Abstract

The wave behavior within a one-dimensional magnetoelectric photonic crystal is studied. A very simple, accurate analytical method of investigating the wave transmission along an arbitrary direction is developed. The method is based on the 2×2 translation matrix instead of the well-known 4×4 matrices. The TE- and TM-wave decomposition of a total wave in a one-dimensional magnetoelectric stack is presented in an accurate analytical form, to our knowledge for the first time, and the physical interpretation of the translation matrix determinant is given, also as far as we know for the first time.

© 2007 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  7. S. R. Kennedy, M. J. Brett, H. Miguez, O. Toader, and S. John, "Optical properties of a three-dimensional silicon square spiral photonic crystal," Photonics Nanostruct. Fundam. Appl. 1, 37-42 (2003).
    [CrossRef]
  8. L. I. Lyubchanskii, N. N. Dadoenkova, M. I. Lyubchanskii, E. A. Shapovalov, and Th. Rasing, "Magnetic photonic crystals," J. Phys. D 36, 277-287 (2003).
    [CrossRef]
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2003 (4)

S. R. Kennedy, M. J. Brett, H. Miguez, O. Toader, and S. John, "Optical properties of a three-dimensional silicon square spiral photonic crystal," Photonics Nanostruct. Fundam. Appl. 1, 37-42 (2003).
[CrossRef]

L. I. Lyubchanskii, N. N. Dadoenkova, M. I. Lyubchanskii, E. A. Shapovalov, and Th. Rasing, "Magnetic photonic crystals," J. Phys. D 36, 277-287 (2003).
[CrossRef]

S. Khorasan and K. Mehrany, "Differential transfer-matrix method for solution of one-dimensional linear nonhomogeneous optical structures," J. Opt. Soc. Am. B 20, 91-96 (2003).
[CrossRef]

C. Jamois, R. B. Wehrspohn, L. C. Andreani, C. Hermann, O. Hess, and U. Gosele, "Silicon-based two-dimensional photonic crystal waveguides," Photonics Nanostruct. Fundam. Appl. 1, 1-13 (2003).
[CrossRef]

2002 (1)

H. Wohler and M. E. Becker, "Numerical modeling of LCD electro-optical performance," Opto-Electron. Rev. 10, 23-33 (2002).

2001 (1)

A. Figotin and I. Vitebsky, "Nonreciprocal magnetic photonic crystals," Phys. Rev. E 63, 066609-1-066609-17 (2001).
[CrossRef]

2000 (2)

M. Qiu and S. He, "Optimal design of a two-dimensional photonic crystal of square lattice with a large complete two-dimensional bandgap," J. Opt. Soc. Am. B 17, 1027-1030 (2000).
[CrossRef]

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

1998 (2)

A. Chutinan and S. Noda, "Spiral three-dimensional photonic-band-gap structure," Phys. Rev. B 57, R2006-R2008 (1998).
[CrossRef]

D. Felbacq, B. Guizal, and F. Zolla, "Wave propagation in one-dimensional photonic crystals," Opt. Commun. 152, 119-126 (1998).
[CrossRef]

1997 (1)

E. B. McDaniel, J. W. P. Hsu, L. S. Goldner, R. J. Jonucci, E. L. Shirley, and G. W. Bryant, "Local characterization of transmission properties of a two-dimensional crystal," Phys. Rev. B 55, 10878-10882 (1997).
[CrossRef]

1994 (1)

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, "Photonic band gaps in three dimensions: new layer-by-layer periodic structures," Solid State Commun. 89, 413-416 (1994).
[CrossRef]

1993 (1)

I. H. Zabel and D. Stroud, "Photonic band structures of optically anisotropic periodic arrays," Phys. Rev. B 48, 5004-5021 (1993).
[CrossRef]

1991 (2)

D. Ager and H. P. Hughes, "Optical properties of stratified systems including lamellar gratings," Phys. Rev. B 44, 13452-13465 (1991).
[CrossRef]

H. L. Ong, "Electro-optics of a twisted nematic liquid crystal display by 2×2 propagation matrix at oblique incidence," Jpn. J. Appl. Phys., Part 1 30, L1028-L1031 (1991).
[CrossRef]

1990 (2)

