## A modular implementation of dispersive materials for time-domain simulations with application to gold nanospheres at optical frequencies

Optics Express, Vol. 17, Issue 17, pp. 15186-15200 (2009)

http://dx.doi.org/10.1364/OE.17.015186

Acrobat PDF (600 KB)

### Abstract

The development of photonic nano-structures can strongly benefit from full-field electromagnetic (EM) simulations. To this end, geometrical flexibility and accurate material modelling are crucial requirements set on the simulation method. This paper introduces a modular implementation of dispersive materials for time-domain EM simulations with focus on the Finite-Volume Time-Domain (FVTD) method. The proposed treatment can handle electric and magnetic dispersive materials exhibiting multi-pole Debye, Lorentz and Drude models, which can be mixed and combined without restrictions. The presented technique is verified in several illustrative examples, where the backscattering from dispersive spheres is calculated. The amount of flexibility and freedom gained from the proposed implementation will be demonstrated in the challenging simulation of the plasmonic resonance behavior of two gold nanospheres coupled in close proximity, where the dispersive characteristic of gold is approximated by realistic values in the optical frequency range.

© 2009 Optical Society of America

5. P. Sewell, T. Benson, C. Christopoulos, D. Thomas, A. Vokuvic, and J. Wykes, “Transmission-Line Modeling (TLM) Based Upon Unstructured Tetrahedral Meshes,” IEEE Trans. Microwave Theory Tech. **53**, 1919–1928 (2005). [CrossRef]

6. J.-F. Lee, R. Lee, and A. Cangellaris, “Time-Domain Finite-Element Methods,” IEEE Trans. Antennas Propag. **45**, 430–442 (1997). [CrossRef]

8. R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, and M. Schneider, “A Frequency-Dependent Finite-Difference Time-Domain Formulation for Dispersive Materials,” IEEE Trans. Electromagn. Compat. **32**, 222–227 (1990). [CrossRef]

12. F. L. Teixeira, “Time-Domain Finite-Difference and Finite-Element Methods for Maxwell Equations in Complex Media,” IEEE Trans. Antennas Propag. **56**, 2150–2166 (2008). [CrossRef]

8. R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, and M. Schneider, “A Frequency-Dependent Finite-Difference Time-Domain Formulation for Dispersive Materials,” IEEE Trans. Electromagn. Compat. **32**, 222–227 (1990). [CrossRef]

9. D. F. Kelley and R. J. Luebbers, “Piecewise Linear Recursive Convolution for Dispersive Media using FDTD,” IEEE Trans. Antennas Propag. **44**, 792–797 (1996). [CrossRef]

10. M. Okoniewski, M. Mrozowski, and M. A. Stuchly, “Simple Treatment of Multi-Term Dispersion in FDTD,” IEEE Microwave Guided Wave Lett. **7**, 121–123 (1997). [CrossRef]

11. Y. Takayama and W. Klaus, “Reinterpretation of the Auxiliary Differential Equation Method for FDTD,” IEEE Microwave Wireless Comp. Lett. **12**, 102–104 (2002). [CrossRef]

## 2. Dispersive Materials for the FVTD Method

_{13}], FVTD has demonstrated attractive features for the solution of the Maxwell’s equations for complex, real-world problems [14

14. D. Baumann, C. Fumeaux, P. Leuchtmann, and R. Vahldieck, “Finite-Volume Time-Domain (FVTD) Modeling of a Broadband Double-Ridged Horn Antenna,” Int. J. Numer. Model. **17**, 285–298 (2004). [CrossRef]

17. Y. Shi and C.-H. Liang, “The Finite-Volume Time-Domain Algorithm using Least Square Method in Solving Maxwell’s Equations,” J. Comput. Phys. **226**, 1444–1457 (2007). [CrossRef]

**U**

*=(*

_{i}*E*)

_{x},E_{y},E_{z},H_{x},H_{y},H_{z}^{T}denotes the collocated electromagnetic field vector in unit cell

*i*with volume

*V*. A tetrahedral mesh is used in the present implementation. The flux

_{i}*k*of cell

*i*is constructed from the tangential field components located in the barycenter of

*k*. Details of the standard flux evaluation can be found in [3]. Based on a local plane wave approach, the fluxes can be split into outgoing

