Abstract

An efficient discontinuous Galerkin time domain (DGTD) method with a generalized dispersive material (GDM) model and periodic boundary conditions (PBCs), hereto referred to as DGTD-GDM-PBCs, is proposed to analyze the electromagnetic scattering from dispersive periodic nanostructures. The GDM model is utilized to achieve a robust and accurate universal model for arbitrary dispersive materials. Using a transformed field variable technique, PBCs are introduced to efficiently truncate the computational domain in the periodic directions for both normally and obliquely incident illumination cases. Based on the transformed Maxwell’s equations with PBCs, the formulation of the DGTD method with a GDM model is derived. Furthermore, a Runge-Kutta time-stepping scheme is proposed to update the semi-discrete transformed Maxwell’s equations and auxiliary differential equations (ADEs) with high order accuracy. Numerical examples for periodic nanostructures with dispersive elements, such as reflection and transmission of a thin film, surface plasmon at the interfaces of a metallic hole array structure, and absorption properties of a dual-band infrared absorber are presented to demonstrate the accuracy and capability of the proposed method.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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    [Crossref]

2018 (3)

Q. Ren, H. Bao, S. D. Campbell, L. J. Prokopeva, A. V. Kildishev, and D. H. Werner, “Continuous-discontinuous Galerkin time domain (CDGTD) method with generalized dispersive material (GDM) model for computational photonics,” Opt. Express 26(22), 29005–29016 (2018).
[Crossref] [PubMed]

K. Sirenko, Y. Sirenko, and H. Bağcı, “Exact absorbing boundary conditions for periodic three-dimensional structures: Derivation and implementation in discontinuous Galerkin time-domain method,” IEEE J. Multiscale Multiphys. Comput. Techn. 3, 108–120 (2018).
[Crossref]

S. Yan, A. D. Greenwood, and J.-M. Jin, “Simulation of high-power microwave air breakdown modeled by a coupled Maxwell–Euler system with a non-Maxwellian EEDF,” IEEE. Trans. Antennas Propagat. 66(4), 1882–1893 (2018).
[Crossref]

2017 (3)

Q. Ren, Y. Bian, L. Kang, P. L. Werner, and D. H. Werner, “Leap-frog continuous–discontinuous Galerkin time domain method for nanoarchitectures with the Drude model,” J. Lit. Technol. 35(22), 4888–4896 (2017).
[Crossref]

S. Yan, C.-P. Lin, R. R. Arslanbekov, V. I. Kolobov, and J.-M. Jin, “A discontinuous Galerkin time-domain method with dynamically adaptive Cartesian mesh for computational electromagnetics,” IEEE. Trans. Antennas Propagat. 65(6), 3122–3133 (2017).
[Crossref]

H. Xu, D. Ding, J. Bi, and R. S. Chen, “A 3-D continuous-discontinuous Galerkin finite-element time-domain method for Maxwell’s equations,” IEEE Antennas Wirel. Propag. Lett. 16, 908–911 (2017).
[Crossref]

2016 (1)

H. T. Chen, A. J. Taylor, and N. Yu, “A review of metasurfaces: Physics and applications,” Rep. Prog. Phys. 79(7), 076401 (2016).
[Crossref] [PubMed]

2015 (2)

N. K. Emani, D. Wang, T.-F. Chung, L. J. Prokopeva, A. V. Kildishev, V. M. Shalaev, Y. P. Chen, and A. Boltasseva, “Plasmon resonance in multilayer graphene nanoribbons,” Laser Photonics Rev. 9(6), 650–655 (2015).
[Crossref]

J. Alvarez, L. D. Angulo, A. R. Bretones, C. M. de Jong van Coevorden, and S. G. Garcia, “Efficient antenna modeling by DGTD: Leap-frog discontinuous Galerkin time-domain method,” IEEE Antennas Propag. Mag. 57(3), 95–106 (2015).
[Crossref]

2014 (1)

N. C. Miller, A. D. Baczewski, J. D. Albrecht, and B. Shanker, “A discontinuous Galerkin time domain framework for periodic structures subject to oblique excitation,” IEEE Trans. Antenn. Propag. 62(8), 4386–4391 (2014).
[Crossref]

2013 (2)

J. Chen and Q. H. Liu, “Discontinuous Galerkin time domain methods for multiscale electromagnetic simulations: A review,” Proc. IEEE 101(2), 242–254 (2013).
[Crossref]

