Abstract

We have developed a new formulation of the coupled-wave method (CWM) to handle aperiodic lamellar structures, and it will be referred to as the aperiodic coupled-wave method (ACWM). The space is still divided into three regions, but the fields are written by use of their Fourier integrals instead of the Fourier series. In the modulated region the relative permittivity is represented by its Fourier transform, and then a set of integrodifferential equations is derived. Discretizing the last system leads to a set of ordinary differential equations that is reduced to an eigenvalue problem, as is usually done in the CWM. To assess the method, we compare our results with three independent formalisms: the Rayleigh perturbation method for small samples, the volume integral method, and the finite-element method.

© 2003 Optical Society of America

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References

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  1. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. A 72, 1385–1392 (1982).
    [CrossRef]
  2. P. Vincent, Differential Methods (Springer-Verlag, Berlin, 1980), Chap. 4.
  3. D. Maystre, Integral Methods (Springer-Verlag, Berlin, 1980), Chap. 3.
  4. A. Sentenac, J. Greffet, “Scattering by deep inhomogeneous gratings,” J. Opt. Soc. Am. A 9, 996–1006 (1992).
    [CrossRef]
  5. F. Montiel, M. Nevière, “Electromagnetic theory of Bragg–Fresnel linear zone plates,” J. Opt. Soc. Am. A 12, 2672–2678 (1995).
    [CrossRef]
  6. H. Giovannini, M. Saillard, A. Sentenac, “Numerical study of scattering from rough inhomogeneous films,” J. Opt. Soc. Am. A 15, 1182–1191 (1998).
    [CrossRef]
  7. K. Hirayama, E. N. Glytis, T. K. Gaylord, “Rigorous electromagnetic analysis of diffraction by finite-number-of-periods gratings,” J. Opt. Soc. Am. A 14, 907–917 (1997).
    [CrossRef]
  8. D. Van Labeke, D. Barchiesi, “Scanning-tunneling optical microscopy: a theoretical macroscopic approach,” J. Opt. Soc. Am. A 9, 732–739 (1992).
    [CrossRef]
  9. D. Barchiesi, “A 3-D multilayer model of scattering by nano-structures. Application to the optimisation of thin coated nano-sources,” Opt. Commun. 126, 7–13 (1996).
    [CrossRef]
  10. D. W. Prather, M. S. Mirotznik, J. N. Mait, “Boundary integral methods applied to the analysis of diffractive optical elements,” J. Opt. Soc. Am. A 14, 34–43 (1997).
    [CrossRef]
  11. 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]
  12. 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]
  13. S. Davy, D. Barchiesi, M. Spajer, D. Courjon, “Spectroscopic study of resonant dielectric structures in near-field,” Eur. Phys. J. Appl. Phys. 5, 277–281 (1999).
    [CrossRef]
  14. J.-J. Greffet, “Scattering of electromagnetic waves by rough dielectric surfaces,” Phys. Rev. B 37, 6436–6441 (1988).
    [CrossRef]
  15. J. Jin, The Finite Element Method in Electromagnetics (Wiley, New York, 1993).

1999 (1)

S. Davy, D. Barchiesi, M. Spajer, D. Courjon, “Spectroscopic study of resonant dielectric structures in near-field,” Eur. Phys. J. Appl. Phys. 5, 277–281 (1999).
[CrossRef]

1998 (1)

1997 (2)

1996 (3)

1995 (1)

1992 (2)

1988 (1)

J.-J. Greffet, “Scattering of electromagnetic waves by rough dielectric surfaces,” Phys. Rev. B 37, 6436–6441 (1988).
[CrossRef]

1982 (1)

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

Barchiesi, D.

S. Davy, D. Barchiesi, M. Spajer, D. Courjon, “Spectroscopic study of resonant dielectric structures in near-field,” Eur. Phys. J. Appl. Phys. 5, 277–281 (1999).
[CrossRef]

D. Barchiesi, “A 3-D multilayer model of scattering by nano-structures. Application to the optimisation of thin coated nano-sources,” Opt. Commun. 126, 7–13 (1996).
[CrossRef]

D. Van Labeke, D. Barchiesi, “Scanning-tunneling optical microscopy: a theoretical macroscopic approach,” J. Opt. Soc. Am. A 9, 732–739 (1992).
[CrossRef]

Courjon, D.

S. Davy, D. Barchiesi, M. Spajer, D. Courjon, “Spectroscopic study of resonant dielectric structures in near-field,” Eur. Phys. J. Appl. Phys. 5, 277–281 (1999).
[CrossRef]

Davy, S.

S. Davy, D. Barchiesi, M. Spajer, D. Courjon, “Spectroscopic study of resonant dielectric structures in near-field,” Eur. Phys. J. Appl. Phys. 5, 277–281 (1999).
[CrossRef]

Gaylord, T. K.

K. Hirayama, E. N. Glytis, T. K. Gaylord, “Rigorous electromagnetic analysis of diffraction by finite-number-of-periods gratings,” J. Opt. Soc. Am. A 14, 907–917 (1997).
[CrossRef]

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

Giovannini, H.

Glytis, E. N.

Granet, G.

Greffet, J.

Greffet, J.-J.

J.-J. Greffet, “Scattering of electromagnetic waves by rough dielectric surfaces,” Phys. Rev. B 37, 6436–6441 (1988).
[CrossRef]

Guizal, B.

Hirayama, K.

Jin, J.

J. Jin, The Finite Element Method in Electromagnetics (Wiley, New York, 1993).

Li, L.

Mait, J. N.

Maystre, D.

