Abstract

An extended and refined scattering-matrix method is proposed for the efficient full parallel implementation of rigorous coupled-wave analysis of multilayer structures. The total electromagnetic field distribution in the rigorous coupled-wave analysis is represented by the linear combination of the eigenmodes with their own coupling coefficients. In the proposed scheme, a refined recursion relation of the coupling coefficients of the eigenmodes is defined for complete parallel computation of the electromagnetic field distributions within multilayer structures.

© 2007 Optical Society of America

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  1. L. Li, "Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings," J. Opt. Soc. Am. A 13, 1024-1035 (1996).
    [CrossRef]
  2. E. L. Tan, "Note on formulation of the enhanced scattering- (transmittance-) matrix approach," J. Opt. Soc. Am. A 19, 1157-1161 (2002).
    [CrossRef]
  3. L. Li, "Note on the S-matrix propagation algorithm," J. Opt. Soc. Am. A 20, 655-660 (2003).
    [CrossRef]
  4. M. G. Moharam and A. B. Greenwell, "Efficient rigorous calculations of power flow in grating coupled surface-emitting devices," Proc. SPIE 5456, 57-67 (2004).
  5. M. G. Moharam, E. B. Grann, and D. A. Pommet, "Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings," J. Opt. Soc. Am. A 12, 1068-1076 (1995).
    [CrossRef]
  6. P. Lalanne and G. M. Morris, "Highly improved convergence of the coupled-wave method for TM polarization," J. Opt. Soc. Am. A 13, 779-784 (1996).
    [CrossRef]
  7. P. Lalanne, "Improved formulation of the coupled-wave method for two-dimensional gratings," J. Opt. Soc. Am. A 14, 1592-1598 (1997).
    [CrossRef]
  8. E. Silberstein, P. Lalanne, J. P. Hugonin, and Q. Cao, "Use of grating theories in integrated optics," J. Opt. Soc. Am. A 18, 2865-2875 (2001).
    [CrossRef]
  9. P. Lalanne and E. Silberstein, "Fourier-modal methods applied to waveguide computational problems," Opt. Lett. 25, 1092-1094 (2000).
    [CrossRef]
  10. H. Kim, S. Kim, I.-M. Lee, and B. Lee, "Pseudo-Fourier modal analysis on dielectric slabs with arbitrary longitudinal permittivity and permeability profiles," J. Opt. Soc. Am. A 23, 2177-2191 (2006).
    [CrossRef]
  11. H. Kim and B. Lee, "Analysis of TIR holography using pseudo-Fourier modal analysis method," Proc. SPIE 6314, 63141C (2006).
    [CrossRef]
  12. M. G. Moharam, D. A. Pommet, and E. B. Grann, "Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach," J. Opt. Soc. Am. A 12, pp. 1077-1086 (1995).
    [CrossRef]

2006 (2)

2004 (1)

M. G. Moharam and A. B. Greenwell, "Efficient rigorous calculations of power flow in grating coupled surface-emitting devices," Proc. SPIE 5456, 57-67 (2004).

2003 (1)

2002 (1)

2001 (1)

2000 (1)

1997 (1)

1996 (2)

1995 (2)

Cao, Q.

Grann, E. B.

Greenwell, A. B.

M. G. Moharam and A. B. Greenwell, "Efficient rigorous calculations of power flow in grating coupled surface-emitting devices," Proc. SPIE 5456, 57-67 (2004).

Hugonin, J. P.

Kim, H.

Kim, S.

Lalanne, P.

Lee, B.

Lee, I.-M.

Li, L.

Moharam, M. G.

Morris, G. M.

Pommet, D. A.

Silberstein, E.

Tan, E. L.

J. Opt. Soc. Am. A (9)

P. Lalanne, "Improved formulation of the coupled-wave method for two-dimensional gratings," J. Opt. Soc. Am. A 14, 1592-1598 (1997).
[CrossRef]

P. Lalanne and G. M. Morris, "Highly improved convergence of the coupled-wave method for TM polarization," J. Opt. Soc. Am. A 13, 779-784 (1996).
[CrossRef]

L. Li, "Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings," J. Opt. Soc. Am. A 13, 1024-1035 (1996).
[CrossRef]

M. G. Moharam, E. B. Grann, and D. A. Pommet, "Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings," J. Opt. Soc. Am. A 12, 1068-1076 (1995).
[CrossRef]

M. G. Moharam, D. A. Pommet, and E. B. Grann, "Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach," J. Opt. Soc. Am. A 12, pp. 1077-1086 (1995).
[CrossRef]

