Abstract

The rigorous Fourier modal method for crossed anisotropic gratings is extended to the analysis of second-harmonic generation in two-dimensionally periodic structures. The method takes the undepleted-pump approximation that uncouples the calculations of the fundamental and the second-harmonic fields. The effectiveness of the method is verified by solving a one-dimensional problem, which has been analyzed by two previously developed methods, and by comparing the simulation results of the L-shaped gold nanoparticle arrays with the previous experimental measurements.

© 2007 Optical Society of America

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    [CrossRef]
  6. B. K. Canfield, S. Kujala, K. Jefimovs, J. Turunen, and M. Kauranen, 'Linear and nonlinear optical responses influenced by broken symmetry in an array of gold nanoparticles,' Opt. Express 12, 5418-5423 (2004).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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2006 (1)

B. K. Canfield, S. Kujala, K. Jefimovs, Y. Svirko, J. Turunen, and M. Kauranen, 'A macroscopic formalism to describe the second-order nonlinear optical response of nanostructures,' J. Opt. A, Pure Appl. Opt. 8, S278-S284 (2006).
[CrossRef]

2005 (2)

B. K. Canfield, S. Kujala, K. Jefimovs, T. Vallius, J. Turunen, and M. Kauranen, 'Polarization effects in the linear and nonlinear optical responses of gold nanoparticle arrays,' J. Opt. A, Pure Appl. Opt. 7, S110-S117 (2005).
[CrossRef]

B. Maes, P. Bienstman, and R. Baets, 'Modeling second-harmonic generation by use of mode expansion,' J. Opt. Soc. Am. B 22, 1378-1383 (2005).
[CrossRef]

2004 (2)

2003 (5)

2002 (3)

F. Raineri, Y. Dumeige, A. Levenson, and X. Letartre, 'Nonlinear decoupled FDTD code: phase-matching in 2D defective photonic crystal,' Electron. Lett. 38, 1704-1706 (2002).
[CrossRef]

H. Tuovinen, M. Kauranen, K. Jefimovs, P. Vahimaa, T. Vallius, J. Turunen, N. V. Tkachenko, and H. Lemmetyinen, 'Linear and second-order nonlinear optical properties of arrays of noncentrosymmetric gold nanoparticles,' J. Nonlinear Opt. Phys. Mater. 11, 421-432 (2002).
[CrossRef]

W. Nakagawa, R.-C. Tyan, and Y. Fainman, 'Analysis of enhanced second-harmonic generation in periodic nanostructures using modified rigorous coupled-wave analysis in the undepleted-pump approximation,' J. Opt. Soc. Am. A 19, 1919-1928 (2002).
[CrossRef]

2001 (1)

2000 (1)

1996 (2)

1994 (1)

1988 (2)

1983 (1)

R. Reinisch and M. Nevière, 'Electromagnetic theory of diffraction in nonlinear optics and surface enhanced nonlinear optical effects,' Phys. Rev. B 28, 1870-1885 (1983).
[CrossRef]

Aitchison, J. S.

Baets, R.

Bienstman, P.

Canfield, B. K.

B. K. Canfield, S. Kujala, K. Jefimovs, Y. Svirko, J. Turunen, and M. Kauranen, 'A macroscopic formalism to describe the second-order nonlinear optical response of nanostructures,' J. Opt. A, Pure Appl. Opt. 8, S278-S284 (2006).
[CrossRef]

B. K. Canfield, S. Kujala, K. Jefimovs, T. Vallius, J. Turunen, and M. Kauranen, 'Polarization effects in the linear and nonlinear optical responses of gold nanoparticle arrays,' J. Opt. A, Pure Appl. Opt. 7, S110-S117 (2005).
[CrossRef]

B. K. Canfield, S. Kujala, K. Jefimovs, J. Turunen, and M. Kauranen, 'Linear and nonlinear optical responses influenced by broken symmetry in an array of gold nanoparticles,' Opt. Express 12, 5418-5423 (2004).
[CrossRef] [PubMed]

Capobianco, A.-D.

Coutaz, J. L.

Cowan, A. R.

