Abstract

A pseudo-Fourier modal analysis method for analyzing finite-sized dielectric slabs with arbitrary longitudinal permittivity and permeability profiles is proposed. In the proposed method, the permittivity and permeability profiles are represented by the Fourier expansion without using the conventional staircase approximation. The total electromagnetic field distribution inside a dielectric slab is a linear superposition of extracted pseudo-Fourier eigenmodes with specific coupling coefficients selected to satisfy given boundary conditions. The proposed pseudo-Fourier modal analysis method shows excellent agreement with the conventional rigorous coupled-wave analysis with the S-matrix method.

© 2006 Optical Society of America

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  1. D. F. Felbacq and F. Zolla, "Scattering theory of photonic crystals," in Introduction to Complex Mediums for Optics and Electromagnetics, Vol. PM123 of the SPIE Press Monographs, W. S. Weiglhofer and A. Lakhtakia, eds. (SPIE Press, 2003).
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    [CrossRef]
  5. 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]
  6. L. Li, "Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings," J. Opt. Soc. Am. A 13, 1024-1035 (1996).
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  7. M. Neviere and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, 2002).
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  9. E. Popov, M. Neviere, B. Gralak, and G. Tayeb, "Staircase approximation validity for arbitrary-shaped gratings," J. Opt. Soc. Am. A 19, 33-42 (2002).
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  10. E. Popov and M. Neviére, "Differential theory for diffraction gratings: a new formulation for TM polarization with rapid convergence," Opt. Lett. 25, 598-600 (2000).
    [CrossRef]
  11. K. Sakoda, "Optical transmittance of a two-dimensional triangular photonic lattice," Phys. Rev. B 51, 4672-4675 (1995).
    [CrossRef]
  12. K. Sakoda, Optical Properties of Photonic Crystals (Springer, 2001).
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    [CrossRef]

2003 (2)

D. F. Felbacq and F. Zolla, "Scattering theory of photonic crystals," in Introduction to Complex Mediums for Optics and Electromagnetics, Vol. PM123 of the SPIE Press Monographs, W. S. Weiglhofer and A. Lakhtakia, eds. (SPIE Press, 2003).
[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]

2002 (2)

M. Neviere and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, 2002).

E. Popov, M. Neviere, B. Gralak, and G. Tayeb, "Staircase approximation validity for arbitrary-shaped gratings," J. Opt. Soc. Am. A 19, 33-42 (2002).
[CrossRef]

2001 (1)

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

2000 (1)

1999 (1)

Y. Jeong and B. Lee, "Nonlinear property analysis of long-period fiber gratings using discretized coupled-mode theory," IEEE J. Quantum Electron. 35, 1284-1292 (1999).
[CrossRef]

1997 (1)

1996 (3)

1995 (2)

K. Sakoda, "Optical transmittance of a two-dimensional triangular photonic lattice," Phys. Rev. B 51, 4672-4675 (1995).
[CrossRef]

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystal: Molding the Flow of Light (Princeton U. Press, 1995).

1981 (1)

Felbacq, D. F.

D. F. Felbacq and F. Zolla, "Scattering theory of photonic crystals," in Introduction to Complex Mediums for Optics and Electromagnetics, Vol. PM123 of the SPIE Press Monographs, W. S. Weiglhofer and A. Lakhtakia, eds. (SPIE Press, 2003).
[CrossRef]

Gaylord, T. K.

Gralak, B.

Jeong, Y.

Y. Jeong and B. Lee, "Nonlinear property analysis of long-period fiber gratings using discretized coupled-mode theory," IEEE J. Quantum Electron. 35, 1284-1292 (1999).
[CrossRef]

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystal: Molding the Flow of Light (Princeton U. Press, 1995).

Lalanne, P.

Lee, B.

Y. Jeong and B. Lee, "Nonlinear property analysis of long-period fiber gratings using discretized coupled-mode theory," IEEE J. Quantum Electron. 35, 1284-1292 (1999).
[CrossRef]

Li, L.

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystal: Molding the Flow of Light (Princeton U. Press, 1995).

Moharam, M. G.

Morris, G. M.

Neviere, M.

M. Neviere and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, 2002).

E. Popov, M. Neviere, B. Gralak, and G. Tayeb, "Staircase approximation validity for arbitrary-shaped gratings," J. Opt. Soc. Am. A 19, 33-42 (2002).
[CrossRef]

Neviére, M.

