Abstract

A new formulation of the differential method in TM polarization, based on correct representation of truncated Fourier series of products of discontinuous functions, is proposed. Although the derived equations are more complicated than in the classical formulation, the convergence rate with respect to the truncation parameter (with the number of diffraction orders taken into account) is much faster for arbitrary grating profiles, approaching the convergence rate in TE polarization. Numerical examples are presented for dielectric and metallic sinusoidal gratings with a 100% modulation ratio.

© 2000 Optical Society of America

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References

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  1. P. Vincent, in Electromagnetic Theory on Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 4.
  2. L. Li, J. Opt. Soc. Am. A 13, 1024 (1996).
    [CrossRef]
  3. L. Li, J. Opt. Soc. Am. A 13, 1870 (1996).
    [CrossRef]
  4. P. Lalanne and G. Morris, J. Opt. Soc. Am. A 13, 779 (1996).
    [CrossRef]
  5. P. Lalanne, J. Opt. Soc. Am. A 14, 1583 (1997).
    [CrossRef]
  6. D. Maystre, in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 3.

1997

1996

Lalanne, P.

Li, L.

Maystre, D.

D. Maystre, in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 3.

Morris, G.

Vincent, P.

P. Vincent, in Electromagnetic Theory on Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 4.

J. Opt. Soc. Am. A

Other

D. Maystre, in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 3.

P. Vincent, in Electromagnetic Theory on Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), Chap. 4.

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Figures (2)

Fig. 1
Fig. 1

Logarithm of the relative error in the propagating diffraction orders as a function of truncation parameter N (the total number of diffraction orders taken into account is 2N+1). Filled squares, results obtained with Eqs. (2); filled circles, Eqs. (3); open triangles, Eqs. (4)–(6). For comparison, the filled triangles show the convergence for TE polarization. The parameters of the sinusoidal metallic grating used are period d=1 µm; depth h=1 µm; complex refractive index, 1.3+i7.6; incidence from air at a 30° angle; and wavelength, 0.6328 µm. The error is calculated as the difference between the numerical results and the results of the integral method.6 The reference efficiencies are order -2, 0.2120; order -1, 0.1598; and order 0, 0.2638.

Fig. 2
Fig. 2

Same as in Fig. 1, except that the grating material is dielectric, with refractive index 2.5. The results for Eqs. (3) are not presented. The reference efficiencies in transmission are order -3, 0.1472; order -2, 0.2261; order -1, 0.2830; and order +2, 0.2205.

Equations (7)

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ddyHzx,yn=-ik2x,yExx,yn,  ddyExx,yn=-iHzx,yn+iαnEyx,yn,
k2Exn=mk2n-mExm,  Eyn=mk-2n-mαmHzm.
k2Exn=mk-2-1n-mExm,  Eyn=mk2-1n-mαmHzm.
tx=1/1+f2,  ty=f/1+f2,
k2Ex=k-2-1+tx2ΔEx+txtyΔEy,
Ey=k2-tx2Δ-1αHz-txtyΔEx.
Δ=k2-k-2-1,

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