E. Silberstein, P. Lalanne, J. P. Hugonin, and Q. Cao, "Use of grating theories in integrated optics," J. Opt. Soc. Am. A 18, 2865-2875 (2001).
[CrossRef]

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

L. Li, "Note on the S-matrix propagation algorithm," J. Opt. Soc. Am. A 20, 655-660 (2003).
[CrossRef]

H. Kim, S. Kim, I.-M. Lee, and B. Lee, "Pseudo-Fourier modal analysis on dielectric slabs with arbitrary longitudinal permittivity and permeability profiles," J. Opt. Soc. Am. A 23, 2177-2191 (2006).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (2)

M. G. Moharam and A. B. Greenwell, "Efficient rigorous calculations of power flow in grating coupled surface-emitting devices," Proc. SPIE 5456, 57-67 (2004).

H. Kim and B. Lee, "Analysis of TIR holography using pseudo-Fourier modal analysis method," Proc. SPIE 6314, 63141C (2006).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Multilayer structure for the RCWA.

Fig. 2
Fig. 2

Bidirectional characterization of a multilayer for obtaining the layer S-matrices: (a) left-to-right directional characterization, (b) right-to-left directional characterization.

Fig. 3
Fig. 3

Intuitive derivation of the Redheffer star product relation through (a) the left-to-right directional characterization, (b) the right-to-left directional characterization of the combined multilayer.

Fig. 4
Fig. 4

Bidirectional characterization of (a) the boundary B 0 and (b) the boundary B N .

Fig. 5
Fig. 5

Coupling coefficient calculation in the conventional SMM.

Fig. 6
Fig. 6

Functional block-based FMM.

Fig. 7
Fig. 7

Multilayer structure divided into M blocks containing K layers.

Fig. 8
Fig. 8

Example target structure: (a) dielectric fiber tip, (b) multilayer structure modeling of the fiber tip with the staircase approximation.

Fig. 9
Fig. 9

(a) First selected input mode profile (in the x y plane), (b) excited electric field distribution (in the z x plane), (c) second selected input mode profile (in the x y plane), (d) excited electric field distribution (in the z x plane).

Fig. 10
Fig. 10

(a) Coupling coefficients of the positive eigenmodes and (b) those of the negative eigenmodes excited by the first selected input mode; (c) the deviation between the coupling coefficients of the positive modes obtained by the proposed SMM and by the ETMM; (d) the deviation between the coupling coefficients of the negative modes obtained by the proposed SMM and by the ETMM.

Fig. 11
Fig. 11

(a) Coupling coefficients of the positive eigenmodes and (b) those of the negative eigenmodes excited by the second selected input mode; (c) the deviation between the coupling coefficients of the positive modes obtained by the proposed SMM and by the ETMM; (d) the deviation between the coupling coefficients of the negative modes obtained by the proposed SMM and by the ETMM.

Equations (123)

