Abstract

The rigorous coupled-wave analysis (RCWA) is a method to compute diffraction of a field by a given grating structure. Within various applications, such as metrology, it is important to know how the field reacts to small perturbations in the grating. This behavior can be expressed by the field derivatives with respect to a certain parameter. Approximations of these derivatives can be found by using finite differences where the field is computed for a neighboring value of the parameter, and the difference gives the derivative. Unfortunately, RCWA involves solving eigenvalue systems that are computationally expensive. Therefore, a faster alternative is given that computes the derivatives by straightforward differentiation of the relations within RCWA. Solving additional eigensystems is replaced by finding derivatives of eigenvalues and eigenvectors, which is less computationally expensive.

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References

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  1. M. G. Moharam and T. K. Gaylord, "Rigorous coupled-wave analysis of planar-grating diffraction," J. Opt. Soc. Am. 71, 811-818 (1981).
    [CrossRef]
  2. L. Li, J. Chandezon, G. Granet, and J. Plumey, "Rigorous and efficient grating-analysis method made easy for optical engineers," Appl. Opt. 28, 304-313 (1999).
    [CrossRef]
  3. R.Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, 1980).
    [CrossRef]
  4. M. G. Moharam 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]
  5. M. G. Moharam 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]
  6. L. Li, "Use of Fourier series in the analysis of discontinuous periodic structures," J. Opt. Soc. Am. A 13, 1870-1876 (1996).
    [CrossRef]
  7. D. V. Murthy and R. T. Haftka, "Derivatives of eigenvalues and eigenvectors of a general complex matrix," Int. J. Numer. Methods Eng. 26, 293-311 (1988).
    [CrossRef]
  8. N. P. van der Aa, "Computation of eigenvalue and eigenvector derivatives for a general complex-valued eigensystem" (submitted to Electronic J. Linear Algebra ).
  9. E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users' Guide (Society for Industrial and Applied Mathematics, 1999).
    [CrossRef]

1999 (1)

L. Li, J. Chandezon, G. Granet, and J. Plumey, "Rigorous and efficient grating-analysis method made easy for optical engineers," Appl. Opt. 28, 304-313 (1999).
[CrossRef]

1996 (1)

1995 (2)

1988 (1)

D. V. Murthy and R. T. Haftka, "Derivatives of eigenvalues and eigenvectors of a general complex matrix," Int. J. Numer. Methods Eng. 26, 293-311 (1988).
[CrossRef]

1981 (1)

Anderson, E.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users' Guide (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Bai, Z.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users' Guide (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Bischof, C.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users' Guide (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Blackford, S.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users' Guide (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Chandezon, J.

L. Li, J. Chandezon, G. Granet, and J. Plumey, "Rigorous and efficient grating-analysis method made easy for optical engineers," Appl. Opt. 28, 304-313 (1999).
[CrossRef]

Demmel, J.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users' Guide (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Dongarra, J.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users' Guide (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Du Croz, J.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users' Guide (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Gaylord, T. K.

Granet, G.

L. Li, J. Chandezon, G. Granet, and J. Plumey, "Rigorous and efficient grating-analysis method made easy for optical engineers," Appl. Opt. 28, 304-313 (1999).
[CrossRef]

Greenbaum, A.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users' Guide (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Haftka, R. T.

D. V. Murthy and R. T. Haftka, "Derivatives of eigenvalues and eigenvectors of a general complex matrix," Int. J. Numer. Methods Eng. 26, 293-311 (1988).
[CrossRef]

Hammarling, S.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users' Guide (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Li, L.

L. Li, J. Chandezon, G. Granet, and J. Plumey, "Rigorous and efficient grating-analysis method made easy for optical engineers," Appl. Opt. 28, 304-313 (1999).
[CrossRef]

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

McKenney, A.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users' Guide (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

Moharam, M. G.

Murthy, D. V.

D. V. Murthy and R. T. Haftka, "Derivatives of eigenvalues and eigenvectors of a general complex matrix," Int. J. Numer. Methods Eng. 26, 293-311 (1988).
[CrossRef]

Plumey, J.

L. Li, J. Chandezon, G. Granet, and J. Plumey, "Rigorous and efficient grating-analysis method made easy for optical engineers," Appl. Opt. 28, 304-313 (1999).
[CrossRef]

Sorensen, D.

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users' Guide (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

van der Aa, N. P.

N. P. van der Aa, "Computation of eigenvalue and eigenvector derivatives for a general complex-valued eigensystem" (submitted to Electronic J. Linear Algebra ).