Popov, E.

Sakoda, K.

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

K. Sakoda, "Optical transmittance of a two-dimensional triangular photonic lattice," Phys. Rev. B 51, 4672-4675 (1995).
[CrossRef]

Tayeb, G.

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystal: Molding the Flow of Light (Princeton U. Press, 1995).

Zolla, F.

D. F. Felbacq and F. Zolla, "Scattering theory of photonic crystals," in Introduction to Complex Mediums for Optics and Electromagnetics, Vol. PM123 of the SPIE Press Monographs, W. S. Weiglhofer and A. Lakhtakia, eds. (SPIE Press, 2003).
[CrossRef]

IEEE J. Quantum Electron. (1)

Y. Jeong and B. Lee, "Nonlinear property analysis of long-period fiber gratings using discretized coupled-mode theory," IEEE J. Quantum Electron. 35, 1284-1292 (1999).
[CrossRef]

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

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. (1)

J. Opt. Soc. Am. A (5)

Opt. Lett. (1)

Phys. Rev. B (1)

K. Sakoda, "Optical transmittance of a two-dimensional triangular photonic lattice," Phys. Rev. B 51, 4672-4675 (1995).
[CrossRef]

Other (4)

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

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystal: Molding the Flow of Light (Princeton U. Press, 1995).

D. F. Felbacq and F. Zolla, "Scattering theory of photonic crystals," in Introduction to Complex Mediums for Optics and Electromagnetics, Vol. PM123 of the SPIE Press Monographs, W. S. Weiglhofer and A. Lakhtakia, eds. (SPIE Press, 2003).
[CrossRef]

M. Neviere and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, 2002).

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Figures (14)

Fig. 1
Fig. 1

Dielectric slab with arbitrary one-dimensional permittivity and permeability profiles.

Fig. 2
Fig. 2

Periodic extension of a finite-sized dielectric slab with one-dimensional arbitrary permittivity profile.

Fig. 3
Fig. 3

Classification of 2 N + 1 pseudo-Fourier representations of the pseudo-Fourier eigenmode according to the convergence condition.

Fig. 4
Fig. 4

Dielectric structure with thickness of 4 λ and (a) longitudinal continuous permittivity and permeability profiles and (b) longitudinal discrete permittivity and permeability profiles.

Fig. 5
Fig. 5

Analyzed eigenvalue distributions in the first Brillouin zone of the dielectric slab with continuous permittivity and permeability profiles when (a) N = 10 and (b) N = 64 .

Fig. 6
Fig. 6

Extracted pseudo-Fourier eigenmodes of the dielectric slab with continuous permittivity and permittivity profiles: (a) ( E ̱ 1 ( z ) , H ̱ 1 ( z ) ) with j k z ( 1 ) k 0 = 0.0767 j , (b) ( E ̱ 2 ( z ) , H ̱ 2 ( z ) ) with j k z ( 2 ) k 0 = 0.0483 j , (c) ( E ̱ 3 ( z ) , H ̱ 3 ( z ) ) with j k z ( 3 ) k 0 = + 0.0483 j , (d) ( E ̱ 4 ( z ) , H ̱ 4 ( z ) ) with j k z ( 4 ) k 0 = + 0.0767 j .

Fig. 7
Fig. 7

Analyzed eigenvalue distributions in the first Brillouin zone of the dielectric slab with discrete permittivity and permeability profiles when (a) N = 10 and (b) N = 64 .

Fig. 8
Fig. 8

Extracted pseudo-Fourier eigenmodes of the dielectric slab with discrete permittivity and permeability profiles: (a) ( E ̱ 1 ( z ) , H ̱ 1 ( z ) ) with j k z ( 1 ) k 0 = 0.0205 , (b) ( E ̱ 2 ( z ) , H ̱ 2 ( z ) ) with j k z ( 2 ) k 0 = 0.0163 , (c) ( E ̱ 3 ( z ) , H ̱ 3 ( z ) ) with j k z ( 3 ) k 0 = 0.0163 , (d) ( E ̱ 4 ( z ) , H ̱ 4 ( z ) ) with j k z ( 4 ) k 0 = + 0.0205 .

