Abstract

The Fourier modal method (FMM) is a method for efficiently solving Maxwell’s equations with periodic boundary conditions. In order to apply the FMM to nonperiodic structures, perfectly matched layers need to be placed at the periodic boundaries, and the Maxwell equations have to be formulated in terms of a contrast (scattered) field. This reformulation modifies the structure of the resulting linear systems and makes the direct application of available stable recursion algorithms impossible. We adapt the well-known S-matrix algorithm for use with the aperiodic FMM in contrast-field formulation. To this end, stable recursive relations are derived for linear systems with nonhomogeneous structure. The stability of the algorithm is confirmed by numerical results.

© 2011 Optical Society of America

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References

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  1. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “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]
  2. E. Popov and M. Nevière, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A 17, 1773–1784 (2000).
    [CrossRef]
  3. M. Pisarenco, J. Maubach, I. Setija, and R. Mattheij, “Aperiodic Fourier modal method in contrast-field formulation for simulation of scattering from finite structures,” J. Opt. Soc. Am. A 27, 2423–2431 (2010).
    [CrossRef]
  4. L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
    [CrossRef]
  5. D. M. Pai and K. A. Awada, “Analysis of dielectric gratings of arbitrary profiles and thicknesses,” J. Opt. Soc. Am. A 8, 755–762(1991).
    [CrossRef]
  6. L. F. DeSandre and J. M. Elson, “Extinction-theorem analysis of diffraction anomalies in overcoated gratings,” J. Opt. Soc. Am. A 8, 763–777 (1991).
    [CrossRef]
  7. N. Chateau and J.-P. Hugonin, “Algorithm for the rigorous coupled-wave analysis of grating diffraction,” J. Opt. Soc. Am. A 11, 1321–1331 (1994).
    [CrossRef]
  8. D. Y. K. Ko and J. R. Sambles, “Scattering matrix method for propagation of radiation in stratified media: attenuated total reflection studies of liquid crystals,” J. Opt. Soc. Am. A 5, 1863–1866 (1988).
    [CrossRef]
  9. M. G. Moharam, D. A. Pommet, E. B. Grann, and T. K. Gaylord, “Stable implementation of the rigorous coupled-wave analysis for surface-relief gratings: enhanced transmittance matrix approach,” J. Opt. Soc. Am. A 12, 1077–1086 (1995).
    [CrossRef]
  10. E. L. Tan, “Note on formulation of the enhanced scattering- (transmittance-) matrix approach,” J. Opt. Soc. Am. A 19, 1157–1161 (2002).
    [CrossRef]
  11. M. van Kraaij, “Forward diffraction modelling: analysis and application to grating reconstruction,” Ph.D. thesis (Eindhoven University of Technology, 2011).
  12. 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]
  13. H. Kim, I.-M. Lee, and B. Lee, “Extended scattering-matrix method for efficient full parallel implementation of rigorous coupled-wave analysis,” J. Opt. Soc. Am. A 24, 2313–2327(2007).
    [CrossRef]
  14. J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200(1994).
    [CrossRef]
  15. W. C. Chew, J. M. Jin, and E. Michielssen, “Complex coordinate stretching as a generalized absorbing boundary condition,” Microw. Opt. Technol. Lett. 15, 363–369 (1997).
    [CrossRef]
  16. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876(1996).
    [CrossRef]
  17. J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189–195(1955).
    [CrossRef]
  18. E. Popov, M. Nevière, B. Gralak, and G. Tayeb, “Staircase approximation validity for arbitrary-shaped gratings,” J. Opt. Soc. Am. A 19, 33–42 (2002).
    [CrossRef]
  19. T. Schuster, J. Ruoff, N. Kerwien, S. Rafler, and W. Osten, “Normal vector method for convergence improvement using the RCWA for crossed gratings,” J. Opt. Soc. Am. A 24, 2880–2890 (2007).
    [CrossRef]

2010

2007

2002

2000

1997

W. C. Chew, J. M. Jin, and E. Michielssen, “Complex coordinate stretching as a generalized absorbing boundary condition,” Microw. Opt. Technol. Lett. 15, 363–369 (1997).
[CrossRef]

1996

1995

1994

N. Chateau and J.-P. Hugonin, “Algorithm for the rigorous coupled-wave analysis of grating diffraction,” J. Opt. Soc. Am. A 11, 1321–1331 (1994).
[CrossRef]

J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200(1994).
[CrossRef]

1993

1991

1988

1955

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189–195(1955).
[CrossRef]

Awada, K. A.