Appl. Opt. (1)

L. Li, J. Chandezon, G. Granet, and J. Plumey, "Rigorous and efficient grating-analysis method made easy for optical engineers," Appl. Opt. 28, 304-313 (1999).
[CrossRef]

Electronic J. Linear Algebra (1)

N. P. van der Aa, "Computation of eigenvalue and eigenvector derivatives for a general complex-valued eigensystem" (submitted to Electronic J. Linear Algebra ).

Int. J. Numer. Methods Eng. (1)

D. V. Murthy and R. T. Haftka, "Derivatives of eigenvalues and eigenvectors of a general complex matrix," Int. J. Numer. Methods Eng. 26, 293-311 (1988).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Other (2)

E. Anderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S. Hammarling, A. McKenney, and D. Sorensen, LAPACK Users' Guide (Society for Industrial and Applied Mathematics, 1999).
[CrossRef]

R.Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, 1980).
[CrossRef]

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Figures (4)

Fig. 1
Fig. 1

Three-dimensional representation of a multilayer 1D periodic grating structure (left), which is reduced to a 2D problem for only one period (right).

Fig. 2
Fig. 2

Shape parameter definition in the physical model (top) and in RCWA (bottom).

Fig. 3
Fig. 3

Factor f as a function of the number of harmonics ( N ) for a binary grating using both the matrix–matrix and matrix–vector approach for planar (TE and TM) and conical diffraction.

Fig. 4
Fig. 4

Factor f as a function of the number of harmonics ( N ) for a symmetric trapezoidal grating, which is approximated by 10 and 25 layers. Both the matrix–matrix and matrix–vector approach has been used for planar (TE and TM) and conical diffraction.

Tables (1)

Tables Icon

Table 1 Results of the Analytical Approach Versus Finite Differences for a Binary Grating a with 51 Harmonics ( N = 25 ) for Both Planar (TE and TM) and Conical Diffraction

Equations (53)