Fig. 9
Fig. 9

Total electric field distributions in the dielectric slab with continuous permittivity and permeability profiles (a) E y (z) and (b) E x (z) obtained by the PFMA.

Fig. 10
Fig. 10

Staircase approximation of the continuous permittivity and permeability profiles in the dielectric slab obtained by the RCWA with the S-matrix method.

Fig. 11
Fig. 11

Total electric field distributions in the dielectric slab with continuous permittivity and permeability profiles (a) E y ( z ) and (b) E x ( z ) obtained by the S-matrix method and the (one-dimensional version) RCWA.

Fig. 12
Fig. 12

Total electric field distributions in the dielectric slab with discrete permittivity and permeability profiles (a) E y ( z ) and (b) E x ( z ) obtained by the PFMA.

Fig. 13
Fig. 13

Total electric field distributions in the dielectric slab with discrete permittivity and permeability profiles (a) E y ( z ) and (b) E x ( z ) obtained by the S-matrix method and the (one-dimensional version) RCWA.

Fig. 14
Fig. 14

Discrete permittivity profile with L homogeneous layers.

Equations (109)

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E ̱ ( x , y , z ) = m = M M n = N N [ S x , m n ( z ) x ̱ + S y , m n ( z ) y ̱ + S z , m n ( z ) z ̱ ] exp [ j ( k x , m x + k y , n y ) ] ,
H ̱ ( x , y , z ) = j ε 0 μ 0 m = M M n = N N [ U x , m n ( z ) x ̱ + U y , m n ( z ) y ̱ + U z , m n ( z ) z ̱ ] exp [ j ( k x , m x + k y , n y ) ] ,
k x , m = k x + m G x for M m M ,
k y , n = k y + n G y for ( N n N ) ,
× E ̱ = j ω μ 0 μ ( x , y , z ) ( H x x ̱ + H y y ̱ + H z z ̱ ) ,
× H ̱ = j ω ε 0 ε ( x , y , z ) ( E x x ̱ + E y y ̱ + E z z ̱ ) ,
ε ( x , y , z ) = m = 2 M 2 M n = 2 N 2 N ε ̃ m , n ( z ) exp [ j ( m G x x + n G y y ) ] ,
μ ( x , y , z ) = m = 2 M 2 M n = 2 N 2 N μ ̃ m , n ( z ) exp [ j ( m G x x + n G y y ) ] .
d S y , m n ( z ) d z = k 0 s , t μ ̃ m s , n t ( z ) U x , s t ( z ) + j k y , n S z , m n ( z ) ,
d S x , m n ( z ) d z = k 0 s , t μ ̃ m s , n t ( z ) U y , s t ( z ) + j k x , m S z , m n ( z ) ,
s , t ε ̃ m s , n t ( z ) S z , s , t ( z ) = j k 0 [ k x , m U y , m n ( z ) k y , n U x , m n ( z ) ] ,
d U y , m n ( z ) d z = j k y , n U z , m n ( z ) + k 0 s , t ε ̃ m s , n t ( z ) S x , s t ( z ) ,
