Abstract

The conditions of occurrence of both hybrid surface modes and hybrid guided modes in a periodic stratified dielectric film in contact with a positive birefringent uniaxial material and a dielectric isotropic medium are studied by means of the 4 × 4 transfer matrix formalism. The dispersion relations of these hybrid modes are exactly calculated. The allowed values of the propagation constant of both surface modes and guided modes form a band structure. The same is true for the allowed values of the direction of propagation required for the existence of these modes. These band structures are explored with the help of numerical solutions of the eigenvalue equation. The lower cutoff thicknesses for various hybrid guided modes are obtained. The TE-dominant–TM-dominant guided-mode conversion effect is suggested.

© 1995 Optical Society of America

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References

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  1. D. P. Gia Russo, J. H. Harris, “Wave propagation in anisotropic thin-film optical waveguides,” J. Opt. Soc. Am. 63, 138–145 (1973).
    [CrossRef]
  2. W. K. Burns, J. Warner, “Mode dispersion in uniaxial optical waveguides,” J. Opt. Soc. Am. 64, 441–446 (1974).
    [CrossRef]
  3. J. Čtyroký, M. Cada, “Guided and semileaky modes in anisotropic waveguides of the LiNbO3type,” Opt. Commun. 27, 353–357 (1978).
    [CrossRef]
  4. D. Marcuse, “Modes of a symmetric slab optical waveguide in birefringent media—Part I: Optical axis not in plane of slab,” IEEE J. Quantum Electron. QE-14, 736–741 (1978).
    [CrossRef]
  5. D. Marcuse, I. P. Kaminow, “Modes of a symmetric slab optical waveguide in birefringent media—Part II: Slab with coplanar optical axis,” IEEE J. Quantum Electron. QE-15, 92–101 (1979).
    [CrossRef]
  6. P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69, 742–756 (1979).
    [CrossRef]
  7. T. K. Gaylord, A. Knoessen, “Passive integrated optical anisotropy-based devices,” J. Mod. Opt. 35, 925–946 (1988).
    [CrossRef]
  8. A. Knoessen, T. K. Gaylord, M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar waveguides,” J. Lightwave Technol. 6, 1083–1104 (1988).
    [CrossRef]
  9. L. Torner, J. Recolons, J. P. Torres, “Guided-to-leaky mode transition in uniaxial optical slab waveguides,” J. Lightwave Technol. 11, 1592–1600 (1993).
    [CrossRef]
  10. M. I. Dyakonov, “New type of electromagnetic wave propagation at an interface,” Sov. Phys. JETP 67, 714–716 (1988).
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    [CrossRef]
  12. L. Torner, J. P. Torres, C. Ojeda, D. Mihalache, “Hybrid waves guided by ultrathin films,” J. Lightwave Technol. (to be published).
  13. D. Mihalache, D.-M. Baboiu, M. Ciumac, L. Torner, J. P. Torres, “Guided waves in anisotropic antiguide structures,” Opt. Commun. 108, 239–242 (1994).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  21. Yi-Fan Li, J. W. Y. Lit, “Guided even and odd modes in symmetric periodic stratified dielectric waveguides,” J. Opt. Soc. Am. A 5, 1050–1057 (1988).
    [CrossRef]
  22. P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

1994 (2)

D. Mihalache, D.-M. Baboiu, M. Ciumac, L. Torner, J. P. Torres, “Guided waves in anisotropic antiguide structures,” Opt. Commun. 108, 239–242 (1994).
[CrossRef]

D. Mihalache, D.-M. Baboiu, M. Ciumac, L. Torner, J. P. Torres, “Hybrid surface plasmon polaritons guided by ultrathin metal films,” Opt. Quantum Electron. 26, 857–863 (1994).
[CrossRef]

1993 (4)

L. Torner, J. Recolons, J. P. Torres, “Guided-to-leaky mode transition in uniaxial optical slab waveguides,” J. Lightwave Technol. 11, 1592–1600 (1993).
[CrossRef]

L. Torner, J. P. Torres, D. Milhalache, “New type of guided waves in birefringent media,” IEEE Photon. Technol. Lett. 5, 201–203 (1993).
[CrossRef]