M. Mansuripur, "Analysis of multilayered thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2×2 matrices," J. Appl. Phys. 67, 6466-6475 (1990).
[CrossRef]

A. Lien, "The general and simplified Jones matrix representations for the high pretilt twisted nematic cell," J. Appl. Phys. 67, 2853-2856 (1990).
[CrossRef]

1987 (1)

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

1982 (1)

1979 (1)

1976 (1)

F. Gharadjedaghi and J. Robert, "Comportement electro-optique d'une structure nématique en hélice--application à l'affichage," Rev. Phys. Appl. 11, 467-473 (1976).
[CrossRef]

1972 (1)

1970 (1)

1941 (1)

J. Appl. Phys. (2)

A. Lien, "The general and simplified Jones matrix representations for the high pretilt twisted nematic cell," J. Appl. Phys. 67, 2853-2856 (1990).
[CrossRef]

M. Mansuripur, "Analysis of multilayered thin-film structures containing magneto-optic and anisotropic media at oblique incidence using 2×2 matrices," J. Appl. Phys. 67, 6466-6475 (1990).
[CrossRef]

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

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

J. Opt. Soc. Am. (5)

J. Opt. Soc. Am. B (2)

J. Phys. D (1)

L. I. Lyubchanskii, N. N. Dadoenkova, M. I. Lyubchanskii, E. A. Shapovalov, and Th. Rasing, "Magnetic photonic crystals," J. Phys. D 36, 277-287 (2003).
[CrossRef]

Jpn. J. Appl. Phys., Part 1 (1)

H. L. Ong, "Electro-optics of a twisted nematic liquid crystal display by 2×2 propagation matrix at oblique incidence," Jpn. J. Appl. Phys., Part 1 30, L1028-L1031 (1991).
[CrossRef]

Opt. Commun. (1)

D. Felbacq, B. Guizal, and F. Zolla, "Wave propagation in one-dimensional photonic crystals," Opt. Commun. 152, 119-126 (1998).
[CrossRef]

Opto-Electron. Rev. (1)

H. Wohler and M. E. Becker, "Numerical modeling of LCD electro-optical performance," Opto-Electron. Rev. 10, 23-33 (2002).

Photonics Nanostruct. Fundam. Appl. (2)

S. R. Kennedy, M. J. Brett, H. Miguez, O. Toader, and S. John, "Optical properties of a three-dimensional silicon square spiral photonic crystal," Photonics Nanostruct. Fundam. Appl. 1, 37-42 (2003).
[CrossRef]

C. Jamois, R. B. Wehrspohn, L. C. Andreani, C. Hermann, O. Hess, and U. Gosele, "Silicon-based two-dimensional photonic crystal waveguides," Photonics Nanostruct. Fundam. Appl. 1, 1-13 (2003).
[CrossRef]

Phys. Rev. B (4)

E. B. McDaniel, J. W. P. Hsu, L. S. Goldner, R. J. Jonucci, E. L. Shirley, and G. W. Bryant, "Local characterization of transmission properties of a two-dimensional crystal," Phys. Rev. B 55, 10878-10882 (1997).
[CrossRef]

A. Chutinan and S. Noda, "Spiral three-dimensional photonic-band-gap structure," Phys. Rev. B 57, R2006-R2008 (1998).
[CrossRef]

I. H. Zabel and D. Stroud, "Photonic band structures of optically anisotropic periodic arrays," Phys. Rev. B 48, 5004-5021 (1993).
[CrossRef]

D. Ager and H. P. Hughes, "Optical properties of stratified systems including lamellar gratings," Phys. Rev. B 44, 13452-13465 (1991).
[CrossRef]

Phys. Rev. E (1)

A. Figotin and I. Vitebsky, "Nonreciprocal magnetic photonic crystals," Phys. Rev. E 63, 066609-1-066609-17 (2001).
[CrossRef]

Phys. Rev. Lett. (1)

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

Rev. Phys. Appl. (1)

F. Gharadjedaghi and J. Robert, "Comportement electro-optique d'une structure nématique en hélice--application à l'affichage," Rev. Phys. Appl. 11, 467-473 (1976).
[CrossRef]

Solid State Commun. (1)

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, "Photonic band gaps in three dimensions: new layer-by-layer periodic structures," Solid State Commun. 89, 413-416 (1994).
[CrossRef]

Other (12)

J. K. Hale, Oscillations in Nonlinear Systems (McGraw-Hill, 1963).

G. Kauderer, Nichtlineare Mechanik (Springer-Verlag, 1958).
[CrossRef]

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

L. D. Landau, E. M. Lifshitz, and L. P. Pitaevskii, Electrodynamics of Continuous Media (Pergamon, 1984).

N. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference, and Diffraction of Light, 5th ed. (Pergamon, 1975), Sec. 11.3.