d U x , m n ( z ) d z = k 0 s , t ε m s , n t ( z ) S y , s , t ( z ) + j k x , m U z , m n ( z ) ,
s , t μ ̃ m s , n t ( z ) U z , m n ( z ) = j k 0 [ k x , m S y , m n ( z ) k y , n S x , m n ( z ) ] .
E ̱ inc = ( U x x ̱ + U y y ̱ + U z z ̱ ) exp [ j ( k I , x x + k I , y y + k I , z z ) ] ,
( U x , U y , U z ) = ( cos ψ cos θ cos ϕ sin ψ sin ϕ , cos ψ cos θ sin ϕ + sin ψ cos ϕ , cos ψ sin θ ) ,
( k I , x , k I , y , k I , z ) = ( k 0 n I sin θ cos ϕ , k 0 n I sin θ sin ϕ , k 0 n I cos θ ) ,
E ̱ I = E inc + ( R x x ̱ + R y y ̱ + R z z ̱ ) exp [ j ( k I , x x + k I , y y k I , z z ) ] ,
E ̱ II = ( T x x ̱ + T y y ̱ + T z z ̱ ) exp [ j ( k II , x x + k II , y y + k II , z ( z d ) ) ] ,
k I , x = k II , x = k 0 n I sin θ cos ϕ ,
k I , y = k II , y = k 0 n I sin θ sin ϕ ,
k I , z = k 0 n I cos θ ,
k II , z = ( k 0 2 n II 2 k 0 2 n I 2 sin 2 θ ) 1 2 ,
E ̱ ( x , y , z ) = [ S x ( z ) x ̱ + S y ( z ) y ̱ + S z ( z ) z ̱ ] × exp [ j ( k x x + k y y ) ] ,
H ̱ ( x , y , z ) = j ε 0 μ 0 [ U x ( z ) x ̱ + U y ( z ) y ̱ + U z ( z ) z ̱ ] × exp [ j ( k x x + k y y ) ] .
ε ̂ ( z ) = ε ( z ) n = + δ ( z n d ) ,
μ ̂ ( z ) = μ ( z ) n = + δ ( z n d ) ,
E ̱ k ̱ ( z ) = exp [ j ( k x x + k y y + k z z ) ] E ̱ ̂ k ̱ ( z ) ,
H ̱ k ̱ ( z ) = exp [ j ( k x x + k y y + k z z ) ] H ̱ ̂ k ̱ ( z ) ,
E ̱ ̂ k ̱ ( N , m ) ( z ) = p = N + m N + m ( E ̃ x , p x ̱ + E ̃ y , p y ̱ + E ̃ z , p z ̱ ) exp ( j p G z z ) ,
H ̱ ̂ k ̱ ( N , m ) ( z ) = j ε 0 μ 0 p = N + m N + m ( H ̃ x , p x ̱ + H ̃ y , p y ̱ + H ̃ z , p z ̱ ) exp ( j p G z z ) ,
E ̱ k ̱ ( N , m ) ( z ) = exp [ j ( k x x + k y y ) ] p = N + m N + m ( E ̃ x , p x ̱ + E ̃ y , p y ̱ + E ̃ z , p z ̱ ) × exp [ j ( p G z + k z ( N , m ) ) z ] ,
H ̱ k ̱ ( N , m ) ( z ) = exp [ j ( k x x + k y y ) ] j ε 0 μ 0 p = N + m N + m ( H ̃ x , p x ̱ + H ̃ y , p y ̱ + H ̃ z , p z ̱ ) exp [ j ( p G z + k z ( N , m ) ) z ] ,
E ̱ k ̱ ( z ) E ̱ k ̱ ( N , m ) ( z ) < σ E , H ̱ k ̱ ( z ) H ̱ k ̱ ( N , m ) ( z ) < σ H ,
k z k z ( N , m ) < σ k ,
× E ̱ = j ω μ 0 μ ̂ ( z ) ( H x x ̱ + H y y ̱ + H z z ̱ ) ,
× H ̱ = j ω ε 0 ε ̂ ( z ) ( E x x ̱ + E y y ̱ + E z z ̱ ) ,
ε ̂ ( z ) = g = 2 N 2 N ε ̃ g exp ( j G z g z ) ,