De Angelis, C.

Dumeige, Y.

F. Raineri, Y. Dumeige, A. Levenson, and X. Letartre, 'Nonlinear decoupled FDTD code: phase-matching in 2D defective photonic crystal,' Electron. Lett. 38, 1704-1706 (2002).
[CrossRef]

Fainman, Y.

Gringoli, F.

Jefimovs, K.

B. K. Canfield, S. Kujala, K. Jefimovs, Y. Svirko, J. Turunen, and M. Kauranen, 'A macroscopic formalism to describe the second-order nonlinear optical response of nanostructures,' J. Opt. A, Pure Appl. Opt. 8, S278-S284 (2006).
[CrossRef]

B. K. Canfield, S. Kujala, K. Jefimovs, T. Vallius, J. Turunen, and M. Kauranen, 'Polarization effects in the linear and nonlinear optical responses of gold nanoparticle arrays,' J. Opt. A, Pure Appl. Opt. 7, S110-S117 (2005).
[CrossRef]

B. K. Canfield, S. Kujala, K. Jefimovs, J. Turunen, and M. Kauranen, 'Linear and nonlinear optical responses influenced by broken symmetry in an array of gold nanoparticles,' Opt. Express 12, 5418-5423 (2004).
[CrossRef] [PubMed]

H. Tuovinen, M. Kauranen, K. Jefimovs, P. Vahimaa, T. Vallius, J. Turunen, N. V. Tkachenko, and H. Lemmetyinen, 'Linear and second-order nonlinear optical properties of arrays of noncentrosymmetric gold nanoparticles,' J. Nonlinear Opt. Phys. Mater. 11, 421-432 (2002).
[CrossRef]

Jiang, W.

Kauranen, M.

B. K. Canfield, S. Kujala, K. Jefimovs, Y. Svirko, J. Turunen, and M. Kauranen, 'A macroscopic formalism to describe the second-order nonlinear optical response of nanostructures,' J. Opt. A, Pure Appl. Opt. 8, S278-S284 (2006).
[CrossRef]

B. K. Canfield, S. Kujala, K. Jefimovs, T. Vallius, J. Turunen, and M. Kauranen, 'Polarization effects in the linear and nonlinear optical responses of gold nanoparticle arrays,' J. Opt. A, Pure Appl. Opt. 7, S110-S117 (2005).
[CrossRef]

B. K. Canfield, S. Kujala, K. Jefimovs, J. Turunen, and M. Kauranen, 'Linear and nonlinear optical responses influenced by broken symmetry in an array of gold nanoparticles,' Opt. Express 12, 5418-5423 (2004).
[CrossRef] [PubMed]

H. Tuovinen, M. Kauranen, K. Jefimovs, P. Vahimaa, T. Vallius, J. Turunen, N. V. Tkachenko, and H. Lemmetyinen, 'Linear and second-order nonlinear optical properties of arrays of noncentrosymmetric gold nanoparticles,' J. Nonlinear Opt. Phys. Mater. 11, 421-432 (2002).
[CrossRef]

Kujala, S.

B. K. Canfield, S. Kujala, K. Jefimovs, Y. Svirko, J. Turunen, and M. Kauranen, 'A macroscopic formalism to describe the second-order nonlinear optical response of nanostructures,' J. Opt. A, Pure Appl. Opt. 8, S278-S284 (2006).
[CrossRef]

B. K. Canfield, S. Kujala, K. Jefimovs, T. Vallius, J. Turunen, and M. Kauranen, 'Polarization effects in the linear and nonlinear optical responses of gold nanoparticle arrays,' J. Opt. A, Pure Appl. Opt. 7, S110-S117 (2005).
[CrossRef]

B. K. Canfield, S. Kujala, K. Jefimovs, J. Turunen, and M. Kauranen, 'Linear and nonlinear optical responses influenced by broken symmetry in an array of gold nanoparticles,' Opt. Express 12, 5418-5423 (2004).
[CrossRef] [PubMed]

Lemmetyinen, H.