S. Dosopoulos, B. Zhao, and J. F. Lee, “Non-conformal and parallel discontinuous Galerkin time domain method for Maxwell’s equations: EM analysis of IC packages,” J. Comput. Phys. 238, 48–70 (2013).
[Crossref]

2011 (3)

L. J. Prokopeva, J. D. Borneman, and A. V. Kildishev, “Optical dispersion models for time-domain modeling of metal-dielectric nanostructures,” IEEE Trans. Magn. 47(5), 1150–1153 (2011).
[Crossref]

M. D. Thoreson, J. Fang, A. V. Kildishev, L. J. Prokopeva, P. Nyga, U. K. Chettiar, V. M. Shalaev, and V. P. Drachev, “Fabrication and realistic modeling of three-dimensional metal-dielectric composites,” J. Nanophotonics 5(1), 051513 (2011).
[Crossref]

B. Zhang, Y. Zhao, Q. Hao, B. Kiraly, I.-C. Khoo, S. Chen, and T. J. Huang, “Polarization-independent dual-band infrared perfect absorber based on a metal-dielectric-metal elliptical nanodisk array,” Opt. Express 19(16), 15221–15228 (2011).
[Crossref] [PubMed]

2009 (2)

K. Lopata and D. Neuhauser, “Multiscale Maxwell-Schrodinger modeling: A split field finite-difference time-domain approach to molecular nanopolaritonics,” J. Chem. Phys. 130(10), 104707 (2009).
[Crossref] [PubMed]

K. Stannigel, M. König, J. Niegemann, and K. Busch, “Discontinuous Galerkin time-domain computations of metallic nanostructures,” Opt. Express 17(17), 14934–14947 (2009).
[Crossref] [PubMed]

2008 (1)

A. Vial and T. Laroche, “Comparison of gold and silver dispersion laws suitable for FDTD simulations,” Appl. Phys. B 93(1), 139–143 (2008).
[Crossref]

2007 (1)

D. Sármány, M. A. Botchev, and J. J. W. van der Vegt, “Dispersion and dissipation error in high-order Runge–Kutta discontinuous Galerkin discretisations of the Maxwell equations,” J. Sci. Comput. 33(1), 47–74 (2007).
[Crossref]

2006 (3)

L. E. R. Petersson and J. M. Jin, “A three-dimensional time-domain finite-element formulation for periodic structures,” IEEE Trans. Antenn. Propag. 54(1), 12–19 (2006).
[Crossref]

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125(16), 164705 (2006).
[Crossref] [PubMed]

A. Aminian and Y. Rahmat-Samii, “Spectral FDTD: A novel technique for the analysis of oblique incident plane wave on periodic structures,” IEEE Trans. Antenn. Propag. 54(6), 1818–1825 (2006).
[Crossref]

2005 (2)

H. Chen and W. Z. Shen, “Perspectives in the characteristics and applications of Tauc-Lorentz dielectric function model,” Eur. Phys. J. B Cond. Matter Complex Syst. 43(4), 503–507 (2005).
[Crossref]

L. Fezoui, S. Lanteri, S. Lohrengel, and S. Piperno, “Convergence and stability of a discontinuous Galerkin time-domain method for the 3-D heterogeneous Maxwell equations on unstructured meshes,” ESAIM: M2AN 39(6), 1149–1176 (2005).
[Crossref]

2004 (1)

S. Jeon, J.-U. Park, R. Cirelli, S. Yang, C. E. Heitzman, P. V. Braun, P. J. Kenis, and J. A. Rogers, “Fabricating complex three-dimensional nanostructures with high-resolution conformable phase masks,” Proc. Natl. Acad. Sci. U.S.A. 101(34), 12428–12433 (2004).
[Crossref] [PubMed]

2003 (3)

A. Bruner, D. Eger, M. B. Oron, P. Blau, M. Katz, and S. Ruschin, “Temperature-dependent Sellmeier equation for the refractive index of stoichiometric lithium tantalate,” Opt. Lett. 28(3), 194–196 (2003).
[Crossref] [PubMed]

C. Genet, M. P. van Exter, and J. P. Woerdman, “Fano-type interpretation of red shifts and red tails in hole array transmission spectra,” Opt. Express Commun. 225(4-6), 331–336 (2003).
[Crossref]

M. M. Ilic and B. M. Notaros, “Higher order hierarchical curved hexahedral vector finite elements for electromagnetic modeling,” IEEE Trans. Antenn. Propag. 51(3), 1026–1033 (2003).