μ ̂ ( z ) = g = 2 N 2 N μ ̃ g exp ( j G z g z ) .
j k z , p E ̃ y , p = k 0 s = N + m N + m μ ̃ p s H ̃ x , s + j k y E ̃ z , p ,
j k z , p E ̃ x , p = k 0 s = N + m N + m μ ̃ p s H ̃ y , s + j k x E ̃ z , p ,
s = N + m N + m ε ̃ p s E ̃ z , s = j k x k 0 H ̃ y , p + j k y k 0 H ̃ x , p ,
j k z , p k 0 H ̃ y , p = s = N + m N + m ε ̃ p s E ̃ x , s + j k y k 0 H ̃ z , p ,
j k z , p k 0 H ̃ x , p = s = N + m N + m ε ̃ p s E ̃ y , s + j k x k 0 H ̃ z , p ,
s = N + m N + m μ ̃ p s H ̃ z , s = j k x k 0 E ̃ y , p + j k y k 0 E ̃ x , p ,
ε ͇ = [ ε ̃ 0 ε ̃ 1 ε ̃ 2 N ε ̃ 1 ε ̃ 0 ε ̃ 2 N + 1 ε ̃ 2 N ε ̃ 2 N 1 ε ̃ 0 ] ,
μ ͇ = [ μ ̃ 0 μ ̃ 1 μ ̃ 2 N μ ̃ 1 μ ̃ 0 μ ̃ 2 N + 1 μ ̃ 2 N μ ̃ 2 N 1 μ ̃ 0 ] .
G ͇ z N + m N + m = [ ( N + m ) G z k 0 0 0 0 ( N + m + 1 ) G z k 0 0 0 0 0 0 ( N + m ) G z k 0 ] ,
K ͇ z N + m N + m = ( k z ( N , m ) k 0 ) I ͇ + G ͇ z N + m N + m ,
K ͇ y = ( k y k 0 ) I ͇ ,
K ͇ x = ( k x k 0 ) I ͇ ,
E ̱ ̃ y N + m N + m = [ E ̃ y , N + m E ̃ y , N + m + 1 E ̃ y , N + m ] t ,
E ̱ ̃ x N + m N + m = [ E ̃ x , N + m E ̃ x , N + m + 1 E ̃ x , N + m ] t ,
H ̱ ̃ x N + m N + m = [ H ̃ x , N + m H ̃ x , N + m + 1 H ̃ x , N + m ] t ,
H ̱ ̃ y N + m N + m = [ H ̃ y , N + m H ̃ y , N + m + 1 H ̃ y , N + m ] t .
j K ͇ z N + m N + m E ̱ ̃ y N + m N + m = μ ͇ H ̱ ̃ x N + m N + m + j K ͇ y E ̱ ̃ z N + m N + m ,
j K ͇ z N + m N + m E ̱ ̃ x N + m N + m = μ ͇ H ̱ ̃ y N + m N + m + j K ͇ x E ̱ ̃ z N + m N + m ,
ε ͇ E ̃ ̱ z N + m N + m = j K ͇ x H ̱ ̃ y N + m N + m + j K ͇ y H ̱ ̃ x N + m N + m ,
j K ͇ z N + m N + m H ̱ ̃ y N + m N + m = j K ͇ y H ̱ ̃ z N + m N + m + ε ͇ E ̃ ̱ x N + m N + m ,
j K ͇ z N + m N + m H ̱ ̃ x N + m N + m = ε ͇ E ̃ ̱ y N + m N + m + j K ͇ x H ̱ ̃ z N + m N + m ,
μ ͇ H ̱ ̃ z N + m N + m = j K ͇ x E ̱ ̃ y N + m N + m + j K ͇ y E ̱ ̃ x N + m N + m .
[ j G ͇ z N + m N + m 0 K ͇ y ε ͇ 1 K ͇ x μ ͇ K ͇ y ε ͇ 1 K ͇ y 0 j G ͇ z N + m N + m μ ͇ + K ͇ x ε ͇ 1 K ͇ x K ͇ x ε ͇ 1 K ͇ y K ͇ y μ ͇ 1 K ͇ x ε ͇ K ͇ y μ ͇ 1 K ͇ y j G ͇ z N + m N + m 0 ε ͇ + K ͇ x μ ͇ 1 K ͇ x K ͇ x μ ͇ 1 K ͇ y 0 j G ͇ z N + m N + m ] ( E ̱ ̃ y N + m N + m E ̱ ̃ x N + m N + m H ̱ ̃ y N + m N + m H ̱ ̃ x N + m N + m ) = j k z ( N , m ) k 0 ( E ̱ ̃ y N + m N + m E ̱ ̃ x N + m N + m H ̱ ̃ y N + m N + m H ̱ ̃ x N + m N + m ) .
