Abstract

A new formulation of the coupled-wave method for two-dimensional gratings is proposed. It is based on mathematical and physical results recently obtained for one-dimensional gratings. Numerical evidence obtained for many different diffraction problems, including dielectric, metallic, volume, and surface-relief gratings, shows that the new formulation outperforms the conventional one in terms of convergence rates. The specific case of gratings with very small thickness, for which opposite conclusions on the convergence performance are obtained, is studied and explained. The methodology can be applied to other numerical techniques that rely on Fourier expansions of the electromagnetic fields and on grating parameters such as the permittivity and the permeability.

© 1997 Optical Society of America

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References

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  1. Ph. Lalanne, G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
    [CrossRef]
  2. G. Granet, B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).
    [CrossRef]
  3. Ph. Lalanne, “Convergence performance of the coupled-wave and the differential methods for thin gratings,” J. Opt. Soc. Am. A 14, 1583–1591 (1997).
    [CrossRef]
  4. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  5. L. Li, J. Chandezon, “Improvement of the coordinate transformation method for surface-relief gratings with sharp edges,” J. Opt. Soc. Am. A 13, 2247–2255 (1996).
    [CrossRef]
  6. J. Chandezon, D. Maystre, G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
    [CrossRef]
  7. M. G. Moharam, “Coupled-wave analysis of two-dimensional gratings,” in Holographic Optics: Design and Applications, I. Cindrich, ed., Proc. SPIE883, 8–11 (1988).
    [CrossRef]
  8. 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, 1068–1076 (1995).
    [CrossRef]
  9. S. Peng, G. M. Morris, “Efficient implementation of rigorous coupled-wave analysis for surface relief gratings,” J. Opt. Soc. Am. A 12, 1087–1096 (1995).
    [CrossRef]
  10. E. Noponen, J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A 11, 2494–2502 (1994).
    [CrossRef]
  11. 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]
  12. This was confirmed by numerical computation results. Also, it can be understood by considering the work of Ref. 4 and by noting that the submatrix Ω1 operates over continuous variables, namely, the magnetic-field vectors.
  13. Ph. Lalanne, D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43, 2063–2085 (1996).
    [CrossRef]
  14. M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Res. Opt. 5, 65–77 (1974).
  15. J. Turunen, “Form-birefringence limits of Fourier-expansion methods in grating theory,” J. Opt. Soc. Am. A 13, 1013–1018 (1996).
    [CrossRef]
  16. E. B. Grann, M. G. Moharam, D. A. Pommet, “Artificial uniaxial and biaxial dielectrics with use of two-dimensional subwavelength binary grating,” J. Opt. Soc. Am. A 11, 2695–2703 (1994).
    [CrossRef]
  17. It can be shown that, in the quasi-static limit, β/k0 is larger than 1/a0,0 and smaller than ∊0,0. Thus, for any arbitrary grating geometry, a solution for α of ηx(α)=β/k0 [or ηy(α)=β/k0] exists in the interval [0, 1]. For more details see S. R. Coriel, J. L. Jackson, “Bound on transport coefficients of two-phase materials,” J. Appl. Phys. 39, 4733–4736 (1968).
    [CrossRef]
  18. Ph. Lalanne, D. Lemercier-Lalanne, “Depth dependence of the effective properties of subwavelength gratings,” J. Opt. Soc. Am. A 14, 450–458 (1997).
    [CrossRef]

1997 (2)

1996 (6)

1995 (2)

1994 (2)

1992 (1)

1980 (1)

J. Chandezon, D. Maystre, G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
[CrossRef]

1974 (1)

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Res. Opt. 5, 65–77 (1974).

1968 (1)

It can be shown that, in the quasi-static limit, β/k0 is larger than 1/a0,0 and smaller than ∊0,0. Thus, for any arbitrary grating geometry, a solution for α of ηx(α)=β/k0 [or ηy(α)=β/k0] exists in the interval [0, 1]. For more details see S. R. Coriel, J. L. Jackson, “Bound on transport coefficients of two-phase materials,” J. Appl. Phys. 39, 4733–4736 (1968).
[CrossRef]

Becker, M. F.

Chandezon, J.