1998 (1)

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394(6690), 251–253 (1998).
[Crossref]

1996 (1)

S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antenn. Propag. 44(12), 1630–1639 (1996).
[Crossref]

1993 (1)

M. E. Veysoglu, R. T. Shin, and J. A. Kong, “A finite-difference time domain analysis of wave scattering from periodic structures: Oblique incidence case,” J. Electromagn. Waves Appl. 7(12), 1595–1607 (1993).
[Crossref]

Albrecht, J. D.

N. C. Miller, A. D. Baczewski, J. D. Albrecht, and B. Shanker, “A discontinuous Galerkin time domain framework for periodic structures subject to oblique excitation,” IEEE Trans. Antenn. Propag. 62(8), 4386–4391 (2014).
[Crossref]

Alvarez, J.

J. Alvarez, L. D. Angulo, A. R. Bretones, C. M. de Jong van Coevorden, and S. G. Garcia, “Efficient antenna modeling by DGTD: Leap-frog discontinuous Galerkin time-domain method,” IEEE Antennas Propag. Mag. 57(3), 95–106 (2015).
[Crossref]

Aminian, A.

A. Aminian and Y. Rahmat-Samii, “Spectral FDTD: A novel technique for the analysis of oblique incident plane wave on periodic structures,” IEEE Trans. Antenn. Propag. 54(6), 1818–1825 (2006).
[Crossref]

Angulo, L. D.

J. Alvarez, L. D. Angulo, A. R. Bretones, C. M. de Jong van Coevorden, and S. G. Garcia, “Efficient antenna modeling by DGTD: Leap-frog discontinuous Galerkin time-domain method,” IEEE Antennas Propag. Mag. 57(3), 95–106 (2015).
[Crossref]

Arslanbekov, R. R.

S. Yan, C.-P. Lin, R. R. Arslanbekov, V. I. Kolobov, and J.-M. Jin, “A discontinuous Galerkin time-domain method with dynamically adaptive Cartesian mesh for computational electromagnetics,” IEEE. Trans. Antennas Propagat. 65(6), 3122–3133 (2017).
[Crossref]

Baczewski, A. D.

N. C. Miller, A. D. Baczewski, J. D. Albrecht, and B. Shanker, “A discontinuous Galerkin time domain framework for periodic structures subject to oblique excitation,” IEEE Trans. Antenn. Propag. 62(8), 4386–4391 (2014).
[Crossref]

Bagci, H.

K. Sirenko, Y. Sirenko, and H. Bağcı, “Exact absorbing boundary conditions for periodic three-dimensional structures: Derivation and implementation in discontinuous Galerkin time-domain method,” IEEE J. Multiscale Multiphys. Comput. Techn. 3, 108–120 (2018).
[Crossref]

Bao, H.

Bi, J.

H. Xu, D. Ding, J. Bi, and R. S. Chen, “A 3-D continuous-discontinuous Galerkin finite-element time-domain method for Maxwell’s equations,” IEEE Antennas Wirel. Propag. Lett. 16, 908–911 (2017).
[Crossref]

Bian, Y.

Q. Ren, Y. Bian, L. Kang, P. L. Werner, and D. H. Werner, “Leap-frog continuous–discontinuous Galerkin time domain method for nanoarchitectures with the Drude model,” J. Lit. Technol. 35(22), 4888–4896 (2017).
[Crossref]

Biswas, R.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394(6690), 251–253 (1998).
[Crossref]

Blau, P.

Boltasseva, A.

N. K. Emani, D. Wang, T.-F. Chung, L. J. Prokopeva, A. V. Kildishev, V. M. Shalaev, Y. P. Chen, and A. Boltasseva, “Plasmon resonance in multilayer graphene nanoribbons,” Laser Photonics Rev. 9(6), 650–655 (2015).
[Crossref]

Borneman, J. D.

L. J. Prokopeva, J. D. Borneman, and A. V. Kildishev, “Optical dispersion models for time-domain modeling of metal-dielectric nanostructures,” IEEE Trans. Magn. 47(5), 1150–1153 (2011).
[Crossref]

Botchev, M. A.

D. Sármány, M. A. Botchev, and J. J. W. van der Vegt, “Dispersion and dissipation error in high-order Runge–Kutta discontinuous Galerkin discretisations of the Maxwell equations,” J. Sci. Comput. 33(1), 47–74 (2007).
[Crossref]

Braun, P. V.

S. Jeon, J.-U. Park, R. Cirelli, S. Yang, C. E. Heitzman, P. V. Braun, P. J. Kenis, and J. A. Rogers, “Fabricating complex three-dimensional nanostructures with high-resolution conformable phase masks,” Proc. Natl. Acad. Sci. U.S.A. 101(34), 12428–12433 (2004).
[Crossref] [PubMed]

Bretones, A. R.

