Abstract

We present a stable method for analysis of light propagation in one-dimensional photonic crystals that may be composed of general bianisotropic media. Our method is based on the solutions to a generalized eigenproblem of hybrid matrix. It enables Bloch–Floquet waves to be determined reliably and overcomes the numerical instability in the standard eigenproblem of transfer matrix. When the unit cell is lossy and electrically thick, the transfer matrix or its characteristic polynomial may become ill-conditioned, whereas the hybrid matrix is always well-conditioned. Using the imaginary part of Bloch–Floquet wavenumbers, we demonstrate that it is convenient to determine (if any) the frequency range of omnidirectional reflection. Some numerical results are illustrated to investigate the effects of chirality, loss, and tunable anisotropy.

© 2009 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2008 (2)

2007 (2)

I. Nusinsky and A. A. Hardy, “Omnidirectional reflection in several frequency ranges of one-dimensional photonic crystals,” Appl. Opt. 46, 3510-3517 (2007).
[CrossRef] [PubMed]

M. V. Davidovich, J. V. Stephuk, and I. V. Shilin, “Waves in active and dissipative flat-layered periodic and pseudo-periodic structures,” Proc. SPIE 6537, 1-6 (2007).

2006 (3)

C. Vandenbem, J.-P. Vigneron, and J.-M. Vigoureux, “Tunable band structures in uniaxial multilayer stacks,” J. Opt. Soc. Am. B 23, 2366-2376 (2006).
[CrossRef]

E. L. Tan, “Recursive asymptotic impedance matrix method for electromagnetic waves in bianisotropic Media,” IEEE Microw. Wirel. Compon. Lett. 16, 351-353 (2006).
[CrossRef]

E. L. Tan, “Hybrid-matrix algorithm for rigorous coupled-wave analysis of multilayered diffraction gratings,” J. Mod. Opt. 4, 417-428 (2006).
[CrossRef]

2004 (1)

2003 (1)

2002 (2)

E. L. Tan, “Vector wave function expansions of dyadic Green's functions for bianisotropic media,” IEE Proc. Microwaves, Antennas Propag. 149, 57-63 (2002).
[CrossRef]

E. L. Tan, “Note on formulation of the enhanced scattering-(transmittance-) matrix approach,” J. Opt. Soc. Am. A 19, 1157-1161 (2002).
[CrossRef]

2001 (1)

2000 (2)

I. Abdulhalim, “Analytic propagation matrix method for anisotropic magneto-optic layered media,” J. Opt. A, Pure Appl. Opt. 2, 557-564 (2000).
[CrossRef]

V. Galdi and I. M. Pinto, “Derivation of high-order impedance boundary conditions for stratified coatings composed of inhomogeneous-dielectric and homogeneous-bianisotropic layers,” Radio Sci. 35, 287-303 (2000).
[CrossRef]

1999 (2)

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, “Observation of total omnidirectional reflection from a one-dimensional dielectric lattice,” Appl. Phys. A 68, 25-28 (1999).
[CrossRef]

P. St. J. Russell, S. Tredwell, and P. J. Roberts, “Full photonic bandgaps and spontaneous emission control in 1D multilayer dielectric structures,” Opt. Commun. 160, 66-71 (1999).
[CrossRef]

1998 (1)

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679-1682 (1998).
[CrossRef] [PubMed]

1997 (1)

H. D. Yang, “A spectral recursive transformation method for electromagnetic waves in generalized anisotropic media,” IEEE Trans. Antennas Propag. 45, 520-526 (1997).
[CrossRef]

1995 (1)

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press, 1995).

1994 (1)

1992 (1)

1987 (3)

L. M. Barkovskii, G. N. Borzdov, and A. V. Lavrinenko, “Fresnel's reflection and transmission operators for stratified gyroanisotropic media,” J. Phys. A 20, 1095-1106 (1987).
[CrossRef]

E. Yablonovitch, “Inhibited spontaneous emission in solid state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

1985 (1)

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).

1979 (1)

1972 (2)

D. W. Berreman, “Optics in stratified and anisotropic media: 4×4 matrix formulation,” J. Opt. Soc. Am. 62, 502-510 (1972).
[CrossRef]

C. B. Moler and G. W. Stewart, “An algorithm for generalized matrix eigenvalue problems,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 10, 241-256 (1972).
[CrossRef]

Abdulhalim, I.