18. D. Baumann, C. Fumeaux, and R. Vahldieck, “Field-Based Scattering-Matrix Extraction Scheme for the FVTD Method Exploiting a Flux-Splitting Algorithm,” IEEE Trans. Microwave Theory Tech. **53**, 3595–3605 (2005). [CrossRef]

_{e},σ

_{e},σ

_{e},0,0,0} is the conductivity matrix. The material matrix Λ contains the permittivity and permeability of the medium. L

^{PML}

_{i}corresponds to the loss vector for an unsplit perfectly matched layer (PML) formulation, which is described in detail for radial and conformal formulations in [19

19. C. Fumeaux, K. Sankaran, and R. Vahldieck, “Spherical Perfectly Matched Absorber for Finite-Volume 3-D Domain Truncation,” IEEE Trans. Microwave Theory Tech. **55**, 2773–2781 (2007). [CrossRef]

21. R. F. Warming and R. M. Beam, “Upwind second-order difference schemes and applications in aerodynamic flows,” AIAA J. **14**, 1241–1249 (1976). [CrossRef]

*n*denotes the time step index. The inhomogeneity of a tetrahedral mesh can be exploited in terms of efficiency by employing a local-time stepping (LTS) scheme, where different time steps Δ

*t*are used in different domains of the mesh Ω

_{ℓ}_{ℓ}depending on the cell size [22

22. C. Fumeaux, D. Baumann, and R. Vahldieck, “A generalized local time-step scheme for efficient FVTD simulations in strongly inhomogeneous meshes,” IEEE Trans. Microwave Theory Tech. **52**, 1067–1076 (2004). [CrossRef]

*κ*translate into a time advancement of a half step 0.5Δ

*t*in the predictor step and accordingly a full step 1.0Δ

_{ℓ}*t*in the corrector step. Obviously, a Lax-Wendroff time-stepping algorithm makes the implementation of dispersive materials more challenging compared to e.g. the standard leap-frog time iteration of the FDTD approach.

_{ℓ}^{κ}=diag{

*ε*} contains

^{κ},ε^{κ},ε^{κ},µ^{κ},µ^{κ},µ^{κ}*β*=

^{κ}_{p}*ℜ*{

*β*̆

*} and*

^{κ}_{p}*α*=ℜ{

^{κ}_{p}*α*̆

*} defined as the real part of complex parameters*

^{κ}_{p}*β*̆

*and*

^{κ}_{p}*α*̆

^{κ}

_{p}. These parameters specifically depend both on the material model (Debye, Lorentz and Drude) as well as on the implementation scheme (RC, PLRC and ADE) and will be explicitly given in paragraphs 2.1 to 2.3. The second term of the permittivity equation in Eq. (3) contains the electric losses σ

_{e}and appears due to a semi-implicit approximation of the electro-magnetic field in Eq. (2) at time step

*n*+0.5. The loss vector

**L**

*can be generally written as*

^{DM,n}_{κ}**U**as well as on a (yet to be defined) loss current

**J**̆

^{n}. The update parameters

*l*=ℜ{

^{ν}*l*̆

_{ν}} with

*l*̆

_{ν}=(

*l*̆

*,*

^{e}_{ν}*l*̆

*) (*

^{m}_{ν}*ν*=1…4) will be determined in paragraphs 2.1 to 2.3 for each material model and implementation scheme. The update parameters are different for the electric field components l̆eν and the magnetic field components

*l*̆

*, and may be of complex nature. The complex loss current*

^{m}_{ν}**J**̆

^{ν}conveniently can be computed iteratively as

*j*̆

_{ν}=(

*j*̆

*,*

^{e}_{ν}*j*̆

*) (*

_{m}_{ν}*ν*=1…4) will be determined in paragraphs 2.1 to 2.3. The emphasis of the FVTD approach presented here is placed on a modular implementation of dispersive materials with the aim of achieving the highest possible flexibility for generating and investigating lossy and dispersive materials. For example, in order to investigate surface plasmons of gold nanospheres, a mixed double-pole Lorentz/single-pole Drude model is able to approximate with reasonable accuracy the measured behavior of gold’s permittivity at optical frequencies. Therefore, in order to describe the properties of dispersive materials in terms of permittivity and permeability, the following general approach is chosen to define the material parameters