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k x , s t = k x + 2 π s Λ x ,
k y , s t = k y + 2 π t Λ y ,
k z , s t = ( ( 2 π λ ) 2 ( k x , s t ) 2 ( k y , s t ) 2 ) 1 2 ,
U s t = ( x ̱ u x , s t + y ̱ u y , s t + z ̱ u z , s t ) exp [ j ( k x , s t x + k y , s t y + k z , s t z ) ]
U s t , ( x ) = ( x ̱ u x , s t + ( u x , s t k x , s t k z , s t ) z ̱ ) exp [ j ( k x , s t x + k y , s t y + k z , s t z ) ] ,
U s t , ( y ) = ( y ̱ u y , s t + ( u y , s t k y , s t k z , s t ) z ̱ ) exp [ j ( k x , s t x + k y , s t y + k z , s t z ) ] .
f = ( s + P ) ( 2 Q + 1 ) + t + Q + 1 for 1 f H ,
f = ( s + P ) ( 2 Q + 1 ) + t + Q + H + 1 for H + 1 f 2 H ,
U f = ( y ̱ u y , s t + ( u y , s t k y , s t k z , s t ) z ̱ ) exp [ j ( k x , s t x + k y , s t y + k z , s t z ) ] for 1 f H ,
U f = ( x ̱ u x , s t + ( u x , s t k x , s t k z , s t ) z ̱ ) exp [ j ( k x , s t x + k y , s t y + k z , s t z ) ] for H + 1 f 2 H .
E ̱ L , f = U f + h ( x ̱ r x , h f ( n , n + m ) + y ̱ r y , h f ( n , n + m ) + z ̱ r z , h f ( n , n + m ) ) exp [ j ( k x , h x + k y , h y k z , h z ) ] .
E ̱ R , f = h ( x ̱ t x , h f ( n , n + m ) + y ̱ t y , h f ( n , n + m ) + z ̱ t z , h f ( n , n + m ) ) exp [ j ( k x , h x + k y , h y + k z , h ( z l n , n + m ) ) ] ,
E ̱ ( n + k ) , f ( n , n + m ) = p = P P q = Q Q [ x ̱ S ( n + k ) , x , p q f ( n , n + m ) ( z ) + y ̱ S ( n + k ) , y , p q f ( n , n + m ) ( z ) + z ̱ S ( n + k ) , z , p q f ( n , n + m ) ( z ) ] exp [ j ( k x , p q x + k y , p q y ) ] ,
H ̱ ( n + k ) , f ( n , n + m ) = j ε 0 μ 0 p = P P q = Q Q [ x ̱ U ( n + k ) , x , p q f ( n , n + m ) ( z ) + y ̱ U ( n + k ) , y , p q f ( n , n + m ) ( z ) + z ̱ U ( n + k ) , z , p q f ( n , n + m ) ( z ) ] exp [ j ( k x , p q x + k y , p q y ) ] .
E ̱ ( n + k ) , f ( n , n + m ) = h = 1 H [ x ̱ S ( n + k ) , x , h f ( n , n + m ) ( z ) + y ̱ S ( n + k ) , y , h f ( n , n + m ) ( z ) + z ̱ S ( n + k ) , z , h f ( n , n + m ) ( z ) ] exp [ j ( k x , h x + k y , h y ) ] ,
H ̱ ( n + k ) , f ( n , n + m ) = j ε 0 μ 0 h = 1 H [ x ̱ U ( n + k ) , x , h f ( n , n + m ) ( z ) + y ̱ U ( n + k ) , y , h f ( n , n + m ) ( z ) + z ̱ U ( n + k ) , z , h f ( n , n + m ) ( z ) ] exp [ j ( k x , h x + k y , h y ) ] ,
h = ( p + P ) ( 2 Q + 1 ) + q + Q + 1 for 1 h H = ( 2 P + 1 ) ( 2 Q + 1 )
S ( n + k ) , h f ( n , n + m ) ( z ) = g W h , g ( n + k ) { c ( n + k ) , g f ( n , n + m ) + exp [ q g ( n + k ) ( z l 1 , n + k 1 ) ] + c ( n + k ) , g f ( n , n + m ) exp [ q g ( n + k ) ( z l 1 , n + k ) ] } ,
U ( n + k ) , h f ( n , n + m ) ( z ) = g V h , g ( n + k ) { c ( n + k ) , g f ( n , n + m ) + exp [ q g ( n + k ) ( z l 1 , n + k 1 ) ] c ( n + k ) , g f ( n , n + m ) exp [ q g ( n + k ) ( z l 1 , n + k ) ] } ,