C. Zmudzinski, D. Botez, L. J. Mawst, C. Tu, L. Frantz, “Coherent 1-W continuous wave operation of large-aperture resonant arrays of antiguide diode lasers,” Appl. Phys. Lett. 62, 2914–2916 (1993).
[CrossRef]

P. Yeh, C. Gu, D. Botez, “Matrix analysis of wave propagation in real-index antiguided arrays,” J. Opt. Soc. Am. B 10, 709–715 (1993).
[CrossRef]

1988 (4)

Yi-Fan Li, J. W. Y. Lit, “Guided even and odd modes in symmetric periodic stratified dielectric waveguides,” J. Opt. Soc. Am. A 5, 1050–1057 (1988).
[CrossRef]

M. I. Dyakonov, “New type of electromagnetic wave propagation at an interface,” Sov. Phys. JETP 67, 714–716 (1988).

T. K. Gaylord, A. Knoessen, “Passive integrated optical anisotropy-based devices,” J. Mod. Opt. 35, 925–946 (1988).
[CrossRef]

A. Knoessen, T. K. Gaylord, M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar waveguides,” J. Lightwave Technol. 6, 1083–1104 (1988).
[CrossRef]

1979 (2)

D. Marcuse, I. P. Kaminow, “Modes of a symmetric slab optical waveguide in birefringent media—Part II: Slab with coplanar optical axis,” IEEE J. Quantum Electron. QE-15, 92–101 (1979).
[CrossRef]

P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69, 742–756 (1979).
[CrossRef]

1978 (2)

J. Čtyroký, M. Cada, “Guided and semileaky modes in anisotropic waveguides of the LiNbO3type,” Opt. Commun. 27, 353–357 (1978).
[CrossRef]

D. Marcuse, “Modes of a symmetric slab optical waveguide in birefringent media—Part I: Optical axis not in plane of slab,” IEEE J. Quantum Electron. QE-14, 736–741 (1978).
[CrossRef]

1977 (2)

1974 (1)

1973 (1)

1972 (1)

1950 (1)

F. Abelès, “Recherches sur la propagation des ondes électromagnetiques sinusoidales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640 (1950).

Abelès, F.

F. Abelès, “Recherches sur la propagation des ondes électromagnetiques sinusoidales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640 (1950).

Baboiu, D.-M.

D. Mihalache, D.-M. Baboiu, M. Ciumac, L. Torner, J. P. Torres, “Hybrid surface plasmon polaritons guided by ultrathin metal films,” Opt. Quantum Electron. 26, 857–863 (1994).
[CrossRef]

D. Mihalache, D.-M. Baboiu, M. Ciumac, L. Torner, J. P. Torres, “Guided waves in anisotropic antiguide structures,” Opt. Commun. 108, 239–242 (1994).
[CrossRef]

Berreman, D. W.

Botez, D.

P. Yeh, C. Gu, D. Botez, “Matrix analysis of wave propagation in real-index antiguided arrays,” J. Opt. Soc. Am. B 10, 709–715 (1993).
[CrossRef]

C. Zmudzinski, D. Botez, L. J. Mawst, C. Tu, L. Frantz, “Coherent 1-W continuous wave operation of large-aperture resonant arrays of antiguide diode lasers,” Appl. Phys. Lett. 62, 2914–2916 (1993).
[CrossRef]

Burns, W. K.

Cada, M.

J. Čtyroký, M. Cada, “Guided and semileaky modes in anisotropic waveguides of the LiNbO3type,” Opt. Commun. 27, 353–357 (1978).
[CrossRef]

Ciumac, M.

D. Mihalache, D.-M. Baboiu, M. Ciumac, L. Torner, J. P. Torres, “Hybrid surface plasmon polaritons guided by ultrathin metal films,” Opt. Quantum Electron. 26, 857–863 (1994).
[CrossRef]

D. Mihalache, D.-M. Baboiu, M. Ciumac, L. Torner, J. P. Torres, “Guided waves in anisotropic antiguide structures,” Opt. Commun. 108, 239–242 (1994).
[CrossRef]

Ctyroký, J.