J. Alvarez, L. D. Angulo, A. R. Bretones, C. M. de Jong van Coevorden, and S. G. Garcia, “Efficient antenna modeling by DGTD: Leap-frog discontinuous Galerkin time-domain method,” IEEE Antennas Propag. Mag. 57(3), 95–106 (2015).
[Crossref]

Bruner, A.

Bur, J.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394(6690), 251–253 (1998).
[Crossref]

Busch, K.

Campbell, S. D.

Chen, H.

H. Chen and W. Z. Shen, “Perspectives in the characteristics and applications of Tauc-Lorentz dielectric function model,” Eur. Phys. J. B Cond. Matter Complex Syst. 43(4), 503–507 (2005).
[Crossref]

Chen, H. T.

H. T. Chen, A. J. Taylor, and N. Yu, “A review of metasurfaces: Physics and applications,” Rep. Prog. Phys. 79(7), 076401 (2016).
[Crossref] [PubMed]

Chen, J.

J. Chen and Q. H. Liu, “Discontinuous Galerkin time domain methods for multiscale electromagnetic simulations: A review,” Proc. IEEE 101(2), 242–254 (2013).
[Crossref]

Chen, R. S.

H. Xu, D. Ding, J. Bi, and R. S. Chen, “A 3-D continuous-discontinuous Galerkin finite-element time-domain method for Maxwell’s equations,” IEEE Antennas Wirel. Propag. Lett. 16, 908–911 (2017).
[Crossref]

Chen, S.

Chen, Y. P.

N. K. Emani, D. Wang, T.-F. Chung, L. J. Prokopeva, A. V. Kildishev, V. M. Shalaev, Y. P. Chen, and A. Boltasseva, “Plasmon resonance in multilayer graphene nanoribbons,” Laser Photonics Rev. 9(6), 650–655 (2015).
[Crossref]

Chettiar, U. K.

M. D. Thoreson, J. Fang, A. V. Kildishev, L. J. Prokopeva, P. Nyga, U. K. Chettiar, V. M. Shalaev, and V. P. Drachev, “Fabrication and realistic modeling of three-dimensional metal-dielectric composites,” J. Nanophotonics 5(1), 051513 (2011).
[Crossref]

Chung, T.-F.

N. K. Emani, D. Wang, T.-F. Chung, L. J. Prokopeva, A. V. Kildishev, V. M. Shalaev, Y. P. Chen, and A. Boltasseva, “Plasmon resonance in multilayer graphene nanoribbons,” Laser Photonics Rev. 9(6), 650–655 (2015).
[Crossref]

Cirelli, R.

S. Jeon, J.-U. Park, R. Cirelli, S. Yang, C. E. Heitzman, P. V. Braun, P. J. Kenis, and J. A. Rogers, “Fabricating complex three-dimensional nanostructures with high-resolution conformable phase masks,” Proc. Natl. Acad. Sci. U.S.A. 101(34), 12428–12433 (2004).
[Crossref] [PubMed]

de Jong van Coevorden, C. M.

J. Alvarez, L. D. Angulo, A. R. Bretones, C. M. de Jong van Coevorden, and S. G. Garcia, “Efficient antenna modeling by DGTD: Leap-frog discontinuous Galerkin time-domain method,” IEEE Antennas Propag. Mag. 57(3), 95–106 (2015).
[Crossref]

Ding, D.

H. Xu, D. Ding, J. Bi, and R. S. Chen, “A 3-D continuous-discontinuous Galerkin finite-element time-domain method for Maxwell’s equations,” IEEE Antennas Wirel. Propag. Lett. 16, 908–911 (2017).
[Crossref]

Dosopoulos, S.

S. Dosopoulos, B. Zhao, and J. F. Lee, “Non-conformal and parallel discontinuous Galerkin time domain method for Maxwell’s equations: EM analysis of IC packages,” J. Comput. Phys. 238, 48–70 (2013).
[Crossref]

Drachev, V. P.

M. D. Thoreson, J. Fang, A. V. Kildishev, L. J. Prokopeva, P. Nyga, U. K. Chettiar, V. M. Shalaev, and V. P. Drachev, “Fabrication and realistic modeling of three-dimensional metal-dielectric composites,” J. Nanophotonics 5(1), 051513 (2011).
[Crossref]

Eger, D.

Emani, N. K.

