Abstract

An enhanced, numerically stable transmittance matrix approach is developed and is applied to the implementation of the rigorous coupled-wave analysis for surface-relief and multilevel gratings. The enhanced approach is shown to produce numerically stable results for excessively deep multilevel surface-relief dielectric gratings. The nature of the numerical instability for the classic transmission matrix approach in the presence of evanescent fields is determined. The finite precision of the numerical representation on digital computers results in insufficient accuracy in numerically representing the elements produced by inverting an ill-conditioned transmission matrix. These inaccuracies will result in numerical instability in the calculations for successive field matching between the layers. The new technique that we present anticipates and preempts these potential numerical problems. In addition to the full-solution approach whereby all the reflected and the transmitted amplitudes are calculated, a simpler, more efficient formulation is proposed for cases in which only the reflected amplitudes (or the transmitted amplitudes) are required. Incorporating this enhanced approach into the implementation of the rigorous coupled-wave analysis, we obtain numerically stable and convergent results for excessively deep (50 wavelengths), 16-level, asymmetric binary gratings. Calculated results are presented for both TE and TM polarization and for conical diffraction.

© 1995 Optical Society of America

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References

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  1. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
    [CrossRef]
  2. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar grating diffraction–E-mode polarization and losses,” J. Opt. Soc. Am. 73, 451–455 (1983).
    [CrossRef]
  3. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  4. M. G. Moharam, T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar grating diffraction,” J. Opt. Soc. Am. 73, 1105–1112 (1983).
    [CrossRef]
  5. W. E. Baird, M. G. Moharam, T. K. Gaylord, “Diffraction characteristics of planar absorption gratings,” Appl. Phys. B 32, 15–20 (1983).
    [CrossRef]
  6. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. A 3, 1780–1796 (1986).
    [CrossRef]
  7. M. G. Moharam, “Diffraction analysis of multiplexed holographic gratings,” in Digest of Topical Meeting on Holography (Optical Society of America, Washington, D.C., 1986), pp. 100–103.
  8. M. G. Moharam, “Coupled-wave analysis of two-dimensional gratings,” in Holographic Optics: Design and Applications, I. Cindrich, ed., Proc. Soc. Photo-Opt. Instrum. Eng.883, 8–11 (1988).
    [CrossRef]
  9. E. N. Glytsis, T. K. Gaylord, “Rigorous three-dimensional coupled-wave diffraction analysis of single and cascaded anisotropic gratings,” J. Opt. Soc. Am. A 4, 2061–2080 (1987).
    [CrossRef]
  10. L. M. Brekhovshikh, Waves in Layered Media (Academic, New York, 1960).
  11. D. M. Pai, K. A. Awada, “Analysis of dielectric gratings of arbitrary profiles and thickness,” J. Opt. Soc. Am. A 8, 755–762 (1991).
    [CrossRef]
  12. S. T. Han, Y.-L. Tsao, R. M. Walser, M. F. Becker, “Electromagnetic scattering of two-dimensional surface-relief dielectric gratings,” Appl. Opt. 31, 2343–2352 (1992).
    [CrossRef] [PubMed]
  13. L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
    [CrossRef]
  14. N. Chateau, J.-P. Hugonin, “Algorithm for the rigorous coupled-wave analysis of grating diffraction,” J. Opt. Soc. Am. A 11, 1321–1331 (1994).
    [CrossRef]
  15. C. Schwartz, L. F. DeSandre, “New calculational approach for multilayer stacks,” Appl. Opt. 26, 3140–3144 (1987).
    [CrossRef] [PubMed]
  16. D. Y. K. Ko, J. R. Sambles, “Scattering matrix method for propagation of radiation in stratified media: attenuated reflection studies of liquid crystals,” J. Opt. Soc. Am. A 5, 1863–1866 (1988).
    [CrossRef]
  17. A. K. Cousins, S. C. Gottschalk, “Application of the impedance formalism to diffraction grating with multiple coating layers,” Appl. Opt. 29, 4268–4271 (1990).
    [CrossRef] [PubMed]
  18. M. G. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, (1995).