α ̂ ( z ) = 1 ε ̂ ( z ) = g = 2 N 2 N α ̃ g exp ( j G z g x ) ,
β ̂ ( z ) = 1 μ ̂ ( z ) = g = 2 N 2 N β ̃ g exp ( j G z g x ) .
α ͇ = [ α ̃ 0 α ̃ 1 α ̃ 2 N α ̃ 1 α ̃ 0 α ̃ 2 N + 1 α ̃ 2 N α ̃ 2 N 1 α ̃ 0 ] ,
β ͇ = [ β ̃ 0 β ̃ 1 β ̃ 2 N β ̃ 1 β ̃ 0 β ̃ 2 N + 1 β ̃ 2 N β ̃ 2 N 1 β ̃ 0 ] .
[ j G ͇ z N + m N + m 0 K ͇ y α ͇ K ͇ x μ ͇ K ͇ y α ͇ K ͇ y 0 j G ͇ z N + m N + m μ ͇ + K ͇ x α ͇ K ͇ x K ͇ x α ͇ K ͇ y K ͇ y β ͇ K ͇ x ε ͇ K ͇ y β ͇ K ͇ y j G ͇ z N + m N + m 0 ε ͇ + K ͇ x β ͇ K ͇ x K ͇ x β ͇ K ͇ y 0 j G ͇ z N + m N + m ] ( E ̱ ̃ y N + m N + m E ̱ ̃ x N + m N + m H ̱ ̃ y N + m N + m H ̱ ̃ x N + m N + m ) = j k z ( N , m ) k 0 ( E ̱ ̃ y N + m N + m E ̱ ̃ x N + m N + m H ̱ ̃ y N + m N + m H ̱ ̃ x N + m N + m ) .
[ j G ͇ z N N 0 K ͇ y α ͇ K ͇ x μ ͇ K ͇ y α ͇ K ͇ y 0 j G ͇ z N N μ ͇ + K ͇ x α ͇ K ͇ x K ͇ x α ͇ K ͇ y K ͇ y β ͇ K ͇ x ε ͇ K ͇ y β ͇ K ͇ y j G ͇ z N N 0 ε ͇ + K ͇ x β ͇ K ͇ x K ͇ x β ͇ K ͇ y 0 j G ͇ z N N ] ( E ̱ ̃ y N + m N + m E ̱ ̃ x N + m N + m H ̱ ̃ y N + m N + m H ̱ ̃ x N + m N + m ) = j ( k z ( N , m ) + m G z k 0 ) ( E ̱ ̃ y N + m N + m E ̱ ̃ x N + m N + m H ̱ ̃ y N + m N + m H ̱ ̃ x N + m N + m ) .
[ j G ͇ z N N 0 K ͇ y α ͇ K ͇ x μ ͇ K ͇ y α ͇ K ͇ y 0 j G ͇ z N N μ ͇ + K ͇ x α ͇ K ͇ x K ͇ x α ͇ K ͇ y K ͇ y β ͇ K ͇ x ε ͇ K ͇ y β ͇ K ͇ y j G ͇ z N N 0 ε ͇ + K ͇ x β ͇ K ͇ x K ͇ x β ͇ K ͇ y 0 j G ͇ z N N ] ( E ̱ y E ̱ x H ̱ y H ̱ x ) = β ( E ̱ y E ̱ x H ̱ y H ̱ x ) .
j k 0 β = k z ( N , N ) + N G z , , k z ( N , N ) N G z .
β ̃ 1 st Brill = β G z [ ( Im ( β ) + 0.5 G z ) mod ( G z ) ] ,
( Im ( β ) + 0.5 G z ) mod ( G z ) = 0 .
E ̱ = g = 1 4 C g E ̱ g ( z ) = exp [ j ( k x x + k y y ) ] g = 1 4 C g [ p = N N ( E ̃ x , p ( g ) x ̱ + E ̃ y , p ( g ) y ̱ + E ̃ z , p ( g ) z ̱ ) exp ( j p G z z ) ] exp ( j k z , g z ) ,
H ̱ = g = 1 4 C g H ̱ g ( z ) = exp [ j ( k x x + k y y ) ] j ε 0 μ 0 g = 1 4 C g [ p = N N ( H ̃ x , p ( g ) x ̱ + H ̃ y , p ( g ) y ̱ + H ̃ z , p ( g ) z ̱ ) exp ( j p G z z ) ] exp ( j k z , g z ) ,
u y + R y = g = 1 4 C g ( p = N N E ̃ y , p ( g ) ) ,
u x + R x = g = 1 4 C g ( p = N N E ̃ x , p ( g ) ) ,
( k I , z u x k x u z ) k 0 + ( k I , z R x k x R z ) k 0 = j g = 1 4 C g ( p = N N H ̃ y , p ( g ) ) ,