Berenger, J.

J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200(1994).
[CrossRef]

Chateau, N.

Chew, W. C.

W. C. Chew, J. M. Jin, and E. Michielssen, “Complex coordinate stretching as a generalized absorbing boundary condition,” Microw. Opt. Technol. Lett. 15, 363–369 (1997).
[CrossRef]

DeSandre, L. F.

Elson, J. M.

Gaylord, T. K.

Gralak, B.

Grann, E. B.

Hugonin, J.-P.

Jin, J. M.

W. C. Chew, J. M. Jin, and E. Michielssen, “Complex coordinate stretching as a generalized absorbing boundary condition,” Microw. Opt. Technol. Lett. 15, 363–369 (1997).
[CrossRef]

Kerwien, N.

Kim, H.

Ko, D. Y. K.

Lee, B.

Lee, I.-M.

Li, L.

Mattheij, R.

Maubach, J.

Michielssen, E.

W. C. Chew, J. M. Jin, and E. Michielssen, “Complex coordinate stretching as a generalized absorbing boundary condition,” Microw. Opt. Technol. Lett. 15, 363–369 (1997).
[CrossRef]

Moharam, M. G.

Nevière, M.

Osten, W.

Pai, D. M.

Pisarenco, M.

Pommet, D. A.

Popov, E.

Rafler, S.

Ruoff, J.

Sambles, J. R.

Schuster, T.

Setija, I.

Tan, E. L.

Tayeb, G.

van Kraaij, M.

M. van Kraaij, “Forward diffraction modelling: analysis and application to grating reconstruction,” Ph.D. thesis (Eindhoven University of Technology, 2011).

Wait, J. R.

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189–195(1955).
[CrossRef]

Can. J. Phys.

J. R. Wait, “Scattering of a plane wave from a circular dielectric cylinder at oblique incidence,” Can. J. Phys. 33, 189–195(1955).
[CrossRef]

J. Comput. Phys.

J. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200(1994).
[CrossRef]

J. Opt. Soc. Am. A

N. Chateau and J.-P. Hugonin, “Algorithm for the rigorous coupled-wave analysis of grating diffraction,” J. Opt. Soc. Am. A 11, 1321–1331 (1994).
[CrossRef]

D. Y. K. Ko and J. R. Sambles, “Scattering matrix method for propagation of radiation in stratified media: attenuated total reflection studies of liquid crystals,” J. Opt. Soc. Am. A 5, 1863–1866 (1988).
[CrossRef]

D. M. Pai and K. A. Awada, “Analysis of dielectric gratings of arbitrary profiles and thicknesses,” J. Opt. Soc. Am. A 8, 755–762(1991).
[CrossRef]

L. F. DeSandre and J. M. Elson, “Extinction-theorem analysis of diffraction anomalies in overcoated gratings,” J. Opt. Soc. Am. A 8, 763–777 (1991).
[CrossRef]

L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
[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]

L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876(1996).
[CrossRef]

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

E. Popov and M. Nevière, “Grating theory: new equations in Fourier space leading to fast converging results for TM polarization,” J. Opt. Soc. Am. A 17, 1773–1784 (2000).
[CrossRef]

E. Popov, M. Nevière, B. Gralak, and G. Tayeb, “Staircase approximation validity for arbitrary-shaped gratings,” J. Opt. Soc. Am. A 19, 33–42 (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]

H. Kim, I.-M. Lee, and B. Lee, “Extended scattering-matrix method for efficient full parallel implementation of rigorous coupled-wave analysis,” J. Opt. Soc. Am. A 24, 2313–2327(2007).
[CrossRef]