H. Tuovinen, M. Kauranen, K. Jefimovs, P. Vahimaa, T. Vallius, J. Turunen, N. V. Tkachenko, and H. Lemmetyinen, 'Linear and second-order nonlinear optical properties of arrays of noncentrosymmetric gold nanoparticles,' J. Nonlinear Opt. Phys. Mater. 11, 421-432 (2002).
[CrossRef]

Letartre, X.

F. Raineri, Y. Dumeige, A. Levenson, and X. Letartre, 'Nonlinear decoupled FDTD code: phase-matching in 2D defective photonic crystal,' Electron. Lett. 38, 1704-1706 (2002).
[CrossRef]

Levenson, A.

F. Raineri, Y. Dumeige, A. Levenson, and X. Letartre, 'Nonlinear decoupled FDTD code: phase-matching in 2D defective photonic crystal,' Electron. Lett. 38, 1704-1706 (2002).
[CrossRef]

Li, L.

Locatelli, A.

Maes, B.

Magnusson, R.

Maldonado, T. A.

Maystre, D.

Midrio, M.

Modotto, D.

Mondia, J. P.

Nakagawa, W.

Nalesso, G.

Nalesso, G. G.

Nevière, M.

Pigozzo, F. M.

Pomerantz, M.

Popov, E.

Priambodo, P. S.

Purvinis, G.

Raineri, F.

F. Raineri, Y. Dumeige, A. Levenson, and X. Letartre, 'Nonlinear decoupled FDTD code: phase-matching in 2D defective photonic crystal,' Electron. Lett. 38, 1704-1706 (2002).
[CrossRef]

Reinisch, R.

Sun, P.-C.

Svirko, Y.

B. K. Canfield, S. Kujala, K. Jefimovs, Y. Svirko, J. Turunen, and M. Kauranen, 'A macroscopic formalism to describe the second-order nonlinear optical response of nanostructures,' J. Opt. A, Pure Appl. Opt. 8, S278-S284 (2006).
[CrossRef]

Tkachenko, N. V.

H. Tuovinen, M. Kauranen, K. Jefimovs, P. Vahimaa, T. Vallius, J. Turunen, N. V. Tkachenko, and H. Lemmetyinen, 'Linear and second-order nonlinear optical properties of arrays of noncentrosymmetric gold nanoparticles,' J. Nonlinear Opt. Phys. Mater. 11, 421-432 (2002).
[CrossRef]

Tuovinen, H.

H. Tuovinen, M. Kauranen, K. Jefimovs, P. Vahimaa, T. Vallius, J. Turunen, N. V. Tkachenko, and H. Lemmetyinen, 'Linear and second-order nonlinear optical properties of arrays of noncentrosymmetric gold nanoparticles,' J. Nonlinear Opt. Phys. Mater. 11, 421-432 (2002).
[CrossRef]

Turunen, J.

B. K. Canfield, S. Kujala, K. Jefimovs, Y. Svirko, J. Turunen, and M. Kauranen, 'A macroscopic formalism to describe the second-order nonlinear optical response of nanostructures,' J. Opt. A, Pure Appl. Opt. 8, S278-S284 (2006).
[CrossRef]

B. K. Canfield, S. Kujala, K. Jefimovs, T. Vallius, J. Turunen, and M. Kauranen, 'Polarization effects in the linear and nonlinear optical responses of gold nanoparticle arrays,' J. Opt. A, Pure Appl. Opt. 7, S110-S117 (2005).
[CrossRef]

B. K. Canfield, S. Kujala, K. Jefimovs, J. Turunen, and M. Kauranen, 'Linear and nonlinear optical responses influenced by broken symmetry in an array of gold nanoparticles,' Opt. Express 12, 5418-5423 (2004).
[CrossRef] [PubMed]

H. Tuovinen, M. Kauranen, K. Jefimovs, P. Vahimaa, T. Vallius, J. Turunen, N. V. Tkachenko, and H. Lemmetyinen, 'Linear and second-order nonlinear optical properties of arrays of noncentrosymmetric gold nanoparticles,' J. Nonlinear Opt. Phys. Mater. 11, 421-432 (2002).
[CrossRef]

Tyan, R.-C.