J. Čtyroký, M. Cada, “Guided and semileaky modes in anisotropic waveguides of the LiNbO3type,” Opt. Commun. 27, 353–357 (1978).
[CrossRef]

Dyakonov, M. I.

M. I. Dyakonov, “New type of electromagnetic wave propagation at an interface,” Sov. Phys. JETP 67, 714–716 (1988).

Frantz, L.

C. Zmudzinski, D. Botez, L. J. Mawst, C. Tu, L. Frantz, “Coherent 1-W continuous wave operation of large-aperture resonant arrays of antiguide diode lasers,” Appl. Phys. Lett. 62, 2914–2916 (1993).
[CrossRef]

Gaylord, T. K.

T. K. Gaylord, A. Knoessen, “Passive integrated optical anisotropy-based devices,” J. Mod. Opt. 35, 925–946 (1988).
[CrossRef]

A. Knoessen, T. K. Gaylord, M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar waveguides,” J. Lightwave Technol. 6, 1083–1104 (1988).
[CrossRef]

Gia Russo, D. P.

Gu, C.

Harris, J. H.

Hong, C.-S.

Kaminow, I. P.

D. Marcuse, I. P. Kaminow, “Modes of a symmetric slab optical waveguide in birefringent media—Part II: Slab with coplanar optical axis,” IEEE J. Quantum Electron. QE-15, 92–101 (1979).
[CrossRef]

Knoessen, A.

T. K. Gaylord, A. Knoessen, “Passive integrated optical anisotropy-based devices,” J. Mod. Opt. 35, 925–946 (1988).
[CrossRef]

A. Knoessen, T. K. Gaylord, M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar waveguides,” J. Lightwave Technol. 6, 1083–1104 (1988).
[CrossRef]

Li, Yi-Fan

Lit, J. W. Y.

Marcuse, D.

D. Marcuse, I. P. Kaminow, “Modes of a symmetric slab optical waveguide in birefringent media—Part II: Slab with coplanar optical axis,” IEEE J. Quantum Electron. QE-15, 92–101 (1979).
[CrossRef]

D. Marcuse, “Modes of a symmetric slab optical waveguide in birefringent media—Part I: Optical axis not in plane of slab,” IEEE J. Quantum Electron. QE-14, 736–741 (1978).
[CrossRef]

Mawst, L. J.

C. Zmudzinski, D. Botez, L. J. Mawst, C. Tu, L. Frantz, “Coherent 1-W continuous wave operation of large-aperture resonant arrays of antiguide diode lasers,” Appl. Phys. Lett. 62, 2914–2916 (1993).
[CrossRef]

Mihalache, D.

D. Mihalache, D.-M. Baboiu, M. Ciumac, L. Torner, J. P. Torres, “Guided waves in anisotropic antiguide structures,” Opt. Commun. 108, 239–242 (1994).
[CrossRef]

D. Mihalache, D.-M. Baboiu, M. Ciumac, L. Torner, J. P. Torres, “Hybrid surface plasmon polaritons guided by ultrathin metal films,” Opt. Quantum Electron. 26, 857–863 (1994).
[CrossRef]

L. Torner, J. P. Torres, C. Ojeda, D. Mihalache, “Hybrid waves guided by ultrathin films,” J. Lightwave Technol. (to be published).

Milhalache, D.

L. Torner, J. P. Torres, D. Milhalache, “New type of guided waves in birefringent media,” IEEE Photon. Technol. Lett. 5, 201–203 (1993).
[CrossRef]

Moharam, M. G.

A. Knoessen, T. K. Gaylord, M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar waveguides,” J. Lightwave Technol. 6, 1083–1104 (1988).
[CrossRef]

Ojeda, C.

L. Torner, J. P. Torres, C. Ojeda, D. Mihalache, “Hybrid waves guided by ultrathin films,” J. Lightwave Technol. (to be published).

Recolons, J.

L. Torner, J. Recolons, J. P. Torres, “Guided-to-leaky mode transition in uniaxial optical slab waveguides,” J. Lightwave Technol. 11, 1592–1600 (1993).
[CrossRef]

Torner, L.