r f ( n , n + m ) = ( [ r y , h f ( n , n + m ) ] [ r x , h f ( n , n + m ) ] ) ,
t f ( n , n + m ) = ( [ t y , h f ( n , n + m ) ] [ t x , h f ( n , n + m ) ] ) ,
[ r y , h f ( n , n + m ) ] = ( r y , 1 f ( n , n + m ) , r y , 2 f ( n , n + m ) , , r y , H f ( n , n + m ) ) t .
R ( n , n + m ) = ( r 1 ( n , n + m ) , r 2 ( n , n + m ) , , r 2 H ( n , n + m ) ) ,
T ( n , n + m ) = ( t 1 ( n , n + m ) , t 2 ( n , n + m ) , , t 2 H ( n , n + m ) ) .
r f ( n , n + m ) = R ( n , n + m ) u f ,
t f ( n , n + m ) = T ( n , n + m ) u f ,
C a , ( n + k ) ( n , n + m ) + = ( c a , ( n + k ) , 1 ( n , n + m ) + , c a , ( n + k ) , 2 ( n , n + m ) + , , c a , ( n + k ) , 2 H ( n , n + m ) + ) ,
C a , ( n + k ) ( n , n + m ) = ( c a , ( n + k ) , 1 ( n , n + m ) , c a , ( n + k ) , 2 ( n , n + m ) , , c a , ( n + k ) , H ( n , n + m ) ) ,
c a , ( n + k ) , f ( n , n + m ) + = [ c ( n + k ) , g f ( n , n + m ) + ] ,
c a , ( n + k ) , f ( n , n + m ) = [ c ( n + k ) , g f ( n , n + m ) ] ,
c a , ( n + k ) , f ( n , n + m ) + = C a , ( n + k ) ( n , n + m ) + u f ,
c a , ( n + k ) , f ( n , n + m ) = C a , ( n + k ) ( n , n + m ) u f .
C a , ( n + k ) ( n , n + m ) = ( C a , ( n + k ) ( n , n + m ) + C a , ( n + k ) ( n , n + m ) ) .
C ̃ a , ( n , n + m ) ( n , n + m ) = { C a , ( n ) ( n , n + m ) , C a , ( n + 1 ) ( n , n + m ) , , C a , ( n + m ) ( n , n + m ) } .
U = [ u 1 u 2 u 2 H ] = [ I 0 0 I ] ,
U f = ( x ̱ u x , f + y ̱ u y , f + z ̱ u z , f ) exp [ j ( k x , f x + k y , f y k z , f ( z d n , n + m ) ) ] .
E ̱ R , f = U f + h ( x ̱ r x , h f ( n , n + m ) + y ̱ r y , h f ( n , n + m ) + z ̱ r z , h f ( n , n + m ) ) exp [ j ( k x , h x + k y , h y + k z , h ( z d n , n + m ) ) ] .
E L , f = h ( x ̱ t x , h f ( n , n + m ) + y ̱ t y , h f ( n , n + m ) + z ̱ t z , h f ( n , n + m ) ) exp [ j ( k x , h x + k y , h y k z , h z ) ] .
C b , ( n + k ) ( n , n + m ) = ( C b , ( n + k ) ( n , n + m ) C b , ( n + k ) ( n , n + m ) + ) .
C ̃ b , ( n , n + m ) ( n , n + m ) = { C b , ( n ) ( n , n + m ) , C b , ( n + 1 ) ( n , n + m ) , , C b , ( n + m ) ( n , n + m ) } .
S ( n , n + m ) = [ T ( n , n + m ) R ( n , n + m ) R ( n , n + m ) T ( n , n + m ) ] .
R ( n , n + m + l ) = R ( n , n + m ) + T ( n , n + m ) [ k = 0 ( R ( n + m + 1 , n + m + l ) R ( n , n + m ) ) k ] R ( n + m + 1 , n + m + l ) T ( n , n + m ) = R ( n , n + m ) + T ( n , n + m ) ( I R ( n + m + 1 , n + m + l ) R ( n , n + m ) ) 1 R ( n + m + 1 , n + m + l ) T ( n , n + m ) ,
