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Analytical model of the optical response of periodically structured metallic films

A. Benabbas, V. Halté, and J.-Y. Bigot

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Abstract

In this paper we investigate the optical response of periodically structured metallic films constituted of sub-wavelength apertures. Our approach consists in studying the diffraction of transverse magnetic polarized electromagnetic waves by a one-dimensional grating. The method that we use is the Rigorous Coupled Waves Analysis allowing us to obtain an analytical model to calculate the diffraction efficiencies. The zero and first order terms allow determining the transmission, reflectivity and absorption of symmetric or asymmetric nanostructures surrounded either by identical or different dielectric media. For both type of nanostructures the spectral shape of the enhanced resonant transmission associated to surface plasmons displays a Fano profile. In the case of symmetric nanostructures, we study the conditions of formation of coupled surface plasmon-polaritons as well as their effect on the optical response of the modulated structure. For asymmetric nanostructures, we discuss the non-reciprocity of the reflectivity and we investigate the spectral dependency of the enhanced resonant transmission on the refractive index of the dielectric surrounding the metal film.

©2005 Optical Society of America

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Figures (7)
Fig. 1.
Fig. 1. Schematic representation of a periodically modulated metallic film. It consists in a one-dimensional metal grating with a period a0 , a slit diameter d, and a film thickness h.

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Fig. 2.
Fig. 2. Wavelength dependence of the modulus of the determinant of the matrix A defined in Eq. (40) for a nanostructure with the following parameters: a0 = 700 nm, d = 200 nm, h = 110 nm, ΩI = 1 and ΩIII = 2.28 . The spectral positions of the two resonances RI (metal/air) and RIII (metal/glass) are located at λ = 728 nm and λ = 1094 nm respectively.

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Fig. 3.
Fig. 3. Zero order transmission of the same nanostructure as in Fig. 2 (a0 = 700 nm, d = 200 nm, h = 110 nm, εI =1 and εIII = 2.28). The two resonances RI (metal/air) and RIII (metal/Glass) clearly display Fano profiles. Inset: comparison between the spectral positions of the transmission (red) and SPP resonance(blue).

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Fig.4.
Fig.4. Calculated transmission (a), reflectivity and absorption (b) spectra of the structured silver film obtained by illuminating from both sides: air (full line) and glass (dashed line).

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Fig. 5.
Fig. 5. Transmission (a) and reflectivity (b) spectra of a symmetric sinusoidally modulated silver film (εI =εIII =1), for different thickness h. Structure parameters: a0 = 700 nm, d = 200 nm. The imaginary part of the dielectric function of silver is neglected.

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Fig.6.
Fig.6. Transmission (a), reflectivity (b) and absorption (c) spectra of the symmetric structure described in Fig. 5. for different values of h, the complex dielectric function of the metal being taken into account.

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Fig. 7.
Fig. 7. Calculated zero order transmission spectra of a modulated silver film as a function of the dielectric constant of the medium I. The medium III (substrate) is quartz εIII =2.31. The structure parameters are a0 = 600 nm, d = 200 nm, and h = 150 nm. (a) εI ≤ εIII , (b) εI ≥εIII .

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Equations (51)