D. Mihalache, D.-M. Baboiu, M. Ciumac, L. Torner, J. P. Torres, “Guided waves in anisotropic antiguide structures,” Opt. Commun. 108, 239–242 (1994).
[CrossRef]

D. Mihalache, D.-M. Baboiu, M. Ciumac, L. Torner, J. P. Torres, “Hybrid surface plasmon polaritons guided by ultrathin metal films,” Opt. Quantum Electron. 26, 857–863 (1994).
[CrossRef]

L. Torner, J. P. Torres, D. Milhalache, “New type of guided waves in birefringent media,” IEEE Photon. Technol. Lett. 5, 201–203 (1993).
[CrossRef]

L. Torner, J. Recolons, J. P. Torres, “Guided-to-leaky mode transition in uniaxial optical slab waveguides,” J. Lightwave Technol. 11, 1592–1600 (1993).
[CrossRef]

L. Torner, J. P. Torres, C. Ojeda, D. Mihalache, “Hybrid waves guided by ultrathin films,” J. Lightwave Technol. (to be published).

Torres, J. P.

D. Mihalache, D.-M. Baboiu, M. Ciumac, L. Torner, J. P. Torres, “Guided waves in anisotropic antiguide structures,” Opt. Commun. 108, 239–242 (1994).
[CrossRef]

D. Mihalache, D.-M. Baboiu, M. Ciumac, L. Torner, J. P. Torres, “Hybrid surface plasmon polaritons guided by ultrathin metal films,” Opt. Quantum Electron. 26, 857–863 (1994).
[CrossRef]

L. Torner, J. P. Torres, D. Milhalache, “New type of guided waves in birefringent media,” IEEE Photon. Technol. Lett. 5, 201–203 (1993).
[CrossRef]

L. Torner, J. Recolons, J. P. Torres, “Guided-to-leaky mode transition in uniaxial optical slab waveguides,” J. Lightwave Technol. 11, 1592–1600 (1993).
[CrossRef]

L. Torner, J. P. Torres, C. Ojeda, D. Mihalache, “Hybrid waves guided by ultrathin films,” J. Lightwave Technol. (to be published).

Tu, C.

C. Zmudzinski, D. Botez, L. J. Mawst, C. Tu, L. Frantz, “Coherent 1-W continuous wave operation of large-aperture resonant arrays of antiguide diode lasers,” Appl. Phys. Lett. 62, 2914–2916 (1993).
[CrossRef]

Warner, J.

Yariv, A.

Yeh, P.

Zmudzinski, C.

C. Zmudzinski, D. Botez, L. J. Mawst, C. Tu, L. Frantz, “Coherent 1-W continuous wave operation of large-aperture resonant arrays of antiguide diode lasers,” Appl. Phys. Lett. 62, 2914–2916 (1993).
[CrossRef]

Ann. Phys. (Paris) (1)

F. Abelès, “Recherches sur la propagation des ondes électromagnetiques sinusoidales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640 (1950).

Appl. Phys. Lett. (1)

C. Zmudzinski, D. Botez, L. J. Mawst, C. Tu, L. Frantz, “Coherent 1-W continuous wave operation of large-aperture resonant arrays of antiguide diode lasers,” Appl. Phys. Lett. 62, 2914–2916 (1993).
[CrossRef]

IEEE J. Quantum Electron. (2)

D. Marcuse, “Modes of a symmetric slab optical waveguide in birefringent media—Part I: Optical axis not in plane of slab,” IEEE J. Quantum Electron. QE-14, 736–741 (1978).
[CrossRef]

D. Marcuse, I. P. Kaminow, “Modes of a symmetric slab optical waveguide in birefringent media—Part II: Slab with coplanar optical axis,” IEEE J. Quantum Electron. QE-15, 92–101 (1979).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

L. Torner, J. P. Torres, D. Milhalache, “New type of guided waves in birefringent media,” IEEE Photon. Technol. Lett. 5, 201–203 (1993).
[CrossRef]

