Abstract

In a recent publication [J. Opt. Soc. Am. A 13, 1013 (1996)] a modified formulation of a rigorous Fourier-expansion eigenmode method was presented in TM polarization, which produces the correct effective refractive indices of form-birefringent stratified media in the lowest-order approximation. In this communication it is shown that a further refinement is consistent, for any angle of incidence, with the extraordinary nature of the TM-polarized wave in the quasi-static limit.

© 1997 Optical Society of America

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References

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  1. 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]
  2. Ph. Lalanne, G. M. Morris, “Highly improved convergence of the coupled-wave method in TM polarization,” J. Opt. Soc. Am. A 13, 779–784 (1996).
    [CrossRef]
  3. 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]
  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. K. Knop, “Rigorous diffraction theory for transmission phase gratings with deep rectangular grooves,” J. Opt. Soc. Am. 68, 1206–1210 (1978).
    [CrossRef]
  6. 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, 1077–1086 (1995).
    [CrossRef]
  7. J. Turunen, “Form-birefringence limits of Fourier-expansion methods in grating theory,” J. Opt. Soc. Am. A 13, 1013–1018 (1996). Two misprints have been detected in this paper: Eq. (53) should read as ∊⊥=∊¯2-Δ∊2, and Eq. (56) should read as ∊⊥=(∊high - ∊low)/(ln ∊high- ln ∊low).
    [CrossRef]
  8. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), pp. 705–708.
  9. A. Yariv, P. Yeh, “Electromagnetic propagation in periodic stratified media. II. Birefringence, phase matching, and x-ray lasers,” J. Opt. Soc. Am. 67, 438–448 (1977).
    [CrossRef]
  10. C. W. Haggans, L. Li, R. K. Kostuk, “Effective-medium theory of zeroth-order lamellar gratings in conical mountings,” J. Opt. Soc. Am. A 10, 2217–2225 (1993).
    [CrossRef]

1996 (4)

1995 (1)

1993 (2)

1978 (1)

1977 (1)

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), pp. 705–708.

Gaylord, T. K.

Granet, G.

Grann, E. B.

Guizal, B.

Haggans, C. W.

Knop, K.

Kostuk, R. K.

Lalanne, Ph.

Li, L.

Moharam, M. G.

Morris, G. M.

Pommet, D. A.

Turunen, J.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), pp. 705–708.

Yariv, A.

Yeh, P.

J. Opt. Soc. Am. (2)

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

Other (1)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), pp. 705–708.

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Figures (2)

Fig. 1
Fig. 1

Transmission of an obliquely incident plane wave through a subwavelength-period grating of arbitrary relative-permittivity profile r(x), interpreted by means of the lowest-order theory of effective media. The rays indicate directions of energy flow.

Fig. 2
Fig. 2

Transmittance of a single-layer quarter-wavelength thin film with nin=n1=1.5 and n2=nout=1. Solid curves: result of the lowest-order effective-medium approach. Dashed curves: rigorous results with d/λ=0.45. Dotted curves: rigorous results with d/λ=0.5. (a) TE polarization; (b) TM polarization.

Equations (19)

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εm=1d 0dr(x)exp(i2πmx/d)dx,
ξm=1d 0d 1r(x) exp(i2πmx/d)dx
(k2E-AIA)E=γ2E,
E(k2I-AXA)H=γ2H,
E(k2I-AE-1A)H=γ2H,
X-1(k2I-AXA)H=γ2H,
X-1(k2I-AE-1A)H=γ2H,
γ2=k2(ε0-nin2 sin2 θin)
γI2=k2ε0(1-ξ0nin2 sin2 θin)
γII2=k2(ε0-nin2 sin2 θin)
γIII2=k2(ξ0-1-nin2 sin2 θin)
γIV2=k2ξ0-1(1-ε0-1nin2 sin2 θin)
n¯TE=ε01/2,
n¯TM=ξ0-1/2,
n¯TE=[n12f+n22(1-f )]1/2,
n¯TM=n1n2[n22f+n12(1-f )]-1/2,
α2n¯TE2+γ2n¯TE2=k2,
α2n¯TM2+γIII2n¯TM2=k2,
α2n¯TE2+γIV2n¯TM2=k2.

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