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e 1 , y ( x , z ) = exp [ j k 0 n I ( x sin θ + z cos θ ) ] + n r n exp [ j ( k x n x k I , z n z ) ] ,
e K + 2 , y ( x , z ) = n t n exp [ j ( k x n x + k II , z n ( z D ) ) ] ,
k x n = k 0 [ n I sin θ n ( λ 0 Λ ) ] ,
k I , z n = ( k 0 2 n I 2 k x n 2 ) 1 2 , k II , z n = ( k 0 2 n II 2 k x n 2 ) 1 2 ,
ε i r ( x ) = n ϵ i , n exp [ j ( 2 π Λ ) n x ] ,
e i , y ( x , z ) = n s i , n ( z ) exp [ j k x n x ] .
d 2 d z 2 s i ( z ) = ( K x 2 E i ) s i ( z ) ,
[ I j Y I ] r [ F 2 G 2 ] t 2 = [ d 0 j d 0 n I cos θ ] .
A i 1 2 ( W i 1 F i + 1 + V i 1 G i + 1 ) ,
B i 1 2 ( W i 1 F i + 1 V i 1 G i + 1 ) ,
F i W i ( I + X i B i A i 1 X i ) ,
G i V i ( I X i B i A i 1 X i ) ,
V i W i Q i , X i = exp [ d i k 0 Q i ] ,
t A K + 1 1 X K + 1 A 2 1 X 2 t 2 .
DE r n R [ k I , z n k 0 n I cos θ ] r n r n * , DE t n R [ k II , z n n I k 0 n II 2 cos θ ] t n t n * ,
r p = r ( p + Δ p ) r ( p ) Δ p + O ( Δ p ) ,
t p = t ( p + Δ p ) t ( p ) Δ p + O ( Δ p ) .
[ I j Y 1 ] r p [ F 2 G 2 ] t 2 p = [ F 2 p G 2 p ] t 2 .
A i = A i ( x i , , x K + 1 , d i + 1 , , d K + 1 ) ,
W i = W i ( x i ) , V i = V i ( x i ) ,
F i = F i ( x i , , x K + 1 , d i , , d K + 1 ) ,
Q i = Q i ( x i ) , X i = X i ( x i , d i ) ,
A i d j = { 1 2 ( W i 1 F i + 1 d j + V i 1 G i + 1 d j ) if j > i 0 if j i } ,
F i d j = { W i X i ( B i d j A i 1 + B i A i 1 d j ) X i if j > i W i ( X i d i B i A i 1 X i + X i B i A i 1 X i d i ) if j = i 0 if j < i } ,
X i d i = k 0 Q i X i .
A i x j k = { 1 2 ( W i 1 F i + 1 x j k + V i 1 G i + 1 x j k ) if j > i 1 2 ( W i 1 x i k F i + 1 + V i 1 x i k G i + 1 ) if j = i 0 if j < i } ,
F i x j k = { W i X i ( B i x j k A i 1 + B i A i 1 x j k ) X i if j > i W i x i k ( I + X i B i A i 1 X i ) + W i ( X i x i k B i A i 1 X i + X i B i x i k A i 1 X i + X i B i A i 1 x i k X i + X i B i A i 1 X i x i k ) if j = i 0 if j < i } ,
[ F 2 p j G 2 p j ] t 2 = 1 2 j 2 i = 2 j 1 { [ W i V i ] X i ( [ W i 1 V i 1 ] B i A i 1 [ W i 1 V i 1 ] ) } [ A j B j ] i = j 1 2 { A i 1 X i } t 2 ,
[ A j B j ] [ W j V j ] ( X j d j B j A j 1 X j + X j B j A j 1 X j d j ) .
[ A j B j ] [ W j V j ] { 1 2 X j ( ( I B j A j 1 ) W j 1 x j k F j + 1 ( I + B j A j 1 ) V j 1 x j k G j + 1 ) + X j x j k B j A j 1 X j + X j B j A j 1 X j x j k } + [ W j x j k ( I + X j B j A j 1 X j ) V j x j k ( I X j B j A j 1 X j ) ] .
t p = A K + 1 1 X K + 1 A i + 1 1 X i + 1 ( A i 1 p X i + A i 1 X i p ) A i 1 1 X i 1 A 2 1 X 2 t 2 + A K + 1 1 X K + 1 A i 1 X i ( A i 1 1 p X i 1 A i 2 1 X i 2 A 2 1 X 2 + + A i 1 1 X i 1 A 3 1 X 3 A 2 1 p X 2 ) t 2 + A K + 1 1 X K + 1 A 2 1 X 2 t 2 p ,
DE r n p = 2 R [ k I , z n k 0 n I cos θ ] R ( r n p r n * ) ,
DE t n p = 2 R [ k II , z n n I k 0 n II 2 cos θ ] R ( t n p t n * ) .
G i { K x 2 E i for TE polarization P i 1 ( K x E i 1 K x I ) for TM polarization } .
G i W i = W i Λ i .
Γ Λ i = Λ i Γ .
W i 1 G i W i Λ i = C Λ i Λ i C ,
W ¯ i 1 G i W ¯ i Γ Γ Λ i = Γ ( C Λ i Λ i C ) .
Λ k ( W ¯ i 1 G i W ¯ i ) k k ,
C k l { ( W ¯ i 1 G i W ¯ i ) k l Λ l Λ k if Λ k Λ l 0 otherwise } ,
Q i = 1 2 Q i 1 Λ i .
V i = { W i Q i + W i Q i for TE polarization P i W i Q i + P i W i Q i + P i W i Q i for TM polarization } ,
X i = k 0 d i X i Q i .
( W i 1 ) = W i 1 W i W i 1 .
ε i r ( x ) = { ε i , 1 , r Λ 2 x t i 1 , ε i , 2 , r t i 1 x t i 2 , ε i , M r , t i ( M 1 ) x Λ 2 , } with Λ 2 < t i 1 < < t i ( M 1 ) < Λ 2 ,
ϵ i , m = { j 2 π m ε i , 1 r ( e j ( 2 π Λ ) m t 1 e j π m ) + j 2 π m ε i , 2 r ( e j ( 2 π Λ ) m t 2 e j ( 2 π Λ ) m t 1 ) + + j 2 π m ε i , M r ( e j π m e j ( 2 π Λ ) m t M 1 ) m 0 1 Λ ε i , 1 r ( t 1 + Λ 2 ) + 1 Λ ε i , 2 r ( t 2 t 1 ) + + 1 Λ ε i , M r ( Λ 2 t M 1 ) m = 0 } .
( E i t i r ) n m = 1 Λ e j ( 2 π Λ ) ( n m ) t r ( ϵ i r ϵ i ( r + 1 ) ) .
E i x i 2 = E i t i 2 E i t i 1 .
f t analytical t RCWA t FD t RCWA .
d i h K , x i 2 X + K 2 i + 3 K Δ ,
h = i = 2 K + 1 d i h d i + x i 2 h x i 2 = 1 K i = 2 K + 1 d i ,
X = i = 2 K + 1 x 2 i ,
Δ = i = 2 K + 1 K 2 i + 3 K x 2 i .

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