1995 (1)

M. G. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, (1995).

1994 (1)

1993 (1)

1992 (1)

1991 (1)

1990 (1)

1988 (1)

1987 (2)

1986 (1)

1983 (3)

1982 (1)

1981 (1)

Awada, K. A.

Baird, W. E.

W. E. Baird, M. G. Moharam, T. K. Gaylord, “Diffraction characteristics of planar absorption gratings,” Appl. Phys. B 32, 15–20 (1983).
[CrossRef]

Becker, M. F.

Brekhovshikh, L. M.

L. M. Brekhovshikh, Waves in Layered Media (Academic, New York, 1960).

Chateau, N.

Cousins, A. K.

DeSandre, L. F.

Gaylord, T. K.

Glytsis, E. N.

Gottschalk, S. C.

Grann, E. B.

M. G. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, (1995).

Han, S. T.

Hugonin, J.-P.

Ko, D. Y. K.

Li, L.

Moharam, M. G.

M. G. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, (1995).

M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. A 3, 1780–1796 (1986).
[CrossRef]

W. E. Baird, M. G. Moharam, T. K. Gaylord, “Diffraction characteristics of planar absorption gratings,” Appl. Phys. B 32, 15–20 (1983).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar grating diffraction–E-mode polarization and losses,” J. Opt. Soc. Am. 73, 451–455 (1983).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Three-dimensional vector coupled-wave analysis of planar grating diffraction,” J. Opt. Soc. Am. 73, 1105–1112 (1983).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811–818 (1981).
[CrossRef]

M. G. Moharam, “Diffraction analysis of multiplexed holographic gratings,” in Digest of Topical Meeting on Holography (Optical Society of America, Washington, D.C., 1986), pp. 100–103.

M. G. Moharam, “Coupled-wave analysis of two-dimensional gratings,” in Holographic Optics: Design and Applications, I. Cindrich, ed., Proc. Soc. Photo-Opt. Instrum. Eng.883, 8–11 (1988).
[CrossRef]

Pai, D. M.

Pommet, D. A.

M. G. Moharam, E. B. Grann, D. A. Pommet, T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, (1995).

Sambles, J. R.

Schwartz, C.

Tsao, Y.-L.

Walser, R. M.

Appl. Opt. (3)

Appl. Phys. B (1)

W. E. Baird, M. G. Moharam, T. K. Gaylord, “Diffraction characteristics of planar absorption gratings,” Appl. Phys. B 32, 15–20 (1983).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (7)

Other (3)

M. G. Moharam, “Diffraction analysis of multiplexed holographic gratings,” in Digest of Topical Meeting on Holography (Optical Society of America, Washington, D.C., 1986), pp. 100–103.

M. G. Moharam, “Coupled-wave analysis of two-dimensional gratings,” in Holographic Optics: Design and Applications, I. Cindrich, ed., Proc. Soc. Photo-Opt. Instrum. Eng.883, 8–11 (1988).
[CrossRef]

L. M. Brekhovshikh, Waves in Layered Media (Academic, New York, 1960).

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Figures (4)

Fig. 1
Fig. 1

Geometry for the reflection and the transmission from a stack of uniform homogeneous layers.

Fig. 2
Fig. 2

Geometry for the surface-relief grating diffraction problem analyzed herein.

Fig. 3
Fig. 3

Diffraction efficiency dependency on the normalized grating of a 16-level (15 layers) asymmetric binary dielectric grating (ng = nII = 2.04, nI = 1). The angle of incidence is θ = 10°. TE-polarization, TM-polarization, and conical-mount results are shown for two grating periods of 1 and 10 wavelengths, respectively.

Fig. 4
Fig. 4

Diffraction efficiency dependency on the number of space harmonics for the grating shown in Fig. 3 for two values of the grating depth (1 and 49 wavelengths, respectively) and for two grating periods of 1 and 10 wavelengths, respectively).