I. Abdulhalim, “Analytic propagation matrix method for anisotropic magneto-optic layered media,” J. Opt. A, Pure Appl. Opt. 2, 557-564 (2000).
[CrossRef]

Ali, S. M.

Barkovskii, L. M.

L. M. Barkovskii, G. N. Borzdov, and A. V. Lavrinenko, “Fresnel's reflection and transmission operators for stratified gyroanisotropic media,” J. Phys. A 20, 1095-1106 (1987).
[CrossRef]

Berreman, D. W.

Borzdov, G. N.

L. M. Barkovskii, G. N. Borzdov, and A. V. Lavrinenko, “Fresnel's reflection and transmission operators for stratified gyroanisotropic media,” J. Phys. A 20, 1095-1106 (1987).
[CrossRef]

Boucher, Y. G.

Chen, C.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679-1682 (1998).
[CrossRef] [PubMed]

Chigrin, D. N.

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, “Observation of total omnidirectional reflection from a one-dimensional dielectric lattice,” Appl. Phys. A 68, 25-28 (1999).
[CrossRef]

Cojocaru, E.

Davidovich, M. V.

M. V. Davidovich, J. V. Stephuk, and I. V. Shilin, “Waves in active and dissipative flat-layered periodic and pseudo-periodic structures,” Proc. SPIE 6537, 1-6 (2007).

Fan, S.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679-1682 (1998).
[CrossRef] [PubMed]

Fink, Y.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679-1682 (1998).
[CrossRef] [PubMed]

Galdi, V.

V. Galdi and I. M. Pinto, “Derivation of high-order impedance boundary conditions for stratified coatings composed of inhomogeneous-dielectric and homogeneous-bianisotropic layers,” Radio Sci. 35, 287-303 (2000).
[CrossRef]

Gaponenko, S. V.

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, “Observation of total omnidirectional reflection from a one-dimensional dielectric lattice,” Appl. Phys. A 68, 25-28 (1999).
[CrossRef]

Habashy, T. M.

Han, P.

Hardy, A. A.

Joannopoulos, J. D.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679-1682 (1998).
[CrossRef] [PubMed]

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press, 1995).

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

Kong, J. A.

Lavrinenko, A. V.

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, “Observation of total omnidirectional reflection from a one-dimensional dielectric lattice,” Appl. Phys. A 68, 25-28 (1999).
[CrossRef]

L. M. Barkovskii, G. N. Borzdov, and A. V. Lavrinenko, “Fresnel's reflection and transmission operators for stratified gyroanisotropic media,” J. Phys. A 20, 1095-1106 (1987).
[CrossRef]

Le Rouzo, J.

Lekner, J.

Liu, C. P.

Mao, D.

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press, 1995).

Michel, J.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679-1682 (1998).
[CrossRef] [PubMed]

Mohamed, I. R.

Moler, C. B.

C. B. Moler and G. W. Stewart, “An algorithm for generalized matrix eigenvalue problems,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 10, 241-256 (1972).
[CrossRef]

Nusinsky, I.

Ouyang, Z. B.

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).

Pinto, I. M.

V. Galdi and I. M. Pinto, “Derivation of high-order impedance boundary conditions for stratified coatings composed of inhomogeneous-dielectric and homogeneous-bianisotropic layers,” Radio Sci. 35, 287-303 (2000).
[CrossRef]

Roberts, P. J.

P. St. J. Russell, S. Tredwell, and P. J. Roberts, “Full photonic bandgaps and spontaneous emission control in 1D multilayer dielectric structures,” Opt. Commun. 160, 66-71 (1999).
[CrossRef]

Russell, P. St. J.

P. St. J. Russell, S. Tredwell, and P. J. Roberts, “Full photonic bandgaps and spontaneous emission control in 1D multilayer dielectric structures,” Opt. Commun. 160, 66-71 (1999).
[CrossRef]

Shilin, I. V.

M. V. Davidovich, J. V. Stephuk, and I. V. Shilin, “Waves in active and dissipative flat-layered periodic and pseudo-periodic structures,” Proc. SPIE 6537, 1-6 (2007).