N. K. Emani, D. Wang, T.-F. Chung, L. J. Prokopeva, A. V. Kildishev, V. M. Shalaev, Y. P. Chen, and A. Boltasseva, “Plasmon resonance in multilayer graphene nanoribbons,” Laser Photonics Rev. 9(6), 650–655 (2015).
[Crossref]

Etchegoin, P. G.

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125(16), 164705 (2006).
[Crossref] [PubMed]

Fang, J.

M. D. Thoreson, J. Fang, A. V. Kildishev, L. J. Prokopeva, P. Nyga, U. K. Chettiar, V. M. Shalaev, and V. P. Drachev, “Fabrication and realistic modeling of three-dimensional metal-dielectric composites,” J. Nanophotonics 5(1), 051513 (2011).
[Crossref]

Fezoui, L.

L. Fezoui, S. Lanteri, S. Lohrengel, and S. Piperno, “Convergence and stability of a discontinuous Galerkin time-domain method for the 3-D heterogeneous Maxwell equations on unstructured meshes,” ESAIM: M2AN 39(6), 1149–1176 (2005).
[Crossref]

Fleming, J. G.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394(6690), 251–253 (1998).
[Crossref]

Garcia, S. G.

J. Alvarez, L. D. Angulo, A. R. Bretones, C. M. de Jong van Coevorden, and S. G. Garcia, “Efficient antenna modeling by DGTD: Leap-frog discontinuous Galerkin time-domain method,” IEEE Antennas Propag. Mag. 57(3), 95–106 (2015).
[Crossref]

Gedney, S. D.

S. D. Gedney, “An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices,” IEEE Trans. Antenn. Propag. 44(12), 1630–1639 (1996).
[Crossref]

Genet, C.

C. Genet, M. P. van Exter, and J. P. Woerdman, “Fano-type interpretation of red shifts and red tails in hole array transmission spectra,” Opt. Express Commun. 225(4-6), 331–336 (2003).
[Crossref]

Greenwood, A. D.

S. Yan, A. D. Greenwood, and J.-M. Jin, “Simulation of high-power microwave air breakdown modeled by a coupled Maxwell–Euler system with a non-Maxwellian EEDF,” IEEE. Trans. Antennas Propagat. 66(4), 1882–1893 (2018).
[Crossref]

Hao, Q.

Heitzman, C. E.

S. Jeon, J.-U. Park, R. Cirelli, S. Yang, C. E. Heitzman, P. V. Braun, P. J. Kenis, and J. A. Rogers, “Fabricating complex three-dimensional nanostructures with high-resolution conformable phase masks,” Proc. Natl. Acad. Sci. U.S.A. 101(34), 12428–12433 (2004).
[Crossref] [PubMed]

Hetherington, D. L.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394(6690), 251–253 (1998).
[Crossref]

Ho, K. M.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394(6690), 251–253 (1998).
[Crossref]

Huang, T. J.

Ilic, M. M.

M. M. Ilic and B. M. Notaros, “Higher order hierarchical curved hexahedral vector finite elements for electromagnetic modeling,” IEEE Trans. Antenn. Propag. 51(3), 1026–1033 (2003).

Jeon, S.

S. Jeon, J.-U. Park, R. Cirelli, S. Yang, C. E. Heitzman, P. V. Braun, P. J. Kenis, and J. A. Rogers, “Fabricating complex three-dimensional nanostructures with high-resolution conformable phase masks,” Proc. Natl. Acad. Sci. U.S.A. 101(34), 12428–12433 (2004).
[Crossref] [PubMed]

Jin, J. M.

L. E. R. Petersson and J. M. Jin, “A three-dimensional time-domain finite-element formulation for periodic structures,” IEEE Trans. Antenn. Propag. 54(1), 12–19 (2006).
[Crossref]

Jin, J.-M.

S. Yan, A. D. Greenwood, and J.-M. Jin, “Simulation of high-power microwave air breakdown modeled by a coupled Maxwell–Euler system with a non-Maxwellian EEDF,” IEEE. Trans. Antennas Propagat. 66(4), 1882–1893 (2018).
[Crossref]

S. Yan, C.-P. Lin, R. R. Arslanbekov, V. I. Kolobov, and J.-M. Jin, “A discontinuous Galerkin time-domain method with dynamically adaptive Cartesian mesh for computational electromagnetics,” IEEE. Trans. Antennas Propagat. 65(6), 3122–3133 (2017).
[Crossref]

Kang, L.