Vahimaa, P.

H. Tuovinen, M. Kauranen, K. Jefimovs, P. Vahimaa, T. Vallius, J. Turunen, N. V. Tkachenko, and H. Lemmetyinen, 'Linear and second-order nonlinear optical properties of arrays of noncentrosymmetric gold nanoparticles,' J. Nonlinear Opt. Phys. Mater. 11, 421-432 (2002).
[CrossRef]

Vallius, T.

B. K. Canfield, S. Kujala, K. Jefimovs, T. Vallius, J. Turunen, and M. Kauranen, 'Polarization effects in the linear and nonlinear optical responses of gold nanoparticle arrays,' J. Opt. A, Pure Appl. Opt. 7, S110-S117 (2005).
[CrossRef]

H. Tuovinen, M. Kauranen, K. Jefimovs, P. Vahimaa, T. Vallius, J. Turunen, N. V. Tkachenko, and H. Lemmetyinen, 'Linear and second-order nonlinear optical properties of arrays of noncentrosymmetric gold nanoparticles,' J. Nonlinear Opt. Phys. Mater. 11, 421-432 (2002).
[CrossRef]

van Driel, H. M.

Vincent, P.

Young, J. F.

Zhou, M.

Electron. Lett. (1)

F. Raineri, Y. Dumeige, A. Levenson, and X. Letartre, 'Nonlinear decoupled FDTD code: phase-matching in 2D defective photonic crystal,' Electron. Lett. 38, 1704-1706 (2002).
[CrossRef]

J. Nonlinear Opt. Phys. Mater. (1)

H. Tuovinen, M. Kauranen, K. Jefimovs, P. Vahimaa, T. Vallius, J. Turunen, N. V. Tkachenko, and H. Lemmetyinen, 'Linear and second-order nonlinear optical properties of arrays of noncentrosymmetric gold nanoparticles,' J. Nonlinear Opt. Phys. Mater. 11, 421-432 (2002).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (3)

B. K. Canfield, S. Kujala, K. Jefimovs, T. Vallius, J. Turunen, and M. Kauranen, 'Polarization effects in the linear and nonlinear optical responses of gold nanoparticle arrays,' J. Opt. A, Pure Appl. Opt. 7, S110-S117 (2005).
[CrossRef]

B. K. Canfield, S. Kujala, K. Jefimovs, Y. Svirko, J. Turunen, and M. Kauranen, 'A macroscopic formalism to describe the second-order nonlinear optical response of nanostructures,' J. Opt. A, Pure Appl. Opt. 8, S278-S284 (2006).
[CrossRef]

L. Li, 'Fourier modal method for crossed anisotropic gratings with arbitrary permittivity and permeability tensors,' J. Opt. A, Pure Appl. Opt. 5, 345-355 (2003).
[CrossRef]

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

J. Opt. Soc. Am. B (7)

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. B (1)

R. Reinisch and M. Nevière, 'Electromagnetic theory of diffraction in nonlinear optics and surface enhanced nonlinear optical effects,' Phys. Rev. B 28, 1870-1885 (1983).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Schematic illustration of a 2D periodic structure and the incidence configuration. A 2D nonlinear periodic layer with periods d 1 × d 2 and thickness h is attached to a homogeneous linear substrate. The incident pump field is a linearly polarized plane wave with wave vector k 0 , whose incident direction is described by a polar angle θ and an azimuthal angle φ in the rectangular coordinate system O x y z .

Fig. 2
Fig. 2

Reciprocal lattices (with each point representing a diffraction order) of the pump field (●) and the SH field (엯) in the reciprocal Fourier space, which are positioned with respect to points ( α 0 , β 0 ) and ( 2 α 0 , 2 β 0 ) , respectively. The two lattices have the same point density with lattice constants K 1 and K 2 . The solid and dashed circles (with radii k ( ± 1 ) and k ̃ ( ± 1 ) ) include the propagating orders of the pump field and those of the SH field, respectively.

Fig. 3
Fig. 3

Total transmitted SHG intensities of the nanograting shown in Fig. 1 of Ref. [13] and the bulk nonlinear material.