*ε*and

_{∞}*µ*are the relative permittivity and permeability at infinite frequency and

_{∞}*χ*̆

_{e}and

*χ*̆

_{m}are the (complex) electric and magnetic susceptibilities. In the following subsections, the electric susceptibilities

*χ*̆

_{e}for Debye (De), Lorentz (Lo) and Drude (Dr) materials will be specified in detail. On this basis, the update parameters

*β*,

^{κ}_{p}*l*̆

*and*

^{e}_{ν}*j*̆

*for the RC, PLRC and ADE method can be derived. The magnetic susceptibilities*

^{e}_{ν}*χ*̆

_{m}and the parameters

*α*,

^{κ}_{p}*l*̆

*and*

^{m}_{ν}*j*̆

*can be easily derived accordingly, and therefore will not be specified explicitly here.*

^{m}_{ν}8. R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, and M. Schneider, “A Frequency-Dependent Finite-Difference Time-Domain Formulation for Dispersive Materials,” IEEE Trans. Electromagn. Compat. **32**, 222–227 (1990). [CrossRef]

9. D. F. Kelley and R. J. Luebbers, “Piecewise Linear Recursive Convolution for Dispersive Media using FDTD,” IEEE Trans. Antennas Propag. **44**, 792–797 (1996). [CrossRef]

10. M. Okoniewski, M. Mrozowski, and M. A. Stuchly, “Simple Treatment of Multi-Term Dispersion in FDTD,” IEEE Microwave Guided Wave Lett. **7**, 121–123 (1997). [CrossRef]

11. Y. Takayama and W. Klaus, “Reinterpretation of the Auxiliary Differential Equation Method for FDTD,” IEEE Microwave Wireless Comp. Lett. **12**, 102–104 (2002). [CrossRef]

### 2.1. Debye material

*ε*̆

^{De}as a function of angular frequency w and pole relaxation time

*γ*

^{De}

_{p}

*Δ*

*ε*

^{De}

_{p}=

*ε*

^{De}

_{s,p}-

*ε*

^{De}

_{s,p-1}, where

*ε*

^{De}

_{s,p}is the static relative permittivity of the

*p*-th pole and

*γ*

*De*

_{p}is the pole relaxation time. It is important to note that according to the definition employed here,

*ε*

^{De}

_{s,0}=

*ε*. After derivation of the RC, PLRC and ADE method for a Debye medium in the formalism of Eq. (2), the necessary parameters for the permittivity equation, Eq. (3), as well as for the update equations, Eq. (5) and Eq. (6), can be determined. The resulting parameters are summarized in Table 1.

_{∞}### 2.2. Lorentz material

*ε*̌

^{Lo}of a Lorentz media is commonly defined as

*ε*

^{Lo}

_{p}=

*ε*

^{Lo}

_{s,p}-

*ε*

^{Lo}

_{s,p-1}(with

*ε*

^{Lo}

_{s,0}=

*ε*),

_{∞}*γ*

^{Lo}

_{p}is the damping coefficient and wLo p is the undamped resonant angular frequency of the pole pair. Following parameters are introduced for the sake of simplicity:

### 2.3. Drude material

*P*-pole susceptibility function can be written as

### 2.4. Note on time-dependency convention

*iωt*), engineers commonly employ exp(+

*jωt*). Although this is well known, it is still a source of confusion. As a consequence, the dispersive models for Debye, Lorentz and Drude materials can either exhibit a positive imaginary part (when using the physicists’ models) or a negative imaginary part of

*χ*̆, as it is the case in Eq. (9)–Eq. (12), which stick to the engineers’ convention. Of course no matter which convention is used, the physics behind the models does not change. The crucial point is that the material model employed must describe an exponentially decaying behavior of the electro-magnetic field after a transient excitation. This constraint dictates the sign of the imaginary part of the permittivity (and permeability respectively). If this is not satisfied, an active medium is created and instabilities in the simulation are very likely to arise.

