Abstract

I present enhanced R-matrix algorithms for analysis of general multilayered diffraction gratings. The previous R-matrix algorithms are enhanced in three aspects: computational efficiency, numerical stability, and application of half R-matrix in addition to full and quarter R-matrix recursions. On the basis of the eigensolutions of rigorous coupled-wave analysis, the enhanced R-matrix algorithms deal with eigen-submatrices directly and bypass the auxiliary layer R matrix. Such exclusion of a layer matrix leads to improvements in efficiency and algorithm robustness particularly for zero or small layer thickness relative to wavelength. Application of the enhanced algorithms to grating diffraction is exploited especially for the half and quarter R-matrix recursions. Comparison of various R-matrix algorithms via a table of flop counts shows that the enhanced algorithms are more efficient apart from being well conditioned.

© 2006 Optical Society of America

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  1. C. Schwartz and L. F. DeSandre, "New calculational technique for multilayer stacks," Appl. Opt. 26, 3140-3144 (1987).
    [CrossRef] [PubMed]
  2. A. K. Cousins and S. C. Gottschalk, "Application of the impedance formalism to diffraction gratings with multiple coating layers," Appl. Opt. 29, 4268-4271 (1990).
    [CrossRef] [PubMed]
  3. L. Li, "Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity," J. Opt. Soc. Am. A 10, 2581-2591 (1993).
    [CrossRef]
  4. F. Montiel and M. Neviere, "Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity through the R-matrix propagation algorithm," J. Opt. Soc. Am. A 11, 3241-3250 (1994).
    [CrossRef]
  5. D. C. Skigin and R. A. Depine, "The multilayer modal method for electromagnetic scattering from surfaces with several arbitrarily shaped grooves," J. Mod. Opt. 44, 1023-1036 (1997).
    [CrossRef]
  6. 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]
  7. N. P. K. Cotter, T. W. Preist, and J. R. Sambles, "Scattering-matrix approach to multilayer diffraction," J. Opt. Soc. Am. A 12, 1097-1103 (1995).
    [CrossRef]
  8. 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]
  9. L. Li, "New formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. Soc. Am. A 14, 2758-2767 (1997).
    [CrossRef]
  10. M. Neviere and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, 2003).
  11. 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]
  12. 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]
  13. E. L. Tan, "Note on formulation of the enhanced scattering- (transmittance-) matrix approach," J. Opt. Soc. Am. A 19, 1157-1161 (2002).
    [CrossRef]
  14. L. Li, "Note on the S-matrix propagation algorithm," J. Opt. Soc. Am. A 20, 655-660 (2003).
    [CrossRef]
  15. The ill conditioning of Rl does not appear in RCWA only. For example, in the differential method (beyond the present scope), the transfer matrix may be simply approximated from the system matrix as Tl ≈ I + dhLambdal for sufficiently small numerical thickness dl. However, the corresponding Rl at such thickness could be very ill conditioned and inaccurate also.

2003 (1)

2002 (1)

1997 (2)

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

D. C. Skigin and R. A. Depine, "The multilayer modal method for electromagnetic scattering from surfaces with several arbitrarily shaped grooves," J. Mod. Opt. 44, 1023-1036 (1997).
[CrossRef]

1996 (1)

1995 (3)

1994 (1)

1993 (1)

1990 (1)

1988 (1)

1987 (1)

Cotter, N. P. K.

Cousins, A. K.

Depine, R. A.

D. C. Skigin and R. A. Depine, "The multilayer modal method for electromagnetic scattering from surfaces with several arbitrarily shaped grooves," J. Mod. Opt. 44, 1023-1036 (1997).
[CrossRef]

DeSandre, L. F.

Gaylord, T. K.

Gottschalk, S. C.

Grann, E. B.

Ko, D. Y. K.

Li, L.

Moharam, M. G.

Montiel, F.

Neviere, M.

Pommet, D. A.

Popov, E.

M. Neviere and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, 2003).

Preist, T. W.

Sambles, J. R.

Schwartz, C.

Skigin, D. C.