P. Lancaster and M. Tismenetsky, The Theory of Matrices With Applications, 2nd ed. (Academic, 1985).

J. A. Kong, Electromagnetic Wave Theory, 2nd ed. (Wiley, 1990).

M. Mansuripur, The Principles of Magneto-Optical Recording (Cambridge U. Press, 1995).
[CrossRef]

Ph. Russell, T. Birks, and F. D. Lloyd-Lucas, Photonic Bloch Waves and Photonic Band Gaps (Plenum, 1995).

B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics (Wiley, 1991).
[CrossRef]

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley-Interscience, 1984).

K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001).

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Figures (2)

Fig. 1
Fig. 1

Illustration of translations of the vector R 0 ( E x 0 , H y 0 ) for the case of a unimodular translation matrix ( R 1 ) and for the case in which the determinant of this matrix is an increasing function ( R 2 ) .

Fig. 2
Fig. 2

Illustration of translations of the vector R: (a) det M ( z ) = 1 , (b) det M ( z ) is an increasing function of z, (c) det M ( z ) is a decreasing function of z, (d) det M ( z ) is a periodic function of z.

Equations (74)

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2 U 1 z 2 + c 11 2 U 1 z + c 12 2 U 2 z + c 13 U 1 + c 14 U 2 = 0 ,
2 U 2 z 2 + c 21 2 U 2 z + c 22 2 U 1 z + c 23 U 2 + c 24 U 1 = 0 ,
ε ̿ = ε x x ( z ) ε x y ( z ) 0 ε x y ( z ) ε x x ( z ) 0 0 0 ε z z ( z ) ,
μ ̿ = μ x x ( z ) μ x y ( z ) 0 μ x y ( z ) μ x x ( z ) 0 0 0 μ z z ( z ) .
E x z = a 11 H x + a 12 H y , H x z = b 11 E x + b 12 E y ,
E y z = a 21 H x + a 22 H y , H y z = b 21 E x + b 22 E y .
E x = E 1 + E 2 ,
E y = ξ 1 E 1 + ξ 2 E 2 ,
H x = ς 1 H 1 + ς 2 H 2 ,
H y = H 1 + H 2 ,
E 1 z = γ 1 H 1 , H 1 z = χ 1 E 1 ,
E 2 z = γ 2 H 2 , H 2 z = χ 2 E 2 ,
γ 1 = a 11 ξ 2 + a 12 ς 1 ξ 2 a 21 a 22 ς 1 ξ 2 ξ 1 ,
γ 2 = a 11 ξ 1 + a 12 ς 2 ξ 1 a 21 a 22 ς 2 ξ 2 ξ 1 ,
χ 1 = b 21 ς 2 + b 22 ξ 1 ς 2 b 11 b 12 ξ 1 ς 2 ς 1 ,