T ( n , n + m + l ) = T ( n + m + 1 , n + m + l ) [ k = 0 ( R ( n , n + m ) R ( n + m + 1 , n + m + l ) ) k ] T ( n , n + m ) = T ( n + m + 1 , n + m + l ) ( I R ( n , n + m ) R ( n + m + 1 , n + m + l ) ) 1 T ( n , n + m ) ,
R ( n , n + m + l ) = R ( n + m + 1 , n + m + l ) + T ( n + m + 1 , n + m + l ) [ k = 0 ( R ( n , n + m ) R ( n + m + 1 , n + m + l ) ) k ] R ( n , n + m ) T ( n + m + 1 , n + m + l ) = R ( n + m + 1 , n + m + l ) + T ( n + m + 1 , n + m + l ) ( I R ( n , n + m ) R ( n + m + 1 , n + m + l ) ) 1 R ( n , n + m ) T ( n + m + 1 , n + m + l ) ,
T ( n , n + m + l ) = T ( n , n + m ) [ k = 0 ( R ( n + m + 1 , n + m + l ) R ( n , n + m ) ) k ] T ( n + m + 1 , n + m + l ) = T ( n , n + m ) ( I R ( n + m + 1 , n + m + l ) R ( n , n + m ) ) 1 T ( n + m + 1 , n + m + l ) .
S ( n , n + m + l ) = S ( n , n + m ) S ( n + m + 1 , n + m + l ) .
C ̃ a , ( n , n + m + l ) ( n , n + m + l ) = { C ̃ a , ( n , n + m ) ( n , n + m + l ) , C ̃ a , ( n + m + 1 , n + m + l ) ( n , n + m + l ) } ,
C ̃ b , ( n , n + m + l ) ( n , n + m + l ) = { C ̃ b , ( n , n + m ) ( n , n + m + l ) , C ̃ b , ( n + m + 1 , n + m + l ) ( n , n + m + l ) } ,
C ̃ a , ( n , n + m ) ( n , n + m + l ) = { C a , ( n ) ( n , n + m + l ) , C a , ( n + 1 ) ( n , n + m + l ) , , C a , ( n + m ) ( n , n + m + l ) } ,
C ̃ a , ( n + m + 1 , n + m + l ) ( n , n + m + l ) = { C a , ( n + m + 1 ) ( n , n + m + l ) , C a , ( n + m + 2 ) ( n , n + m + l ) , , C a , ( n + m + l ) ( n , n + m + l ) } ,
C ̃ b , ( n , n + m ) ( n , n + m + l ) = { C b , ( n ) ( n , n + m + l ) , C b , ( n + 1 ) ( n , n + m + l ) , , C b , ( n + m ) ( n , n + m + l ) } ,
C ̃ b , ( n + m + 1 , n + m + l ) ( n , n + m + l ) = { C b , ( n + m + 1 ) ( n , n + m + l ) , C b , ( n + m + 2 ) ( n , n + m + l ) , , C b , ( n + m + l ) ( n , n + m + l ) } .
C a , ( k ) ( n , n + m + l ) = C a , ( k ) ( n , n + m ) + C b , ( k ) ( n , n + m ) ( I R ( n + m + 1 , n + m + l ) R ( n , n + m ) ) 1 R ( n + m + 1 , n + m + l ) T ( n , n + m ) ,
C b , ( k ) ( n , n + m + l ) = C b , ( k ) ( n , n + m ) ( I R ( n + m + 1 , n + m + l ) R ( n , n + m ) ) 1 T ( n + m + 1 , n + m + l ) .
C a , ( k ) ( n , n + m + l ) = C a , ( k ) ( n + m + 1 , n + m + l ) ( I R ( n , n + m ) R ( n + m + 1 , n + m + l ) ) 1 T ( n , n + m ) ,
C b , ( k ) ( n , n + m + l ) = C b , ( k ) ( n + m + 1 , n + m + l ) + C a , ( k ) ( n + m + 1 , n + m + l ) ( I R ( n , n + m ) R ( n + m + 1 , n + m + l ) ) 1 R ( n , n + m ) T ( n + m + 1 , n + m + l ) .
( C ̃ a , ( n , n + m + l ) ( n , n + m + l ) , C ̃ b , ( n , n + m + l ) ( n , n + m + l ) ) = ( C ̃ a , ( n , n + m ) ( n , n + m ) , C ̃ b , ( n , n + m ) ( n , n + m ) ) ( C ̃ a , ( n + m + 1 , n + m + l ) ( n + m + 1 , n + m + l ) , C ̃ b , ( n + m + 1 , n + m + l ) ( n + m + 1 , n + m + l ) ) .