Stephuk, J. V.

M. V. Davidovich, J. V. Stephuk, and I. V. Shilin, “Waves in active and dissipative flat-layered periodic and pseudo-periodic structures,” Proc. SPIE 6537, 1-6 (2007).

Stewart, G. W.

C. B. Moler and G. W. Stewart, “An algorithm for generalized matrix eigenvalue problems,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 10, 241-256 (1972).
[CrossRef]

Tan, E. L.

E. L. Tan, “Recursive asymptotic impedance matrix method for electromagnetic waves in bianisotropic Media,” IEEE Microw. Wirel. Compon. Lett. 16, 351-353 (2006).
[CrossRef]

E. L. Tan, “Hybrid-matrix algorithm for rigorous coupled-wave analysis of multilayered diffraction gratings,” J. Mod. Opt. 4, 417-428 (2006).
[CrossRef]

E. L. Tan, “Vector wave function expansions of dyadic Green's functions for bianisotropic media,” IEE Proc. Microwaves, Antennas Propag. 149, 57-63 (2002).
[CrossRef]

E. L. Tan, “Note on formulation of the enhanced scattering-(transmittance-) matrix approach,” J. Opt. Soc. Am. A 19, 1157-1161 (2002).
[CrossRef]

Thomas, E. L.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679-1682 (1998).
[CrossRef] [PubMed]

Tredwell, S.

P. St. J. Russell, S. Tredwell, and P. J. Roberts, “Full photonic bandgaps and spontaneous emission control in 1D multilayer dielectric structures,” Opt. Commun. 160, 66-71 (1999).
[CrossRef]

Vandenbem, C.

Vigneron, J.-P.

Vigoureux, J.-M.

Wang, H.

Wang, J. C.

Winn, J. N.

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679-1682 (1998).
[CrossRef] [PubMed]

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press, 1995).

Wu, T. X.

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

Yang, H. D.

H. D. Yang, “A spectral recursive transformation method for electromagnetic waves in generalized anisotropic media,” IEEE Trans. Antennas Propag. 45, 520-526 (1997).
[CrossRef]

Yang, X.

Yarotsky, D. A.

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, “Observation of total omnidirectional reflection from a one-dimensional dielectric lattice,” Appl. Phys. A 68, 25-28 (1999).
[CrossRef]

Yeh, P.

Appl. Opt. (2)

Appl. Phys. A (1)

D. N. Chigrin, A. V. Lavrinenko, D. A. Yarotsky, and S. V. Gaponenko, “Observation of total omnidirectional reflection from a one-dimensional dielectric lattice,” Appl. Phys. A 68, 25-28 (1999).
[CrossRef]

IEE Proc. Microwaves, Antennas Propag. (1)

E. L. Tan, “Vector wave function expansions of dyadic Green's functions for bianisotropic media,” IEE Proc. Microwaves, Antennas Propag. 149, 57-63 (2002).
[CrossRef]

IEEE Microw. Wirel. Compon. Lett. (1)

E. L. Tan, “Recursive asymptotic impedance matrix method for electromagnetic waves in bianisotropic Media,” IEEE Microw. Wirel. Compon. Lett. 16, 351-353 (2006).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

H. D. Yang, “A spectral recursive transformation method for electromagnetic waves in generalized anisotropic media,” IEEE Trans. Antennas Propag. 45, 520-526 (1997).
[CrossRef]

J. Mod. Opt. (1)

E. L. Tan, “Hybrid-matrix algorithm for rigorous coupled-wave analysis of multilayered diffraction gratings,” J. Mod. Opt. 4, 417-428 (2006).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

I. Abdulhalim, “Analytic propagation matrix method for anisotropic magneto-optic layered media,” J. Opt. A, Pure Appl. Opt. 2, 557-564 (2000).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

J. Opt. Soc. Am. B (4)

J. Phys. A (1)

L. M. Barkovskii, G. N. Borzdov, and A. V. Lavrinenko, “Fresnel's reflection and transmission operators for stratified gyroanisotropic media,” J. Phys. A 20, 1095-1106 (1987).
[CrossRef]

Opt. Commun. (1)

P. St. J. Russell, S. Tredwell, and P. J. Roberts, “Full photonic bandgaps and spontaneous emission control in 1D multilayer dielectric structures,” Opt. Commun. 160, 66-71 (1999).
[CrossRef]

Phys. Rev. Lett. (2)

E. Yablonovitch, “Inhibited spontaneous emission in solid state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486-2489 (1987).
[CrossRef] [PubMed]

Proc. SPIE (1)

M. V. Davidovich, J. V. Stephuk, and I. V. Shilin, “Waves in active and dissipative flat-layered periodic and pseudo-periodic structures,” Proc. SPIE 6537, 1-6 (2007).