J. Lightwave Technol. (2)

A. Knoessen, T. K. Gaylord, M. G. Moharam, “Hybrid guided modes in uniaxial dielectric planar waveguides,” J. Lightwave Technol. 6, 1083–1104 (1988).
[CrossRef]

L. Torner, J. Recolons, J. P. Torres, “Guided-to-leaky mode transition in uniaxial optical slab waveguides,” J. Lightwave Technol. 11, 1592–1600 (1993).
[CrossRef]

J. Mod. Opt. (1)

T. K. Gaylord, A. Knoessen, “Passive integrated optical anisotropy-based devices,” J. Mod. Opt. 35, 925–946 (1988).
[CrossRef]

J. Opt. Soc. Am. (6)

J. Opt. Soc. Am. A (1)

J. Opt. Soc. Am. B (1)

Opt. Commun. (2)

D. Mihalache, D.-M. Baboiu, M. Ciumac, L. Torner, J. P. Torres, “Guided waves in anisotropic antiguide structures,” Opt. Commun. 108, 239–242 (1994).
[CrossRef]

J. Čtyroký, M. Cada, “Guided and semileaky modes in anisotropic waveguides of the LiNbO3type,” Opt. Commun. 27, 353–357 (1978).
[CrossRef]

Opt. Quantum Electron. (1)

D. Mihalache, D.-M. Baboiu, M. Ciumac, L. Torner, J. P. Torres, “Hybrid surface plasmon polaritons guided by ultrathin metal films,” Opt. Quantum Electron. 26, 857–863 (1994).
[CrossRef]

Sov. Phys. JETP (1)

M. I. Dyakonov, “New type of electromagnetic wave propagation at an interface,” Sov. Phys. JETP 67, 714–716 (1988).

Other (2)

L. Torner, J. P. Torres, C. Ojeda, D. Mihalache, “Hybrid waves guided by ultrathin films,” J. Lightwave Technol. (to be published).

P. Yeh, Optical Waves in Layered Media (Wiley, New York, 1988).

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Figures (8)

Fig. 1
Fig. 1

d2/λ dependence of the propagation constant β. Here no = 1.52, ne = 1.68, d1/λ = 0.5, n1 = 1.56, and n2 = 1.57.

Fig. 2
Fig. 2

Allowed optical axis orientations as a function of d2/λ for N = 1, corresponding to the propagation constants within the shaded bands in Fig. 1 labeled 0 (dashed curve) and 1 (solid curves).

Fig. 3
Fig. 3

Allowed optical axis orientations as functions of d2/λ for N = 1, corresponding to the propagation constants within the unshaded region labeled 0 in Fig. 1.

Fig. 4
Fig. 4

(a) Allowed optical axis orientations as functions of d2/λ for N = 2, corresponding to the propagation constants within the unshaded region labeled 0 in Fig. 1, (b) detailed region of (a).

Fig. 5
Fig. 5

Allowed optical axis orientations as functions of d2/λ for the unshaded region labeled 1 in Fig. 1: (a) N = 1, (b) N = 2.

Fig. 6
Fig. 6

Electric field distribution for the TE1-dominant hybrid surface mode for N = 2, d1/λ = 0.5, d2/λ = 1.3, θ = 22.5°, and β = 1.540611.

Fig. 7
Fig. 7

Electric field distribution for the TE-dominant hybrid guided mode for N = 2, d1/λ = 0.5, d2/λ = 0.8, and θ = 10°: (a) TE0-dominant hybrid guided mode corresponding to β = 1.5604471, (b) TE1-dominant hybrid guided mode corresponding to β = 1.54486091.

Fig. 8
Fig. 8

Ex (solid curve) and Ez (dashed curve) field distribution of the hybrid TE1-dominant guided mode for the same parameters as in Fig. 7.