( k y u z k I , z u y ) k 0 + ( k y R z + k I , z R y ) k 0 = j g = 1 4 C g ( p = N N H ̃ x , p ( g ) ) .
k x R x + k y R y k I , z R z = 0 ,
( u y u x ( k I , z u x k x u z ) k 0 ( k y u z k I , z u y ) k 0 ) + [ 1 0 0 1 k x k y k 0 k I , z ( k I , z 2 + k x 2 ) k 0 k I , z ( k y 2 + k I , z 2 ) k 0 k I , z k y k x k 0 k I , z ] ( R y R x ) = [ p = N N E ̃ y , p ( 1 ) p = N N E ̃ y , p ( 2 ) p = N N E ̃ y , p ( 3 ) p = N N E ̃ y , p ( 4 ) p = N N E ̃ x , p ( 1 ) p = N N E ̃ x , p ( 2 ) p = N N E ̃ x , p ( 3 ) p = N N E ̃ x , p ( 4 ) j p = N N H ̃ y , p ( 1 ) j p = N N H ̃ y , p ( 2 ) j p = N N H ̃ y , p ( 3 ) j p = N N H ̃ y , p ( 4 ) j p = N N H ̃ x , p ( 1 ) j p = N N H ̃ x , p ( 2 ) j p = N N H ̃ x , p ( 3 ) j p = N N H ̃ x , p ( 4 ) ] ( C 1 C 2 C 3 C 4 ) .
T y = g = 1 4 C g ( p = N N E ̃ y , p ( g ) exp ( j ( k z , g + p G z ) d ) ) ,
T x = g = 1 4 C g ( p = N N E ̃ x , p ( g ) exp ( j ( k z , g + p G z ) d ) ) ,
( k II , z T x k x T z ) k 0 = j g = 1 4 C g ( p = N N H ̃ y , p ( g ) exp ( j ( k z , g + p G z ) d ) ) ,
( k y T z k II , z T y ) k 0 = j g = 1 4 C g ( p = N N H ̃ x , p ( g ) exp ( j ( k z , g + p G z ) d ) ) .
k x T x + k y T y + k II , z T z = 0 ,
[ I 0 0 I k y k x k 0 k II , z ( k II , z 2 + k x 2 ) k 0 k II , z ( k y 2 + k II , z 2 ) k 0 k II , z k y k x k 0 k II , z ] ( T y T x ) = [ p = N N E ̃ y , p ( 1 ) exp ( j ( k z , 1 + p G z ) d ) p = N N E ̃ y , p ( 2 ) exp ( j ( k z , 2 + p G z ) d ) p = N N E ̃ y , p ( 3 ) exp ( j ( k z , 3 + p G z ) d ) p = N N E ̃ y , p ( 4 ) exp ( j ( k z , 4 + p G z ) d ) p = N N E ̃ x , p ( 1 ) exp ( j ( k z , 1 + p G z ) d ) p = N N E ̃ x , p ( 2 ) exp ( j ( k z , 2 + p G z ) d ) p = N N E ̃ x , p ( 3 ) exp ( j ( k z , 3 + p G z ) d ) p = N N E ̃ x , p ( 4 ) exp ( j ( k z , 4 + p G z ) d ) j p = N N H ̃ y , p ( 1 ) exp ( j ( k z , 1 + p G z ) d ) j p = N N H ̃ y , p ( 2 ) exp ( j ( k z , 2 + p G z ) d ) j p = N N H ̃ y , p ( 3 ) exp ( j ( k z , 3 + p G z ) d ) j p = N N H ̃ y , p ( 4 ) exp ( j ( k z , 4 + p G z ) d ) j p = N N H ̃ x , p ( 1 ) exp ( j ( k z , 1 + p G z ) d ) j p = N N H ̃ x , p ( 2 ) exp ( j ( k z , 2 + p G z ) d ) j p = N N H ̃ x , p ( 3 ) exp ( j ( k z , 3 + p G z ) d ) j p = N N H ̃ x , p ( 4 ) exp ( j ( k z , 4 + p G z ) d ) ] ( C 1 C 2 C 3 C 4 ) .
