Abstract

The utilization of symmetries in rigorous coupled wave analysis in order to reduce the matrix sizes and thus the computation time is an appropriate measure for an effective numerical implementation with no loss of accuracy. Another method to improve the convergence is the so-called normal vector method. This method is based on the transformation of the components of the electromagnetic fields in a lateral plane (or slice) from global coordinates into local normal and tangential coordinates relative to the boundaries between two different materials. In this way, the lateral boundary conditions of the electromagnetic field can be imposed more correctly as opposed to the traditional approximation introduced by Li, resulting in a better convergence in many cases. This paper shows how both methods can be combined to attain an optimum solution.

© 2010 Optical Society of America

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References

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2007 (1)

2006 (1)

2005 (2)

B. Bai and L. Li, “Group-theoretic approach to enhancing the Fourier modal method for crossed gratings with one or two reflection symmetries,” J. Opt. A, Pure Appl. Opt.  7, 271–278 (2005).
[CrossRef]

B. Bai and L. Li, “Group-theoretic approach to enhancing the Fourier modal method for crossed gratings of plane group p3,” J. Mod. Opt.  52, 1619–1634 (2005).
[CrossRef]

2004 (2)

C. Zhou and L. Li, “Formulation of the Fourier modal method for symmetric crossed gratings in symmetric mountings,” J. Opt.  6, 43–50 (2004).
[CrossRef]

B. Bai and L. Li, “Reduction of computation time for crossed-grating problems: a group-theoretic approach,” J. Opt. Soc. Am. A  21, 1886–1894 (2004).
[CrossRef]

2000 (1)

1997 (2)

1996 (1)

1993 (1)

1983 (1)

Bai, B.

B. Bai and L. Li, “Group-theoretic approach to enhancing the Fourier modal method for crossed gratings with square symmetry,” J. Opt. Soc. Am. A  23, 572–580 (2006).
[CrossRef]

B. Bai and L. Li, “Group-theoretic approach to enhancing the Fourier modal method for crossed gratings with one or two reflection symmetries,” J. Opt. A, Pure Appl. Opt.  7, 271–278 (2005).
[CrossRef]

B. Bai and L. Li, “Group-theoretic approach to enhancing the Fourier modal method for crossed gratings of plane group p3,” J. Mod. Opt.  52, 1619–1634 (2005).
[CrossRef]

B. Bai and L. Li, “Reduction of computation time for crossed-grating problems: a group-theoretic approach,” J. Opt. Soc. Am. A  21, 1886–1894 (2004).
[CrossRef]

Gaylord, T. K.

Kerwien, N.

Lalanne, P.

Li, L.

Moharam, M. G.

Nevière, M.

Osten, W.

Popov, E.

Raffler, S.

Ruoff, J.

Schuster, T.

Zhou, C.

C. Zhou and L. Li, “Formulation of the Fourier modal method for symmetric crossed gratings in symmetric mountings,” J. Opt.  6, 43–50 (2004).
[CrossRef]

J. Mod. Opt. (1)

B. Bai and L. Li, “Group-theoretic approach to enhancing the Fourier modal method for crossed gratings of plane group p3,” J. Mod. Opt.  52, 1619–1634 (2005).
[CrossRef]

J. Opt. (1)

C. Zhou and L. Li, “Formulation of the Fourier modal method for symmetric crossed gratings in symmetric mountings,” J. Opt.  6, 43–50 (2004).
[CrossRef]

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

B. Bai and L. Li, “Group-theoretic approach to enhancing the Fourier modal method for crossed gratings with one or two reflection symmetries,” J. Opt. A, Pure Appl. Opt.  7, 271–278 (2005).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (8)

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Figures (4)

Fig. 1
Fig. 1

Comparison of the simulated diffraction efficiencies of zeroth reflection order for a symmetric grating made of circular metallic posts with symmetric oblique incidence. The curves show the results for the Li method versus. NV the method. A reference simulation was provided by Anne-Laure Fehrembach (AL-FFF)/Marseille (F).

Fig. 2
Fig. 2

Relative error related to the maximum order value of the results in Fig. 1.

Fig. 3
Fig. 3

Speed-up factor for symmetry application in example 1.

Fig. 4
Fig. 4

Comparison of the simulated diffraction efficiencies of zeroth reflection order for a symmetric grating made of quadratic, dielectric metallic posts rotated by 45° relative to the grating at normal incidence.

