Abstract

The new formulation of the coupled-wave analysis recently proposed by Lalanne and Morris [J. Opt. Soc. Am. A 13, 779 (1996)] and by Granet and Guizal [J. Opt. Soc. Am. A 13, 1019 (1996)] that drastically improves the convergence performance of the method for lamellar gratings and for TM polarization is shown to be badly conditioned for gratings with a small thickness. Numerical evidence obtained with the coupled-wave analysis and with the differential methods for several grating diffraction problems shows that, in some cases that we identify, the convergence of the conventional formulation can be faster than that of the new one. The discussion includes lamellar, multilevel binary, and continuous-profile geometries.

© 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. 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]
  4. J. Turunen, “Form-birefringence limits of Fourier-expansion methods in grating theory,” J. Opt. Soc. Am. A 13, 1013–1018 (1996).
    [CrossRef]
  5. L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A 13, 1870–1876 (1996).
    [CrossRef]
  6. R. Petit, G. Tayeb, “Theoretical and numerical study of gratings consisting of periodic arrays of thin and lossy strips,” J. Opt. Soc. Am. A 7, 1686–1692 (1990).
    [CrossRef]
  7. When lamellar gratings are considered, in general, matrices E and A-1 are different. However, a0,0(-1) is equal to ε0. For an enlightening discussion, see Ref. 5.
  8. L. Li, C. W. Haggans, “Convergence of the coupled-wave method for metallic lamellar diffraction gratings,” J. Opt. Soc. Am. A 10, 1184–1189 (1993).
    [CrossRef]
  9. M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).
  10. F. Montiel, M. Nevière, “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]
  11. M. G. Moharam, D. A. Pommet, E. B. Grann, 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]
  12. Ph. Lalanne, D. Lemercier-Lalanne, “Depth dependence of the effective properties of subwavelength gratings,” J. Opt. Soc. Am. A 14, 450–458 (1997).
    [CrossRef]
  13. 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]
  14. S. Peng, G. M. Morris, “An efficient implementation of rigorous coupled-wave analysis for surface-relief gratings,” J. Opt. Soc. Am. A 12, 1087–1096 (1995).
    [CrossRef]
  15. Ph. Lalanne, D. Lemercier-Lalanne, “On the effective medium theory of subwavelength periodic structures,” J. Mod. Opt. 43, 2063–2085 (1996).
    [CrossRef]

1997 (1)

1996 (5)

1995 (3)

1994 (2)

1993 (1)

1990 (1)

1974 (1)

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

Gaylord, T. K.

Granet, G.

Grann, E. B.

Guizal, B.

Haggans, C. W.

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.

Moharam, M. G.

Montiel, F.

Morris, G. M.

Nevière, M.

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

Noponen, E.

Peng, S.

Petit, R.

R. Petit, G. Tayeb, “Theoretical and numerical study of gratings consisting of periodic arrays of thin and lossy strips,” J. Opt. Soc. Am. A 7, 1686–1692 (1990).
[CrossRef]

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

Pommet, D. A.

Tayeb, G.

Turunen, J.

Vincent, P.

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

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. Soc. Am. A (12)

L. Li, C. W. Haggans, “Convergence of the coupled-wave method for metallic lamellar diffraction gratings,” J. Opt. Soc. Am. A 10, 1184–1189 (1993).
[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]

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]

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

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

R. Petit, G. Tayeb, “Theoretical and numerical study of gratings consisting of periodic arrays of thin and lossy strips,” J. Opt. Soc. Am. A 7, 1686–1692 (1990).
[CrossRef]

F. Montiel, M. Nevière, “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, D. A. Pommet, E. B. Grann, 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]

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

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

Nouv. Rev. Opt. (1)

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

Other (1)

When lamellar gratings are considered, in general, matrices E and A-1 are different. However, a0,0(-1) is equal to ε0. For an enlightening discussion, see Ref. 5.

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Figures (8)

Fig. 1
Fig. 1

Geometry for the nonconical grating diffraction problem analyzed in Sections 2 and 3 for TM polarization. The relative permittivity is assumed to be independent of the z variable. The refractive indices of the incident medium and of the substrate are n1 and n3, respectively. The grating depth is h, and θ denotes the angle of incidence.

