Abstract

A set of full-matrix recursion formulas for the WS variant of the S-matrix algorithm is derived, which includes the recent results of some other authors as a subset. In addition, a special type of symmetry that is often found in the structure of coefficient matrices (W matrices) that appear in boundary-matching conditions is identified and fully exploited for the purpose of increasing computation efficiency. Two tables of floating-point operation (flop) counts for both the new WS variant and the old WtS variant of the S-matrix algorithm are given. Comparisons of flop counts show that in performing S-matrix recursions in the absence of the symmetry, it is more efficient to go directly from W matrices to S matrices. In the presence of the symmetry, however, using t matrices is equally and sometimes more advantageous, provided that the symmetry is utilized.

© 2003 Optical Society of America

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References

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    [CrossRef]
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  3. K. Fu, Z. Wang, D. Zhang, J. Zhang, Q. Zhang, “A modal theory and recursion RTCM algorithm for gratings of deep grooves and arbitrary profile,” Sci. China, Ser. A 42, 636–645 (1999).
    [CrossRef]
  4. K. Fu, Z. Wang, J. Zhang, Q. Zhang, “Fast processing of Fourier modal method for perpendicularly crossed surface-relief binary-period gratings,” Acta Opt. Sin. 21, 236–241 (2001) (in Chinese).
  5. X. Tang, K. Fu, Z. Wang, X. Liu, “Analysis of rigorous modal theory for arbitrary dielectric gratings made withanisotropic materials,” Acta Opt. Sin. 22, 774–779 (2002) (in Chinese).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2002 (2)

X. Tang, K. Fu, Z. Wang, X. Liu, “Analysis of rigorous modal theory for arbitrary dielectric gratings made withanisotropic materials,” Acta Opt. Sin. 22, 774–779 (2002) (in Chinese).

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

2001 (1)

K. Fu, Z. Wang, J. Zhang, Q. Zhang, “Fast processing of Fourier modal method for perpendicularly crossed surface-relief binary-period gratings,” Acta Opt. Sin. 21, 236–241 (2001) (in Chinese).

1999 (1)

K. Fu, Z. Wang, D. Zhang, J. Zhang, Q. Zhang, “A modal theory and recursion RTCM algorithm for gratings of deep grooves and arbitrary profile,” Sci. China, Ser. A 42, 636–645 (1999).
[CrossRef]

1997 (2)

K. Fu, Z. Wang, D. Zhang, J. Wen, J. Tang, “A vector analytical method of phase diffraction grating,” Acta Opt. Sin. 17, 1652–1659 (1997) (in Chinese).

L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997).
[CrossRef]

1996 (2)

1995 (1)

1994 (1)

1993 (1)

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
[CrossRef]

Fu, K.

X. Tang, K. Fu, Z. Wang, X. Liu, “Analysis of rigorous modal theory for arbitrary dielectric gratings made withanisotropic materials,” Acta Opt. Sin. 22, 774–779 (2002) (in Chinese).

K. Fu, Z. Wang, J. Zhang, Q. Zhang, “Fast processing of Fourier modal method for perpendicularly crossed surface-relief binary-period gratings,” Acta Opt. Sin. 21, 236–241 (2001) (in Chinese).

K. Fu, Z. Wang, D. Zhang, J. Zhang, Q. Zhang, “A modal theory and recursion RTCM algorithm for gratings of deep grooves and arbitrary profile,” Sci. China, Ser. A 42, 636–645 (1999).
[CrossRef]

K. Fu, Z. Wang, D. Zhang, J. Wen, J. Tang, “A vector analytical method of phase diffraction grating,” Acta Opt. Sin. 17, 1652–1659 (1997) (in Chinese).

Gaylord, T. K.

Golub, G. H.

In this paper the flop counts of matrix operations are based on information provided in G. H. Golub, C. F. Van Loan, Matrix Computations (John Hopkins University Press, Baltimore, Md., 1983, 1989, and 1996). To be consistent with Table 1 of Ref. 1, the meaning of a flop follows the original definition given by the authors in the first edition of their book. See the footnote on page 18 of the third edition.

Grann, E. B.

Li, L.

Liu, X.

X. Tang, K. Fu, Z. Wang, X. Liu, “Analysis of rigorous modal theory for arbitrary dielectric gratings made withanisotropic materials,” Acta Opt. Sin. 22, 774–779 (2002) (in Chinese).

Moharam, M. G.

Pommet, D. A.

Tan, E. L.

Tang, J.