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ε II ( x ) = ∑ n ε n exp ( ingx )
{ ε 0 = f ε s + ( 1 − f ) ε m , for n = 0 ε n = ( ε s − ε m ) sin ⁡ ( nπf ) nπ , for n ≠ 0
H 0 y = exp [ − i ( k Ix x + k Iz z ) ]
k x n = k Ix + ng
k spp = k x n = k Ix + ng
H Iy = H 0 y + ∑ n r n exp [ − i ( k x n x − k Iz n z ) ]
H IIIy = ∑ n t n exp [ − i ( k x n x + k IIIz n ( z − h ) ) ]
E → = ( − i ω ε f ε ) ∇ → × H →
∂ H IIy ∂ z = − iω ε f ε ( x ) E IIx
∂ E IIx ∂ z = − iω μ 0 H IIy + ∂ E IIz ∂ x
H IIy = ∑ n H II n ( z ) exp ( − i k x n x )
E IIx = i ( μ 0 ε f ) 1 2 ∑ n E II n ( z ) exp ( − i k x n x )
∂ [ H II ] ∂ z = k 0 [ ε ] [ E II ]
∂ [ E II ] ∂ z = k 0 ( K x [ ε ] − 1 K x − I d ) [ H II ]
∂ 2 [ E II ] ∂ z 2 = k 0 2 M [ E II ]
∂ 2 [ H II ] ∂ z 2 = k 0 2 M ′ [ H II ]
M = ( K x [ ε ] − 1 K x − I d ) [ ε ]
M ′ = [ ε ] ( K x [ ε ] − 1 K x − I d )
E II n ( z ) = ∑ m N E n , m { ψ m + exp ( − k 0 q m z ) + ψ m − exp ( k 0 q m ( z − h ) ) }
H II n ( z ) = ∑ m N H n , m { − ψ m + exp ( − k 0 q m z ) + ψ m − exp ( k 0 q m ( z − h ) ) }
δ n 0 + r n = ∑ m = 1 N H n , m [ − ψ m + + ψ m − exp ( − k 0 q m h ) ]
− k Iz 0 ε I δ n 0 + k Iz n ε I r n = ∑ m = 1 N − iE n , m [ ψ m + + ψ m − exp ( − k 0 q m h ) ]
∑ m = 1 N H n , m [ − ψ m + exp ( − k 0 q m h ) + ψ m − ] = t n
∑ m = 1 N iE n , m [ ψ m + exp ( − k 0 q m h ) + ψ m − ] = k IIIz n ε III t n
∑ m = 1 N [ − k I z n ε I H n , m + i E n , m ] ψ m + + ∑ m = 1 N [ k I z n ε I H n , m + i E n , m ] exp ( − k 0 q m h ) ψ m − = 2 k I ε I δ n 0
∑ m = 1 N [ k IIIz n ε III H n , m + i E n , m ] exp ( − k 0 q m h ) ψ m + + ∑ m = 1 N [ − k IIIz n ε III H n , m + i E n , m ] ψ m − = 0
[ − K IZ [ H ] + i [ E ] { K IZ [ H ] + i [ E ] } X { K IIIZ [ H ] + i [ E ] } X − K IIIZ [ H ] + i [ E ] ] [ ψ m + ψ m − ] = [ INC 0 ]
R n = r n r n * Re ( k Iz n k 0 n I cos ⁡ θ )
T n = t n t n * Re ( n I k IIIz n k 0 n III 2 cos ⁡ θ )
ε II ( x ) = ε 0 + ε − 1 exp ( − igx ) + ε 1 exp ( igx )
ε II ( x ) = ε 0 ( 1 + α exp ( − igx ) + α exp ( igx ) )
K x = ( − g k 0 0 0 0 0 0 0 0 g k 0 )
[ ε ] = ε 0 ( 1 α 0 α 1 α 0 α 1 )
M = ( [ g 2 ε 0 k 0 2 ( − 1 + α 2 − 1 + 2 α 2 ) − 1 ] ε 0 α ( g 2 − ε 0 k 0 2 ) k 0 2 α 2 − 1 + 2 α 2 g 2 k 0 2 − ε 0 α − ε 0 − ε 0 α α 2 − 1 + 2 α 2 g 2 k 0 2 α ( g 2 − ε 0 k 0 2 ) k 0 2 [ g 2 ε 0 k 0 2 ( − 1 + α 2 − 1 + 2 α 2 ) − 1 ] ε 0 )
q 1 2 = 1 2 k 0 2 ( − 2 ε 0 k 0 2 + g 2 + g 4 + 8 ε 0 k 0 2 α 2 ( ε 0 k 0 2 − g 2 ) )
q 2 2 = 1 2 k 0 2 ( − 2 ε 0 k 0 2 + g 2 − g 4 + 8 ε 0 k 0 2 α 2 ( ε 0 k 0 2 − g 2 ) )
q 3 2 = 1 k 0 2 ( − ε 0 k 0 2 + g 2 1 − 2 α 2 )
[ E ] = ( 1 1 − 1 E 1 E 2 0 1 1 1 )
E 1 = k 0 2 ε 0 + q 2 2 α ( ε 0 k 0 2 − g 2 )
E 2 = k 0 2 ε 0 + q 1 2 α ( ε 0 k 0 2 − g 2 )
[ H ] = − ( H 1 H 2 − ε k 0 q 3 H 3 H 4 0 H 1 H 2 − ε 0 k 0 q 3 )
H 1 = − ε 0 k 0 q 1 ( 1 + α E 1 ) H 2 = − ε 0 k 0 q 2 ( 1 + α E 2 )
H 3 = − ε 0 k 0 q 1 ( E 1 + 2 α ) H 4 = − ε 0 k 0 q 2 ( E 2 + 2 α )
A ( ψ 3 + ψ 1 + ψ 3 − ψ 1 − ) = ( 2 k I ε I 0 0 0 )
A = ( − k I ε I H 3 + i E 1 − k I ε I H 4 + i E 2 ( k I ε I H 3 + i E 1 ) e ( − k 0 q 1 h ) ( k I ε I H 4 + i E 2 ) e ( − k 0 q 2 h ) − k Iz ε I H 1 + i − k Iz ε I H 2 + i ( k Iz ε I H 1 + i ) e ( − k 0 q 1 h ) ( k Iz ε I H 2 + i ) e ( − k 0 q 2 h ) ( k III ε III H 3 + i E 1 ) e ( − k 0 q 1 h ) ( k III ε III H 4 + i E 2 ) e ( − k 0 q 2 h ) ( − k III ε III H 3 + i E 1 ) ( − k III ε III H 4 + i E 2 ) ( k IIIz ε III H 1 + i ) e ( − k 0 q 1 h ) ( k IIIz ε III H 2 + i ) e ( − k 0 q 2 h ) ( − k IIIz ε III H 1 + i ) ( − k IIIz ε III H 2 + i ) )
k ( I , III ) z = k ( I , III ) 2 − g 2
t 0 = 1 Δ [ − Δ 1 H 3 exp ( − k 0 q 1 h ) − Δ 2 H 4 exp ( − k 0 q 2 h ) + Δ 3 H 3 + Δ 4 H 4 ]
t 1 = 1 Δ [ − Δ 1 H 1 exp ( − k 0 q 1 h ) − Δ 2 H 2 exp ( − k 0 q 2 h ) + Δ 3 H 1 + Δ 4 H 2 ]
r 0 = 1 Δ [ − Δ 1 H 3 − Δ 2 H 4 + Δ 3 H 3 exp ( − k 0 q 1 h ) + Δ 4 H 4 exp ( − k 0 q 2 h ) ] − 1
r 1 = 1 Δ [ − Δ 1 H 1 − Δ 2 H 2 + Δ 3 H 1 exp ( − k 0 q 1 h ) + Δ 4 H 2 exp ( − k 0 q 2 h ) ]
A = 1 − R 0 − 2 R 1 − T 0 − 2 T 1
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