χ 2 = b 21 ς 1 + b 22 ξ 2 ς 1 b 11 b 12 ξ 2 ς 2 ς 1 ,
ξ 1 , 2 = a 22 b 22 a 11 b 11 + a 21 b 12 a 12 b 21 2 ( a 11 b 12 + a 12 b 22 ) [ 1 ± 1 + 4 ( a 22 b 21 + a 21 b 11 ) ( a 11 b 12 + a 12 b 22 ) ( a 11 b 11 a 22 b 22 + a 12 b 21 a 21 b 12 ) 2 ] ,
ς 1 , 2 = b 12 ξ 1 , 2 + b 11 b 22 ξ 1 , 2 + b 21 .
2 E 1 z 2 γ 1 χ 1 E 1 = 0 ,
2 E 2 z 2 γ 2 χ 2 E 2 = 0 .
M 1 , 2 i = cos k 1 , 2 z γ 1 , 2 k 1 , 2 sin k 1 , 2 z k 1 , 2 γ 1 , 2 sin k 1 , 2 z cos k 1 , 2 z .
E x 0 H y 0 = E 10 + E 20 H 10 + H 20 = 1 0 0 1 E 10 H 10 + 1 0 0 1 E 20 H 20 = M 1 ( 0 ) E 10 H 10 + M 2 ( 0 ) E 20 H 20 ,
E x ( z ) H y ( z ) = E 1 ( z ) H 1 ( z ) + E 2 ( z ) H 2 ( z ) = M 1 ( z ) E 10 H 10 + M 2 ( z ) E 20 H 20 .
M ( z ) = E x 0 H y 0 = M 1 ( z ) E 10 H 10 + M 2 ( z ) E 20 H 20 .
M ( z ) = χ 2 γ 1 exp ( j k 1 z ) χ 1 γ 2 exp ( j k 2 z ) χ 2 γ 1 χ 1 γ 2 γ 1 γ 2 exp ( j k 2 z ) exp ( j k 1 z ) χ 2 γ 1 χ 1 γ 2 j χ 1 γ 1 χ 2 γ 1 exp ( j k 1 z ) χ 1 γ 2 exp ( j k 2 z ) χ 2 γ 1 χ 1 γ 2 χ 2 γ 1 exp ( j k 2 z ) χ 1 γ 2 exp ( j k 1 z ) χ 2 γ 1 χ 1 γ 2 .
M 1 ( z ) = 1 χ 2 γ 1 χ 1 γ 2 χ 2 γ 1 exp ( j k 1 z ) γ 1 γ 2 exp ( j k 1 z ) j χ 1 χ 2 exp ( j k 1 z ) χ 1 γ 2 exp ( j k 1 z ) ,
M 2 ( z ) = 1 χ 2 γ 1 χ 1 γ 2 χ 1 γ 2 exp ( j k 2 z ) γ 1 γ 2 exp ( j k 2 z ) j χ 1 γ 2 γ 1 exp ( j k 2 z ) χ 2 γ 1 exp ( j k 2 z ) .
L ( z ) = M ( z ) 0 0 N ( z ) .
E y = 1 b 22 H y z b 21 b 22 E x ,
H x = 1 a 11 E x z a 12 a 11 H y .
E x ( z ) H y ( z ) = M ( z ) E x 0 H y 0
L Σ = i = 1 N M i 0 0 i = 1 N N i
a 11 = j ( ω μ x y + k x k y ω ε z z ) , a 12 = j ( ω μ x x k x 2 ω ε z z ) ,
a 21 = j ( ω μ x x k y 2 ω ε z z ) , a 22 = j ( ω μ x y k x k y ω ε z z ) ,
b 11 = j ( ω ε x y + k x k y ω μ z z ) , b 12 = j ( ω ε x x k x 2 ω μ z z )
b 21 = j ( ω ε x x k y 2 ω μ z z ) , b 22 = j ( ω ε x y k x k y ω μ z z ) .
E 1 z + E 2 z = ( a 11 ς 1 + a 12 ) H 1 + ( a 11 ς 2 + a 12 ) H 2 ,