### 2.5. Implementation

## 3. Total-Field Scattered-Field (TF/SF) Formulation

^{+}and incoming Ψ

^{-}fluxes for the cells adjacent to the Huygens’ source surface. The location of the depticted boundary cells is arbitrary, in the sense that they could as well be adjacent cells. For an incident plane wave, the source flux on the source surface is defined as

*c*=(

*εµ*)

^{-1/2}is the velocity of light in the medium. As source terms are always added to the incoming flux, the source surface can even be placed on the computational boundary and e.g. combined with a radiation boundary condition. As an example, Fig. 1(b) depicts four snapshots of the incident and scattered field around a perfectly electric conducting (PEC) sphere (

*R*=7.5mm) located in the center of a spherical TF region (

*r*=27.0mm) and illuminated by a pulsed plane wave in the frequency range of 10 to 50 GHz.

## 4. Mie Scattering

*ε*≠1 and

_{r}*µ*=1 can be found in literature, but rarely for simultaneously electric

_{r}*ε*≠1 and magnetic

_{r}*µ*≠1 materials [25

_{r}25. R. F. Harrington, *Time-Harmonic Electromagnetic Fields* (John Wiley & Sons, Inc., 2001). [CrossRef]

26. M. Kerker, W. D.S., and C. Giles, “Electromagnetic Scattering by Magnetic Spheres,” J. Opt. Soc. Am. **73**, 765–767 (1983). [CrossRef]

27. H. Du, “Mie-Scattering Calculation,” Appl. Opt. **43**, 1951–1956 (2004). [CrossRef] [PubMed]

*ψ*(

_{n}*x*)=

*xj*(

_{n}*x*) and

*ζ*=

_{n}*xh*(

_{n}*x*) are derived from the Spherical Bessel functions of first

*j*(

_{n}*x*) and third kind

*h*(

_{n}*x*). The ratio of the Riccati-Bessel functions are defined as

*r*(

_{n}*mx*)=

*ψ*

_{n-1}(

*mx*)/

*ψ*(

_{n}*mx*) and the ratio of the material parameters are

*ε*2 and

*µ*2, and the

*ε*1 and

*µ*1 represents the background material (commonly vacuum). The normalized backscattering radar cross section σ

_{n}used in the following section is computed using

*k*=2

*π/λ*the angular wave number. The function

*S*(

_{1}*π*) is given by

## 5. Validation

4. F. Edelvik and B. Strand, “Frequency Dispersive Materials for 3-D Hybrid Solvers in Time Domain,” IEEE Trans. Antennas Propag. **51**, 1199–1205 (2003). [CrossRef]

28. R. Luebbers, D. Steich, and K. Kunz, “FDTD Calculation of Scattering from Frequency-Dependent Materials,” IEEE Trans. Antennas Propag. **41**, 1249–1257 (1993). [CrossRef]

*λ*

_{rmin}/15 on the surface of the scattering sphere to

*λ*

_{0min}/7 at the computational boundary, where

*λ*

_{rmin}and

*λ*

*0min*are the wavelengths at the highest frequency of interest in the material and in vacuum, respectively.

### 5.1. Backscattering from single-pole electric Debye sphere

*R*=420

*µ*m is calculated at microwave frequencies. The parameters which describe the dielectric characteristic of water are

*ε*

_{∞}=5.9,

*ε*

^{De}

_{s,1}=80.2, and

*γ*

^{De}

_{1}=9.5ps [29]. Figure 2(a) depicts the value of the complex permittivity in the frequency range from 10 to 50 GHz, as well as the comparison between the analytical Mie solution (line with diamond-shaped markers) and the numerical results obtained with FVTD (solid line) in Fig. 2(b). The agreement between the Mie series and FVTD is excellent over the whole frequency range.

### 5.2. Backscattering from single-pole electric Lorentz sphere

*R*=900

*µ*m. The employed Lorentz parameters are

*ε*∞=4.1,

*ε*

^{Lo}

_{s,1}=5.8,

*ω*

^{Lo}

_{1}=40

*π*· 10

^{9}s

^{-1}and

*γ*

^{Lo}

_{1}=30

*π*· 10

^{8}s

^{-1}. The characteristic of the permittivity is shown in Fig. 3(a) and the comparison between analytical Mie scattering (line with diamond-shaped markers) and FVTD simulation results (solid line) is plotted in Fig. 3(b). Again, the agreement between the Mie series and FVTD is very good in the frequency range from 10 to 50 GHz where the Lorentz pole shows its resonance.