D. Maystre, Integral Methods (Springer-Verlag, Berlin, 1980), Chap. 3.

Mirotznik, M. S.

Moharam, M. G.

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

Montiel, F.

Nevière, M.

Prather, D. W.

Saillard, M.

Sentenac, A.

Spajer, M.

S. Davy, D. Barchiesi, M. Spajer, D. Courjon, “Spectroscopic study of resonant dielectric structures in near-field,” Eur. Phys. J. Appl. Phys. 5, 277–281 (1999).
[CrossRef]

Van Labeke, D.

Vincent, P.

P. Vincent, Differential Methods (Springer-Verlag, Berlin, 1980), Chap. 4.

Eur. Phys. J. Appl. Phys. (1)

S. Davy, D. Barchiesi, M. Spajer, D. Courjon, “Spectroscopic study of resonant dielectric structures in near-field,” Eur. Phys. J. Appl. Phys. 5, 277–281 (1999).
[CrossRef]

J. Opt. Soc. Am. A (9)

Opt. Commun. (1)

D. Barchiesi, “A 3-D multilayer model of scattering by nano-structures. Application to the optimisation of thin coated nano-sources,” Opt. Commun. 126, 7–13 (1996).
[CrossRef]

Phys. Rev. B (1)

J.-J. Greffet, “Scattering of electromagnetic waves by rough dielectric surfaces,” Phys. Rev. B 37, 6436–6441 (1988).
[CrossRef]

Other (3)

J. Jin, The Finite Element Method in Electromagnetics (Wiley, New York, 1993).

P. Vincent, Differential Methods (Springer-Verlag, Berlin, 1980), Chap. 4.

D. Maystre, Integral Methods (Springer-Verlag, Berlin, 1980), Chap. 3.

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Figures (6)

Fig. 1
Fig. 1

Geometry of the problem.

Fig. 2
Fig. 2

Discretization of the spectral space. The parameter η is used to control the number of evanescent waves accounted for.

Fig. 3
Fig. 3

Intensity below the defect in the plane y=-λ/20. (a) TE or s polarization, (b) TM or p polarization.

Fig. 4
Fig. 4

(a) Intensity below the λ-square cylinder in the plane y=-λ/10. (b) Convergence of the intensity at (x=λ/2, y=-λ/10) for the λ-square cylinder. The polarization of the incoming light is TE.

Fig. 5
Fig. 5

(a) Geometry of the Fresnel-like lens. (b) Intensity in the focal plane of the lens (y=-10λ). (c) Map of the intensity around the lens. The polarization of the incoming light is TE.

Fig. 6
Fig. 6

(a) Geometry of the Fresnel beam splitter. (b) Intensity in the focal plane of the two Fresnel lenses (y=-10λ). The polarization of the incoming light is TE.

Equations (24)

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u1(x, y)=-+I(α)exp(-iαx)exp[iβ1(y-h)]dα+-+R(α)exp(-iαx)exp[-iβ1(y-h)]dα,
u3(x, y)=-+T(α)exp(-iαx)exp(iβ3y)dα,
E2(x, y)=-+e(α, y)exp(-iαx)dαforTEpolarization,
H2(x, y)=-+h(α, y)exp(-iαx)dαforTMpolarization.
ΔE2(x, y)+k02ε2(x)E2(x, y)=0,
ΔE2(x, y)+k02[ε2(x)-εs]×E2(x, y)+k02εsE2(x, y)=0.
α,2e(α, y)y2-α2e(α, y)+k02εse(α, y)+k02εˆ(α)*e(α, y)=0,
α,2e(α, y)y2=(α2-k02εs)e(α, y)-k02-+εˆ(α-γ)e(γ, y)dγ.
2e(αn, y)y2=(αn2-k02εs)e(αn, y)-k02me(αm, y)×αm-δα/2αm+δα/2εˆ(αn-γ)dγ,
d2E/dy2=AE(y),
x1ε2(x)H2(x, y)x+y1ε2(x)H2(x, y)y+k02H2(x, y)=0
xζ(x) H2(x, y)x+1εs2H2(x, y)x2+ζ(x)+1εs 2H2(x, y)y2+k02H2(x, y)=0.
-+1εsδ(α-γ)+ζˆ(α-γ) 2h(γ, y)y2dγ=α2εs-k02h(α, y)+α-+γζˆ(α-γ)h(γ, y)dγ
m 2h(αm, y)y2×αm-δα/2αm+δα/21εsδ(αn-γ)+ηˆ(αn-γ)dγ=αn2εs-k02h(αn, y)+αnmh(αm, y)×αm-δα/2αm+δα/2γηˆ(αn-γ)dγ.
d2H/dy2=AH(y),
ζ(x)=n ζn exp-in 2πDx.
ζˆ(α)=12π-+ζ(x)exp(iαx)dx
=12πnζn-+×expiα-n 2πDxdx
=nζnδα-n 2πD.
m 2h(αm, y)y21εsδ(m-n)+ζm-n=αn2εs-k02h(αn, y)+αnmζn-mαmh(αm, y).
m 2h(αm, y)y21εsδ(m-n)+ζm-n=-k02h(αn, y)+αnm×1εsδ(m-n)+ζm-nαmh(αm, y),
I(α)=w2πexp(iαx0+iβy0)exp-w24(α-α0)2,
I=|Ez(x, y)|2|Ez0(x, y)|2forTEpolarization|Ex(x, y)|2+|Ey(x, y)|2|Ex0(x, y)|2+|Ey0(x, y)|2forTMpolarization.
Δ=log10IN-I601I601,

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