ξ 1 E 1 z + ξ 2 E 2 z = ( a 21 ς 1 + a 22 ) H 1 + ( a 21 ς 2 + a 22 ) H 2 .
( ξ 2 ξ 1 ) E 1 z = ( a 11 ς 1 ξ 2 + a 12 ξ 2 a 21 ς 1 a 22 ) H 1 + ( a 11 ς 2 ξ 2 + a 21 ξ 2 a 21 ς 2 a 22 ) H 2 .
( ξ 1 ξ 2 ) E 2 z = ( a 11 ς 1 ξ 1 + a 12 ξ 1 a 21 ς 1 a 22 ) H 1 + ( a 11 ς 2 ξ 1 + a 21 ξ 1 a 21 ς 2 a 22 ) H 2 .
( ς 2 ς 1 ) H 1 z = ( b 22 ς 2 ξ 1 + b 21 ς 2 b 12 ξ 1 b 11 ) E 1 + ( b 22 ς 2 ξ 2 + b 21 ς 2 b 12 ξ 2 b 11 ) E 2 ,
( ς 1 ς 2 ) H 2 z = ( b 22 ς 1 ξ 1 + b 21 ς 1 b 12 ξ 1 b 11 ) E 1 + ( b 22 ς 1 ξ 2 + b 21 ς 1 b 12 ξ 2 b 11 ) E 2 .
a 11 ς 2 ξ 2 + a 12 ξ 2 a 21 ς 2 a 22 = 0 ,
a 11 ς 1 ξ 1 + a 12 ξ 1 a 21 ς 1 a 22 = 0 ;
b 22 ς 2 ξ 2 + b 21 ς 2 b 12 ξ 2 b 11 = 0 ,
b 22 ς 1 ξ 1 + b 21 ς 1 b 12 ξ 1 b 11 = 0 .
ς 1 , 2 = a 12 ξ 1 , 2 a 22 a 11 ξ 1 , 2 a 21
ς 1 , 2 = b 12 ξ 1 , 2 + b 11 b 22 ξ 1 , 2 + b 21 .
M 11 E x 0 + M 12 H y 0 = m 11 ( 1 ) E 10 + m 12 ( 1 ) H 10 + m 11 ( 2 ) E 20 + m 12 ( 2 ) H 20 ,
M 21 E x 0 + M 22 H y 0 = m 21 ( 1 ) E 10 + m 22 ( 1 ) H 10 + m 21 ( 2 ) E 20 + m 22 ( 2 ) H 20 .
M 11 E 10 + M 11 E 20 + M 12 H 10 + M 12 H 20 = m 11 ( 1 ) E 10 + m 12 ( 1 ) H 10 + m 11 ( 2 ) E 20 + m 12 ( 2 ) H 20 .
H 1 , 2 = j k 1 , 2 γ 1 , 2 E 1 , 2 = χ 1 , 2 γ 1 , 2 E 1 , 2 .
M 11 E 10 + M 11 E 20 + j k 1 γ 1 M 12 E 10 + j k 2 γ 2 M 12 E 20 = m 11 ( 1 ) E 10 + j k 1 γ 1 m 12 ( 1 ) E 10 + m 11 ( 2 ) E 20 + j k 2 γ 2 m 12 ( 2 ) E 20 .
M 11 + j k 1 γ 1 M 12 = m 11 ( 1 ) + j k 1 γ 1 m 12 ( 1 ) ,
M 11 + j k 2 γ 2 M 12 = m 11 ( 2 ) + j k 2 γ 2 m 12 ( 2 ) .
M 11 = m 11 ( 1 ) + j k 1 γ 1 m 12 ( 1 ) k 1 γ 1 m 11 ( 2 ) m 11 ( 1 ) + j k 2 γ 2 m 12 ( 2 ) j k 1 γ 1 m 12 ( 1 ) k 2 γ 2 k 1 γ 1 ,
M 12 = m 11 ( 2 ) m 11 ( 1 ) + j k 2 γ 2 m 12 ( 2 ) j k 1 γ 1 m 12 ( 1 ) k 2 γ 2 k 1 γ 1 .
M 21 = m 21 ( 1 ) + j k 1 γ 1 m 22 ( 1 ) k 1 γ 1 m 21 ( 2 ) m 21 ( 1 ) + j k 2 γ 2 m 22 ( 2 ) j k 1 γ 1 m 22 ( 1 ) k 2 γ 2 k 1 γ 1 ,
M 22 = m 21 ( 2 ) m 21 ( 1 ) + j k 2 γ 2 m 22 ( 2 ) j k 1 γ 1 m 22 ( 1 ) k 2 γ 2 k 1 γ 1 .
