der Wellen in der drahtlosen Telegraphie,” Ann. Phys. (Leipzig) 28, 665–736 (1909).

J. Mod. Opt. (3)

M. C. Simon, L. I. Perez, “Reflection and transmission coefficients in uniaxial crystals,” J. Mod. Opt. 38, 503–518 (1991).
[CrossRef]

J. M. Simon, V. A. Presa, “Surface electromagnetic waves at the interface with anisotropic media,” J. Mod. Opt. 42, 2201–2211 (1995).
[CrossRef]

J. M. Simon, V. A. Presa, “Behaviour of the phases in the observation of surface electromagnetic waves,” J. Mod. Opt. 36, 649–657 (1989).
[CrossRef]

Opt. News (1)

A. Lakthakia, “Would Brewster recognize today’s Brewster angle?” Opt. News 15(6), 14–18 (1994).
[CrossRef]

Optik (Stuttgart) (3)

V. Presa, J. M. Simon, “Reflection and transmission coefficients of multilayer on a uniaxial substrate,” Optik (Stuttgart) 98(4) 181–185 (1995).

T. Tamir, “Inhomogeneous wave types at planar interfaces. II. Surface waves,” Optik (Stuttgart) 37(2), 204–228 (1973).

T. Tamir, “Inhomogeneous wave types at planar interfaces. III. Leaky waves,” Optik (Stuttgart) 37(3), 269–299 (1973).

Phys. Rev. B (1)

D. Rogovin, T. P. Shen, “Excitation of surface polaritons via optical nonlinearities: threshold behavior,” Phys. Rev. B 37, 1121–1135 (1988).
[CrossRef]

Pure Appl. Opt. (1)

R. M. Echarri, M. T. Garea, “Behaviour of the Poynting vector in uniaxial absorbent media,” Pure Appl. Opt. 3, 931–941 (1994).
[CrossRef]

Surf. Sci. (1)

A. Hartstein, E. Burstein, J. J. Brion, R. F. Wallis, “Surface polaritons on semi-infinite anisotropic media,” Surf. Sci. 34, 81–89 (1973).
[CrossRef]

Z. Phys. (2)

A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,”Z. Phys. 216, 398–410 (1968).
[CrossRef]

E. Kretschmann, “Die Bestimmung Optischer Konstanten von Metallen durch Anregung von Oberflächenplasma–schwingungen,”Z. Phys. 241, 313–324 (1971).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Coordinate system. The plane yz is the interface, the plane zx is the incidence one, and the plane zx is the one that contains the normal to the interface and the optical axis z3. For x < 0 the medium is isotropic and absorbing, and for x > 0 it is uniaxial and nonabsorbing. The direction of the optical axis is characterized by the angles θ and δ.

Fig. 2
Fig. 2

Modulus of the magnetic vector on both sides of the interface as a function of the distance to it for aluminum–calcite, δ = 0° and fixed θ = 30°, component of the magnetic vector in the direction perpendicular to the interface equal to zero, and incidence angle equal to Brewster’s angle (semi-infinite media).

Fig. 3
Fig. 3

Modulus of the magnetic vector on both sides of the interface as a function of the distance to it for aluminum–calcite, δ = 0° and fixed θ = 30°, component of the electric field in the direction perpendicular to the interface equal to zero, and incidence angle equal to Brewster’s angle (semi-infinite media).

Fig. 4
Fig. 4

Determinant of the reflection matrix versus incidence angle in (a) modulus and (b) phase for the case in which the prism has n1 = 1.79, the metal has nm = 1.22 + i6.92 and width 16.5 nm, and the crystal has no = 1.6584 and ne = 1.4865, with (δ = 0° and θ = 30°).

Fig. 5
Fig. 5

Electric field in the crystal at different distances from the metal–crystal interface.

Fig. 6
Fig. 6

Modulus of the magnetic vector on both sides of the interface as a function of the distance to it for aluminum–calcite, δ = 0° and θ = 30°, component of the electric field in the direction perpendicular to the interface equal to zero, and incidence angle corresponding to Brewster’s angle (Kretschmann configuration). In the inset is detail in the metal (16.5 nm in width).

Fig. 7
Fig. 7

Determinant of the reflection matrix versus incidence angle in (a) modulus and (b) phase for the case in which the prism has np = 1.79, the metal has nm = 1.22 + i6.92 and width 17 nm, and the crystal has no = 1.6584 and ne = 1.4865, with δ = 90° and θ = 0°.