Fig. 4
Fig. 4

Top view of the type-I (a) and type-II (b) L-particle arrays. The L-particle has two equal arms with length l and width w.

Fig. 5
Fig. 5

Discretization of the unit cell of the L-shaped periodic pattern by a mosaic pattern.

Fig. 6
Fig. 6

Simulation of the (0,0)th-order transmitted SHG intensity as a function of the input intensity for both type-I and type-II structures and for the polarization directions A, B, and C. The results are comparable with the measurements shown in Fig. 5 in Ref. [5].

Equations (61)

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s ̂ = l 2 + F 2 l 2 l 1 + F 1 l 1 ( s ) ,
s ̱ = l 3 ( s ̂ ) .
D = ϵ E ,
D ̃ = ϵ ̃ E ̃ + 4 π P ̃ NL ,
P ̃ i NL = j , k χ i j k E j E k ( i , j , k = 1 , 2 , 3 )
× E = i k 0 μ H ,
× H = i k 0 ϵ E ,
× E ̃ = i k ̃ 0 μ ̃ H ̃ ,
× H ̃ = i k ̃ 0 ϵ ̃ E ̃ i 4 π k ̃ 0 P ̃ NL ,
E σ ( + 1 ) ( r ) = I σ exp [ i ( α 0 x + β 0 y γ 00 ( + 1 ) z ) ] + m , n R σ m n exp [ i ( α m x + β n y + γ m n ( + 1 ) z ) ] ,
E σ ( 1 ) ( r ) = m , n T σ m n exp [ i ( α m x + β n y γ m n ( 1 ) z ) ] ,
α 0 = k ( + 1 ) sin θ cos φ , β 0 = k ( + 1 ) sin θ sin φ ,
γ 00 ( + 1 ) = k ( + 1 ) cos θ ,
α m = α 0 + m K 1 , β n = β 0 + n K 2 ,
γ m n ( ± 1 ) = [ k ( ± 1 ) 2 α m 2 β n 2 ] 1 2 ,
K i = 2 π d i , k ( ± 1 ) = ( ϵ ± 1 μ ± 1 ) 1 2 k 0 ,
Re [ γ m n ( ± 1 ) ] + Im [ γ m n ( ± 1 ) ] > 0 .
( E 1 m n ( ± 1 ) ( z ) E 2 m n ( ± 1 ) ( z ) H 1 m n ( ± 1 ) ( z ) H 2 m n ( ± 1 ) ( z ) ) = ( I 0 I 0 0 I 0 I C m n ( ± 1 ) A m n ( ± 1 ) C m n ( ± 1 ) A m n ( ± 1 ) B m n ( ± 1 ) C m n ( ± 1 ) B m n ( ± 1 ) C m n ( ± 1 ) ) ( u 1 m n ( ± 1 ) ( z ) u 2 m n ( ± 1 ) ( z ) d 1 m n ( ± 1 ) ( z ) d 2 m n ( ± 1 ) ( z ) ) ,
A m n ( ± 1 ) = α m 2 k ( ± 1 ) 2 μ ± 1 k 0 γ m n ( ± 1 ) , B m n ( ± 1 ) = k ( ± 1 ) 2 β n 2 μ ± 1 k 0 γ m n ( ± 1 ) ,
C m n ( ± 1 ) = α m β n μ ± 1 k 0 γ m n ( ± 1 ) ,