Q. Ren, Y. Bian, L. Kang, P. L. Werner, and D. H. Werner, “Leap-frog continuous–discontinuous Galerkin time domain method for nanoarchitectures with the Drude model,” J. Lit. Technol. 35(22), 4888–4896 (2017).
[Crossref]

H. Bao, L. Kang, S. D. Campbell, and D. H. Werner, “PML implementation in a non-conforming mixed-element DGTD method for periodic structure analysis,” (unpublished).

Katz, M.

Kenis, P. J.

S. Jeon, J.-U. Park, R. Cirelli, S. Yang, C. E. Heitzman, P. V. Braun, P. J. Kenis, and J. A. Rogers, “Fabricating complex three-dimensional nanostructures with high-resolution conformable phase masks,” Proc. Natl. Acad. Sci. U.S.A. 101(34), 12428–12433 (2004).
[Crossref] [PubMed]

Khoo, I.-C.

Kildishev, A. V.

Q. Ren, H. Bao, S. D. Campbell, L. J. Prokopeva, A. V. Kildishev, and D. H. Werner, “Continuous-discontinuous Galerkin time domain (CDGTD) method with generalized dispersive material (GDM) model for computational photonics,” Opt. Express 26(22), 29005–29016 (2018).
[Crossref] [PubMed]

N. K. Emani, D. Wang, T.-F. Chung, L. J. Prokopeva, A. V. Kildishev, V. M. Shalaev, Y. P. Chen, and A. Boltasseva, “Plasmon resonance in multilayer graphene nanoribbons,” Laser Photonics Rev. 9(6), 650–655 (2015).
[Crossref]

L. J. Prokopeva, J. D. Borneman, and A. V. Kildishev, “Optical dispersion models for time-domain modeling of metal-dielectric nanostructures,” IEEE Trans. Magn. 47(5), 1150–1153 (2011).
[Crossref]

M. D. Thoreson, J. Fang, A. V. Kildishev, L. J. Prokopeva, P. Nyga, U. K. Chettiar, V. M. Shalaev, and V. P. Drachev, “Fabrication and realistic modeling of three-dimensional metal-dielectric composites,” J. Nanophotonics 5(1), 051513 (2011).
[Crossref]

Kiraly, B.

Kolobov, V. I.

S. Yan, C.-P. Lin, R. R. Arslanbekov, V. I. Kolobov, and J.-M. Jin, “A discontinuous Galerkin time-domain method with dynamically adaptive Cartesian mesh for computational electromagnetics,” IEEE. Trans. Antennas Propagat. 65(6), 3122–3133 (2017).
[Crossref]

Kong, J. A.

M. E. Veysoglu, R. T. Shin, and J. A. Kong, “A finite-difference time domain analysis of wave scattering from periodic structures: Oblique incidence case,” J. Electromagn. Waves Appl. 7(12), 1595–1607 (1993).
[Crossref]

König, M.

Kurtz, S. R.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394(6690), 251–253 (1998).
[Crossref]

Lanteri, S.

L. Fezoui, S. Lanteri, S. Lohrengel, and S. Piperno, “Convergence and stability of a discontinuous Galerkin time-domain method for the 3-D heterogeneous Maxwell equations on unstructured meshes,” ESAIM: M2AN 39(6), 1149–1176 (2005).
[Crossref]

Laroche, T.

A. Vial and T. Laroche, “Comparison of gold and silver dispersion laws suitable for FDTD simulations,” Appl. Phys. B 93(1), 139–143 (2008).
[Crossref]

Le Ru, E. C.

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125(16), 164705 (2006).
[Crossref] [PubMed]

Lee, J. F.

S. Dosopoulos, B. Zhao, and J. F. Lee, “Non-conformal and parallel discontinuous Galerkin time domain method for Maxwell’s equations: EM analysis of IC packages,” J. Comput. Phys. 238, 48–70 (2013).
[Crossref]

Lin, C.-P.

S. Yan, C.-P. Lin, R. R. Arslanbekov, V. I. Kolobov, and J.-M. Jin, “A discontinuous Galerkin time-domain method with dynamically adaptive Cartesian mesh for computational electromagnetics,” IEEE. Trans. Antennas Propagat. 65(6), 3122–3133 (2017).
[Crossref]

Lin, S. Y.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394(6690), 251–253 (1998).
[Crossref]

Liu, Q. H.

J. Chen and Q. H. Liu, “Discontinuous Galerkin time domain methods for multiscale electromagnetic simulations: A review,” Proc. IEEE 101(2), 242–254 (2013).
[Crossref]

Lohrengel, S.

