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Finite checkerboards of dissipative negative refractive index

Sangeeta Chakrabarti, S. Anantha Ramakrishna, and S. Guenneau

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Abstract

The electromagnetic properties of finite checkerboards consisting of alternating rectangular cells of positive refractive index (ε=+1, μ=+1) and negative refractive index (ε=-1, μ=-1) have been investigated numerically. We show that the numerical calculations have to be carried out with very fine discretization to accurately model the highly singular behaviour of these checkerboards. Our solutions show that, within the accuracy of the numerical calculations, the focussing properties of these checkerboards are reasonably robust in the presence of moderate levels of dissipation. We also show that even small systems of checkerboards can display focussing effects to some extent.

©2006 Optical Society of America

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Figures (4)
Fig. 1.
Fig. 1. A pair of complementary checkerboard layers with ε=±1 and μ=±1. Positive and negative refractive index are schematically depicted by white and coloured regions. A ray can be transmitted with no change in direction or retro- reflected.

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Fig. 2.
Fig. 2. The transmission and reflection coefficients for a pair of complementary checker-board layers with ε=±1 and μ=±1. (a) and (b) are the results for inadequate differencing with 202 points across d, while (c) and (d) are the results for better differencing with 262 points across d.

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Fig. 3.
Fig. 3. The transmittivity and reflectivity for dissipative checkerboard systems with ε=μ=-1+i0.1 (top panel) and a silver checkerboard with ε=-1+i0.4 and μ=+1 everywhere (bottom panel). Resonant excitation of surface plasmon modes at subwavelength kx can be seen for the silver checkerboard.

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Fig. 4.
Fig. 4. Left panel: P-polarized eigenfield associated with a line source of wavelength 0.5d (d is about 9µm in section 3) placed at a distance 0.2d from a silver checkerboard slab consisting of 30 cells of side length 0.1d, alternating air and weakly dissipative silver cells (ε=-1+i0.01 and μ=+1). The scale on the right is in arbitrary units. (a) 2D plot of the field; (c) Profile of the field computed along a vertical segment with endpoints (0, 1)d and (0,-1)d. Right panel: Same with two pairs of complementary checkerboard layers (60 cells); (b) 2D plot of the field; (d) 3D plot of the field.

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Equations (6)

Equations on this page are rendered with MathJax. Learn more.

𝒯 ( k x ) = 1 , 𝓡 ( k x ) = 0 ,
ω 2 ( k ) = c 2 ε μ a 2 4 sin 2 ( 1 2 k a ) ≃ c 2 ε μ k 2 [ 1 − 1 12 ( k a ) 2 ] ,
k 2 = ε μ ω 2 c 2 [ 1 + 1 12 ( k a ) 2 ] ,
δ μ = μ k 2 a 2 12 ,
k x t = − ln ( δ μ 2 ) = − ln ( k x 2 a 2 24 )
w e = λ i ∇ λ j − λ j ∇ λ i
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