T. Schuster, J. Ruoff, N. Kerwien, S. Rafler, and W. Osten, “Normal vector method for convergence improvement using the RCWA for crossed gratings,” J. Opt. Soc. Am. A 24, 2880–2890 (2007).
[CrossRef]

M. Pisarenco, J. Maubach, I. Setija, and R. Mattheij, “Aperiodic Fourier modal method in contrast-field formulation for simulation of scattering from finite structures,” J. Opt. Soc. Am. A 27, 2423–2431 (2010).
[CrossRef]

Microw. Opt. Technol. Lett.

W. C. Chew, J. M. Jin, and E. Michielssen, “Complex coordinate stretching as a generalized absorbing boundary condition,” Microw. Opt. Technol. Lett. 15, 363–369 (1997).
[CrossRef]

Other

M. van Kraaij, “Forward diffraction modelling: analysis and application to grating reconstruction,” Ph.D. thesis (Eindhoven University of Technology, 2011).

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Figures (5)

Fig. 1
Fig. 1

Sliced geometry. The dashed line represents the smooth profile being approximated. The hatched areas indicate PMLs.

Fig. 2
Fig. 2

Stack of interfaces with upward- and downward-traveling waves in between.

Fig. 3
Fig. 3

Sliced geometry of a cylinder. Different colors correspond to different refraction indices.

Fig. 4
Fig. 4

Absolute values of the magnetic field: (left) exact solution and (right) solution computed with the aFMM-CFF. Hatched areas indicate PMLs.

Fig. 5
Fig. 5

Convergence of the aFMM-CFF with the adapted S-matrix and T-matrix approaches for a cylinder with radius ρ = 50 nm approximated by M = 79 slices.

Tables (1)

Tables Icon

Table 1 Magnitudes of the Magnetic Field in a Fixed Point above the Cylinder Computed with the T-Matrix and S-Matrix Algorithms Adapted for Contrast-Field Formulation for Increasing Radius ρ and Truncation Order N

Equations (98)