K. Fu, Z. Wang, D. Zhang, J. Wen, J. Tang, “A vector analytical method of phase diffraction grating,” Acta Opt. Sin. 17, 1652–1659 (1997) (in Chinese).

Tang, X.

X. Tang, K. Fu, Z. Wang, X. Liu, “Analysis of rigorous modal theory for arbitrary dielectric gratings made withanisotropic materials,” Acta Opt. Sin. 22, 774–779 (2002) (in Chinese).

Van Loan, C. F.

In this paper the flop counts of matrix operations are based on information provided in G. H. Golub, C. F. Van Loan, Matrix Computations (John Hopkins University Press, Baltimore, Md., 1983, 1989, and 1996). To be consistent with Table 1 of Ref. 1, the meaning of a flop follows the original definition given by the authors in the first edition of their book. See the footnote on page 18 of the third edition.

Wang, Z.

X. Tang, K. Fu, Z. Wang, X. Liu, “Analysis of rigorous modal theory for arbitrary dielectric gratings made withanisotropic materials,” Acta Opt. Sin. 22, 774–779 (2002) (in Chinese).

K. Fu, Z. Wang, J. Zhang, Q. Zhang, “Fast processing of Fourier modal method for perpendicularly crossed surface-relief binary-period gratings,” Acta Opt. Sin. 21, 236–241 (2001) (in Chinese).

K. Fu, Z. Wang, D. Zhang, J. Zhang, Q. Zhang, “A modal theory and recursion RTCM algorithm for gratings of deep grooves and arbitrary profile,” Sci. China, Ser. A 42, 636–645 (1999).
[CrossRef]

K. Fu, Z. Wang, D. Zhang, J. Wen, J. Tang, “A vector analytical method of phase diffraction grating,” Acta Opt. Sin. 17, 1652–1659 (1997) (in Chinese).

Wen, J.

K. Fu, Z. Wang, D. Zhang, J. Wen, J. Tang, “A vector analytical method of phase diffraction grating,” Acta Opt. Sin. 17, 1652–1659 (1997) (in Chinese).

Zhang, D.

K. Fu, Z. Wang, D. Zhang, J. Zhang, Q. Zhang, “A modal theory and recursion RTCM algorithm for gratings of deep grooves and arbitrary profile,” Sci. China, Ser. A 42, 636–645 (1999).
[CrossRef]

K. Fu, Z. Wang, D. Zhang, J. Wen, J. Tang, “A vector analytical method of phase diffraction grating,” Acta Opt. Sin. 17, 1652–1659 (1997) (in Chinese).

Zhang, J.

K. Fu, Z. Wang, J. Zhang, Q. Zhang, “Fast processing of Fourier modal method for perpendicularly crossed surface-relief binary-period gratings,” Acta Opt. Sin. 21, 236–241 (2001) (in Chinese).

K. Fu, Z. Wang, D. Zhang, J. Zhang, Q. Zhang, “A modal theory and recursion RTCM algorithm for gratings of deep grooves and arbitrary profile,” Sci. China, Ser. A 42, 636–645 (1999).
[CrossRef]

Zhang, Q.

K. Fu, Z. Wang, J. Zhang, Q. Zhang, “Fast processing of Fourier modal method for perpendicularly crossed surface-relief binary-period gratings,” Acta Opt. Sin. 21, 236–241 (2001) (in Chinese).

K. Fu, Z. Wang, D. Zhang, J. Zhang, Q. Zhang, “A modal theory and recursion RTCM algorithm for gratings of deep grooves and arbitrary profile,” Sci. China, Ser. A 42, 636–645 (1999).
[CrossRef]

Acta Opt. Sin. (3)

K. Fu, Z. Wang, J. Zhang, Q. Zhang, “Fast processing of Fourier modal method for perpendicularly crossed surface-relief binary-period gratings,” Acta Opt. Sin. 21, 236–241 (2001) (in Chinese).

X. Tang, K. Fu, Z. Wang, X. Liu, “Analysis of rigorous modal theory for arbitrary dielectric gratings made withanisotropic materials,” Acta Opt. Sin. 22, 774–779 (2002) (in Chinese).

K. Fu, Z. Wang, D. Zhang, J. Wen, J. Tang, “A vector analytical method of phase diffraction grating,” Acta Opt. Sin. 17, 1652–1659 (1997) (in Chinese).