Equations (43)

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E 0 = [ exp ( j k I , z z ) + R exp ( j k I , z z ) ] exp ( j k x x ) , z 0 , E = { P exp [ k 0 γ ( z D 1 ) ] + Q exp [ k 0 γ ( z D ) ] } exp ( j k x x ) , D 1 z D , E t = T exp { j [ k x x + k II , z ( z D L ) ] } , z D L ,
k x = k 0 n I sin ( θ ) , k I , z = k 0 n I cos ( θ ) , k II , z = k 0 [ n II 2 n I 2 sin 2 ( θ ) ] 1 / 2 , γ = j [ n 2 n I 2 sin 2 ( θ ) ] 1 / 2 , = 1 , . . . , L , D = p = 1 d p ,
1 + R = P 1 + Q 1 exp ( k 0 γ 1 d 1 ) , j ( k I , z / k 0 ) ( 1 R ) = γ 1 [ P 1 Q 1 exp ( k 0 γ 1 d 1 ) ] ;
P 1 exp ( k 0 γ 1 d 1 ) + Q 1 = P + Q exp ( k 0 γ d ) , γ 1 [ P 1 exp ( k 0 γ 1 d 1 ) Q 1 ] = γ [ P Q exp ( k 0 γ d ) ] ;
P L exp ( k 0 γ L d L ) + Q L = T , γ L [ P L exp ( k 0 γ L d L ) Q L ] = j ( k II , z / k 0 ) T .
[ P L Q L ] = [ exp ( k 0 γ L d L ) 1 γ L exp ( k 0 γ L d L ) γ L ] 1 [ 1 j ( k II , z / k 0 ) ] T ,
[ P L 1 Q L 1 ] = [ exp ( k 0 γ L 1 d L 1 ) 1 γ L 1 exp ( k 0 γ L 1 d L 1 ) γ L 1 ] 1 × [ 1 exp ( k 0 γ L d L ) γ L γ L exp ( k 0 γ L d L ) ] × [ exp ( k 0 γ L d L ) 1 γ L exp ( k 0 γ L d L ) γ L ] 1 × [ 1 j ( k II , z / k 0 ) ] T .
[ 1 j ( k I , z / k 0 ) ] + [ 1 j ( k I , z / k 0 ) ] R = = 1 L [ 1 exp ( k 0 γ d ) γ γ exp ( k 0 γ d ) ] × [ exp ( k 0 γ d ) 1 γ exp ( k 0 γ d ) γ ] 1 × [ 1 j ( k II , z / k 0 ) ] T .
[ 1 exp ( k 0 γ L d L ) γ L γ L exp ( k 0 γ L d L ) ] [ exp ( k 0 γ L d L ) 1 γ L exp ( k 0 γ L d L ) γ L ] 1 × [ f L + 1 g L + 1 ] = [ 1 exp ( k 0 γ L d L ) γ L γ L exp ( k 0 γ L d L ) ] × [ exp ( k 0 γ L d L ) 0 0 1 ] 1 [ 1 1 γ L γ L ] 1 [ f L + 1 g L + 1 ] ,
[ 1 exp ( k 0 γ L d L ) γ L γ L exp ( k 0 γ L d L ) ] [ exp ( k 0 γ L d L ) 0 0 1 ] 1 × [ a L b L ] T ,
[ a L b L ] = [ 1 1 γ L γ L ] 1 [ f L + 1 g L + 1 ] .
[ f L g L ] T L = [ 1 exp ( k 0 γ L d L ) γ L γ L exp ( k 0 γ L d L ) ] × [ a L b L exp ( k 0 γ L d L ) ] T L = [ a L + b L exp ( 2 k 0 γ L d L ) γ L [ a L b L exp ( 2 k 0 γ L d L ) ] ] T L .
[ 1 j ( k I , z / k 0 ) ] + [ 1 j ( k I , z / k 0 ) ] R = [ f 1 g 1 ] T 1 ,
T = exp ( k 0 γ L d L ) exp ( k 0 γ d ) exp ( k 0 γ 1 d 1 ) T 1
ɛ ( x , z ) = h ɛ h ( z ) exp [ j ( 2 π h / Λ ) ] ,
ɛ ( x ) = h ɛ , h exp [ j ( 2 π h / Λ ) ] , D d < z < D = p = 1 d p .