L. Li, J. Chandezon, “Improvement of the coordinate transformation method for surface-relief gratings with sharp edges,” J. Opt. Soc. Am. A 13, 2247–2255 (1996).
[CrossRef]

J. Chandezon, D. Maystre, G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
[CrossRef]

Coriel, S. R.

It can be shown that, in the quasi-static limit, β/k0 is larger than 1/a0,0 and smaller than ∊0,0. Thus, for any arbitrary grating geometry, a solution for α of ηx(α)=β/k0 [or ηy(α)=β/k0] exists in the interval [0, 1]. For more details see S. R. Coriel, J. L. Jackson, “Bound on transport coefficients of two-phase materials,” J. Appl. Phys. 39, 4733–4736 (1968).
[CrossRef]

Gaylord, T. K.

Granet, G.

Grann, E. B.

Guizal, B.

Han, S. T.

Jackson, J. L.

It can be shown that, in the quasi-static limit, β/k0 is larger than 1/a0,0 and smaller than ∊0,0. Thus, for any arbitrary grating geometry, a solution for α of ηx(α)=β/k0 [or ηy(α)=β/k0] exists in the interval [0, 1]. For more details see S. R. Coriel, J. L. Jackson, “Bound on transport coefficients of two-phase materials,” J. Appl. Phys. 39, 4733–4736 (1968).
[CrossRef]

Lalanne, Ph.

Lemercier-Lalanne, D.

Ph. Lalanne, D. Lemercier-Lalanne, “Depth dependence of the effective properties of subwavelength gratings,” J. Opt. Soc. Am. A 14, 450–458 (1997).
[CrossRef]

Ph. Lalanne, D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43, 2063–2085 (1996).
[CrossRef]

Li, L.

Maystre, D.

J. Chandezon, D. Maystre, G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
[CrossRef]

Moharam, M. G.

Morris, G. M.

Nevière, M.

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Res. Opt. 5, 65–77 (1974).

Noponen, E.

Peng, S.

Petit, R.

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Res. Opt. 5, 65–77 (1974).

Pommet, D. A.

Raoult, G.

J. Chandezon, D. Maystre, G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
[CrossRef]

Tsao, Y. L

Turunen, J.

Vincent, P.

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Res. Opt. 5, 65–77 (1974).

Walser, R. M.

Appl. Opt. (1)

J. Appl. Phys. (1)

It can be shown that, in the quasi-static limit, β/k0 is larger than 1/a0,0 and smaller than ∊0,0. Thus, for any arbitrary grating geometry, a solution for α of ηx(α)=β/k0 [or ηy(α)=β/k0] exists in the interval [0, 1]. For more details see S. R. Coriel, J. L. Jackson, “Bound on transport coefficients of two-phase materials,” J. Appl. Phys. 39, 4733–4736 (1968).
[CrossRef]

J. Mod. Opt. (1)

Ph. Lalanne, D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43, 2063–2085 (1996).
[CrossRef]

J. Opt. (Paris) (1)

J. Chandezon, D. Maystre, G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J. Opt. (Paris) 11, 235–241 (1980).
[CrossRef]

J. Opt. Soc. Am. A (11)

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, 1068–1076 (1995).
[CrossRef]

S. Peng, G. M. Morris, “Efficient implementation of rigorous coupled-wave analysis for surface relief gratings,” J. Opt. Soc. Am. A 12, 1087–1096 (1995).
[CrossRef]

E. Noponen, J. Turunen, “Eigenmode method for electromagnetic synthesis of diffractive elements with three-dimensional profiles,” J. Opt. Soc. Am. A 11, 2494–2502 (1994).
[CrossRef]

Ph. Lalanne, G. M. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
[CrossRef]

G. Granet, B. Guizal, “Efficient implementation of the coupled-wave method for metallic lamellar gratings in TM polarization,” J. Opt. Soc. Am. A 13, 1019–1023 (1996).
[CrossRef]

Ph. Lalanne, “Convergence performance of the coupled-wave and the differential methods for thin gratings,” J. Opt. Soc. Am. A 14, 1583–1591 (1997).
[CrossRef]

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

L. Li, J. Chandezon, “Improvement of the coordinate transformation method for surface-relief gratings with sharp edges,” J. Opt. Soc. Am. A 13, 2247–2255 (1996).
[CrossRef]