u σ m n ( + 1 ) ( z ) = R σ m n exp ( i γ m n ( + 1 ) z ) ,
d σ m n ( + 1 ) ( z ) = I σ δ m 0 δ n 0 exp ( i γ m n ( + 1 ) z ) ,
u σ m n ( 1 ) ( z ) = 0 , d σ m n ( 1 ) ( z ) = T σ m n exp ( i γ m n ( 1 ) z ) .
u σ ( 0 ) ( r ) = m n u σ m n ( 0 ) ( z ) exp [ i ( α m x + β n y ) ] ,
M X = γ X ,
M = ( μ ̱ 23 β α ϵ ̱ 31 μ ̱ 23 α α ϵ ̱ 32 k 0 μ ̱ 21 + 1 k 0 α ϵ ̱ 33 β k 0 μ ̱ 22 1 k 0 α ϵ ̱ 33 α μ ̱ 13 β β ϵ ̱ 31 μ ̱ 13 α β ϵ ̱ 32 k 0 μ ̱ 11 + 1 k 0 β ϵ ̱ 33 β k 0 μ ̱ 12 1 k 0 β ϵ ̱ 33 α k 0 ϵ ̱ 21 1 k 0 α μ ̱ 33 β k 0 ϵ ̱ 22 + 1 k 0 α μ ̱ 33 α ϵ ̱ 23 β α μ ̱ 31 ϵ ̱ 23 α α μ ̱ 32 k 0 ϵ ̱ 11 1 k 0 β μ ̱ 33 β k 0 ϵ ̱ 12 + 1 k 0 β μ ̱ 33 α ϵ ̱ 13 β β μ ̱ 31 ϵ ̱ 13 α β μ ̱ 32 ) .
( E 1 m n ( 0 ) ( z ) E 2 m n ( 0 ) ( z ) H 1 m n ( 0 ) ( z ) H 2 m n ( 0 ) ( z ) ) = ( E 1 m n q + E 1 m n q E 2 m n q + E 2 m n q H 1 m n q + H 1 m n q H 2 m n q + H 2 m n q ) ( exp ( i γ q + z ) 0 0 exp ( i γ q z ) ) ( u q d q ) ,
X ( + 1 ) ( h ) = ( I I D ( + 1 ) D ( + 1 ) ) ( R I 0 ) ,
X ( 1 ) ( 0 ) = ( I I D ( 1 ) D ( 1 ) ) ( 0 T ) .
X ( 0 ) ( h ) = ( E + E H + H ) ( e + 0 0 e ) ( u d ) ,
X ( 0 ) ( 0 ) = ( E + E H + H ) ( u d ) ,
X ( + 1 ) ( h ) = X ( 0 ) ( h ) , X ( 0 ) ( 0 ) = X ( 1 ) ( 0 ) ,
( E + e + E I 0 H + e + H D ( + 1 ) 0 E + E ( e ) 1 0 I H + H ( e ) 1 0 D ( 1 ) ) ( u e d R T ) = ( I 0 D ( + 1 ) I 0 0 0 ) .
χ i j k = m , n χ m n i j k exp [ i ( m K 1 x + n K 2 y ) ] .
P ̃ i NL ( r ) = m , n P ̃ i m n ( z ) exp [ i ( α ̃ m x + β ̃ n y ) ] ,
α ̃ m = 2 α 0 + m K 1 , β ̃ n = 2 β 0 + n K 2 ,
P ̃ i m n ( z ) = s , t P ̃ i m n s t exp [ i ( γ s + γ t ) z ] ,
P ̃ i m n s t v s v t j , k , p , q , p , q χ m p p , n q q i j k E j p q s E k p q t .
E 3 = 1 k 0 ϵ ̱ 33 ( α H 2 β H 1 ) ϵ ̱ 31 E 1 ϵ ̱ 33 E 2 ,
u ̃ σ ( 0 ) ( r ) = m , n u ̃ σ m n ( 0 ) ( z ) exp [ i ( α ̃ m x + β ̃ n y ) ] ( u = E , H )
E ̃ σ ( ± 1 ) ( r ) = m , n u ̃ σ m n ( ± 1 ) exp [ i ( α ̃ m x + β ̃ n y ± γ ̃ m n ( ± 1 ) z ) ]