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× e l t ( x ) = k 0 h l t ( x ) , × h l t ( x ) = k 0 ϵ l ( x ) e l t ( x ) ,
× e l b ( x ) = k 0 h l b ( x ) , × h l b ( x ) = k 0 ϵ l b e l b ( x ) .
x 1 f ( x ) x ,
f ( x ) = { x + i σ 0 | x x l | ( p + 1 ) / ( p + 1 ) , 0 x x l , x , x l < x < x r , x i σ 0 | x x r | ( p + 1 ) / ( p + 1 ) , x r x Λ .
ϕ n ( x , y ) = e i ( k x n x + k y y ) ,
k x n = k x inc + n 2 π Λ , k y = k y inc , for     n = N + N .
e α , l t / b ( x , y , z ) = n = N N s α , l , n t / b ( z ) ϕ n ( x , y ) = ( s α , l t / b ( z ) ) T · ϕ ( x , y ) ,
h α , l t / b ( x , y , z ) = n = N N u α , l , n t / b ( z ) ϕ n ( x , y ) = ( u α , l t / b ( z ) ) T · ϕ ( x , y ) .
i K y s z , l c ( z ) k 0 1 d d z s y , l c ( z ) = u x , l c ( z ) ,
k 0 1 d d z s x , l c ( z ) + i F K x s z , l c ( z ) = u y , l c ( z ) ,
i F K x s y , l c ( z ) + i K y s x , l c ( z ) = u z , l c ( z ) ,
i K y u z , l c ( z ) k 0 1 d d z u y , l c ( z ) = P l 1 s x , l c ( z ) ( P l 1 ( P l b ) 1 ) s x , l b ( z ) ,
k 0 1 d d z u x , l c ( z ) + i F K x u z , l c ( z ) = E l s y , l c ( z ) ( E l E l b ) s y , l b ( z ) ,
i F K x u y , l c ( z ) + i K y u x , l c ( z ) = E l s z , l c ( z ) ( E l E l b ) s z , l b ( z ) .
( K x ) m n = ( k x n / k 0 ) δ m n ,
( K y ) m n = ( k y / k 0 ) δ m n ,
( E l ) m n = ϵ ^ l , n m ,
( P l ) m n = p ^ l , n m ,
( E l b ) m n = ϵ l b δ m n ,
( P l b ) m n = ( ϵ l b ) 1 δ m n ,
( F ) m n = γ ^ n m ,
s α , l b ( z ) = d 0 s α , l b ( z ) = d 0 ( a s , α , l e k 0 q l ( z h l ) + r s , α , l e k 0 q l ( z h l ) ) ,
u α , l b ( z ) = d 0 u α , l b ( z ) = d 0 ( a u , α , l e k 0 q l ( z h l ) + r u , α , l e k 0 q l ( z h l ) ) ,
q l = i ϵ l b ( k x inc k 0 ) 2 ( k y inc k 0 ) 2 .
d 2 d z 2 s y , l c ( z ) = k 0 2 A l s y , l c ( z ) k 0 2 ( E l E l b ) s y , l b ( z ) ,
s y , l c = s y , hom , l c + s y , part , l c .
s y , hom , l c ( z ) = W l ( e k 0 Q l ( z h l ) c l + + e k 0 Q l ( z h l + 1 ) c l ) ,
s y , part , l c ( z ) = p l s y , l b ( z ) ,
( A l q l 2 I ) p l = ( E l E l b ) d 0 .
s y , l c ( h l + 1 ) = s y , l + 1 c ( h l + 1 ) ,
u x , l c ( h l + 1 ) = u x , l + 1 c ( h l + 1 ) .
s y , l c ( h l + 1 ) = s y , l + 1 c ( h l + 1 ) ,
k 0 1 d d z s y , l c ( h l + 1 ) = k 0 1 d d z s y , l + 1 c ( h l + 1 ) .
[ W l X l W l V l X l V l ] [ c l + c l ] + g l ( h l + 1 ) = [ W l + 1 W l + 1 X l + 1 V l + 1 V l + 1 X l + 1 ] [ c l + 1 + c l + 1 ] + g l + 1 ( h l + 1 ) ,
g l ( z ) = [ p l s y , l b p l u x , l b ] .
d 2 d z 2 u y , l c = P l 1 B l u y , l c + ( P l 1 B l ( P l b ) 1 B l b ) u y , l b ,
