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Extraordinary optical reflection from sub-wavelength cylinder arrays

Raquel Gomez-Medina, Marine Laroche, and Juan José Sáenz

Open Access Open Access

Abstract

A multiple scattering analysis of the reflectance of a periodic array of sub-wavelength cylinders is presented. The optical properties and their dependence on wavelength, geometrical parameters and cylinder dielectric constant are analytically derived for both s- and p-polarized waves. In absence of Mie resonances and surface (plasmon) modes, and for positive cylinder polarizabilities, the reflectance presents sharp peaks close to the onset of new diffraction modes (Rayleigh frequencies). At the lowest resonance frequency, and in the absence of absorption, the wave is perfectly reflected even for vanishingly small cylinder radii.

©2006 Optical Society of America

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Figures (3)
Fig. 1.
Fig. 1. (s-polarization) Calculated reflectance R in a frequency ω versus in-plane wave number Q 0 = (ω/c)sin(θ). The reflectance along the vertical lines is shown in the inset.

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Fig. 2.
Fig. 2. (s-polarization) (a) Plot of ℜ{Gb } and ℜ{1/(k 2α zz } versus frequency along the constant Q 0 line in Fig. 1 (Q 0 = 0.8π/D)). The crossing points (open circles) correspond to different resonant frequencies ω 0m . (b) Calculated reflectance R versus ω. The inset shows a zoom-out of the ω 01 resonance. Dashed lines corresponds to the approximate expression given in eq. 15 with no fitting parameters.

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Fig. 3.
Fig. 3. (p-polarization) Calculated reflectance R in a frequency ω versus in-plane wave number Q 0 = (ω/c)sin(θ). The reflectance along the vertical lines is shown in the inset.

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Equations (28)

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k 0 = k sin ⁡ θ u x + k cos ⁡ θ u y ≡ Q 0 u x + q 0 u y ,
E n scatt ( r ) = α zz E in ( r n ) k 2 G 0 r r n
α zz ≈ π a 2 ( ε − 1 ) [ 1 − i π 4 ( ka ) 2 ( ε − 1 ) ] − 1 ,
E scatt ( r ) = ( α zz E in ( r 0 ) ) k 2 G ( r )
G ( r ) ≡ ∑ n = − ∞ + ∞ e i Q 0 x n G 0 r r n = 1 D ∑ m = − ∞ ∞ e i ( Q 0 − K m ) x ( i 2 q m e i q m ∣ y ∣ )
E in ( r 0 ) = E 0 + α zz k 2 ∑ m ≠ 0 E in ( r m ) G 0 r 0 r m = E 0 + α zz k 2 E in ( r 0 ) G b
= ( 1 − α zz k 2 G b ) − 1 E 0
G b = i { 1 2 D q 0 − 1 4 } + 1 2 D ∑ m = 1 ∞ ( i q m + i q − m − 2 K m ) + 1 2 π ( ln { kD 4 π } + γ E ) .
E scatt ( r ) = α ̂ zz E 0 k 2 G ( r )
k 2 α ̂ zz = ( 1 k 2 α zz − G b ) − 1 .
ψ ( r ) = ψ 0 e i Q 0 x e i q 0 y + ψ 0 ∑ m Prop i 4 πf ̂ q m q 0 2 D q m e i ( Q 0 − K m ) x e i q m ∣ y ∣
T = 1 − 2 ℑ { 4 πf ̂ q 0 q 0 } 2 D q 0 + 1 D 2 ∑ n Prop ( ∣ 4 πf ̂ q 0 q 0 ∣ 2 4 q n q 0 )
R = 1 D 2 ∑ n Prop ( ∣ 4 πf ̂ − q n q 0 ∣ 2 4 q n q 0 ) .
R = ℑ { G ( 0 ) } 2 D q 0 ∣ α ̂ zz k 2 ∣ 2 .
R = ℑ G ( 0 ) } 2 D q 0 ( ℜ 2 { 1 k 2 α zz − G b } + ℑ 2 G ( 0 ) } ) − 1
R ( ω ≲ ω m ) ≈ R max ( 1 γ 2 ( 1 − ω m 2 − ω m 2 ω m 2 − ω 2 ) 2 + 1 ) − 1
H n scatt ( r ) = − { α yy ∂ x H in ( r ) } r = r n ∂ x G 0 r r n − { α xx ∂ y H in ( r ) } r = r n ∂ y G 0 r r n
α xx = α yy ≈ 2 π a 2 ε − 1 ε + 1 [ 1 − i π 4 ( ka ) 2 ε − 1 ε + 1 ] − 1
α xy = α yx = 0
lim r → r 0 ∂ x H in ( r ) = i Q 0 H 0 − α yy ∂ x H in ( r ∣ r = r 0 ∂ x 2 G b = i Q 0 H 0 ( 1 + α yy ∂ x 2 G b ) − 1
∂ x 2 G b = − 1 2 D ∑ m = 1 ∞ { i ( K m − Q 0 ) 2 q m + i ( K m + Q 0 ) 2 q − m − 2 K m − k 2 K m }
− k 2 4 π ( ln { kD 4 π } + γ E − 1 2 ) + 1 6 π D 2 + i ( k 2 8 − Q 0 2 2 D q 0 )
∂ y 2 G b = − 1 2 D ∑ m = 1 ∞ { i q m + i q − m + 2 K m − k 2 K m }
− k 2 4 π ( ln { kD 4 π } + γ E + 1 2 ) + 1 6 π D 2 + i ( k 2 8 − q 0 2 2 D q 0 )
H n scatt ( r ) = − i H 0 α ̂ yy Q 0 ∂ x G ( r ) − i H 0 α ̂ xx q 0 ∂ y G ( r )
α ̂ xx = ( 1 + α xx ∂ y 2 G b ) − 1 α xx
α ̂ yy = ( 1 + α yy ∂ x 2 G b ) − 1 α yy .
4 πf ̂ q m q 0 = α ̂ yy Q 0 ( Q 0 − K m ) + α ̂ xx q 0 q m .
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