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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ACM T. Math Software (1)

T.A. Davis and I.S. Duff, �??A combined unifrontal multifrontal method for unsymmetric sparse matrices,�?? ACM T. Math Software 25, 1-20 (1999).
[CrossRef]

Ann. Phys. (1)

G. Mie, �??Beitr¨age zur Optik tr¨uber Medien, speziell kolloidaler Metall¨osungen,�?? Ann. Phys. 25, 377-445 (1908).
[CrossRef]

Appl. Opt. (1)

IEEE Microw. Wirel. Compon. Lett. (1)

W.H. Yu, and R. Mittra, �??A conformal finite difference time domain technique for modeling curved dielectric surfaces,�?? IEEE Microw. Wirel. Compon. Lett. 11, 25-27 (2001).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

K.S. Yee, �??Numerical solution of initial boundary value problems involving Maxwell�??s equations in isotropic media,�?? IEEE Trans. Antennas Propag. 16, 302-307 (1966).

IEEE Trans. Geosci. Remote Sens. (1)

F.L. Teixeira, W.C. Chew, M. Straka, M.L. Oristaglio, and T. Wang, �??Finite-difference time-domain simulation of ground penetrating radar on dispersive, inhomogeneous, and conductive soils,�?? IEEE Trans. Geosci. Remote Sens. 36, 1928-1937 (1998).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

S. Dey, and R. Mittra, �??A conformal finite-difference time-domain technique for modeling cylindrical dielectric resonators,�?? IEEE Trans. Microwave Theory Tech. 47, 1737-1739 (1999).
[CrossRef]

J. Chem. Phys. (2)

M.C. Beard, and C.A. Schmuttenmaer, �??Using the finite-difference time-domain pulse propagation method to simulate time-resolved the experiments,�?? J. Chem. Phys. 114, 2903-2909 (2001).
[CrossRef]

J.T. Krug II, E.J. Sanchez, and X.S. Xie, �??Design of near-field optical probes with optimal field enhancement by finite difference time domain electromagnetic simulation,�?? J. Chem. Phys. 116, 10895-10901 (2002).
[CrossRef]

J. Opt. Soc. Am. A (2)

J. Opt. Soc. Am. B (1)

J. Phys. Chem. B (1)

M. Futamata, Y. Maruyama, and M. Ishikawa, �??Local electric field and scattering cross section of Ag nanoparticles under surface plasmon resonance by finite difference time domain method,�?? J. Phys. Chem. B 107, 7607- 7617 (2003).
[CrossRef]

Opt. Commun. (3)

R. Fikri, D. Barchiesi, F. H�??Dhili, R. Bachelot, A. Vial, and P. Royer, �??Modeling recent experiments of apertureless near-field optical microscopy using 2D finite element method,�?? Opt. Commun. 221, 13-22 (2003).
[CrossRef]

R. Fikri, T. Grosges, and D. Barchiesi, �??Apertureless scanning near-field optical microscopy: Numerical modeling of the lock-in detection,�?? Opt. Commun. 232, 15-23 (2004).
[CrossRef]

C. Gr´ehan, G. Gouesbet, and F. Guilloteau, �??Comparison of the diffraction theory and the generalized lorenz-mie theory for a sphere arbitrarily located into a laser beam,�?? Opt. Commun. 90, 1-6 (1992).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

Phys. Rev. (1)

P. Johnson and R. Christy, �??Optical constants of the noble metals,�?? Phys. Rev. 6, 4370-4379 (1972).

Phys. Rev. B (3)

N. F´elidj, J. Aubard, G. L´evi, J.R. Krenn, M. Salerno, G. Schider, B. Lamprecht, A. Leitner, and F.R. Aussenegg, �??Controlling the optical response of regular arrays of gold particles for surface-enhanced Raman scattering,�?? Phys. Rev. B 65, 075419-075427 (2002).
[CrossRef]

S.K. Gray, and T. Kupka, �??Propagation of light in metallic nanowire arrays: Finite-difference time-domain studies of silver cylinders,�?? Phys. Rev. B 68, 045415-045425 (2003).
[CrossRef]

A. Vial, A.S. Grimault, D. Mac´ýas, D. Barchiesi, and M. Lamy de la Chapelle, �??Improved analytical fit of gold dispersion: application to the modelling of extinction spectra with the FDTD method,�?? Phys. Rev. B 71, 085416- 085422 (2005).
[CrossRef]

Phys. Rev. E (1)

D. Barchiesi, C. Girard, O.J.F. Martin, D. Van Labeke, and D. Courjon, �??Computing the optical near-field distributions around complex subwavelength surface structures: A comparative study of different methods,�?? Phys. Rev. E 54, 4285-4292 (1996).
[CrossRef]

Phys. Rev. Lett. (1)

C. Ropers, D.J. Park, G. Stibenz, G. Steinmeyer, J. Kim, D.S. Kim, and C. Lienau, �??Femtosecond Light Transmission and Subradiant Damping in Plasmonic Crystals,�?? Phys. Rev. Lett. 94, 113901-4 (2005).
[CrossRef] [PubMed]

Syn. Metals (1)

J. Grand, S. Kostcheev, J.L. Bijeon, M. Lamy de la Chapelle, P.M. Adam, A. Rumyantseva, G. L´erondel, and P. Royer, �??Optimization of SERS-active substrates for near-field raman spectroscopy,�?? Syn. Metals 139, 621-624 (2003).
[CrossRef]

Other (6)

A. Taflove, and S.C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, Norwood, 2000).

K. Kunz, and R. Luebbers, The Finite Difference Time Domain Method for Electromagnetics (CRC Press, 1993).

A. Taflove, Advances in Computational Electrodynamics, the Finite-Difference Time-Domain Method (Artech House, Norwood, 1998).

C.F. Bohren, and D.R. Huffman, Absorption and scattering of light by small particles (John Wiley and Sons, New York, 1983).

M. Born, and E.Wolf, Principle of Optics (Pergamon Press, Oxford, 1993).

J. Jin, The Finite Element Method in Electromagnetics (John Wiley and Sons, New York, 1993).

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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.

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).

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).

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.

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).

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.

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).

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.

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).

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).

Tables (1)

Tables Icon

Table 1. Computational procedure for the three methods.

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 + D ) Δ ε · Ω L 2 ( ω 2 Ω L 2 ) + j Γ L ω ,

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