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

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  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).
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    [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]
  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]
  13. V. Shankar, W. Hall, and A. Mohammadian, "A time-domain differential solver for electromagnetic scattering problems," Proc. IEEE 77, 709-721 (1989).
    [CrossRef]
  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]
  15. D. K. Firsov and J. LoVetri, "FVTD - Integral Equation Hybrid for Maxwell’s Equations," Int. J. Numer. Model. 21, 29-42 (2007).
    [CrossRef]
  16. C. Fumeaux, D. Baumann, and R. Vahldieck, "Finite-Volume Time-Domain Analysis of a Cavity-Backed Archimedean Spiral Antenna," IEEE Trans. Antennas Propag. 54, 844-851 (2006).
    [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]
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    [CrossRef]
  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).
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  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).
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    [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).
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2009 (1)

2008 (2)

D. Pinto and S. Obayya, "Accurate Perfectly Matched Layer Finite-Volume Time-Domain Method for Photonic Bandgap Devices," IEEE Photon. Technol. Lett. 20, 339-341 (2008).
[CrossRef]

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]

2007 (4)

D. K. Firsov and J. LoVetri, "FVTD - Integral Equation Hybrid for Maxwell’s Equations," Int. J. Numer. Model. 21, 29-42 (2007).
[CrossRef]

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]

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]

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

2006 (1)

C. Fumeaux, D. Baumann, and R. Vahldieck, "Finite-Volume Time-Domain Analysis of a Cavity-Backed Archimedean Spiral Antenna," IEEE Trans. Antennas Propag. 54, 844-851 (2006).
[CrossRef]

2005 (2)

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]

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]

2004 (3)

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]

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]

H. Du, "Mie-Scattering Calculation," Appl. Opt. 43, 1951-1956 (2004).
[CrossRef] [PubMed]

2003 (1)

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]

2002 (1)

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

1997 (2)

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]

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

1996 (2)

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]

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]

1993 (1)

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

1990 (1)

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]

1989 (1)

V. Shankar, W. Hall, and A. Mohammadian, "A time-domain differential solver for electromagnetic scattering problems," Proc. IEEE 77, 709-721 (1989).
[CrossRef]

1983 (1)

1976 (1)

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

1972 (1)

P. B. Johnson and R. W. Christy, "Optical Constants of the Noble Metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Baumann, D.

C. Fumeaux, D. Baumann, and R. Vahldieck, "Finite-Volume Time-Domain Analysis of a Cavity-Backed Archimedean Spiral Antenna," IEEE Trans. Antennas Propag. 54, 844-851 (2006).
[CrossRef]

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]

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]

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]

Beam, R. M.

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

Benson, T.

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]

Cangellaris, A.

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

Christopoulos, C.

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]

Christy, R. W.

P. B. Johnson and R. W. Christy, "Optical Constants of the Noble Metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Danckwerts, M.

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

Dhawan, A.

Du, H.

Edelvik, F.

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]

Firsov, D. K.

D. K. Firsov and J. LoVetri, "FVTD - Integral Equation Hybrid for Maxwell’s Equations," Int. J. Numer. Model. 21, 29-42 (2007).
[CrossRef]

Fumeaux, C.

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]

C. Fumeaux, D. Baumann, and R. Vahldieck, "Finite-Volume Time-Domain Analysis of a Cavity-Backed Archimedean Spiral Antenna," IEEE Trans. Antennas Propag. 54, 844-851 (2006).
[CrossRef]

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]

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]

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]

Gabriel, C.

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]

Gabriel, S.

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]

Gerhold, M. D.

Hall, W.

V. Shankar, W. Hall, and A. Mohammadian, "A time-domain differential solver for electromagnetic scattering problems," Proc. IEEE 77, 709-721 (1989).
[CrossRef]

Hunsberger, F. P.

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]

Johnson, P. B.

P. B. Johnson and R. W. Christy, "Optical Constants of the Noble Metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Kelley, D. F.

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]

Kerker, M.

Klaus, W.

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

Kunz, K.

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

Kunz, K. S.

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]

Lau, R. W.

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]

Lee, J.-F.

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

Lee, R.

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

Leuchtmann, P.

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]

Liang, C.-H.

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]

LoVetri, J.

D. K. Firsov and J. LoVetri, "FVTD - Integral Equation Hybrid for Maxwell’s Equations," Int. J. Numer. Model. 21, 29-42 (2007).
[CrossRef]

Luebbers, R.

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

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]

Luebbers, R. J.

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]

Mohammadian, A.

V. Shankar, W. Hall, and A. Mohammadian, "A time-domain differential solver for electromagnetic scattering problems," Proc. IEEE 77, 709-721 (1989).
[CrossRef]

Mrozowski, M.

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]

Norton, S. J.

Novotny, L.

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

Obayya, S.

D. Pinto and S. Obayya, "Accurate Perfectly Matched Layer Finite-Volume Time-Domain Method for Photonic Bandgap Devices," IEEE Photon. Technol. Lett. 20, 339-341 (2008).
[CrossRef]

Okoniewski, M.

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]

Pinto, D.

D. Pinto and S. Obayya, "Accurate Perfectly Matched Layer Finite-Volume Time-Domain Method for Photonic Bandgap Devices," IEEE Photon. Technol. Lett. 20, 339-341 (2008).
[CrossRef]

Sankaran, K.