Equations (62)

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× E = μ 0 H t ,
× H = * E t ,
1 ( E 1 + E 1 * ) = [ o f n 2 + 1 ( x ˆ · z ˆ 3 ) 2 ] ( E 1 + E 1 ) + f n ( x ˆ · z ˆ 3 ) ( e 1 ) E 3 ,
E 2 + E 2 * = E 2 + E 2 ,
1 ( E 3 + E 3 * ) = ( x ˆ · z ˆ 3 ) f n ( o 1 ) ( E 1 + E 1 ) + [ e ( x ˆ · z ˆ 3 ) 2 + 1 f n 2 ] E 3 ,
H 1 + H 1 * = H 1 + H 1 ,
H 2 + H 2 * = H 2 + H 2 ,
H 3 + H 3 * = H 3 ,
k x 2 + k z 2 = μ 0 1 ω 2 ,
k x 2 + k x 2 = μ 0 o ω 2 ,
k x 2 [ o f n 2 + e ( x ˆ · z ˆ 3 ) 2 ] + 2 k x k z ( e o ) cos δ ( x ˆ · z ˆ 3 ) f n + k z 2 [ o ( x ˆ · z ˆ 3 ) 2 cos 2 δ + o sin 2 δ + e f n 2 cos 2 δ ] = μ 0 o e ω 2 ,
E 3 = E 1 = H 2 = 0 ,
E 2 [ k x ( x ˆ · z ˆ 3 ) + k z f n ] = μ 0 ω H 1 ,
E 2 [ k x f n k z ( x ˆ · z ˆ 3 ) ] = μ 0 ω H 3 ,
H 1 [ k x ( x ˆ · z ˆ 3 ) + k z f n ] H 3 [ k x f n k z ( x ˆ · z ˆ 3 ) ] = o ω E 2 ,
H 3 = E 2 = H 1 = 0 ,
E 1 [ k x ( x ˆ · z ˆ 3 ) + k z f n ] E 3 [ k x f n k z ( x ˆ · z ˆ 3 ) ] = μ 0 ω H 2 ,
H 2 [ k x ( x ˆ · z ˆ 3 ) + k z f n ] = o ω E 1 ,
H 2 [ k x f n k z ( x ˆ · z ˆ 3 ) ] = e ω E 3 ,
( E 2 E 2 * ) k x f n ( E 2 + E 2 * ) k z ( x ˆ · z ˆ 3 ) = μ 0 ω ( H 3 + H 3 * ) ,
k x ( E 2 E 2 * E 2 ) = k x .
k z 2 ( E 2 E 2 * E 2 ) 2 + k x 2 1 μ 0 ω 2 ( E 2 E 2 * E 2 ) 2 = 0 ,
k z 2 = μ 0 ω 2 1 ( E 2 E 2 * E 2 ) 2 o ( E 2 E 2 * E 2 ) 2 1 ,
k x 2 = μ 0 ω 2 ( o 1 ) 1 ( E 2 E 2 * E 2 ) 2 ( E 2 E 2 * E 2 ) 2 1 ,
( k x + k x ) E 2 = 2 k x E 2 ,
( k x + k x ) E 2 * = ( k x k x ) E 2 .
( H 2 H 2 * ) k x f n + ( H 2 + H 2 * ) k z ( x ˆ · z ˆ 3 ) = 1 ω ( E 3 + E 3 * ) .
k z 2 [ ( e o ) 2 ( x ˆ · z ˆ 3 ) 2 f n 2 + o 2 e 2 1 2 ( H 2 H 2 * H 2 ) 2 ] + k x 2 [ o f n 2 + e ( x ˆ · z ˆ 3 ) 2 ] 2 + 2 k z k x ( e o ) [ o ( z ˆ · z ˆ 3 ) 2 + e ( x ˆ · z ˆ 3 ) 2 ] ( x ˆ · z ˆ 3 ) f n o 2 e 2 1 ω 2 ( H 2 H 2 * H 2 ) 2 = 0.
k z 2 = 1 ω 2 μ 0 o e ( H 2 H 2 * H 2 ) 2 1 [ o f n 2 + e ( x ˆ · z ˆ 3 ) 2 ] o e ( H 2 H 2 * H 2 ) 2 1 2 ,
k x = ( 1 ( e o ) f n ( x ˆ · z ˆ 3 ) { o e ( H 2 H 2 * H 2 ) 2 1 [ o f n 2 + e ( x ˆ · z ˆ 3 ) 2 ] } 1 / 2 [ o f n 2 + e ( x ˆ · z ˆ 3 ) 2 ] [ o e ( H 2 H 2 * H 2 ) 2 1 2 ] 1 / 2 + e o ( H 2 H 2 * H 2 ) { [ o f n 2 + e ( x ˆ · z ˆ 3 ) 2 ] 1 } 1 / 2 [ o f n 2 + e ( x ˆ · z ˆ 3 ) 2 ] [ o e ( H 2 H 2 * H 2 ) 2 1 2 ] 1 / 2 ) ω μ 0 ,
{ o e k x + 1 k x [ e ( x ˆ · z ˆ 3 ) 2 + o f n 2 ] + 1 k z ( x ˆ · z ˆ 3 ) f n ( e o ) } H 2 = 2 o e k x H 2 ,