( u ̃ ( + 1 ) = R ̃ , u ̃ ( 1 ) = T ̃ )
3 i X ̃ ( z ) = M ̃ X ̃ ( z ) + Q ̃ ( z ) ,
Q ̃ ( z ) = s , t 4 π ( α ̃ ϵ ̱ ̃ 33 P ̃ 3 m n s t β ̃ ϵ ̱ ̃ 33 P ̃ 3 m n s t k ̃ 0 ( ϵ ̃ ̱ 23 P ̃ 3 m n s t P ̃ 2 m n s t ) k ̃ 0 ( P ̃ 1 m n s t ϵ ̱ ̃ 13 P ̃ 3 m n s t ) ) exp [ i ( γ s + γ t ) z ] s , t P ̃ s t exp [ i ( γ s + γ t ) z ] .
X ̃ G ( z ) = ( E ̃ + E ̃ H ̃ + H ̃ ) ( exp ( i γ ̃ q + z ) 0 0 exp ( i γ ̃ q z ) ) W ̃ ϕ ̃ ( z ) ,
X ̃ P = X ̃ G Ψ ( z ) .
3 Ψ = i ϕ ̃ 1 W ̃ 1 Q ̃ .
3 Ψ p = i s , t , q W ̃ p q 1 P ̃ q s t exp [ i ( γ s + γ t γ ̃ p ) z ] ,
Ψ p = s , t , q 1 γ s + γ t γ ̃ p W ̃ p q 1 P ̃ q s t exp [ i ( γ s + γ t γ ̃ p ) z ] .
Ω s t , p = 1 γ s + γ t γ ̃ p exp [ i ( γ s + γ t γ ̃ p ) z ] ,
X ̃ ( 0 ) ( h ) = ( E ̃ + E ̃ H ̃ + H ̃ ) ( e ̃ + 0 0 e ̃ ) ( u ̃ q d ̃ q ) + X ̃ P ( h ) ,
X ̃ ( 0 ) ( 0 ) = ( E ̃ + E ̃ H ̃ + H ̃ ) ( u ̃ q d ̃ q ) + X ̃ P ( 0 ) ,
X ̃ ( + 1 ) ( h ) = X ̃ ( 0 ) ( h ) , X ̃ ( 0 ) ( 0 ) = X ̃ ( 1 ) ( 0 ) ,
( E ̃ + e ̃ + E ̃ I 0 H ̃ + e ̃ + H ̃ D ̃ ( + 1 ) 0 E ̃ + E ̃ ( e ̃ ) 1 0 I H ̃ + H ̃ ( e ̃ ) 1 0 D ̃ ( 1 ) ) ( u ̃ e ̃ d ̃ R ̃ T ̃ ) = ( X ̃ P ( h ) X ̃ P ( 0 ) ) ,
η m n ( ± 1 ) = 1 γ m n ( ± 1 ) [ ( k ( ± 1 ) 2 β n 2 ) E 1 m n ( ± 1 ) 2 + ( k ( ± 1 ) 2 α m 2 ) E 2 m n ( ± 1 ) 2 + α m β m ( E 1 m n ( ± 1 ) E 2 m n ( ± 1 ) ¯ + E 2 m n ( ± 1 ) E 1 m n ( ± 1 ) ¯ ) ] ,
η ̃ m n ( ± 1 ) = 1 γ ̃ m n ( ± 1 ) [ ( k ̃ ( ± 1 ) 2 β ̃ n 2 ) E ̃ 1 m n ( ± 1 ) 2 + ( k ̃ ( ± 1 ) 2 α ̃ m 2 ) E ̃ 2 m n ( ± 1 ) 2 + α ̃ m β ̃ n ( E ̃ 1 m n ( ± 1 ) E ̃ 2 m n ( ± 1 ) ¯ + E ̃ 2 m n ( ± 1 ) E ̃ 1 m n ( ± 1 ) ¯ ) ] ,
1 γ 00 ( + 1 ) [ ( k ( + 1 ) 2 β 0 2 ) I 1 2 + ( k ( + 1 ) 2 α 0 2 ) I 2 2 + α 0 β 0 ( I 1 I ¯ 2 + I 2 I ¯ 1 ) ] = 1 .
N [ α 0 K 1 ] m N [ α 0 K 1 ] ,
N [ β 0 K 2 ] n N [ β 0 K 2 ] ,
2 N [ 2 α 0 K 1 ] m 2 N [ 2 α 0 K 1 ] ,
2 N [ 2 β 0 K 2 ] n 2 N [ 2 β 0 K 2 ] .

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