Equations (40)

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z ( S x m n S y m n ) = [ K x E 1 K y 1 K x E 1 K x K y E 1 K y 1 K y E 1 K x ] ( U x j l U y j l ) = F ( U x j l U y j l ) ,
z ( U x m n U y m n ) = ( K x K y K x 2 C B K y 2 K y K x ) ( S x j l S y j l ) = G B C ( S x j l S y j l ) .
z ( S x m ̂ n S y m ¯ n ) = [ K x E o 1 K y 1 K x E o 1 K x K y E o 1 K y 1 K y E o 1 K x ] ( U x j ¯ l U y j ̂ l ) = ( F o ) ( U x j ¯ l U y j ̂ l ) ,
z ( U x m ¯ n U y m ̂ n ) = [ K x K y K x 2 C o B e K y 2 K y K x ] ( S x j ̂ l S y j ¯ l ) = ( G o ) ( S x j ̂ l S y j ¯ l ) ;
z ( S x m ¯ n S y m ̂ n ) = [ K x E e 1 K y 1 K x E e 1 K x K y E e 1 K y 1 K y E e 1 K x ] ( U x j ̂ l U y j ¯ l ) = ( F e ) ( U x j ̂ l U y j ¯ l ) ,
z ( U x m ̂ n U y m ¯ n ) = [ K x K y K x 2 C e B o K y 2 K y K x ] ( S x j ¯ l S y j ̂ l ) = ( G e ) ( S x j ¯ l S y j ̂ l ) ;
Ω e m ̃ n , j ̃ l = { 1 2 ( Ω m ̃ n , 0 l + Ω m ̃ n , 0 l ) , if j ̃ = 0 1 2 ( Ω m ̃ n , j ̃ l + Ω m ̃ n , j ̃ l + Ω m ̃ n , j ̃ l + Ω m ̃ n , j ̃ l ) , if j ̃ 0 } ,
Ω o m ̃ n , j ̃ l = 1 2 ( Ω m ̃ n , j l + Ω m ̃ n , j l Ω m ̃ n , j l Ω m ̃ n , j l ) ,
where m ̃ = m ̂ or m ¯ , and j ̃ = j ̂ or j ¯ .
Ω m ̃ n , j ̃ l = Ω m ̃ n , j ̃ l ,
S x m n = S x m n and S x m 0 = 0 , S y m n = S y m n ,
U x m n = U x m n , U y m n = U y m n and U y m 0 = 0 ;
U x m n = U x m n and U x m 0 = 0 , U y m n = U y m n ,
S x m n = S x m n , S y m n = S y m n and S y m 0 = 0 .
z ( S x m n ¯ S y m n ̂ ) = [ K x E o 1 K y 1 K x E o 1 K x K y E o 1 K y 1 K y E o 1 K x ] ( U x j l ̂ U y j l ¯ ) = ( F o ) ( U x j l ̂ U y j l ¯ ) ,
z ( U x m n ̂ U y m n ¯ ) = [ K x K y K x 2 C o B e K y 2 K y K x ] ( S x j l ¯ S y j l ̂ ) = ( G o ) ( S x j l ¯ S y j l ̂ ) ;
z ( S x m n ̂ S y m n ¯ ) = [ K x E e 1 K y 1 K x E e 1 K x K y E e 1 K y 1 K y E e 1 K x ] ( U x j l ¯ U y j l ̂ ) = ( F e ) ( U x j l ¯ U y j l ̂ ) ,
z ( U x m n ¯ U y m n ̂ ) = [ K x K y K x 2 C e B o K y 2 K y K x ] ( S x j l ̂ S y j l ¯ ) = ( G e ) ( S x j l ̂ S y j l ¯ ) .
Ω e m n ̃ , j l ̃ = { 1 2 ( Ω m n ̃ , j 0 + Ω m n ̃ , j 0 ) , if l ̃ = 0 1 2 ( Ω m n ̃ , j l ̃ + Ω m n ̃ , j l ̃ + Ω m n ̃ , j l ̃ + Ω m n ̃ , j l ̃ ) , if l ̃ 0 } ,
Ω o m n ̃ , j l ̃ = 1 2 ( Ω m n ̃ , j l ̃ + Ω m n ̃ , j l ̃ Ω m n ̃ , j l ̃ Ω m n ̃ , j l ̃ ) .
Ω e m n ̃ , j l ̃ = { Ω m n ̃ , j 0 , if l ̃ = 0 Ω m n ̃ , j l ̃ + Ω m n ̃ , j l ̃ , if l ̃ 0 } ,
Ω o m n ̃ , j l ̃ = Ω m n ̃ , j l ̃ Ω m n ̃ , j l ̃ .