Radio Sci. (1)

V. Galdi and I. M. Pinto, “Derivation of high-order impedance boundary conditions for stratified coatings composed of inhomogeneous-dielectric and homogeneous-bianisotropic layers,” Radio Sci. 35, 287-303 (2000).
[CrossRef]

Science (1)

Y. Fink, J. N. Winn, S. Fan, C. Chen, J. Michel, J. D. Joannopoulos, and E. L. Thomas, “A dielectric omnidirectional reflector,” Science 282, 1679-1682 (1998).
[CrossRef] [PubMed]

SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. (1)

C. B. Moler and G. W. Stewart, “An algorithm for generalized matrix eigenvalue problems,” SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 10, 241-256 (1972).
[CrossRef]

Other (2)

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press, 1995).

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Figures (11)

Fig. 1
Fig. 1

Geometry of one-dimensional photonic crystals.

Fig. 2
Fig. 2

Characteristic polynomial coefficients of cell transfer matrix versus k 0 h F . The cell is made up by stacked Li Nb O 3 and liquid crystal layers, both with permittivity tensors (18), h 1 = 2 h F 3 , h 2 = h F 3 , ε a 1 = 4.12 + 9.38 i , ε c 1 = 6.27 + 6.40 i , ε a 2 = 2.25 , and ε c 2 = 13.2 .

Fig. 3
Fig. 3

Condition numbers of T c , Z c , and H c matrices versus k 0 h F . The cell is the same as the one in Fig. 2.

Fig. 4
Fig. 4

Dispersion relations of k z F and k 0 h F as calculated by (a) standard eigenproblem of transfer matrix and (b) generalized eigenproblem of hybrid matrix methods. The cell is the same as the one in Fig. 2.

Fig. 5
Fig. 5

Relative norm errors of (a) eigenvectors for single layer ( Li Nb O 3 ) and (b) impedance matrices for unit cell (taken from Fig. 2) as functions of k 0 h F .

Fig. 6
Fig. 6

Real and imaginary parts of k z F as functions of k 0 h F for a range of incident angles. The unit cell is composed of two layers of arbitrarily oriented liquid crystals with permittivity tensor (46), ε a = 2.25 , ε c = 13.2 , ϕ 1 = 0 ° , and ϕ 2 = 90 ° .

Fig. 7
Fig. 7

Variation of reflectivity against incident angles at k 0 h F 2 π = 0.20 when omnidirectional reflection occurs, cf. Fig. 6, with n = 10 cells.

Fig. 8
Fig. 8

Variation of reflectivity against incident angles with chirality included at k 0 h F 2 π = 0.43 . Chirality is introduced into the parameters of Fig. 6 via additional magnetoelectric tensors, ξ = ς = i κ μ 0 ε 0 I , κ = 0.3 .

Fig. 9
Fig. 9

Real and imaginary parts of k z F as functions of k 0 h F at normal incidence for lossless “●” and lossy “+” unit cells. The cell parameters are the same as those in Fig. 6 except that for the lossy case, ε a = 2.25 + 0.1 i , ε c = 13.2 + 0.1 i .

Fig. 10
Fig. 10

Dispersion relations of k z F and k 0 h F as calculated by (a) standard eigenproblem of transfer matrix and (b) generalized eigenproblem of hybrid matrix methods for the lossy unit cell in Fig. 9.

Fig. 11
Fig. 11

Variation of total reflectivity against incident angles at k 0 h F 2 π = 0.20 for lossy materials (same as Fig. 9) with tunable anisotropy (a) ϕ 1 = 0 ° , ϕ 2 = 90 ° ; (b) ϕ 1 = 90 ° , ϕ 2 = 0 ° .