Equations (36)

Equations on this page are rendered with MathJax. Learn more.

n ( x ) = { n 1 , m d x m d + d 1 n 2 , m d + d 1 x ( m + 1 ) d ,
^ = [ n o 2 0 0 0 n o 2 cos 2 θ + n e 2 sin 2 θ - ( n e 2 - n o 2 ) sin θ cos θ 0 - ( n e 2 - n o 2 ) sin θ cos θ n o 2 sin 2 θ + n e 2 cos 2 θ ] .
E ( x ) = E o exp ( γ o k 0 x ) + E e exp ( γ e k 0 x ) , H ( x ) = H o exp ( γ o k 0 x ) + H e exp ( γ e k 0 x ) .
n ( θ ) = n o n e [ n o 2 sin 2 ( θ ) + n e 2 cos 2 ( θ ) ] 1 / 2 .
E ( x ) = E s exp [ - γ s k 0 ( x - N d ) ] , H ( x ) = H s exp [ - γ s k 0 ( x - N d ) ] ,
E y ( 1 ) ( x ) = a m cosh [ q 1 k 0 ( x - m d ) ] + b m q 1 sinh [ q 1 k 0 ( x - m d ) ] ,
H y ( 1 ) ( x ) = A m cosh [ q 1 k 0 ( x - m d ) ] + 1 q 1 B m sinh [ q 1 k 0 ( x - m d ) ] ,
E y ( 2 ) ( x ) = c m cosh [ q 2 k 0 ( x - d - m d ) ] + d m q 2 sinh [ q 2 k 0 ( x - d - m d ) ] ,
H y ( 2 ) ( x ) = C m cosh [ q 2 k 0 ( x - d - m d ) ] + 2 q 2 D m sinh [ q 2 k 0 ( x - d - m d ) ] ,
( a m + 1 b m + 1 A m + 1 B m + 1 ) = T ( a m b m A m B m ) ,             m = 0 , 1 , 2 , , N - 1.
T = [ T 11 T 12 0 0 T 21 T 22 0 0 0 0 T 33 T 34 0 0 T 43 T 44 ] ,
T 11 = cosh γ 1 cosh γ 2 + q 1 / q 2 sinh γ 1 sinh γ 2 , T 12 = q 1 - 1 sinh γ 1 cosh γ 2 + q 2 - 1 cosh γ 1 sinh γ 2 , T 21 = q 2 cosh γ 1 sinh γ 2 + q 1 sinh γ 1 cosh γ 2 , T 22 = cosh γ 1 cosh γ 2 + q 2 / q 1 sinh γ 1 sinh γ 2 , T 33 = cosh γ 1 cosh γ 2 + 2 q 1 / 1 q 2 sinh γ 1 sinh γ 2 , T 34 = 1 / q 1 sinh γ 1 cosh γ 2 + 2 / q 2 cosh γ 1 sinh γ 2 , T 43 = q 2 / 2 cosh γ 1 sinh γ 2 + q 1 / 1 sinh γ 1 cosh γ 2 , T 44 = cosh γ 1 cosh γ 2 + 1 q 2 / 2 q 1 sinh γ 1 sinh γ 2 ,
( c m d m C m D m ) = ( a m + 1 b m + 1 A m + 1 B m + 1 ) ,             m = 0 , 1 , 2 , , N - 2.
( a m b m A m B m ) = T m ( a 0 b 0 A 0 B 0 ) ,             m = 0 , 1 , 2 , , N - 1.
λ 1 , 2 = s 1 exp ( ± k 0 p d ) = s 1 ( f 1 ( β ) ± Δ 1 1 / 2 ) ,
λ 3 , 4 = s 2 exp ( ± k 0 r d ) = s 2 ( f 2 ( β ) ± Δ 2 1 / 2 ) ,
f 1 ( β ) = cosh γ 1 cosh γ 2 + 1 2 ( q 1 q 2 + q 2 q 1 ) sinh γ 1 sinh γ 2 ,
f 2 ( β ) = cosh γ 1 cosh γ 2 + 1 2 ( 2 q 1 1 q 2 + 1 q 2 2 q 1 ) × sinh γ 1 sinh γ 2 ,
V 1 , 2 = ( 1 a ± 0 0 ) ,             V 3 , 4 = ( 0 0 1 b ± ) ,
a ± = 1 / 2 ( q 2 / q 1 - q 1 / q 2 ) sinh γ 1 sinh γ 2 ± s 1 Δ 1 1 / 2 1 / q 1 sinh γ 1 cosh γ 2 + 1 / q 2 cosh γ 1 sinh γ 2 ,