H I , y = exp { j k 0 n 0 [ sin ( θ ) x + cos ( θ ) z ] } + i R i exp [ j ( k x i x k I , z i z ) ] , H II , y = i T i exp { j [ k x i x + k II , z i ( D L z ) ] } .
k x i = k 0 [ n I sin ( θ ) i ( λ 0 / Λ ) ] , k , z i = ( k 0 2 n 2 k x i 2 ) 1 / 2 , = I , II .
DE r i = R i R i * Re [ k I , z i / k 0 n I cos ( θ ) ] , DE t i = T i T i * Re ( k II , 2 z i n II 2 ) / [ k 0 cos ( θ ) n I ] .
H , g y = i U , y i ( z ) exp ( j k x i x ) , E , g x = j ( μ 0 / ɛ o ) 1 / 2 i S , x i ( z ) exp ( j k x i x ) .
H , g y / z = j ω ɛ 0 ɛ ( x ) E , g x , E , g x / z = j ω μ 0 H , g y + ( E , g z / x ) ,
[ 2 U , y / ( z ) 2 ] = [ E ] [ B ] [ U , y ] ,
B = K x E 1 K x I ,
[ S , x ] = [ E ] 1 [ U , x / ( z ) ] .
U , y = m = 1 n w , i , m { c , m + exp [ k 0 q , m ( z D + d ) ] + c , m exp [ k 0 q , m ( z D ) ] } , S , x = m = 1 n υ , i , m { c , m + exp [ k 0 q , m ( z D + d ) ] + c , m exp [ k 0 q , m ( z D ) ] } , D d < z < D = p = 1 d p ,
[ δ i 0 j δ i 0 cos ( θ ) / n I ] + [ I j Z I ] R = [ W 1 W 1 X 1 V 1 V 1 X 1 ] [ c 1 + c 1 ] ,
[ W 1 X 1 W 1 V 1 X 1 V 1 ] [ c 1 + c 1 ] = [ W W X V V X ] [ c + c ] ,
[ W L X L W L V L X L V L ] [ c L + c L ] = [ I j Z II ] T ,
[ δ i 0 j δ i 0 cos ( θ ) / n I ] + [ I j Z I ] R = = 1 L [ W W X V V X ] [ W X W V X V ] 1 [ I j Z II ] T .
[ W L W L X L V L V L X L ] [ W L X L W L V L X L V L ] 1 [ f L + 1 g L + 1 ] T = [ W L W L X L V L V L X L ] [ X L 0 0 I ] 1 [ W L W L V L V L ] 1 [ f L + 1 g L + 1 ] T ,
[ W L W L X L V L V L X L ] [ X L 0 0 I ] 1 [ a L b L ] T ,
[ a L b L ] = [ W L W L V L V L ] 1 [ f L + 1 g L + 1 ] .
[ f L g L ] T L = [ W L W L X L V L V L X L ] [ I b L a L 1 X L ] T L = [ W L ( I + X L b L a L 1 X L ) V L ( I X L b L a L 1 X L ) ] T L .
[ δ i 0 j δ i 0 cos ( θ ) / n I ] + [ I j Z I ] R = [ f 1 g 1 ] T 1 ,
T = a L 1 X L a 1 X a 1 1 X 1 T 1 .
[ W L f L + 1 V L g L + 1 ] [ c L c L + 1 + ] = [ W L X L V L X L ] C L + ,
C L = a L C L + ,
[ a L b L ] = [ W L f L + 1 V L g L + 1 ] 1 [ W L X L V L X L ] .
[ W L W L X L V L V L X L ] [ c L + c L ] = [ f L g L ] c L + ,
[ f L g L ] = [ W L ( I + X L a L ) V L ( I X L a L ) ] .
[ W L 1 f L V L 1 g L ] [ c L 1 c L + ] = [ W L 1 X L 1 V L 1 X L 1 ] c L 1 + .
[ δ i 0 j δ i 0 cos ( θ ) / n I ] + [ I j Z I ] R = [ f 1 g 1 ] c 1 + ,
i ( DE r i + DE t i ) = 1

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