J. Mod. Opt. (1)

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553–573 (1993).
[CrossRef]

J. Opt. Soc. Am. A (6)

Sci. China, Ser. A (1)

K. Fu, Z. Wang, D. Zhang, J. Zhang, Q. Zhang, “A modal theory and recursion RTCM algorithm for gratings of deep grooves and arbitrary profile,” Sci. China, Ser. A 42, 636–645 (1999).
[CrossRef]

Other (1)

In this paper the flop counts of matrix operations are based on information provided in G. H. Golub, C. F. Van Loan, Matrix Computations (John Hopkins University Press, Baltimore, Md., 1983, 1989, and 1996). To be consistent with Table 1 of Ref. 1, the meaning of a flop follows the original definition given by the authors in the first edition of their book. See the footnote on page 18 of the third edition.

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Tables (2)

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Table 1 Operation Counts (in N 3 Flops) per Recursion for Two Variants of the S-matrix Algorithm (Nonconical Mountings)

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Table 2 Operation Counts (in N 3 Flops) per Recursion for Two Variants of the S-matrix Algorithm (Conical Mountings)

Equations (58)

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W 11 ( p + 1 ) W 12 ( p + 1 ) W 21 ( p + 1 ) W 22 ( p + 1 )   u ( p + 1 ) d ( p + 1 )
= W 11 ( p ) W 12 ( p ) W 21 ( p ) W 22 ( p )   ϕ + ( p ) 0 0 ϕ - ( p )   u ( p ) d ( p ) ,
u ( p ) d ( 0 ) = S ( p - 1 ) u ( 0 ) d ( p ) = T uu ( p - 1 ) R ud ( p - 1 ) R du ( p - 1 ) T dd ( p - 1 )   u ( 0 ) d ( p ) ,
W 11 ( p + 1 ) u ( p + 1 ) + W 12 ( p + 1 ) d ( p + 1 )
= W 11 ( p ) ϕ + ( p ) u ( p ) + W 12 ( p ) ϕ - ( p ) d ( p ) ,
W 21 ( p + 1 ) u ( p + 1 ) + W 22 ( p + 1 ) d ( p + 1 )
= W 21 ( p ) ϕ + ( p ) u ( p ) + W 22 ( p ) ϕ - ( p ) d ( p ) ,
u ( p ) = T uu ( p - 1 ) u ( 0 ) + R ud ( p - 1 ) d ( p ) ,
d ( 0 ) = R du ( p - 1 ) u ( 0 ) + T dd ( p - 1 ) d ( p ) .
R ud ( p ) = ( Z - 1 X 2 ) 1 ,
T dd ( p ) = T ˜ dd ( p - 1 ) ( Z - 1 X 2 ) 2 ,
T uu ( p ) = ( Z - 1 X 1 ) 1 ,
R du ( p ) = R du ( p - 1 ) + T ˜ dd ( p - 1 ) ( Z - 1 X 1 ) 2 ,
Z = W 11 ( p + 1 ) - W 11 ( p ) R ˜ ud ( p - 1 ) - W 12 ( p ) W 21 ( p + 1 ) - W 21 ( p ) R ˜ ud ( p - 1 ) - W 22 ( p ) ,
X = W 11 ( p ) T ˜ uu ( p - 1 ) - W 12 ( p + 1 ) W 21 ( p ) T ˜ uu ( p - 1 ) - W 22 ( p + 1 ) = [ X 1 ,   X 2 ] ,
R ˜ ud ( p - 1 ) = ϕ + ( p ) R ud ( p - 1 ) ϕ - ( p ) - 1 ,
T ˜ dd ( p - 1 ) = T dd ( p - 1 ) ϕ - ( p ) - 1 ,
