Abstract

Differential theory is said to be difficult to apply to surface-relief gratings made of metals with very high conductivity even though the formulation follows Li’s Fourier factorization rules. Recently, Popov et al. [J. Opt. Soc. Am. 21, 199 (2004) ] pointed out this difficulty and explained that its origin is related to the inversion of Toeplitz matrices constructed by the permittivity distribution inside the groove region. The current paper provides information about the differential theory for highly conducting gratings and considers the numerical instability problems. A stable calculation for lossless gratings is described, based on the extrapolation technique with the assumption of small losses.

© 2006 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  14. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The finitely conducting lamellar diffraction grating," Opt. Acta 28, 1087-1102 (1981).
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  17. T. Hosono and S. Yamaguchi, "Some difficulties in homogeneous multilayer approximation method and their remedy," IEICE Trans. J64-B, 1115-1122 (1981) (in Japanese).
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  19. T. Yamasaki, K. Isono, and T. Hinata, "Analysis of electromagnetic field in inhomogeneous media by Fourier series expansion methods: The case of dielectric constant mixed in positive/negative medium: part II," Papers of Tech. Meeting on Electromagnetic Theory, EMT-05-7 (IEE Japan, 2005) (in Japanese).

2005 (1)

2004 (1)

2002 (2)

2001 (1)

2000 (1)

1998 (1)

L. Li, "Reformulation of the Fourier model method for surface-relief grating made with anisotropic materials," J. Mod. Opt. 45, 1313-1334 (1998).
[CrossRef]

1996 (2)

1982 (1)

1981 (3)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The dielectric lamellar diffraction grating," Opt. Acta 28, 413-428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The finitely conducting lamellar diffraction grating," Opt. Acta 28, 1087-1102 (1981).
[CrossRef]

T. Hosono and S. Yamaguchi, "Some difficulties in homogeneous multilayer approximation method and their remedy," IEICE Trans. J64-B, 1115-1122 (1981) (in Japanese).

1978 (1)

1969 (1)

G. Cerutti-Maori, R. Petit, and M. Cadilhac, "Etude numérique du champ diffracté par un réseau," C. R. Hebd. Seances Acad. Sci. 268, 1060-1063 (1969) (in French).

1966 (1)

R. Petit, "Diffraction d'une onde plane par un réseau métallique," Rev. Opt., Theor. Instrum. 45, 353-370 (1966) (in French).

Adams, J. L.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The dielectric lamellar diffraction grating," Opt. Acta 28, 413-428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The finitely conducting lamellar diffraction grating," Opt. Acta 28, 1087-1102 (1981).
[CrossRef]

Andrewartha, J. R.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The finitely conducting lamellar diffraction grating," Opt. Acta 28, 1087-1102 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The dielectric lamellar diffraction grating," Opt. Acta 28, 413-428 (1981).
[CrossRef]

Bonod, N.

Botten, L. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The dielectric lamellar diffraction grating," Opt. Acta 28, 413-428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The finitely conducting lamellar diffraction grating," Opt. Acta 28, 1087-1102 (1981).
[CrossRef]

Cadilhac, M.

G. Cerutti-Maori, R. Petit, and M. Cadilhac, "Etude numérique du champ diffracté par un réseau," C. R. Hebd. Seances Acad. Sci. 268, 1060-1063 (1969) (in French).

Cerutti-Maori, G.

G. Cerutti-Maori, R. Petit, and M. Cadilhac, "Etude numérique du champ diffracté par un réseau," C. R. Hebd. Seances Acad. Sci. 268, 1060-1063 (1969) (in French).

Chernov, B.

Craig, M. S.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The dielectric lamellar diffraction grating," Opt. Acta 28, 413-428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The finitely conducting lamellar diffraction grating," Opt. Acta 28, 1087-1102 (1981).
[CrossRef]

Foldyna, M.

Gaylord, T. K.

Hinata, T.

T. Yamasaki, K. Isono, and T. Hinata, "Analysis of electromagnetic field in inhomogeneous media by Fourier series expansion methods: The case of dielectric constant mixed in positive/negative medium: part II," Papers of Tech. Meeting on Electromagnetic Theory, EMT-05-7 (IEE Japan, 2005) (in Japanese).

T. Yamasaki, K. Isono, and T. Hinata, "Analysis of electromagnetic field in inhomogeneous media by Fourier series expansion methods: The case of dielectric constant mixed in positive/negative medium," Papers of Tech. Meeting on Electromagnetic Theory, IEE Japan, EMT-04-121, 2004 (in Japanese).

Hosono, T.

T. Hosono and S. Yamaguchi, "Some difficulties in homogeneous multilayer approximation method and their remedy," IEICE Trans. J64-B, 1115-1122 (1981) (in Japanese).

Isono, K.

T. Yamasaki, K. Isono, and T. Hinata, "Analysis of electromagnetic field in inhomogeneous media by Fourier series expansion methods: The case of dielectric constant mixed in positive/negative medium," Papers of Tech. Meeting on Electromagnetic Theory, IEE Japan, EMT-04-121, 2004 (in Japanese).

T. Yamasaki, K. Isono, and T. Hinata, "Analysis of electromagnetic field in inhomogeneous media by Fourier series expansion methods: The case of dielectric constant mixed in positive/negative medium: part II," Papers of Tech. Meeting on Electromagnetic Theory, EMT-05-7 (IEE Japan, 2005) (in Japanese).

Knop, K.

Li, L.