Equations (55)

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

[ D ¯ B ¯ ] = [ ε ξ ζ μ ] [ E ¯ H ¯ ] .
d d z [ E ¯ t H ¯ t ] = A [ E ¯ t H ¯ t ] .
A = [ A 11 A 12 A 21 A 22 ] = i ω Γ v [ ( K t z M t z ) M z z 1 ( K z t M z t ) M t t ] ,
M t t = [ ε x x ε x y ξ x x ξ x y ε y x ε y y ξ y x ξ y y ζ x x ζ x y μ x x μ x y ζ y x ζ y y μ y x μ y y ] , M t z = [ ε x z ξ x z ε y z ξ y z ζ x z μ x z ζ y z μ y z ] ,
M z t = [ ε z x ε z y ξ z x ξ z y ζ z x ζ z y μ z x μ z y ] , M z z = [ ε z z ξ z z ζ z z μ z z ] ,
Γ v = [ 0 0 0 1 0 0 1 0 0 1 0 0 1 0 0 0 ] , K t z = 1 ω [ 0 k y 0 k x k y 0 k x 0 ] ,
K z t = 1 ω [ 0 0 k y k x k y k x 0 0 ] .
[ E ¯ t f ( z ) H ¯ t f ( z ) ] = ψ f P f ( z ) c ¯ f = ψ f w ¯ f ( z ) .
ψ f = [ e f > e f < h f > h f < ] , w ¯ f = [ w ¯ f > w ¯ f < ] , P f = [ P f > 0 0 P f < ] ,
[ E ¯ t ( Z f > ) H ¯ t ( Z f > ) ] = T f [ E ¯ t ( Z f < ) H ¯ t ( Z f < ) ] ,
T f = ψ f P f ( h f ) ψ f 1 ,
T c = T N T 2 T 1 .
T c ψ F ( j ) = p F ( j ) ψ F ( j ) .
ψ F = [ e F > e F < h F > h F < ] , P F = [ P F > 0 0 P F < ] .
T c = ψ F P F ( h F ) ψ F 1 ,
det ( T c p F I ) = p F 4 + a 3 p F 3 + a 2 p F 2 + a 1 p F + a 0 = 0 ,
a 3 = tr ( T c ) ,
a 2 = [ tr 2 ( T c ) tr ( T c 2 ) ] 2 ,
a 1 = tr 3 ( T c ) 6 + tr ( T c ) tr ( T c 2 ) 2 tr ( T c 3 ) 3 ,
a 0 = det ( T c ) .
ϵ = ϵ 0 [ ϵ c 0 0 0 ϵ a 0 0 0 ϵ a ] ,
[ E ¯ t ( Z f < ) H ¯ t ( Z f > ) ] = H f [ H ¯ t ( Z f < ) E ¯ t ( Z f > ) ] , H f = [ H f , 11 H f , 12 H f , 21 H f , 22 ] .
H f = [ e f > e f < P f < ( h f ) h f > P f > ( h f ) h f < ] [ h f > h f < P f < ( h f ) e f > P f > ( h f ) e f < ] 1 .
[ E ¯ t ( Z i < ) H ¯ t ( Z j > ) ] = H ( i , j ) [ H ¯ t ( Z i < ) E ¯ t ( Z j > ) ] ,
H 11 ( f , N ) = H f , 11 + H f , 12 H 11 ( f + 1 , N ) [ I H f , 22 H 11 ( f + 1 , N ) ] 1 H f , 21 ,
H 12 ( f , N ) = H f , 12 [ I H 11 ( f + 1 , N ) H f , 22 ] 1 H 12 ( f + 1 , N ) ,
H 21 ( f , N ) = H 21 ( f + 1 , N ) [ I H f , 22 H 11 ( f + 1 , N ) ] 1 H f , 21 ,
H 22 ( f , N ) = H 22 ( f + 1 , N ) + H 21 ( f + 1 , N ) H f , 22 [ I H 11 ( f + 1 , N ) H f , 22 ] 1 H 12 ( f + 1 , N ) .
H c = H ( 1 , N ) .
H A ψ F ( j ) β ( j ) = H B ψ F ( j ) α ( j ) ,