Ph. Lalanne, D. Lemercier-Lalanne, “Depth dependence of the effective properties of subwavelength gratings,” J. Opt. Soc. Am. A 14, 450–458 (1997).
[CrossRef]

J. Turunen, “Form-birefringence limits of Fourier-expansion methods in grating theory,” J. Opt. Soc. Am. A 13, 1013–1018 (1996).
[CrossRef]

E. B. Grann, M. G. Moharam, D. A. Pommet, “Artificial uniaxial and biaxial dielectrics with use of two-dimensional subwavelength binary grating,” J. Opt. Soc. Am. A 11, 2695–2703 (1994).
[CrossRef]

Nouv. Res. Opt. (1)

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Res. Opt. 5, 65–77 (1974).

Other (2)

This was confirmed by numerical computation results. Also, it can be understood by considering the work of Ref. 4 and by noting that the submatrix Ω1 operates over continuous variables, namely, the magnetic-field vectors.

M. G. Moharam, “Coupled-wave analysis of two-dimensional gratings,” in Holographic Optics: Design and Applications, I. Cindrich, ed., Proc. SPIE883, 8–11 (1988).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Zero-order transmitted intensity of a lamellar grating with cylindrical grooves etched in a substrate of refractive index nII=2. Other grating parameters are nI=1, h=λ, Λx =Λy=0.1λ, D=0.5Λx, θ=ϕ=0, ψ=90°, and α=0.5. The dotted curve is obtained with the new formulation for α =0.237 and for an incident plane wave polarized along the y direction.

Fig. 2
Fig. 2

(0,0) and (1,0) transmitted intensities of a lamellar grating with square grooves etched in a substrate of refractive index nII=1.5. The grating parameters are those given in the caption to Fig. 7 of Ref. 9: nI=1, h=λ, Λx=Λy=1.2λ, fx =fy=0.5, θ=ϕ=0, ψ=90°, and α=0.5.

Fig. 3
Fig. 3

Zero-order reflected intensity of a nonsymmetric lamellar grating with rectangular grooves etched in a substrate of refractive index nII=2. Other grating parameters are nI=1, h=λ, Λx=Λy=0.1λ, fx=0.1, fy=0.8, θ=ϕ=0, and ψ =90°, and α is given by Eq. (3).

Fig. 4
Fig. 4

Absorption of a lamellar grating composed of metallic grooves with refractive index nh=(3.18-4.41i)1/2 and deposited on a glass substrate of refractive index nII=1.5. Other grating parameters are nI=1, h=0.2λ, Λx=Λy=0.5λ, fx =fy=0.6, θ=ϕ=0, ψ=90°, and α=0.5.

Fig. 5
Fig. 5

Unit cell of a 2-D lamellar grating with rectangular grooves. In this top view, the horizontal and vertical lines, S1 and S2, represent two surface boundaries of the rectangular groove.

Fig. 6
Fig. 6

Same as Fig. 4, except that h=0.005λ.

Tables (1)

Tables Icon

Table 1 Zero-Order Reflected Amplitude Coefficients of a Sinusoidally Modulated Grating a

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

k0-1SySxUyUx=ΩSySxUyUx,
Ω=00KyE-1KxI-KyE-1Ky00KxE-1Kx-I-KxE-1KyKxKyαA-1+(1-α)E-Ky200Kx2-αE-(1-α)A-1-KxKy00.
Ω=00KyAKxI-KyAKy00KxAKx-I-KxAKyKxKyE-Ky200Kx2-E-KxKy00.
[KxKyA-1-Ky2Kx2-E-KxKy]
[KxKyE-Ky2Kx2-A-1-KxKy]
r=0+Δ2 cos(2πx/Λx)+Δ2 cos(2πy/Λy),
α=δfy1-δfx2+fyΛyfxΛx+fyΛy(1-δfx)(1-δfy),
ηx(α)=[(1-α)0,0+α/a0,0]1/2,
ηy(α)=[α0,0+(1-α)/a0,0]1/2.
ηx=ηy=(0,0)1/2.
Ω2=KxKyα1A-1+(1-α1)E-Ky2Kx2-α2E-(1-α2)A-1-KxKy,

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