S x m n = S x m n = S x m n , S y m n = S y m n = S y m n ;
U x m n = U x m n = U x m n , U y m n = U y m n = U y m n .
z ( S x m ̂ n ̂ S y m ¯ n ¯ ) = [ K x E o e 1 K y 1 K x E o e 1 K x K y E o e 1 K y 1 K y E o e 1 K x ] ( U x j ¯ l ¯ U y j ̂ l ̂ ) = ( F o e ) ( U x j ¯ l ¯ U y j ̂ l ̂ ) ,
z ( U x m ¯ n ¯ U y m ̂ n ̂ ) = [ K x K y K x 2 C o o B e e K y 2 K y K x ] ( S x j ̂ l ̂ S y j ¯ l ¯ ) = ( G o ) ( S x j ̂ l ̂ S y j ¯ l ¯ ) ,
Ω o e m ̃ n ̃ , j ̃ l ̃ = { ( Ω m ̃ n ̃ , j ̃ 0 Ω m ̃ n ̃ , j ̃ 0 ) , if l ̃ = 0 ( Ω m ̃ n ̃ , j ̃ l ̃ + Ω m ̃ n ̃ , j ̃ l ̃ Ω m ̃ n ̃ , j ̃ l ̃ Ω m ̃ n ̃ , j ̃ l ̃ ) , if l ̃ 0 } .
z ( U x m n U y m n ) = [ K x K y + Δ N x y K x 2 E + Δ N y y E K y 2 Δ N x x K y K x Δ N x y ] ( S x j l S y j l ) = G N V ( S x j l S y j l ) ,
z ( U x m n U y m n ) = [ K x K y + P K x 2 M N K y 2 K y K x P ] ( S x j l S y j l ) = G N V ( S x j l S y j l ) ,
P = Δ N x y , M = E Δ N y y and N = E Δ N x x .
z ( U x m ¯ n U y m ̂ n ) = [ K x K y + P e K x 2 M o N e K y 2 K y K x P o ] ( S x j ̂ l S y j ¯ l ) = ( G s ) ( S x j ̂ l S y j ¯ l ) ;
z ( U x m ̂ n U y m ¯ n ) = [ K x K y + P o K x 2 M e N o K y 2 K y K x P e ] ( S x j ¯ l S y j ̂ l ) = ( G p ) ( S x j ¯ l S y j ̂ l ) ;
P o m ̃ n , j ̃ l = 1 2 ( P m ̃ n , j ̃ l + P m ̃ n , j ̃ l P m ̃ n , j ̃ l P m ̃ n , j ̃ l ) ,
P e m ̃ n , j ̃ l = { 1 2 ( P m ̃ n , j ̃ l + P m ̃ n , j ̃ l P m ̃ n , j ̃ l P m ̃ n , j ̃ l ) , if j ̃ 0 1 2 ( P m ̃ n , 0 l P m ̃ n , 0 l ) , if j ̃ = 0 } .
z ( U x m n ̂ U y m n ¯ ) = [ K x K y + P o K x 2 M e N o K y 2 K y K x P e ] ( S x j l ¯ S y j l ̂ ) = ( G s ) ( S x j l ¯ S y j l ̂ ) ;
z ( U x m n ¯ U y m n ̂ ) = [ K x K y + P e K x 2 M o N e K y 2 K y K x P o ] ( S x j l ̂ S y j l ¯ ) = ( G p ) ( S x j l ̂ S y j l ¯ ) ;
P o m n ̃ , j l ̃ = 1 2 ( P m n ̃ , j l ̃ + P m n ̃ , j l ̃ P m n ̃ , j l ̃ P m n ̃ , j l ̃ ) ,
P e m n ̃ , j l ̃ = { 1 2 ( P m n ̃ , j l ̃ + P m n ̃ , j l ̃ P m n ̃ , j l ̃ P m n ̃ , j l ̃ ) if l ̃ 0 1 2 ( P m n ̃ , j 0 P m n ̃ , j 0 ) if l ̃ = 0 } .
z ( U x m ¯ n ¯ U y m ̂ n ̂ ) = [ K x K y + P e e K x 2 M o o N e e K y 2 K y K x P o o ] ( S x j ̂ l ̂ S y j ¯ l ¯ ) = ( G s ) ( S x j ̂ l ̂ S y j ¯ l ¯ ) ,
z ( U x m ̂ n ̂ U y m ¯ n ¯ ) = [ K x K y + P o o K x 2 M e e N o o K y 2 K y K x P e e ] ( S x j ¯ l ¯ S y j ̂ l ̂ ) = ( G p ) ( S x j ¯ l ¯ S y j ̂ l ̂ ) ,

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