McPhedran, R. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The dielectric lamellar diffraction grating," Opt. Acta 28, 413-428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The finitely conducting lamellar diffraction grating," Opt. Acta 28, 1087-1102 (1981).
[CrossRef]

Moharam, M. G.

Nevière, M.

Petit, R.

K. Watanabe, R. Petit, and M. Nevière, "Differential theory of grating made of anisotropic materials," J. Opt. Soc. Am. A 19, 325-334 (2002).
[CrossRef]

G. Cerutti-Maori, R. Petit, and M. Cadilhac, "Etude numérique du champ diffracté par un réseau," C. R. Hebd. Seances Acad. Sci. 268, 1060-1063 (1969) (in French).

R. Petit, "Diffraction d'une onde plane par un réseau métallique," Rev. Opt., Theor. Instrum. 45, 353-370 (1966) (in French).

Pistora, J.

Popov, E.

Postava, K.

Vincent, P.

P. Vincent, "Differential methods," in Electromagnetic Theory of Gratings, R.Petit, ed. (Springer-Verlag, 1980), pp. 101-121.
[CrossRef]

Vlcek, J.

Watanabe, K.

Yamaguchi, S.

T. Hosono and S. Yamaguchi, "Some difficulties in homogeneous multilayer approximation method and their remedy," IEICE Trans. J64-B, 1115-1122 (1981) (in Japanese).

Yamasaki, T.

T. Yamasaki, K. Isono, and T. Hinata, "Analysis of electromagnetic field in inhomogeneous media by Fourier series expansion methods: The case of dielectric constant mixed in positive/negative medium: part II," Papers of Tech. Meeting on Electromagnetic Theory, EMT-05-7 (IEE Japan, 2005) (in Japanese).

T. Yamasaki, K. Isono, and T. Hinata, "Analysis of electromagnetic field in inhomogeneous media by Fourier series expansion methods: The case of dielectric constant mixed in positive/negative medium," Papers of Tech. Meeting on Electromagnetic Theory, IEE Japan, EMT-04-121, 2004 (in Japanese).

C. R. Hebd. Seances Acad. Sci. (1)

G. Cerutti-Maori, R. Petit, and M. Cadilhac, "Etude numérique du champ diffracté par un réseau," C. R. Hebd. Seances Acad. Sci. 268, 1060-1063 (1969) (in French).

IEICE Trans. (1)

T. Hosono and S. Yamaguchi, "Some difficulties in homogeneous multilayer approximation method and their remedy," IEICE Trans. J64-B, 1115-1122 (1981) (in Japanese).

J. Mod. Opt. (1)

L. Li, "Reformulation of the Fourier model method for surface-relief grating made with anisotropic materials," J. Mod. Opt. 45, 1313-1334 (1998).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Opt. Acta (2)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The dielectric lamellar diffraction grating," Opt. Acta 28, 413-428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, "The finitely conducting lamellar diffraction grating," Opt. Acta 28, 1087-1102 (1981).
[CrossRef]

Rev. Opt., Theor. Instrum. (1)

R. Petit, "Diffraction d'une onde plane par un réseau métallique," Rev. Opt., Theor. Instrum. 45, 353-370 (1966) (in French).

Other (3)

P. Vincent, "Differential methods," in Electromagnetic Theory of Gratings, R.Petit, ed. (Springer-Verlag, 1980), pp. 101-121.
[CrossRef]

T. Yamasaki, K. Isono, and T. Hinata, "Analysis of electromagnetic field in inhomogeneous media by Fourier series expansion methods: The case of dielectric constant mixed in positive/negative medium," Papers of Tech. Meeting on Electromagnetic Theory, IEE Japan, EMT-04-121, 2004 (in Japanese).

T. Yamasaki, K. Isono, and T. Hinata, "Analysis of electromagnetic field in inhomogeneous media by Fourier series expansion methods: The case of dielectric constant mixed in positive/negative medium: part II," Papers of Tech. Meeting on Electromagnetic Theory, EMT-05-7 (IEE Japan, 2005) (in Japanese).

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Figures (8)

Fig. 1
Fig. 1

Geometry of a lamellar grating under consideration.

Fig. 2
Fig. 2

Minus-first-order diffraction efficiency of a lamellar grating with refractive index of the material n s = 0 + i 10 as a function of the groove width g for the following parameters: d = h = 500 nm , λ 0 = 632.8 nm , θ = 30 ° , N = 15 , and TM incident plane wave.

Fig. 3
Fig. 3

Condition number of the Toeplitz matrices ϵ and 1 ϵ as a function of the groove width g. All parameters are the same as for Fig. 2.

Fig. 4
Fig. 4

Same as Fig. 2, but computed by the RCWM with SMPA for N = 100 and M = 100 .

Fig. 5
Fig. 5

Same as Fig. 2, but computed by the DM-IMS with SMPA for N = 100 , M = 100 , and Δ y = h 100 . The arrow indicates one artifact (see text for Fig. 7).

Fig. 6
Fig. 6

Same as Fig. 5, but with slightly lossy substrate of n s = 0.05 + i 10 .

Fig. 7
Fig. 7

Efficiency of the grating with g = 151.6 nm and I ( n s ) = 10 as function of R ( n s ) . The results are computed by the DM-IMS with SMPA for M = 100 , Δ y = h 100 , and various N.

Fig. 8
Fig. 8

Same as Fig. 5, but with N = 60 and estimated by quadratic extrapolation assuming lossy substrate of R ( n s ) = 0.05 , 0.1 , 0.15 .

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