M 11 = exp ( j k 1 z ) k 1 γ 2 k 2 γ 1 k 1 γ 2 [ exp ( j k 2 z ) exp ( j k 1 z ) ] ,
M 12 = j γ 1 γ 2 k 2 γ 1 k 1 γ 2 [ exp ( j k 2 z ) exp ( j k 1 z ) ] ,
M 21 = j { k 1 γ 1 exp ( j k 1 z ) k 1 γ 2 k 2 γ 1 k 1 γ 2 [ k 2 γ 2 exp ( j k 2 z ) k 1 γ 1 exp ( j k 1 z ) ] } ,
M 22 = γ 1 γ 2 k 2 γ 1 k 1 γ 2 [ k 2 γ 2 exp ( j k 2 z ) k 1 γ 1 exp ( j k 1 z ) ] ,
M ( z ) = γ 1 γ 2 k 2 γ 1 k 1 γ 2 k 2 γ 2 exp ( j k 1 z ) k 1 γ 1 exp ( j k 2 z ) j [ exp ( j k 2 z ) exp ( j k 1 z ) ] j k 1 k 2 γ 1 γ 2 [ exp ( j k 1 z ) exp ( j k 2 z ) ] k 2 γ 2 exp ( j k 2 z ) k 1 γ 1 exp ( j k 1 z ) .
L 1 , 2 i = cos k 1 , 2 z γ 1 , 2 k 1 , 2 ξ 1 , 2 ς 1 , 2 sin k 1 , 2 z k 1 , 2 γ 1 , 2 ς 1 , 2 ξ 1 , 2 sin k 1 , 2 z cos k 1 , 2 z ,
N ( z ) E y 0 H x 0 = L 1 ( z ) ξ 1 E 10 ς 1 H 10 + L 1 ( z ) ξ 2 E 20 ς 2 H 20 .
N 11 = l 11 ( 1 ) + j k 1 γ 1 ς 1 ξ 1 l 12 ( 1 ) k 1 γ 1 ς 1 ξ 1 l 11 ( 2 ) l 11 ( 1 ) + j k 2 ς 2 γ 2 ξ 2 l 12 ( 2 ) j k 1 ς 1 γ 1 ξ 1 l 12 ( 1 ) k 2 ς 2 γ 2 ξ 2 k 1 ς 1 γ 1 ξ 1 ,
N 12 = l 11 ( 2 ) l 11 ( 1 ) + j k 2 ς 2 γ 2 ξ 2 l 12 ( 2 ) j k 1 ς 1 γ 1 ξ 1 l 12 ( 1 ) k 2 ς 2 γ 2 ξ 2 k 1 ς 1 γ 1 ξ 1 ;
N 21 = l 21 ( 1 ) + j k 1 γ 1 ς 1 ξ 1 l 22 ( 1 ) k 1 γ 1 ς 1 ξ 1 l 21 ( 2 ) l 21 ( 1 ) + j k 2 ς 2 γ 2 ξ 2 l 22 ( 2 ) j k 1 ς 1 γ 1 ξ 1 l 22 ( 1 ) k 2 ς 2 γ 2 ξ 2 k 1 ς 1 γ 1 ξ 1 ,
N 22 = l 21 ( 2 ) l 21 ( 1 ) + j k 2 ς 2 γ 2 ξ 2 l 22 ( 2 ) j k 1 ς 1 γ 1 ξ 1 l 22 ( 1 ) k 2 ς 2 γ 2 ξ 2 k 1 ς 1 γ 1 ξ 1 .
N 11 = γ 1 γ 2 ξ 1 ξ 2 k 2 γ 1 ς 2 ξ 1 k 1 γ 2 ς 1 ξ 2 [ k 1 ς 1 γ 1 ξ 1 exp ( j k 2 z ) k 2 ς 2 γ 2 ξ 2 exp ( j k 1 z ) ] ,
N 12 = j γ 1 γ 2 ξ 1 ξ 2 k 2 γ 1 ς 2 ξ 1 k 1 γ 2 ς 1 ξ 2 [ exp ( j k 2 z ) exp ( j k 1 z ) ] ,
N 21 = j k 1 k 2 ξ 1 ξ 2 k 2 γ 1 ς 2 ξ 1 k 1 γ 2 ς 1 ξ 2 [ exp ( j k 2 z ) exp ( j k 1 z ) ] ,
N 11 = γ 1 γ 2 ξ 1 ξ 2 k 2 γ 1 ς 2 ξ 1 k 1 γ 2 ς 1 ξ 2 [ k 2 ς 2 γ 2 ξ 2 exp ( j k 2 z ) k 1 ς 1 γ 1 ξ 1 exp ( j k 1 z ) ] .

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