D. C. Skigin and R. A. Depine, "The multilayer modal method for electromagnetic scattering from surfaces with several arbitrarily shaped grooves," J. Mod. Opt. 44, 1023-1036 (1997).
[CrossRef]

Tan, E. L.

Appl. Opt. (2)

J. Mod. Opt. (1)

D. C. Skigin and R. A. Depine, "The multilayer modal method for electromagnetic scattering from surfaces with several arbitrarily shaped grooves," J. Mod. Opt. 44, 1023-1036 (1997).
[CrossRef]

J. Opt. Soc. Am. A (10)

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]

N. P. K. Cotter, T. W. Preist, and J. R. Sambles, "Scattering-matrix approach to multilayer diffraction," J. Opt. Soc. Am. A 12, 1097-1103 (1995).
[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, "New formulation of the Fourier modal method for crossed surface-relief gratings," J. Opt. Soc. Am. A 14, 2758-2767 (1997).
[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]

F. Montiel and M. Neviere, "Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity through the R-matrix propagation algorithm," J. Opt. Soc. Am. A 11, 3241-3250 (1994).
[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. 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]

Other (2)

The ill conditioning of Rl does not appear in RCWA only. For example, in the differential method (beyond the present scope), the transfer matrix may be simply approximated from the system matrix as Tl ≈ I + dhLambdal for sufficiently small numerical thickness dl. However, the corresponding Rl at such thickness could be very ill conditioned and inaccurate also.

M. Neviere and E. Popov, Light Propagation in Periodic Media: Differential Theory and Design (Marcel Dekker, 2003).

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Figures (3)

Fig. 1
Fig. 1

Geometry of a multilayered grating structure embedded in two exterior media. The upper and lower bounding interfaces of each grating layer l are denoted by Zl > and Zl <, respectively.

Fig. 2
Fig. 2

Condition numbers of the matrices to be inverted in Eqs. (7) and (17) versus layer thickness relative to wavelength.

Fig. 3
Fig. 3

Diffraction efficiencies of a 16-level (15 layers) asymmetric binary dielectric grating in a conical mounting as a function of the normalized groove depth. The grating and substrate index of refraction is 2.04, the grating period is one wavelength, the incident angle is θ = 10°, the azimuthal angle is ϕ = 30°, and the polarization angle is ψ = 45°.

Tables (1)

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Table 1 Operation Counts (in N 3 Flops) per Grating Layer

Equations (29)