H A = [ I H c 11 0 H c 21 ] , H B = [ H c 12 0 H c 22 I ] .
H c = [ e F > e F < P F < 1 h F > P F > h F < ] [ h F > h F < P F < 1 e F > P F > e F < ] 1 .
H c h F = [ e F > ( h F > ) 1 0 0 h F < ( e F < ) 1 ] .
T c h F = [ 0 e F < 0 h F < ] [ e F > 0 h F > 0 ] 1 ,
[ E ¯ t ( Z 1 < ) E ¯ t ( Z N > ) ] = Z c [ H ¯ t ( Z 1 < ) H ¯ t ( Z N > ) ] ,
Z c = [ e F > e F < P F < 1 e F > P F > e F < ] [ h F > h F < P F < 1 h F > P F > h F < ] 1 .
Z c h F = 0 = [ e F > e F < e F > e F < ] [ h F > h F < h F > h F < ] 1 ,
H c h F = 0 = [ 0 I I 0 ] .
k z F ( j ) = ( 2 m π + arg p F ( j ) i ln p F ( j ) ) h F = ( 2 m π + arg α ( j ) arg β ( j ) ) h F i ( ln α ( j ) ln β ( j ) ) h F ,
m = 0 , ± 1 , ± 2 , .
w ¯ 0 < ( Z 0 > ) = r 0 , 1 w ¯ 0 > ( Z 0 > ) ,
w ¯ n N + 1 > ( Z n N + 1 < ) = t 0 , n N + 1 w ¯ 0 > ( Z 0 > ) .
H ( 1 , n N ) = [ e F > e F < ( P F < 1 ) n h F > ( P F > ) n h F < ] [ h F > h F < ( P F < 1 ) n e F > ( P F > ) n e F < ] 1 .
r 0 , 1 = [ H S h 0 < e 0 < ] 1 [ e 0 > H S h 0 > ] ,
t 0 , n N + 1 = [ h n N + 1 > H 22 ( 1 , n N + 1 ) e n N + 1 > ] 1 H 21 ( 1 , n N + 1 ) [ h 0 > + h 0 < r 0 , 1 ] ,
H S = H 11 ( 1 , n N + 1 ) + H 12 ( 1 , n N + 1 ) [ h n N + 1 > ( e n N + 1 > ) 1 H 22 ( 1 , n N + 1 ) ] 1 H 21 ( 1 , n N + 1 ) ,
ψ 0 = [ e 0 > e 0 < h 0 > h 0 < ] = [ cos θ i 0 cos θ i 0 0 1 0 1 0 cos θ i η 0 cos θ i η 1 η 0 1 η 0 ] ,
r 0 , 1 = [ e F > ( h F > ) 1 h 0 < e 0 < ] 1 [ e 0 > e F > ( h F > ) 1 h 0 > ] .
r 0 , 1 = [ r p p r p s r s p r s s ] , t 0 , n N + 1 = [ t p p t p s t s p t s s ] .
ε = ε 0 [ ε c cos 2 ϕ + ε a sin 2 ϕ ( ε c ε a ) cos ϕ sin ϕ 0 ( ε c ε a ) cos ϕ sin ϕ ε a cos 2 ϕ + ε c sin 2 ϕ 0 0 0 ε a ] ,
[ E ¯ t ( Z f < ) E ¯ t ( Z f > ) ] = Z f [ H ¯ t ( Z f < ) H ¯ t ( Z f > ) ] .
Z 11 ( f , N ) = Z f , 11 + Z f , 12 [ Z 11 ( f + 1 , N ) Z f , 22 ] 1 Z f , 21 ,
Z 12 ( f , N ) = Z f , 12 [ Z 11 ( f + 1 , N ) Z f , 22 ] 1 Z 12 ( f + 1 , N ) ,
Z 21 ( f , N ) = Z 21 ( f + 1 , N ) [ Z 11 ( f + 1 , N ) Z f , 22 ] 1 Z f , 12 ,
Z 11 ( f , N ) = Z 22 ( f + 1 , N ) Z 21 ( f + 1 , N ) [ Z 11 ( f + 1 , N ) Z f , 22 ] 1 Z 12 ( f + 1 , N ) .

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