L. Fezoui, S. Lanteri, S. Lohrengel, and S. Piperno, “Convergence and stability of a discontinuous Galerkin time-domain method for the 3-D heterogeneous Maxwell equations on unstructured meshes,” ESAIM: M2AN 39(6), 1149–1176 (2005).
[Crossref]

Lopata, K.

K. Lopata and D. Neuhauser, “Multiscale Maxwell-Schrodinger modeling: A split field finite-difference time-domain approach to molecular nanopolaritonics,” J. Chem. Phys. 130(10), 104707 (2009).
[Crossref] [PubMed]

Meyer, M.

P. G. Etchegoin, E. C. Le Ru, and M. Meyer, “An analytic model for the optical properties of gold,” J. Chem. Phys. 125(16), 164705 (2006).
[Crossref] [PubMed]

Miller, N. C.

N. C. Miller, A. D. Baczewski, J. D. Albrecht, and B. Shanker, “A discontinuous Galerkin time domain framework for periodic structures subject to oblique excitation,” IEEE Trans. Antenn. Propag. 62(8), 4386–4391 (2014).
[Crossref]

Neuhauser, D.

K. Lopata and D. Neuhauser, “Multiscale Maxwell-Schrodinger modeling: A split field finite-difference time-domain approach to molecular nanopolaritonics,” J. Chem. Phys. 130(10), 104707 (2009).
[Crossref] [PubMed]

Niegemann, J.

Notaros, B. M.

M. M. Ilic and B. M. Notaros, “Higher order hierarchical curved hexahedral vector finite elements for electromagnetic modeling,” IEEE Trans. Antenn. Propag. 51(3), 1026–1033 (2003).

Nyga, P.

M. D. Thoreson, J. Fang, A. V. Kildishev, L. J. Prokopeva, P. Nyga, U. K. Chettiar, V. M. Shalaev, and V. P. Drachev, “Fabrication and realistic modeling of three-dimensional metal-dielectric composites,” J. Nanophotonics 5(1), 051513 (2011).
[Crossref]

Oron, M. B.

Park, J.-U.

S. Jeon, J.-U. Park, R. Cirelli, S. Yang, C. E. Heitzman, P. V. Braun, P. J. Kenis, and J. A. Rogers, “Fabricating complex three-dimensional nanostructures with high-resolution conformable phase masks,” Proc. Natl. Acad. Sci. U.S.A. 101(34), 12428–12433 (2004).
[Crossref] [PubMed]

Petersson, L. E. R.

L. E. R. Petersson and J. M. Jin, “A three-dimensional time-domain finite-element formulation for periodic structures,” IEEE Trans. Antenn. Propag. 54(1), 12–19 (2006).
[Crossref]

Piperno, S.

L. Fezoui, S. Lanteri, S. Lohrengel, and S. Piperno, “Convergence and stability of a discontinuous Galerkin time-domain method for the 3-D heterogeneous Maxwell equations on unstructured meshes,” ESAIM: M2AN 39(6), 1149–1176 (2005).
[Crossref]

Prokopeva, L. J.

Q. Ren, H. Bao, S. D. Campbell, L. J. Prokopeva, A. V. Kildishev, and D. H. Werner, “Continuous-discontinuous Galerkin time domain (CDGTD) method with generalized dispersive material (GDM) model for computational photonics,” Opt. Express 26(22), 29005–29016 (2018).
[Crossref] [PubMed]

N. K. Emani, D. Wang, T.-F. Chung, L. J. Prokopeva, A. V. Kildishev, V. M. Shalaev, Y. P. Chen, and A. Boltasseva, “Plasmon resonance in multilayer graphene nanoribbons,” Laser Photonics Rev. 9(6), 650–655 (2015).
[Crossref]

M. D. Thoreson, J. Fang, A. V. Kildishev, L. J. Prokopeva, P. Nyga, U. K. Chettiar, V. M. Shalaev, and V. P. Drachev, “Fabrication and realistic modeling of three-dimensional metal-dielectric composites,” J. Nanophotonics 5(1), 051513 (2011).
[Crossref]

L. J. Prokopeva, J. D. Borneman, and A. V. Kildishev, “Optical dispersion models for time-domain modeling of metal-dielectric nanostructures,” IEEE Trans. Magn. 47(5), 1150–1153 (2011).
[Crossref]

Rahmat-Samii, Y.

A. Aminian and Y. Rahmat-Samii, “Spectral FDTD: A novel technique for the analysis of oblique incident plane wave on periodic structures,” IEEE Trans. Antenn. Propag. 54(6), 1818–1825 (2006).
[Crossref]

Ren, Q.