### 5.3. Backscattering from single-pole electric Drude sphere

*R*=3000

*µ*m) is computed. The parameters of the Drude model are

*ε*∞=1.0,

*ω*

^{Dr}

_{1}=80

*π*· 10

^{9}s

^{-1}and

*γ*

^{Dr}

_{1}=1.5 · 10

^{10}s

^{-1}. This example is especially demanding since the radius of the sphere is very large in terms of wavelength for the observed frequency range of 1 to 100 GHz. A discretization corresponding to approximately λ

_{r min}/10 at 100 GHz is deployed on the surface of the sphere. Figure 4(a) plots the frequency dependent behavior of the permittivity of the Drude medium. The normalized radar cross section σ

_{n}is shown in Fig. 4(b), where a very good agreement between the analytical Mie scattering series (line with diamond-shaped markers) and the simulated FVTD results (solid line) can be observed. The small discrepancies observed at the highest frequencies are explained by the relatively coarse discretization.

### 5.4. Backscattering from double-pole electric Debye sphere

*ε*∞=11.05,

*ε*

^{De}

_{s,1}=83.97,

*ε*

^{De}

_{s,2}=43.35,

*τ*

^{De}

_{1}=8.56·10-

^{12}s,

*τ*

^{De}

_{2}=2.33 ·10

^{-10}s. These values have been obtained by a least-square fitting with a Cole-Cole model for muscle tissue in the licensed frequency range of ultra-wideband (UWB) signals (3.1 to 10.6 GHz) [30

30. S. Gabriel, R. W. Lau, and C. Gabriel, “The dielectric properties of biological tissues: III. Parametric models for the dielectric spectrum of tissues,” Phys. Med. Biol. **41**, 2271–2293 (1996). [CrossRef] [PubMed]

*R*=1800

*µ*m).

### 5.5. Backscattering from mixed single-pole electric and single-pole magnetic Lorentz sphere

*R*=900

*µ*m) is calculated. The frequency characteristic of permittivity and permeability of the Lorentz medium are depicted in Fig. 6(a). The electric Lorentz pole is determined by

*ε*∞=4.1,

*ε*

^{Lo}

_{s,1}=8.0,

*ω*

^{Lo}

_{e,1}=40

*π*·10

^{9}s

^{-1}, and

*γ*

^{Lo}

_{e,1}=30

*π*·10

^{9}s

^{-1}. The magnetic pole is characterized by

*µ*∞=4.1,

*µ*

^{Lo}

_{s,1}=5.0,

*ω*

^{Lo}

_{m,1}=60

*π*· 10

^{9}s

^{-1}and

*γ*

^{Lo}

_{m,1}=40

*π*· 10

^{9}s

^{-1}. The normalized radar cross section σ

_{n}is plotted in Fig. 6(b) and a good agreement between the FVTD simulation (solid line) results and the analytical Mie solution (line with diamond-shaped markers) is found. This example nicely demonstrates the flexibility of the FVTD implementation of dispersive materials for simultaneous deployment of electric and magnetic dispersive poles in a single medium.

### 5.6. Backscattering from a gold nanosphere

*R*=40nm) is calculated, illustrating the relevance of the technique for optical nano-structure simulations. The permittivity of gold is approximated by an electric mixed double-pole Lorentz/single-pole Drude model, which parameters have been obtained by a least-square fitting of experimental data [31

31. P. B. Johnson and R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B **6**, 4370–4379 (1972). [CrossRef]

*ε*∞=3.65, two Lorentz poles by

*ε*

^{Lo}

_{s,1}=7.01,

*ε*

^{Lo}

_{s,2}=5.54,

*ω*

^{Lo}

_{1}=4.79 · 10

^{15}s

^{-1},

*ω*

^{Lo}

_{2}=6.45 · 10

^{15}s

^{-1},

*γ*

^{Lo}

_{1}=9.08 · 10

^{14}s

^{-1},

*γ*

^{Lo}

_{2}=1.39 · 10

^{15}s

^{-1}; the Drude pole is modeled with

*ω*

^{Dr}

_{1}=1.28 · 10

^{16}s

^{-1}and

*γ*

^{Dr}

_{1}=2.76 · 10

^{13}s

^{-1}. The wavelength-dependent characteristic of the fitted permittivity values is plotted in Fig. 7(a) in the visible spectrum of wavelengths, where plasma oscillations occur. The measured permittivity characteristic is indicated with diamonds at discrete frequencies, which suggests that the fitted parameters approximate the experimental data very well. The simulated results (solid line) of the backscattering is shown in Fig. 7(b) where they are compared to the reference solution obtained by an analytical Mie series (line with diamond-shaped markers).