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]

Schneider, M.

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]

Sewell, P.

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]

Shankar, V.

V. Shankar, W. Hall, and A. Mohammadian, "A time-domain differential solver for electromagnetic scattering problems," Proc. IEEE 77, 709-721 (1989).
[CrossRef]

Shi, Y.

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]

Standler, R. B.

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]

Steich, D.

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

Strand, B.

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]

Stuchly, M. A.

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]

Takayama, Y.

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

Teixeira, F. L.

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]

Thomas, D.

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]

Vahldieck, R.

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]

C. Fumeaux, D. Baumann, and R. Vahldieck, "Finite-Volume Time-Domain Analysis of a Cavity-Backed Archimedean Spiral Antenna," IEEE Trans. Antennas Propag. 54, 844-851 (2006).
[CrossRef]

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]

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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).
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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).
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Figures (8)

Fig. 1.
Fig. 1.

(a) Source fluxes at the boundary of a TF/SF region (gray). The source flux is added (inside cell) or subtracted (outside) from the original flux term. (b) Snap shots of the near field around a PEC scatterer.

Fig. 2.
Fig. 2.

(a) Frequency characteristic of permittivity ε̆ r of a single-pole electric Debye medium resembling water. (b) Backscattering: The computed FVTD results (solid line) are compared to the analytical Mie series (line with diamond-shaped markers).

Fig. 3.
Fig. 3.

(a) Frequency characteristic of permittivity ε˘ r of a single-pole electric Lorentz medium. (b) Backscattering: The computed FVTD results (solid line) are compared the analytical Mie series results (line with diamond-shaped markers).

Fig. 4.
Fig. 4.

(a) Frequency characteristic of permittivity of a single-pole electric Drude medium. (b) Backscattering: The computed FVTD results (solid line) are compared to the analytical Mie series (line with diamond-shaped markers).

Fig. 5.
Fig. 5.

(a) Frequency characteristic of permittivity of a double-pole electric Debye medium resembling human muscle tissue. (b) Backscattering: The FVTD results (solid line) are compared to the analytical Mie series (line with diamond-shaped markers).

Fig. 6.
Fig. 6.

(a) Frequency characteristic of permittivity ε̆ r and permeability µ̆ r of a mixed Lorentz medium. (b) Backscattering: The computed FVTD results (solid line) are compared to the analytical Mie series (line with diamond-shaped markers).

Fig. 7.
Fig. 7.

(a) Frequency characteristic of permittivity of gold. The measured permittivity values are indicated with diamonds. (b) Backscattering: The FVTD results (solid line) are compared to the analytical Mie series (line with diamond-shaped markers).

Fig. 8.
Fig. 8.

(a) Sketch of two gold nanospheres separated by a narrow gap. (b) Surface and volume mesh employed in the simulation (only TF domain is shown). The gap is shown in a magnified view. (c) Normalized electric field in the center of the gap. The FVTD result (dashed line) is compared to the reference MMP solution (solid line).

Tables (3)

Tables Icon

Table 1. Real update parameters for Debye materials

Tables Icon

Table 2. Complex update parameters for Lorentz materials.

Tables Icon

Table 3. Real update parameters for Drude materials. The coefficients l2=l4=j2=0 are equal to zero for all methods RC, PLRC and ADE.

Equations (19)

Equations on this page are rendered with MathJax. Learn more.

ΛtUi=1ViΣk=14ΨUkσUiLiPMLLiDM.
Uin+κ=Uin(Λκ)1κΔt(1ViΣk=14ΨUkn+κ0.5σUin+Lκ,iPML,n+Lκ,iDM,n)
withκ={0.5inthepredictorstep1.0inthecorrectorstep,
εκ=ε0ε+κΔt2σe+12Σp=1p{β˘pκ}
μκ=μ0μ+12Σp=1p{α˘pκ},
LκDM,n= {l1J˘n+l2J˘nκ+l3Un+l4Unκ}
J˘n+κ=j˘1J˘n+j˘2Jnκ+j˘3Un+κ+j˘4Un.
ε˘r=ε+χ˘eDe+χ˘eLo+χ˘eDr
μ˘r=μ+χ˘mDe+χ˘mLo+χ˘mDr,
χ˘eDe=Σp=1pΔεpDe1+jωγpDe
χ˘eLo=Σp=1pΔεpLo·(ωpLo)2(ωpLo)2+2jωγpLoω2
ρpLo=γpLoj(ωpLo)2(γpLo)2andδpLo=ε0ΔεpLo·(ωpLo)2(ωpLo)2(γpLo)2
χ˘eDr=Σp=1p(ωpDr)2ω2jωγpDr
Ψ=ΨUk±Ψi.
Ψi=12{[nk×1εHink×(nk×cEi)],[nk×1μEi+nk×(nk×cHi)]}T
an=[urn(mx)m+n(1um2)x]ψn(x)ψn1(x)[urn(mx)m+n(1um2)x]ζn(x)ζn1(x)
bn=ψn(x)[mrn(mx)u+n(11u)x]ψn1(x)ζn(x)[mrn(mx)u+n(11u)x]ζn1(x),
σn=4S1(π)(kR)2
S1(π)=Σn=1(n+12)(1)n(anbn).

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