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Models of near-field spectroscopic studies: comparison between Finite-Element and Finite-Difference methods

Thomas Grosges, Alexandre Vial, and Dominique Barchiesi

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Abstract

We compare the numerical results obtained by the Finite Element Method (FEM) and the Finite Difference Time Domain Method (FDTD) for near-field spectroscopic studies and intensity map computations. We evaluate their respective efficiencies and we show that an accurate description of the dispersion and of the geometry of the material must be included for a realistic modeling. In particular for the nano-objects, we show that a grid size around Δρa ≈ 4πa/λ (expressed in λ units) as well as a Drude-Lorentz’ model of dispersion for FDTD should be used in order to describe more accurately the confinement of the light around the nanostructures (i.e. the high gradients of the electromagnetic field) and to assure the convergence to the physical solution.

©2005 Optical Society of America

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Figures (10)
Fig. 1.
Fig. 1. Geometry of the study for (a) the infinite circular cylinder along the z-axis and (b) the infinite square cylinder.

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Fig. 2.
Fig. 2. Comparison of the Real (a), Imaginary (b) part of the permittivity of gold, calculated with Drude’s and Drude-Lorentz’ models and the corresponding relative error with the experimental permittivity (Drude’s (c) and Drude-Lorentz’ (d) models).

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Fig. 3.
Fig. 3. Examples of non-Cartesian mesh used for FEM (a) and the Cartesian-grid used for FDTD (b). Zoom around the curve domain for FEM (c) and FDTD (d).

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Fig. 4.
Fig. 4. Comparison of the spectra of intensities computed by Mie and FEM (a), Mie and FDTD with Drude’s model (b), Mie and FDTD with Drude-Lorentz’ model (c) and computed by FEM and FDTD with Drude-Lorentz’ model (d) for different distances dy from the center of the particle along the y-axis for radius a = 15 nm. The vertical dashed line in (b), (c) and (d) shows the limit of FDTD computations.

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Fig. 5.
Fig. 5. Maps of the total electric field intensity |E|2 in the xy-plane for a gold cylinder of radius a = 15 nm for a p-polarized illumination at λ = 660 nm computed by the Mie’s theory (a), FEM (b), FDTD (c) and the absolute difference maps (Mie-FEM—) (d) and (Mie-FDTD) (e).

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Fig. 6.
Fig. 6. Comparison between (a) the intensity and (b) the relative errors as functions of the distance dx from the center of the nano-object (a = 15 nm), along the x-axis computed by Mie, FEM and FDTD for different grid sizes.

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Fig. 7.
Fig. 7. Maps of the total electric field intensity |E|2 for a gold cylinder of radius a = 120 nm illuminated in a p-polarization source at λ = 550 nm computed by the Mie’s theory (a), FEM (b), FDTD (c) and the absolute difference maps (Me-FEM) (d) and (Mie-FDTD) (e).

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Fig. 8.
Fig. 8. Comparison of the spectrum of the total field intensities |E|2 computed by FEM and FDTD for different distances dy from the center of the particle along the y-axis for the half-length a = 15 nm.

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Fig. 9.
Fig. 9. Maps of the total electric field intensity |E|2 in the xy-plane of a gold nano-square of half-length a = 15 nm for a p-polarized illumination at λ = 660 nm computed by the FEM (a) and FDTD (b).

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Fig. 10.
Fig. 10. Maps of the total electric field intensity |E|2 in the xy-plane of a gold nano-square of half-length a = 120 nm for a p-polarized illumination at λ = 550 nm computed by the FEM (a) and FDTD (b).

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Tables (1)

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Table 1. Computational procedure for the three methods.

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Equations (14)

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E i ( ρ r , ϕ ) = − j ∑ n = − ∞ + ∞ E n M n ( 1 ) ( ρ r , ϕ )
E s ( ρ r , ϕ ) = j ∑ n = − ∞ + ∞ E n a n ( ρ a ) M n ( 3 ) ( ρ r , ϕ )
E n = E 0 ( − j ) n k
a n ( x ) = [ D n ( mx ) m + n x ] J n ( x ) − J n − 1 ( x ) [ D n ( mx ) m + n x ] H n ( 1 ) ( x ) − H n ( 1 ) ( x ) .
M n ( l ) ( ρ r , ϕ ) = k ( jn Z n ( ρ r ) ρ r e r − Z n ′ ( ρ r ) e ϕ ) exp ( jnϕ ) .
D n − 1 ( z ) = n − 1 z − 1 n z + D n ( z )
Z n ' ( x ) = Z n − 1 ( x ) − n x Z n ( x )
[ ∇ · ( 1 ε r ∇ ) + ω 2 c 2 ] H z = 0 in Ω ,
H z = H i on Γ 0 and 1 ε r ∂ H z ∂ n = − j ω c H z , on Γ 1
∫ Ω = [ ∇ · ( 1 ε r ∇ H z ) + ω 2 c 2 H z ] · vd Ω = 0 ,
H z ∣ i , j n + 1 2 = H z ∣ i , j n − 1 2 + Δ t μ 0 Δ x ( E x ∣ i , j + 1 n − E x ∣ i , j n + E y ∣ i , j n − E y ∣ i + i , j n )
E x ∣ i , j n + 1 = E x ∣ i , j n + Δ t ε 0 ε i , j Δ x ( H z ∣ i , j n + 1 2 − H z ∣ i , j − 1 n + 1 2 )
E y ∣ i , j n + 1 = E y ∣ i , j n + Δ t ε 0 ε i , j Δ x ( H z ∣ i − 1 , j n + 1 2 − H z ∣ i , j n + 1 2 )
ε DL ( ω ) = ε ∞ − ω D 2 ω ( ω + jγ D ) − Δ ε · Ω L 2 ( ω 2 − Ω L 2 ) + j Γ L ω ,
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