## 6. Plasmonic Resonance of Two Coupled Gold Nanospheres

1. M. Danckwerts and L. Novotny, “Optical Frequency Mixing at Coupled Gold Nanoparticles,” Phys. Rev. Lett. **98**, 1–4 (2007). [CrossRef]

32. A. Dhawan, S. J. Norton, M. D. Gerhold, and T. Vo-Dinh, “Comparison of FDTD numerical computations and analytical multipole expansion method for plasmonics-active nanosphere dimers,” Opt. Express **17**, 9688–9703 (2009). [CrossRef] [PubMed]

*d*=1nm) in between the two spheres, where a strong plasmonic resonance of the EM field can be achieved. The permittivity of gold is approximated with the same parameters as given in section 5.6, as they provide a very good agreement with experimental data. From a numerical point of view, such an arrangement becomes increasingly challenging as the gap between the spheres becomes smaller. However, an unstructured and strongly inhomogeneous mesh can tackle such multi-scale problems, as illustrated in Fig. 8(b), where the surface and volume meshes of the computational model employed here are depicted. The gap region is magnified in order to display the fine mesh in between the spheres. Overall, the model heavily exploits the inhomogeneity of a tetrahedral mesh and hence minimizes the total number of cells to about 270’000. The LTS scheme allows to apply a time steps 64 times higher in the larger free-space cells compared to the time step in the tiny cells in the gap.

## 7. Conclusion

*P*-pole electric and magnetic Debye, Lorentz and Drude materials, which can be combined in any fashion. This results in a very flexible treatment of dispersive materials, as even complicated frequency-dependent characteristics of permittivities and permeabilities can be approximated very well. As a challenging example, the plasmonic resonance of two gold nanospheres coupled in close proximity has been simulated successfully. Since FVTD is relying on an unstructured mesh, arbitrarily-shaped particles and structures can be simulated without restrictions and without increase in computational costs for a comparable size. This extension has demonstrated the capability of FVTD as a promising CEM tool for complex optical problems exhibiting dispersive characteristics.

## References and links

1. | M. Danckwerts and L. Novotny, “Optical Frequency Mixing at Coupled Gold Nanoparticles,” Phys. Rev. Lett. |

2. | D. Pinto and S. Obayya, “Accurate Perfectly Matched Layer Finite-Volume Time-Domain Method for Photonic Bandgap Devices,” IEEE Photon. Technol. Lett. |

3. | P. Bonnet, X. Ferrieres, B. Michielsen, P. Klotz, and J. Roumiguieres, |

4. | F. Edelvik and B. Strand, “Frequency Dispersive Materials for 3-D Hybrid Solvers in Time Domain,” IEEE Trans. Antennas Propag. |

5. | P. Sewell, T. Benson, C. Christopoulos, D. Thomas, A. Vokuvic, and J. Wykes, “Transmission-Line Modeling (TLM) Based Upon Unstructured Tetrahedral Meshes,” IEEE Trans. Microwave Theory Tech. |

6. | J.-F. Lee, R. Lee, and A. Cangellaris, “Time-Domain Finite-Element Methods,” IEEE Trans. Antennas Propag. |

7. | J. S. Hesthaven and T. Warburton, |

8. | R. Luebbers, F. P. Hunsberger, K. S. Kunz, R. B. Standler, and M. Schneider, “A Frequency-Dependent Finite-Difference Time-Domain Formulation for Dispersive Materials,” IEEE Trans. Electromagn. Compat. |

9. | D. F. Kelley and R. J. Luebbers, “Piecewise Linear Recursive Convolution for Dispersive Media using FDTD,” IEEE Trans. Antennas Propag. |

10. | M. Okoniewski, M. Mrozowski, and M. A. Stuchly, “Simple Treatment of Multi-Term Dispersion in FDTD,” IEEE Microwave Guided Wave Lett. |