Fig. 2
Fig. 2

Total reflected intensity (including the zeroth and the minus-first orders) as a function of the number of retained orders for a metallic grating and TM-polarized light. The solid curve is obtained with the new eigenproblem formulation. The plus signs are data provided by the conventional eigenproblem formulation. The lamellar grating considered for the computation is etched in the substrate (the relative permittivity is either n12 or n32) and has a fill factor of 0.5 (the fill factor is defined as the groove-width-to-period ratio). n1=1, n3=3.18+j4.41, θ =30°, Λ=λ, and h=0.01λ.

Fig. 3
Fig. 3

Same as in Fig. 2, except that h=0.05λ.

Fig. 4
Fig. 4

Same as in Fig. 2, except that h=0.5λ.

Fig. 5
Fig. 5

Diffraction problems analyzed in Section 4. Two grating geometries are shown: the grating with a triangular profile (heavy lines), and its associated multilevel binary version for N =4 grating layers. n1=1 and θ=30°. h denotes the grating depth, and a TM-polarized light is considered.

Fig. 6
Fig. 6

Comparison between the new and the conventional formulations for the dielectric (n3=2.42) gratings shown in Fig. 5 and for h=0.1λ. The solid curves and the plus signs correspond to the new and the conventional formulations, respectively. (a) Results obtained with the coupled-wave method and N=10, (b) results obtained with the differential method and N=10.

Fig. 7
Fig. 7

Same as in Fig. 6, except that h=λ and N=100 slices are used with the differential method.    

Fig. 8
Fig. 8

Total reflected intensity as a function of the number of retained orders for the metallic (n3=3.18+j4.41) gratings shown in Fig. 5 and for h=0.1λ. The solid curves and the plus signs correspond to the new and the conventional formulations, respectively. (a) Results obtained with the coupled-wave method and N=20, (b) results obtained with the differential method and N=20.

Equations (28)

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k0-2[U]=[E(KxE-1Kx-I)][U].
k0-2[U]=[A-1(KxE-1Kx-I)][U].
Ex=mSm(z)exp jKmx,
Hy=1μ0c mUm(z)exp jKmx.
Umk0=-jpεm-pSp,
Smk0=jmα2ppεm,p(-1)Up-jUm.
Umk0=-jpam,p(-1)Sp,
Smk0=jmα2ppεm,p(-1)Up-jUm.
μ0cH1,y=exp[jn1k0(z-h)]+mRm×exp[-jk0(n12-m2α2)1/2(z-h)]×exp(jmKx),
μ0cH3,y=mTm exp[jk0(n32-m2α2)1/2z]×exp(jmKx),
Sm(h)=Sm(0)+hk0 Sm(0)k0+O(h2k02),
Um(h)=Um(0)+hk0 Um(0)k0+O(h2k02).
Rm=Rm(0)+hk0Rm(1)+O(h2k02),
Tm=Tm(0)+hk0Tm(1)+O(h2k02).
Sm(h)=Sm(0)+hk0jmα2ppεm,p(-1)Up(0)-jUm(0)+O(h2k02),
Um(h)=Um(0)+hk0-jpεm-pSp(0)+O(h2k02).
Um(0)+hk0-jpεm-pSp(0)=δm,0+Rm(0)+hk0Rm(1),
Um(0)=Tm(0)+hk0Tm(1),
Sm(0)=-1n32 n32-m2α2[Tm(0)+hk0Tm(1)],
Sm(0)+hk0jmα2ppεm,p(-1)Up(0)-jUm(0)=1n12 {-n1δm,0+n12-m2α2×[Rm(0)+hk0Rm(1)]}.
δm,0+Rm(0)-Tm(0)+hk0×-jpεm-p 1n32 n32-p2α2Tp(0)
+Rm(1)-Tm(1)=0,
-1n32 n32-m2α2Tm(0)+1n1 δm,0-1n12 n12-m2α2Rm(0)+hk0-1n32 n32-m2α2Tm(1)+jmα2ppεm,p(-1)Tp(0)-jTm(0)
-1n12 n12-m2α2Rm(1)=0.
Rm(0)=n3-n1n3+n1 δm,0,
Tm(0)=2n3n3+n1 δm,0.
R0(1)=jε0n3-n3n1+ε0n3+n1T0(0),
T0(1)=-j n3n1+ε0n3+n1 T0(0).

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