Abstract

Li’s Fourier factorization rule [J. Opt. Soc. Am. A 13, 1870 (1996) ] was recently shown to be problematic to apply to highly conducting metallic gratings. We provide further information about the applicability of different differential methods and are concerned with the relation of observed numerical artifacts, the total number of retained space harmonics, the presence of both positive and negative permittivity inside the groove region, and the validity of Li’s inverse rule. Two different cases corresponding to lossless and low-loss binary metallic gratings are considered, and it is shown that an increase in the number of retained space harmonics can relieve the presence of numerical artifacts.

© 2008 Optical Society of America

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References

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2005 (1)

T. Yamasaki, K. Isono, and T. Hinata, IEEE Trans. Image Process. E88-C, 2216 (2005).

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Figures (4)

Fig. 1
Fig. 1

Typical binary grating.

Fig. 2
Fig. 2

(a) Minus-first diffraction efficiency of a lamellar grating with refractive index of the material n s = 10 j as a function of the groove width g for the following parameters: d = Λ G = 500 nm , λ 0 = 632.8 nm , θ = 30 ° , number of retained harmonics = 31 , and TM-incident plane wave stabilized by two different methods: enhanced transmittance matrix approach (solid curve) and S-matrix algorithm (dashed curve). (b) Same as (a) but computed with 201 retained harmonics.

Fig. 3
Fig. 3

Error of using the inverse rule in terms of truncation order N. Insets: truncated Fourier series h N ( x ) (solid curve) calculated with N = 45 , N = 388 , N = 44 , and h ( x ) (dashed line).

Fig. 4
Fig. 4

Same as Fig. 2 but with n s = 0.05 10 j .

Equations (4)

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[ h ] = [ [ 1 f ] ] 1 [ g ] ,
f ( x ) = { 1 x < π 2 100 π 2 < x π } ,
g ( x ) = { ( 1 x π ) x < π 2 0.01 ( 1 x π ) π 2 < x π } .
h ( N ) ( x ) = m = N N h m e j m x

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