S ( 1 , N ) = S ( 1 , 1 ) S ( 2 , 2 ) S ( N 1 , N 1 ) S ( N , N ) ,
( C ̃ a , ( 1 , N ) ( 1 , N ) , C ̃ b , ( 1 , N ) ( 1 , N ) ) = ( C ̃ a , ( 1 , 1 ) ( 1 , 1 ) , C ̃ b , ( 1 , 1 ) ( 1 , 1 ) ) ( C ̃ a , ( 2 , 2 ) ( 2 , 2 ) , C ̃ b , ( 2 , 2 ) ( 2 , 2 ) ) ( C ̃ a , ( N 1 , N 1 ) ( N 1 , N 1 ) , C ̃ b , ( N 1 , N 1 ) ( N 1 , N 1 ) ) ( C ̃ a , ( N , N ) ( N , N ) , C ̃ b , ( N , N ) ( N , N ) ) .
[ W h W h V h V h ] ( U R ( n , n ) ) = [ W ( n ) W ( n ) X ( n ) V ( n ) V ( n ) X ( n ) ] ( C a , ( n ) ( n , n ) + C a , ( n ) ( n , n ) ) ,
[ W ( n ) X ( n ) W ( n ) V ( n ) X ( n ) V ( n ) ] ( C a , ( n ) ( n , n ) + C a , ( n ) ( n , n ) ) = ( W h V h ) T ( n , n ) .
( C a , ( n ) ( n , n ) + C a , ( n ) ( n , n ) ) = [ W h 1 W ( n ) + V h 1 V ( n ) ( W h 1 W ( n ) V h 1 V ( n ) ) X ( n ) ( W h 1 W ( n ) V h 1 V ( n ) ) X ( n ) W h 1 W ( n ) + V h 1 V ( n ) ] 1 ( 2 U 0 ) ,
R ( n , n ) = W h 1 [ W ( n ) C a , ( n ) ( n , n ) + + W ( n ) X ( n ) C a , ( n ) ( n , n ) W h U ] ,
T ( n , n ) = W h 1 [ W ( n ) X ( n ) C a , ( n ) ( n , n ) + + W ( n ) C a , ( n ) ( n , n ) ] ,
W h = [ I 0 0 I ] ,
V h = [ [ k x , s t k y , s t j k 0 k I , z , s t ] [ ( k I , z , s t 2 + k x , s t 2 ) j k 0 k I , z , s t ] [ ( k y , s t 2 + k I , z , s t 2 ) j k 0 k I , z , s t ] [ k y , s t k x , s t j k 0 k I , z , s t ] ] .
( W h V h ) T ( n , n ) = [ W ( n ) X ( n ) W ( n ) V ( n ) X ( n ) V ( n ) ] ( C b , ( n ) ( n , n ) C b , ( n ) ( n , n ) + ) ,
[ W ( n ) W ( n ) X ( n ) V ( n ) V ( n ) X ( n ) ] ( C b , ( n ) ( n , n ) C b , ( n ) ( n , n ) + ) = [ W h W h V h V h ] ( U R ( n , n ) ) .
( C b , ( n ) ( n , n ) C b , ( n ) ( n , n ) + ) = [ W h 1 W ( n ) + V h 1 V ( n ) ( W h 1 W ( n ) V h 1 V ( n ) ) X ( n ) ( W h 1 W ( n ) V h 1 V ( n ) ) X ( n ) W h 1 W ( n ) + V h 1 V ( n ) ] 1 ( 2 U 0 ) ,
R ( n , n ) = W h 1 [ W ( n ) C b , ( n ) ( n , n ) + W ( n ) X ( n ) C b , ( n ) ( n , n ) + W h U ] ,
T ( n , n ) = W h 1 [ W ( n ) X ( n ) C b , ( n ) ( n , n ) + W ( n ) C b , ( n ) ( n , n ) + ] .
R ( n , n ) = R ( n , n ) ,
T ( n , n ) = T ( n , n ) ,
( C a , ( n ) ( n , n ) + C a , ( n ) ( n , n ) ) = ( C b , ( n ) ( n , n ) C b , ( n ) ( n , n ) + ) .
S ( 0 , 0 ) = [ T ( 0 , 0 ) R ( 0 , 0 ) R ( 0 , 0 ) T ( 0 , 0 ) ] ,
T ( 0 , 0 ) = 2 [ ( W ( 0 ) ) 1 W h + ( V ( 0 ) ) 1 V h ] 1 ,
R ( 0 , 0 ) = [ ( W h ) 1 W ( 0 ) + ( V h ) 1 V ( 0 ) ] 1 [ ( W h ) 1 W ( 0 ) + ( V h ) 1 V ( 0 ) ] ,
T ( 0 , 0 ) = 2 [ ( W h ) 1 W ( 0 ) + ( V h ) 1 V ( 0 ) ] 1 ,
R ( 0 , 0 ) = [ ( W ( 0 ) ) 1 W h + ( V ( 0 ) ) 1 V h ] 1 [ ( W ( 0 ) ) 1 W h + ( V ( 0 ) ) 1 V h ] .