11. | Y. Takayama and W. Klaus, “Reinterpretation of the Auxiliary Differential Equation Method for FDTD,” IEEE Microwave Wireless Comp. Lett. |

12. | F. L. Teixeira, “Time-Domain Finite-Difference and Finite-Element Methods for Maxwell Equations in Complex Media,” IEEE Trans. Antennas Propag. |

13. | V. Shankar, W. Hall, and A. Mohammadian, “A time-domain differential solver for electromagnetic scattering problems,” Proc. IEEE |

14. | D. Baumann, C. Fumeaux, P. Leuchtmann, and R. Vahldieck, “Finite-Volume Time-Domain (FVTD) Modeling of a Broadband Double-Ridged Horn Antenna,” Int. J. Numer. Model. |

15. | D. K. Firsov and J. LoVetri, “FVTD - Integral Equation Hybrid for Maxwell’s Equations,” Int. J. Numer. Model. |

16. | C. Fumeaux, D. Baumann, and R. Vahldieck, “Finite-Volume Time-Domain Analysis of a Cavity-Backed Archimedean Spiral Antenna,” IEEE Trans. Antennas Propag. |

17. | Y. Shi and C.-H. Liang, “The Finite-Volume Time-Domain Algorithm using Least Square Method in Solving Maxwell’s Equations,” J. Comput. Phys. |

18. | D. Baumann, C. Fumeaux, and R. Vahldieck, “Field-Based Scattering-Matrix Extraction Scheme for the FVTD Method Exploiting a Flux-Splitting Algorithm,” IEEE Trans. Microwave Theory Tech. |

19. | C. Fumeaux, K. Sankaran, and R. Vahldieck, “Spherical Perfectly Matched Absorber for Finite-Volume 3-D Domain Truncation,” IEEE Trans. Microwave Theory Tech. |

20. | D. Baumann, C. Fumeaux, R. Vahldieck, and E. P. Li, “Conformal Perfectly Matched Absorber for Finite-Volume Simulations,” in |

21. | R. F. Warming and R. M. Beam, “Upwind second-order difference schemes and applications in aerodynamic flows,” AIAA J. |

22. | C. Fumeaux, D. Baumann, and R. Vahldieck, “A generalized local time-step scheme for efficient FVTD simulations in strongly inhomogeneous meshes,” IEEE Trans. Microwave Theory Tech. |

23. | A. Taflove and S. C. Hagness, |

24. | H. C. Van de Hulst, |

25. | R. F. Harrington, |

26. | M. Kerker, W. D.S., and C. Giles, “Electromagnetic Scattering by Magnetic Spheres,” J. Opt. Soc. Am. |

27. | H. Du, “Mie-Scattering Calculation,” Appl. Opt. |

28. | R. Luebbers, D. Steich, and K. Kunz, “FDTD Calculation of Scattering from Frequency-Dependent Materials,” IEEE Trans. Antennas Propag. |

29. | F. S. Barnes and B. Greenebaum, eds., |

30. | S. Gabriel, R. W. Lau, and C. Gabriel, “The dielectric properties of biological tissues: III. Parametric models for the dielectric spectrum of tissues,” Phys. Med. Biol. |

31. | P. B. Johnson and R. W. Christy, “Optical Constants of the Noble Metals,” Phys. Rev. B |

32. | A. Dhawan, S. J. Norton, M. D. Gerhold, and T. Vo-Dinh, “Comparison of FDTD numerical computations and analytical multipole expansion method for plasmonics-active nanosphere dimers,” Opt. Express |

33. | C. Hafner, |

**OCIS Codes**

(240.6680) Optics at surfaces : Surface plasmons

(260.2110) Physical optics : Electromagnetic optics

(290.1350) Scattering : Backscattering

(290.4020) Scattering : Mie theory

(050.1755) Diffraction and gratings : Computational electromagnetic methods

**ToC Category:**

Physical Optics

**History**

Original Manuscript: June 22, 2009

Revised Manuscript: July 17, 2009

Manuscript Accepted: July 23, 2009

Published: August 12, 2009

**Citation**

D. Baumann, C. Fumeaux, C. Hafner, and E. P. Li, "A modular implementation of dispersive materials for time-domain simulations with application to gold nanospheres at optical frequencies," Opt. Express **17**, 15186-15200 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-17-15186

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