{ o e k x + 1 k x [ e ( x ˆ · z ˆ 3 ) 2 + o f n 2 ] + 1 k z ( x ˆ · z ˆ 3 ) f n ( e o ) } H 2 = { o e k x 1 k x × [ e ( x ˆ · z ˆ 3 ) 2 + o f n 2 ] 1 k z ( x ˆ · z ˆ 3 ) f n ( e o ) } H 2 .
| H I | 2 = | E 2 * | 2 μ 0 2 ω 2 exp ( 2 d I z ) | k x k x | 2 { ( | k x | 2 + | k z | 2 ) × [ | k x + k x | 2 exp ( 2 a I x ) + | k x k x | 2 exp ( 2 a I x ) ] ( | k x | 2 | k x | 2 ) ( | k x | 2 | k z | 2 ) × [ exp ( 2 i a R x ) + exp ( 2 i a R x ) ] + ( | k x | 2 | k z | 2 ) ( k x k x ¯ k x ¯ k x ) × [ exp ( 2 i a R x ) exp ( 2 i a R x ) ] } + [ | H 2 | 2 exp ( 2 a I x ) + | H 2 * | 2 exp ( 2 a I x ) + H 2 H 2 * ¯ exp ( 2 i a R x ) + H 2 * H 2 ¯ × exp ( 2 i a R x ) ] exp ( 2 d I z )
| H II | 2 = | E 2 * | 2 μ 0 2 ω 2 4 | k x | 2 ( | k x | 2 + | k z | 2 ) | k x + k x | 2 exp ( 2 b I x ) × exp ( 2 d I z ) + | H 2 | 2 exp ( 2 c I x ) exp ( 2 d I z ) ,
o e k x = 1 k x [ e ( x ˆ · z ˆ 3 ) 2 + o f n 2 ] + 1 k z ( x ˆ · z ˆ 3 ) f n ( e o ) .
k z 2 = 1 ω 2 μ 0 o e 1 [ o f n 2 + e ( x ˆ · z ˆ 3 ) 2 ] o e 1 2 ,
k x = ( 1 ( e o ) f n ( x ˆ · z ˆ 3 ) { o e 1 [ o f n 2 + e ( x ˆ · z ˆ 3 ) 2 ] } 1 / 2 [ o f n 2 + e ( x ˆ · z ˆ 3 ) 2 ] ( o e 1 2 ) 1 / 2 ) + e o { [ o f n 2 + e ( x ˆ · z ˆ 3 ) 2 ] 1 } 1 / 2 [ o f n 2 + e ( x ˆ · z ˆ 3 ) 2 ] ( o e 1 2 ) 1 / 2 ) ω μ 0 ,
k x 2 = 1 ω 2 μ 0 o e ( o 1 ) + 1 2 ( o + e ) ( x ˆ · z ˆ 3 ) 2 o e 1 2 ,
k x 2 = 1 2 ω 2 μ 0 [ o f n 2 + e ( x ˆ · z ˆ 3 ) 2 ] 1 o e 1 2 .
k z 2 k x 2 = tan 2 α B = o e 1 [ e ( x ˆ · z ˆ 3 ) 2 + o f n 2 ] 1 [ e ( x ˆ · z ˆ 3 ) 2 + o f n 2 1 ] ,
E 3 = H 1 = H 2 = 0 ,
H 3 = o ω k x E 2 ,
E 1 = k z k x E 2 ,
E 1 = E 2 = H 3 = 0 ,
E 3 = μ 0 ω k x H 2 ,
E 1 = k z k x H 2 .
k x 2 = μ o 2 ω 2 o 1 o 2 1 2 ( E 2 E 2 * E 2 ) 2 ,
( k x 1 + k x o ) E 2 * = ( k x 1 k x o ) E 2 ,
( k x 1 + k x o ) E 2 = 2 k x 1 E 2 ,
k x 2 = μ 0 ω 2 e 1 1 ( H 2 H 2 * H 2 ) 2 ,
( k x + k x ) H 2 * = ( k x k x ) H 2 ,
( k x + k x ) H 2 = 2 k x H 2 ,
k x 2 = μ 1 2 ω 2 o 1 o 2 1 2 ( E 2 E 2 * E 2 ) 2 ( E 2 E 2 * E 2 ) 2 .
k z 2 = μ 0 o 1 ω 2 o 1 ( E 2 E 2 * E 2 ) 2 o 2 1 2 ( E 2 E 2 * E 2 ) 2 ,
k z 2 = μ 0 ω 2 1 e ( H 2 H 2 * H 2 ) 2 1 ( H 2 H 2 * H 2 ) 2 .
k x 1 k x o = 0.
k x 2 = μ ω 2 1 2 o + 1 ,
k x 2 = μ ω 2 o 2 o + 1 ,
k x 2 = μ ω 2 ( e o 1 o + 1 ) ,
k z 2 = μ ω 2 o 1 o + 1 .
R p p R s s R s p R p s = λ 1 λ 2 = 0.
( E o E e ) = [ T o s T o p T e s T e p ] ( E s E p ) .

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