S ( N + 1 , N + 1 ) = [ T ( N + 1 , N + 1 ) R ( N + 1 , N + 1 ) R ( N + 1 , N + 1 ) T ( N + 1 , N + 1 ) ] ,
T ( N + 1 , N + 1 ) = 2 [ ( W h ) 1 W ( N + 1 ) + ( V h ) 1 V ( N + 1 ) ] 1 ,
R ( N + 1 , N + 1 ) = [ ( W ( N + 1 ) ) 1 W h + ( V ( N + 1 ) ) 1 V h ] 1 [ ( W ( N + 1 ) ) 1 W h + ( V ( N + 1 ) ) 1 V h ] ,
T ( N + 1 , N + 1 ) = 2 [ ( W ( N + 1 ) ) 1 W h + ( V ( N + 1 ) ) 1 V h ] 1 ,
R ( N + 1 , N + 1 ) = [ ( W h ) 1 W ( N + 1 ) + ( V h ) 1 V ( N + 1 ) ] 1 [ ( W h ) 1 W ( N + 1 ) + ( V h ) 1 V ( N + 1 ) ] .
S ( 0 , N ) = S ( 0 , 0 ) S ( 1 , N ) .
( C a , ( k ) ( 0 , N ) , C b , ( k ) ( 0 , N ) ) = ( C a , ( k ) ( 1 , N ) ( I R ( 0 , 0 ) R ( 1 , N ) ) 1 T ( 0 , 0 ) , C b , ( k ) ( 1 , N ) + C a , ( k ) ( 1 , N ) ( I R ( 0 , 0 ) R ( 1 , N ) ) 1 R ( 0 , 0 ) T ( 1 , N ) ) .
S ( 0 , N + 1 ) = S ( 0 , N ) S ( N + 1 , N + 1 ) .
( C a , ( k ) ( 0 , N + 1 ) , C b , ( k ) ( 0 , N + 1 ) ) = ( C a , ( k ) ( 0 , N ) + C b , ( k ) ( 0 , N ) ( I R ( N + 1 , N + 1 ) R ( 0 , N ) ) 1 R ( N + 1 , N + 1 ) T ( 0 , N ) , C b , ( k ) ( 0 , N ) ( I R ( N + 1 , N + 1 ) R ( 0 , N ) ) 1 T ( N + 1 , N + 1 ) ) .
[ W ( n ) X ( n ) W ( n ) V ( n ) X ( n ) V ( n ) ] [ C ( n ) ( 0 , N + 1 ) + C ( n ) ( 0 , N + 1 ) ] = [ W ( n + 1 ) W ( n + 1 ) X ( n + 1 ) V ( n + 1 ) V ( n ) X ( n + 1 ) ] [ C ( n + 1 ) ( 0 , N + 1 ) + C ( n + 1 ) ( 0 , N + 1 ) ] .
[ C ( n + 1 ) ( 0 , N + 1 ) + C ( n ) ( 0 , N + 1 ) ] = S ̂ ( n , n ) [ C ( n ) ( 0 , N + 1 ) + C ( n + 1 ) ( 0 , N + 1 ) ] = [ t ( n , n ) r ( n , n ) r ( n , n ) t ( n , n ) ] [ C ( n ) ( 0 , N + 1 ) + C ( n + 1 ) ( 0 , N + 1 ) ] ,
t ( n , n ) = 2 [ ( W ( n ) ) 1 W ( n + 1 ) + ( V ( n ) ) 1 V ( n + 1 ) ] 1 X ( n ) ,
r ( n , n ) = [ ( W ( n + 1 ) ) 1 W ( n ) + ( V ( n + 1 ) ) 1 V ( n ) ] 1 [ ( W ( n + 1 ) ) 1 W ( n ) + ( V ( n + 1 ) ) 1 V ( n ) ] X ( n ) ,
t ( n , n ) = 2 [ ( W ( n + 1 ) ) 1 W ( n ) + ( V ( n + 1 ) ) 1 V ( n ) ] 1 X ( n + 1 ) ,
r ( n , n ) = [ ( W ( n ) ) 1 W ( n + 1 ) + ( V ( n ) ) 1 V ( n + 1 ) ] 1 [ ( W ( n ) ) 1 W ( n + 1 ) + ( V ( n ) ) 1 V ( n + 1 ) ] X ( n + 1 ) .
S ̂ ( 0 , N ) = S ̂ ( 0 , 0 ) S ̂ ( 2 , 2 ) S ̂ ( N , N )
[ C ( N + 1 ) ( 0 , N + 1 ) + C ( 0 ) ( 0 , N + 1 ) ] = S ̂ ( 0 , N ) [ C ( 0 ) ( 0 , N + 1 ) + C ( N + 1 ) ( 0 , N + 1 ) ] = [ t ( 0 , N ) r ( 0 , N ) r ( 0 , N ) t ( 0 , N ) ] [ C ( 0 ) ( 0 , N + 1 ) + C ( N + 1 ) ( 0 , N + 1 ) ] .
S ̂ ( 0 , n 1 ) = [ t ( 0 , n 1 ) r ( 0 , n 1 ) r ( 0 , n 1 ) t ( 0 , n 1 ) ] ,
S ̂ ( n , N ) = [ t ( n , N ) r ( n , N ) r ( n , N ) t ( n , N ) ] .
C a , ( n ) ( 0 , N + 1 ) + = ( I r ( 0 , n 1 ) r ( n , N ) ) 1 t ( 0 , n 1 ) ,
C a , ( n ) ( 0 , N + 1 ) = r ( n , N ) ( I r ( 0 , n 1 ) r ( n , N ) ) 1 t ( 0 , n 1 ) ,