× [ exp ( j ( k ̱ r ̱ ) ) E ̱ ] = j ω μ 0 [ exp ( j ( k ̱ r ̱ ) ) H ̱ ] ,
× [ exp ( j ( k ̱ r ̱ ) ) H ̱ ] = j ω ε 0 ε ( z ) [ exp ( j ( k ̱ r ̱ ) ) E ̱ ] .
ε ̃ 0 = 0 d ε ( z ) d z ,
E ̱ ̃ 0 = 0 d E ̱ ( z ) d z ,
H ̱ ̃ 0 = 0 d H ̱ ( z ) d z .
× [ exp ( j ( k ̱ r ̱ ) ) ( E ̱ E ̱ ̃ 0 + E ̱ ̃ 0 ) ] = j ω μ 0 [ exp ( j ( k ̱ r ̱ ) ) ( H ̱ H ̱ ̃ 0 + H ̱ ̃ 0 ) ] ,
× [ exp ( j ( k ̱ r ̱ ) ) ( H ̱ H ̱ ̃ 0 + H ̱ ̃ 0 ) ] = j ω ε 0 [ ε ( z ) ε ̃ 0 + ε ̃ 0 ] [ exp ( j ( k ̱ r ̱ ) ) ( E ̱ E ̱ ̃ 0 + E ̱ ̃ 0 ) ] .
× [ exp ( j ( k ̱ r ̱ ) ) ( E ̱ E ̱ ̃ 0 ) ] + × [ exp ( j ( k ̱ r ̱ ) ) E ̱ ̃ 0 ] = j ω μ 0 [ exp ( j ( k ̱ r ̱ ) ) ( H ̱ H ̱ ̃ 0 ) ] + j ω μ 0 [ exp ( j ( k ̱ r ̱ ) ) H ̱ ̃ 0 ] ,
× [ exp ( j ( k ̱ r ̱ ) ) ( H ̱ H ̱ ̃ 0 ) ] + × [ exp ( j ( k ̱ r ̱ ) ) H ̱ ̃ 0 ] = j ω ε 0 ε ̃ 0 [ exp ( j ( k ̱ r ̱ ) ) ( E ̱ E ̱ ̃ 0 ) ] j ω ε 0 ε ̃ 0 [ exp ( j ( k ̱ r ̱ ) ) E ̱ ] j ω ε 0 ( ε ( z ) ε ̃ 0 ) [ exp ( j ( k ̱ r ̱ ) ) E ̱ ] .
( i ) × [ exp ( j ( k ̱ r ̱ ) ) E ̱ ̃ 0 ] = j ω μ 0 [ exp ( j ( k ̱ r ̱ ) ) H ̱ ̃ 0 ] ,
× [ exp ( j ( k ̱ r ̱ ) ) H ̱ ̃ 0 ] = j ω ε 0 ε ̃ 0 [ exp ( j ( k ̱ r ̱ ) ) E ̱ ] ,
( ii ) × [ exp ( j ( k ̱ r ̱ ) ) ( E ̱ E ̱ ̃ 0 ) ] = j ω μ 0 [ exp ( j ( k ̱ r ̱ ) ) ( H ̱ H ̱ ̃ 0 ) ] ,
× [ exp ( j ( k ̱ r ̱ ) ) ( H ̱ H ̱ ̃ 0 ) ] = j ω ε 0 ε ̃ 0 [ exp ( j ( k ̱ r ̱ ) ) ( E ̱ E ̱ ̃ 0 ) ] j ω ε 0 ( ε ( z ) ε ̃ 0 ) [ exp ( j ( k ̱ r ̱ ) ) E ̱ ] .
E ̱ ( N ) ( z ) = p = 0 2 N ( E ̃ x , p x ̱ + E ̃ y , p y ̱ + E ̃ z , p z ̱ ) exp ( j p G z z ) ,
k z = k z , 0 + Δ k ( N ) ,
E ̱ ( N 1 ) ( z ) = p = 1 2 N 1 ( E ̃ x , p x ̱ + E ̃ y , p y ̱ + E ̃ z , p z ̱ ) exp ( j p G z z ) ,
k z = k z , 0 + Δ k ( N 1 ) ,
E ̱ ( N 1 ) ( z ) = p = ( 2 N 1 ) 1 ( E ̃ x , p x ̱ + E ̃ y , p y ̱ + E ̃ z , p z ̱ ) exp ( j p G z z ) ,
k z = k z , 0 + Δ k ( N 1 ) ,
E ̱ N ( z ) = p = 2 N 0 ( E ̃ x , p x ̱ + E ̃ y , p y ̱ + E ̃ z , p z ̱ ) exp ( j p G z z ) ,
k z = k z , 0 + Δ k N .
ε ̃ ( z ) = k = 2 N k = 2 N ε ̃ k exp ( j 2 π k d z ) ,
ε ̃ k = p = 1 L ε ̃ p d p d sinc ( d p k d ) exp ( j π k d ( l p 1 + l p ) ) .

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