u y , l c = u y , hom , l c + u y , part , l c .
u y , hom , l c ( z ) = W l ( e k 0 Q l ( z h l ) c l + + e k 0 Q l ( z h l + 1 ) c l ) ,
u y , part , l c ( z ) = p l u y , l b ( z ) .
( P l 1 B l q l 2 I ) p l = ( ( P l b ) 1 B l b P l 1 B l ) d 0 .
u y , l c ( h l + 1 ) = u y , l + 1 c ( h l + 1 ) ,
s x , l c ( h l + 1 ) = s x , l + 1 c ( h l + 1 ) .
u y , l c ( h l + 1 ) = u y , l + 1 c ( h l + 1 ) ,
k 0 1 P l d d z u y , l c ( h l + 1 ) P l ( P l 1 ( P l b ) 1 ) s x , l b ( h l + 1 ) = k 0 1 P l + 1 d d z u y , l + 1 c ( h l + 1 ) P l + 1 ( P l + 1 1 ( P l + 1 b ) 1 ) s x , l + 1 b ( h l + 1 ) .
[ W l X l W l V l X l V l ] [ c l + c l ] + g l ( h l + 1 ) = [ W l + 1 W l + 1 X l + 1 V l + 1 V l + 1 X l + 1 ] [ c l + 1 + c l + 1 ] + g l + 1 ( h l + 1 ) ,
g l ( z ) = [ p l u y , l b ( P l p l + ( P l ϵ l b I ) d 0 ) s x , l b ] .
d 2 d z 2 s x , l c ( z ) = k 0 2 C l s x , l c ( z ) + k 0 2 ( B l P l 1 B l b ( P l b ) 1 ) s x , l b ( z ) ,
d 2 d z 2 u x , l c ( z ) = k 0 2 D l u x , l c ( z ) k 0 2 ( E l E l b ) u x , l b ( z ) ,
s x , l c = s x , hom , l c + s x , part , l c ,
u x , l c = u x , hom , l c + u x , part , l c .
s x , hom , l c ( z ) = W s , l ( e k 0 Q s , l ( z h l ) c s , l + + e k 0 Q s , l ( z h l + 1 ) c s , l ) ,
u x , hom , l c ( z ) = W u , l ( e k 0 Q u , l ( z h l ) c u , l + + e k 0 Q u , l ( z h l + 1 ) c u , l ) ,
s x , part , l c ( z ) = p s , l s x , l b ( z ) ,
u x , part , l c ( z ) = p u , l u x , l b ( z ) .
( C l q l 2 I ) p s , l = ( B l P l 1 B l b ( P l b ) 1 ) d 0 ,
( D l q l 2 I ) p u , l = ( E l E l b ) d 0 .
s x , l c ( h l + 1 ) = s x , l + 1 c ( h l + 1 ) ,
s y , l c ( h l + 1 ) = s y , l + 1 c ( h l + 1 ) ,
u x , l c ( h l + 1 ) = u x , l + 1 c ( h l + 1 ) ,
u y , l c ( h l + 1 ) = u y , l + 1 c ( h l + 1 ) .
s y , l c = A l 1 ( F K x K y s x , l c + k 0 1 d d z u x , l c + ( E l E l b ) s y , l b ) ,
u y , l c = B l 1 ( F K x E 1 K y u x , l c + k 0 1 d d z s x , l c i F K x ( I E l b E l 1 ) s z , l b ) .
W l = [ 0 W s , l A l 1 W u , l Q u , l A l 1 F K x K y W s , l ] ,
V l = [ W u , l 0 B l 1 F K x E l 1 K y W u , l B l 1 W s , l Q s , l ] ,
c l + = [ c u , l + c s , l + ] , c l = [ c u , l c s , l ] .
[ W l X l W l V l X l V l ] [ c l + c l ] + g l ( h l + 1 ) = [ W l + 1 W l + 1 X l + 1 V l + 1 V l + 1 X l + 1 ] [ c l + 1 + c l + 1 ] + g l + 1 ( h l + 1 ) ,
g l ( z ) = [ p s , l s x , l b A l 1 ( F K x K y p s , l s x , l b + k 0 1 p u , l d d z u x , l b + ( E l ϵ l b I ) d 0 s y , l b ) p u , l u x , l b B l 1 ( F K x E 1 K y p u , l u x , l b + k 0 1 p s , l d d z s x , l b i F K x ( I ϵ l b E l 1 ) d 0 s z , l b ) ] ,