T ˜ uu ( p - 1 ) = ϕ + ( p ) T uu ( p - 1 ) .
W = W 1 W 1 W 2 - W 2 .
Z = W 1 ( p + 1 ) 0 0 W 2 ( p + 1 )   1 - F ( p ) 1 G ( p ) ,
F ( p ) = Q 1 ( p ) ( 1 + R ˜ ud ( p - 1 ) ) ,
G ( p ) = Q 2 ( p ) ( 1 - R ˜ ud ( p - 1 ) ) ,
Q l ( p ) = W l ( p + 1 ) - 1 W l ( p ) , l = 1 ,   2 .
1 A 1 B - 1 = - B A 1 - 1 ( A - B ) - 1
R ud ( p ) = 1 - 2 G ( p ) τ ( p ) ,
T dd ( p ) = 2 T ˜ dd ( p - 1 ) τ ( p ) ,
T uu ( p ) = ( F ( p ) τ ( p ) Q 2 ( p ) + G ( p ) τ ( p ) Q 1 ( p ) ) T ˜ uu ( p - 1 ) ,
R du ( p ) = R du ( p - 1 ) + T ˜ dd ( p - 1 ) τ ( p ) ( Q 2 ( p ) - Q 1 ( p ) ) T ˜ uu ( p - 1 ) ,
τ ( p ) = ( F ( p ) + G ( p ) ) - 1 .
t ( p ) = 1 2 Q 1 ( p ) + Q 2 ( p ) Q 1 ( p ) - Q 2 ( p ) Q 1 ( p ) - Q 2 ( p ) Q 1 ( p ) + Q 2 ( p ) .
W = A J A - J B D - B D C 0 - C 0 0 K 0 - K ,
A mq = E zmq ,
B mq = - k z α m μ k 0 λ q ( e )   E zmq ,
C mq = k z 2 + λ q ( e ) 2 μ k 0 λ q ( e )   E zmq ,
D mq = H zmq ,
J mq = k z k 0 λ q ( h ) n mn - 1 α n H znq ,
K mq = - k z 2 + λ q ( h ) 2 k 0 λ q ( h ) n 1 mn H znq .
( μ k 0 2 - α 2 - k z 2 ) E z = λ ( e ) 2 E z ,
1 - 1 ( μ k 0 2 - α - 1 α ) H z - k z 2 H z = λ ( h ) 2 H z ,
W - 1 = 1 2 A - 1 0 C - 1 - A - 1 JK - 1 0 D - 1 - D - 1 BC - 1 K - 1 A - 1 0 - C - 1 - A - 1 JK - 1 0 D - 1 - D - 1 BC - 1 - K - 1 .
t = z 1 + z 2 z 6 z 1 - z 2 - z 6 z 5 z 3 + z 4 - z 5 z 3 - z 4 z 1 - z 2 z 6 z 1 + z 2 - z 6 z 5 z 3 - z 4 - z 5 z 3 + z 4 ,
z 1 = A 2 - 1 A 1 / 2 , z 2 = C 2 - 1 C 1 / 2 ,
z 3 = D 2 - 1 D 1 / 2 , z 4 = K 2 - 1 K 1 / 2 ,
z 5 = D 2 - 1 ( B 1 - B 2 C 2 - 1 C 1 ) / 2 ,
z 6 = A 2 - 1 ( J 1 - J 2 K 2 - 1 K 1 ) / 2 .
R ud ( p ) = 1 - 2 G ^ ( p ) τ ^ ( p ) U ( p ) ,
T dd ( p ) = 2 T ˜ dd ( p - 1 ) τ ^ ( p ) U ( p ) ,
T uu ( p ) = ( F ^ ( p ) τ ^ ( p ) Q ^ 2 ( p ) + G ^ ( p ) τ ^ ( p ) Q ^ 1 ( p ) ) T ˜ uu ( p - 1 ) ,
R du ( p ) = R du ( p - 1 ) + T ˜ dd ( p - 1 ) τ ^ ( p ) ( Q ^ 2 ( p ) - Q ^ 1 ( p ) ) T ˜ uu ( p - 1 ) ,
F ^ ( p ) = W 1 + ( p + 1 ) - 1 ( W 1 + ( p ) R ˜ ud ( p - 1 ) + W 1 - ( p ) ) ,
G ^ ( p ) = Q ^ 2 ( p ) ( 1 - R ˜ ud ( p - 1 ) ) ,
τ ^ ( p ) = ( F ^ ( p ) + G ^ ( p ) ) - 1 ,
Q ^ 1 ( p ) = W 1 + ( p + 1 ) - 1 W 1 + ( p ) ,
Q ^ 2 ( p ) = W ^ 2 ( p + 1 ) - 1 W ^ 2 ( p ) ,
U ( p ) = W 1 + ( p + 1 ) - 1 W ^ 1 ( p + 1 ) ,
W ^ 1 = A 0 0 D , W ^ 2 = C 0 0 K ,
W 1 ± = A ± J ± B D .
τ ^ ( p ) U ( p ) = [ W 1 - ( p ) + W 1 + ( p + 1 ) Q ^ 2 ( p ) + ( W 1 + ( p ) - W 1 + ( p + 1 ) Q ^ 2 ( p ) ) R ˜ ud ( p - 1 ) ] - 1 W ^ 1 ( p + 1 ) ,

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