b ± = 1 / 2 ( 1 q 2 / 2 q 1 - 2 q 1 / 1 q 2 ) sinh γ 1 sinh γ 2 ± s 2 Δ 2 1 / 2 1 / q 1 sinh γ 1 cosh γ 2 + 2 / q 2 cosh γ 1 sinh γ 2 .
( a m b m A m B m ) = S m D × ( A - exp ( m k 0 p d ) - A + exp ( - m k 0 p d ) a + A - exp ( m k 0 p d ) - a - A + exp ( - m k 0 p d ) B - exp ( m k 0 r d ) - B + exp ( - m k 0 r d ) b + B - exp ( m k 0 r d ) - b - B + exp ( - m k 0 r d ) ) ,
D = [ ( a - - a + ) - 1 0 0 0 0 ( a - - a + ) - 1 0 0 0 0 ( b - - b + ) - 1 0 0 0 0 ( b - - b + ) - 1 ] ,
S = [ s 1 0 0 0 0 s 1 0 0 0 0 s 2 0 0 0 0 s 2 ] .
E o = ( - i β sin θ ,             γ o cos θ ,             γ o sin θ ) A o r ,
H o = ( - β γ o cos θ ,             - i n o 2 sin θ ,             - i γ o 2 cos θ ) A o r
E e = ( i γ e β cos θ ,             - n o 2 sin θ ,             - γ o 2 cos θ ) A e x ,
H e = ( β n o 2 sin θ ,             i γ e n o 2 cos θ ,             i γ e n o 2 sin θ ) A e x
E s = ( i A β ,             B ,             A γ s ) , H s = ( - B β ,             i A n s 2 ,             i B γ s ) ,
γ o cos 2 θ [ ( u 3 - u 4 ) γ o 2 - γ e n o 2 ( u 3 b + - b - u 4 ) ] × [ γ o ( u 1 - u 2 ) - ( a + u 1 - a - u 2 ) ] + n o 2 sin 2 θ [ n o 2 ( u 3 b + - b - u 4 ) - γ o ( u 3 - u 4 ) ] × [ γ e ( u 1 - u 2 ) - ( a + u 1 - a - u 2 ) ] = 0 ,
u 1 = ( a - + γ s ) exp ( - N k 0 p d ) , u 2 = ( a + + γ s ) exp ( N k o p d ) , u 3 = ( b - n s 2 + γ s ) exp ( - N k o r d ) , u 4 = ( b + n s 2 + γ s ) exp ( N k o r d ) .
a = a ± , a i = ± Im a ± , a r = Re a ± , b = b ± , b i = ± Im b ± , b r = Re b ± ,
γ o cos 2 θ ( r 2 γ o 2 - γ e n o 2 r 4 ) ( γ o r 1 - r 3 ) + n o 2 sin 2 θ ( n o 2 r 4 - γ o r 2 ) ( γ e r 1 - r 3 ) = 0 ,
r 1 = - [ a i cos ( N t 1 ) + ( a r + γ s ) sin ( N t 1 ) ] , r 2 = - [ b i n s 2 cos ( N t 2 ) + ( b r n s 2 + γ s ) sin ( N t 2 ) ] , r 3 = a i γ s cos ( N t 1 ) - ( a 2 + γ s a r ) sin ( N t 1 ) , r 4 = b i γ s cos ( N t 2 ) - ( b 2 n s 2 + γ s b r ) sin ( N t 2 ) .
( a m b m A m B m ) = S m D ( a 0 a i cos ( m t 1 ) + ( b 0 - a 0 a r ) sin ( m t 1 ) b 0 a i cos ( m t 1 ) + ( b 0 a r - a 0 a 2 ) sin ( m t 1 ) A 0 b i cos ( m t 2 ) + ( B 0 - A 0 b r ) sin ( m t 2 ) B 0 b i cos ( m t 2 ) + ( B 0 - A 0 b r ) sin ( m t 2 ) ) ,
D = [ a i - 1 0 0 0 0 a i - 1 0 0 0 0 b i - 1 0 0 0 0 b i - 1 ] .

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