R l [ X l c l + c l ] + g l ( h l + 1 ) = R l + 1 [ c l + 1 + X l + 1 c l + 1 ] + g l + 1 ( h l + 1 ) .
[ c l + 1 + X l + 1 c l + 1 ] = [ T l 11 T l 12 T l 21 T l 22 ] [ X l c l + c l ] + [ g l 1 g l 2 ] ,
T l = R l + 1 1 R l = 1 2 [ W l + 1 1 V l + 1 1 W l + 1 1 V l + 1 1 ] [ W l W l V l V l ] ,
g l = R l + 1 1 ( g l ( h l + 1 ) g l + 1 ( h l + 1 ) ) .
[ c l + 1 + c l ] = [ S l 11 S l 12 S l 21 S l 22 ] [ c l + c l + 1 ] + [ f l 1 f l 2 ] .
S l 11 = ( T l 11 T l 12 ( T l 22 ) 1 T l 21 ) X l ,
S l 12 = T l 12 ( T l 22 ) 1 X l + 1 ,
S l 21 = ( T l 22 ) 1 T l 21 X l ,
S l 22 = ( T l 22 ) 1 X l + 1 ,
f l 1 = g l 1 T l 12 ( T l 22 ) 1 g l 2 ,
f l 2 = ( T l 22 ) 1 g l 2 .
[ c l + 1 + c 1 ] = [ S ¯ l 11 S ¯ l 12 S ¯ l 21 S ¯ l 22 ] [ c 1 + c l + 1 ] + [ f ¯ l 1 f ¯ l 2 ] .
( S ¯ l 1 , f ¯ l 1 ) ( S l , f l ) ( S ¯ l , f ¯ l ) .
( S ¯ 1 , f ¯ 1 ) = ( S 1 , f 1 ) .
[ c l + c 1 ] = [ S ¯ l 1 11 S ¯ l 1 12 S ¯ l 1 21 S ¯ l 1 22 ] [ c 1 + c l ] + [ f ¯ l 1 1 f ¯ l 1 2 ] .
c l + = H l S ¯ l 1 11 c 1 + + S ¯ l 1 12 H l S l 22 c l + 1 + H l S ¯ l 1 12 f l 2 + H l f ¯ l 1 1 ,
H l = ( I S ¯ l 1 12 S l 21 ) 1 ,
H l = ( I S l 21 S ¯ l 1 12 ) 1 .
c l + 1 + = S l 11 H l S ¯ l 1 11 c 1 + + ( S l 12 + S l 11 S ¯ l 1 12 H l S l 22 ) c l + 1 + S l 11 ( H l S ¯ l 1 12 f l 2 + H l f ¯ l 1 1 ) + f l 1 .
c 1 = ( S ¯ l 1 21 + S ¯ l 1 22 S l 21 H l S ¯ l 1 11 ) c 1 + + S ¯ l 1 22 H l S l 22 c l + 1 + S ¯ l 1 22 S l 21 ( H l S ¯ l 1 12 f l 2 + H l f ¯ l 1 1 ) + S ¯ l 1 22 f l 2 + f ¯ l 1 2 .
[ S ¯ l 11 S ¯ l 12 S ¯ l 21 S ¯ l 22 ] = [ S l 11 H l S ¯ l 1 11 S l 12 + S l 11 S ¯ l 1 12 H l S l 22 S ¯ l 1 21 + S ¯ l 1 22 S l 21 H l S ¯ l 1 11 S ¯ l 1 22 H l S l 22 ] ,
[ f ¯ l 1 f ¯ l 2 ] = [ S l 11 ( H l S ¯ l 1 12 f l 2 + H l f ¯ l 1 1 ) + f l 1 S ¯ l 1 22 S l 21 ( H l S ¯ l 1 12 f l 2 + H l f ¯ l 1 1 ) + S ¯ l 1 22 f l 2 + f ¯ l 1 2 ] .
[ c M + c 1 ] = [ S ¯ M 1 11 S ¯ M 1 12 S ¯ M 1 21 S ¯ M 1 22 ] [ c 1 + c M ] + [ f ¯ M 1 1 f ¯ M 1 2 ] .
[ c M + c 1 ] = [ f ¯ M 1 1 f ¯ M 1 2 ] .
[ c l + 1 + c l ] = [ S l 11 S l 12 S l 21 S l 22 ] [ X l c l + X l + 1 c l + 1 ] + [ f l 1 f l 2 ] .
X l c l + = ( S l 11 ) 1 ( c l + 1 + S l 12 X l + 1 c l + 1 f l 1 ) ,
c l = S l 21 ( S l 11 ) 1 c l + 1 + + ( S l 21 ( S l 11 ) 1 S l 12 + S l 22 ) X l + 1 c l + 1 S l 21 ( S l 11 ) 1 f l 1 + f l 2 .
c l + = S ¯ l 1 11 c 1 + + S ¯ l 1 12 c l + f ¯ l 1 1 .
c l + = S ¯ l 1 12 c l + f ¯ l 1 1 .
c l + 1 ( S l , f l ) , ( S ¯ l 1 , f ¯ l 1 ) c l .
E = h y t , N ( x , z ) h y t , ref ( x , z ) 2 , on Ω s ,

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