C b , ( n ) ( 0 , N + 1 ) + = r ( 0 , n 1 ) ( I r ( n , N ) r ( 0 , n 1 ) ) 1 t ( n , N ) ,
C b , ( n ) ( 0 , N + 1 ) = ( I r ( n , N ) r ( 0 , n 1 ) ) 1 t ( n , N ) .
cnt ( R ( 1 , 2 ) ) = 4 m M + 2 m A + m I ,
cnt ( T ( 1 , 2 ) ) = 3 m M + m A + m I ,
cnt ( R ( 1 , 2 ) ) = 3 m M + m A ,
cnt ( T ( 1 , 2 ) ) = 2 m M ,
cnt ( S ( 1 , 2 ) ) = 12 m M + 4 m A + 2 m I .
cnt ( C a , ( 1 ) ( 1 , 2 ) ) = 2 m M + 2 m A ,
cnt ( C b , ( 1 ) ( 1 , 2 ) ) = 2 m M ,
cnt ( C a , ( 2 ) ( 1 , 2 ) ) = 2 m M ,
cnt ( C b , ( 2 ) ( 1 , 2 ) ) = 2 m M + 2 m A .
cnt [ ( C ̃ a , ( 1 , 2 ) ( 1 , 2 ) , C ̃ b , ( 1 , 2 ) ( 1 , 2 ) ) ] = 8 m M + 4 m A .
T prop = cnt ( S ( 1 , N ) ) + cnt [ ( C ̃ a , ( 1 , N ) ( 1 , N ) , C ̃ b , ( 1 , N ) ( 1 , N ) ) ] = cnt ( S ( 1 , 2 ) = S ( 1 , 1 ) S ( 2 , 2 ) ) + cnt ( S ( 1 , 3 ) = S ( 1 , 2 ) S ( 3 , 3 ) ) + + cnt ( S ( 1 , N ) = S ( 1 , N 1 ) S ( N , N ) ) + cnt [ ( C ̃ a , ( 1 , 2 ) ( 1 , 2 ) , C ̃ b , ( 1 , 2 ) ( 1 , 2 ) ) ] + cnt [ ( C ̃ a , ( 1 , 3 ) ( 1 , 3 ) , C ̃ b , ( 1 , 3 ) ( 1 , 3 ) ) ] + + cnt [ ( C ̃ a , ( 1 , N ) ( 1 , N ) , C ̃ b , ( 1 , N ) ( 1 , N ) ) ] = ( N 1 ) ( 12 m M + 4 m A + 2 m I ) + [ 2 + 3 + + N ] ( 4 m M + 2 m A ) = ( N 1 ) ( 12 m M + 4 m A + 2 m I ) + ( N 1 ) ( N + 2 ) ( 2 m M + m A ) = ( 2 N 2 + 14 N 16 ) m M + ( N 2 + 5 N 6 ) m A + ( 2 N 2 ) m I .
cnt ( S ̂ ( 1 , 2 ) = S ̂ ( 1 , 1 ) S ̂ ( 2 , 2 ) ) = 12 m M + 4 m A + 2 m I .
cnt ( C a , ( n ) ( 0 , N + 1 ) + ) = 2 m M + m A + m I ,
cnt ( C a , ( n ) ( 0 , N + 1 ) ) = m M ,
cnt ( C b , ( n ) ( 0 , N + 1 ) ) = 2 m M + m A + m I ,
cnt ( C b , ( n ) ( 0 , N + 1 ) + ) = m M .
T conv = cnt ( ( S ̂ ( 0 , 0 ) , S ̂ ( 1 , N ) ) ) + cnt ( ( S ̂ ( 0 , 1 ) , S ̂ ( 2 , N ) ) ) + + cnt ( ( S ̂ ( 0 , N 1 ) , S ̂ ( N , N ) ) ) + cnt [ ( C a , ( 1 ) ( 0 , N + 1 ) + , C a , ( 1 ) ( 0 , N + 1 ) , C b , ( 1 ) ( 0 , N + 1 ) + , C b , ( 1 ) ( 0 , N + 1 ) ) ] + + cnt [ ( C a , ( N ) ( 0 , N + 1 ) + , C a , ( N ) ( 0 , N + 1 ) , C b , ( N ) ( 0 , N + 1 ) + , C b , ( N ) ( 0 , N + 1 ) ) ] = N [ ( N 1 ) ( 12 m M + 4 m A + 2 m I ) + 6 m M + 2 m A + 2 m I ] = ( 12 N 2 6 N ) m M + ( 4 N 2 2 N ) m A + 2 N 2 m I .
T conv T prop = ( 10 N 2 20 N + 16 ) m M + ( 3 N 2 7 N + 6 ) m A + ( 2 N 2 2 N + 2 ) m I 0 .
M p = { m ( p ) m ( p ) = 1 , 2 , 3 , , 2 p } for p = 0 , 1 , , log 2 M ,
S ̃ ( p 1 ) ( m ( p 1 ) ) = S ̃ ( p ) ( 2 m ( p ) 1 ) S ̃ ( p ) ( 2 m ( p ) ) ,
( C ̃ a , ( p 1 ) ( m ( p 1 ) ) , C ̃ b , ( p 1 ) ( m ( p 1 ) ) ) = ( C ̃ a , ( p ) ( 2 m ( p 1 ) 1 ) , C ̃ b , ( p ) ( 2 m ( p 1 ) 1 ) ) ( C ̃ a , ( p ) ( 2 m ( p 1 ) ) , C ̃ b , ( p ) ( 2 m ( p 1 ) ) ) .

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