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f l ( z ) = Ψ l P l ( z ) c l = Ψ l w l ( z ) .
f l = [ E l H l ] , Ψ l = [ e l > e l < h l > h l < ] ,
P l ( z ) = [ P l > ( z ) 0 0 P l < ( z ) ] ,
c l = [ c l > c l < ] , w l ( z ) = [ w l > ( z ) w l < ( z ) ] .
[ E l ( Z l < ) E L ( Z L > ) ] = [ R 11 [ l ] R 12 [ l ] R 21 [ l ] R 22 [ l ] ] [ H l ( Z l < ) H L ( Z L > ) ] .
[ E l ( Z l < ) E l ( Z l > ) ] = [ R 11 l R 12 l R 21 l R 22 l ] [ H l ( Z l < ) H l ( Z l > ) ] .
[ R 11 l R 12 l R 21 l R 22 l ] = E l ( l ) 1 ,
E l = [ e l > e l < P l < ( d l ) e l > P l > ( d l ) e l < ] ,
l = [ h l > h l   < P l   < ( d l ) h l > P l > ( d l ) h l   < ] .
R 11 [ l ] = R 11 l R 12 l [ R 22 l R 11 [ l + 1 ] ] 1 R 21 l ,
R 12 [ l ] = R 12 l [ R 22 l R 11 [ l + 1 ] ] 1 R 12 [ l + 1 ] ,
R 21 [ l ] = R 21 [ l + 1 ] [ R 22 l R 11 [ l + 1 ] ] 1 R 21 l ,
R 22 [ l ] = R 22 [ l + 1 ] + R 21 [ l + 1 ] [ R 22 l R 11 [ l + 1 ] ] 1 R 12 [ l + 1 ] ,
[ R 11         [ l ] R 12         [ l ] R 21         [ l ] R 22         [ l ] R 22         [ l + 1 ] ] = 1 2 [ R 11         l R 12         l 0 R 21         [ l + 1 ] ] [ I 0 - [ R 22         l R 11         [ l + 1 ] ] 1 R 21         l [ R 22         l R 11        [ l + 1 ] ] 1 R 12           [ l + 1 ] ] ,
1 = { [ I 0 0 0 ] E l + [ 0 0 0 R 21         [ l + 1 ] ] l } ( l ) 1 .
2 = [ I 0 R 21 l R 22 l R 11 [ l + 1 ] ] 1 [ I 0 0 R 12 [ l + 1 ] ] .
[ R 11 [ l ] R 12 [ l ] R 21 [ l ] R 22 [ l ] - R 22 [ l + 1 ] ] = [ e l > e l < P l < ( - d l ) R 21 [ l + 1 ] h l > P l > ( d l ) R 21 [ l + 1 ] h l < ] [ h l > h l < P l < ( - d l ) [ e l > - R 11 [ l + 1 ] h l > ] P l > ( d l ) e l < - R 11 [ l + 1 ] h l < ] - 1 × [ I 0 0 R 12 [ l + 1 ] ] .
[ R 11         l R 12         l R 21         l R 22         l ] d l = 0 = [ e l > e l   < e l > e l   < ] [ h l > h l   < h l > h l   < ] 1 ,
[ R 11         [ l ] R 21         [ l ] ] = [ e l     > e l     < P l     < ( - d l ) R 21         [ l + 1 ] h l     > P l     > ( d l ) R 21         [ l + 1 ] h l < ] [ h l     > h l < P l     < ( - d l ) [ e l     > - R 11        [ l + 1 ] h l     > ] P l     > ( d l ) e l     < - R 11         [ l + 1 ] h l < ] - 1 [ I 0 ] .
ρ l = [ e l < - R 11 [ l + 1 ] h l < ] - 1 [ R 11 [ l + 1 ] h l > - e l > ] ,
Γ l = P l < ( - d l ) ρ l P l > ( d l ) ,
R 11 [ l ] = [ e l > + e l < Γ l ] [ h l > + h l < Γ l ] - 1 ,
R 21 [ l ] = R 21 [ l + 1 ] [ h l > + h l < ρ l ] P l > ( d l ) [ h l > + h l < Γ l ] - 1 .
w 0 < ( Z 0 > ) = r 0 , 1 w 0 > ( Z 0 > ) ,
w L + 1 > ( Z L + 1 < ) = t 0 , L + 1 w 0 > ( Z 0 > ) ,
[ R 11         [ 1 ] h 0     < - e 0     < R 12         [ 1 ] h L + 1               > R 21         [ 1 ] h 0     < R 22         [ 1 ] h L + 1               > - e L + 1               > ] [ r 0 , 1 t L + 1 , 0 t 0 , L + 1 r L + 1 , L ] = [ e 0     > - R 11         [ 1 ] h 0     > - R 12         [ 1 ] h L + 1               < - R 21         [ 1 ] h 0     > e L + 1               < - R 22         [ 1 ] h L + 1               < ] .
r 0 , 1 = [ e 0     < - R 11         [ 1 ] h 0     < ] - 1 [ R 11         [ 1 ] h 0     > - e 0     > ] ,
t 0 , L + 1 = ( e L + 1               > ) - 1 R 21         [ 1 ] [ h 0     > + h 0      < r 0 , 1 ] .
E l ( Z l     > ) = R 11         [ l + 1 ] { [ h l > + h l     < ρ l ] P l     > ( d l ) × [ h l     > + h l < Γ l ] - 1 } { [ h 1 > + h 1 < ρ 1 ] P 1     > ( d 1 ) × [ h 1     > + h 1     < Γ l ] - 1 } H 1 ( Z 1     < ) .

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