Q. Ren, H. Bao, S. D. Campbell, L. J. Prokopeva, A. V. Kildishev, and D. H. Werner, “Continuous-discontinuous Galerkin time domain (CDGTD) method with generalized dispersive material (GDM) model for computational photonics,” Opt. Express 26(22), 29005–29016 (2018).
[Crossref] [PubMed]

Q. Ren, Y. Bian, L. Kang, P. L. Werner, and D. H. Werner, “Leap-frog continuous–discontinuous Galerkin time domain method for nanoarchitectures with the Drude model,” J. Lit. Technol. 35(22), 4888–4896 (2017).
[Crossref]

Rogers, J. A.

S. Jeon, J.-U. Park, R. Cirelli, S. Yang, C. E. Heitzman, P. V. Braun, P. J. Kenis, and J. A. Rogers, “Fabricating complex three-dimensional nanostructures with high-resolution conformable phase masks,” Proc. Natl. Acad. Sci. U.S.A. 101(34), 12428–12433 (2004).
[Crossref] [PubMed]

Ruschin, S.

Sármány, D.

D. Sármány, M. A. Botchev, and J. J. W. van der Vegt, “Dispersion and dissipation error in high-order Runge–Kutta discontinuous Galerkin discretisations of the Maxwell equations,” J. Sci. Comput. 33(1), 47–74 (2007).
[Crossref]

Shalaev, V. M.

N. K. Emani, D. Wang, T.-F. Chung, L. J. Prokopeva, A. V. Kildishev, V. M. Shalaev, Y. P. Chen, and A. Boltasseva, “Plasmon resonance in multilayer graphene nanoribbons,” Laser Photonics Rev. 9(6), 650–655 (2015).
[Crossref]

M. D. Thoreson, J. Fang, A. V. Kildishev, L. J. Prokopeva, P. Nyga, U. K. Chettiar, V. M. Shalaev, and V. P. Drachev, “Fabrication and realistic modeling of three-dimensional metal-dielectric composites,” J. Nanophotonics 5(1), 051513 (2011).
[Crossref]

Shanker, B.

N. C. Miller, A. D. Baczewski, J. D. Albrecht, and B. Shanker, “A discontinuous Galerkin time domain framework for periodic structures subject to oblique excitation,” IEEE Trans. Antenn. Propag. 62(8), 4386–4391 (2014).
[Crossref]

Shen, W. Z.

H. Chen and W. Z. Shen, “Perspectives in the characteristics and applications of Tauc-Lorentz dielectric function model,” Eur. Phys. J. B Cond. Matter Complex Syst. 43(4), 503–507 (2005).
[Crossref]

Shin, R. T.

M. E. Veysoglu, R. T. Shin, and J. A. Kong, “A finite-difference time domain analysis of wave scattering from periodic structures: Oblique incidence case,” J. Electromagn. Waves Appl. 7(12), 1595–1607 (1993).
[Crossref]

Sigalas, M. M.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394(6690), 251–253 (1998).
[Crossref]

Sirenko, K.

K. Sirenko, Y. Sirenko, and H. Bağcı, “Exact absorbing boundary conditions for periodic three-dimensional structures: Derivation and implementation in discontinuous Galerkin time-domain method,” IEEE J. Multiscale Multiphys. Comput. Techn. 3, 108–120 (2018).
[Crossref]

Sirenko, Y.

K. Sirenko, Y. Sirenko, and H. Bağcı, “Exact absorbing boundary conditions for periodic three-dimensional structures: Derivation and implementation in discontinuous Galerkin time-domain method,” IEEE J. Multiscale Multiphys. Comput. Techn. 3, 108–120 (2018).
[Crossref]

Smith, B. K.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394(6690), 251–253 (1998).
[Crossref]

Stannigel, K.

Taylor, A. J.

H. T. Chen, A. J. Taylor, and N. Yu, “A review of metasurfaces: Physics and applications,” Rep. Prog. Phys. 79(7), 076401 (2016).
[Crossref] [PubMed]

Thoreson, M. D.

M. D. Thoreson, J. Fang, A. V. Kildishev, L. J. Prokopeva, P. Nyga, U. K. Chettiar, V. M. Shalaev, and V. P. Drachev, “Fabrication and realistic modeling of three-dimensional metal-dielectric composites,” J. Nanophotonics 5(1), 051513 (2011).
[Crossref]

van der Vegt, J. J. W.

D. Sármány, M. A. Botchev, and J. J. W. van der Vegt, “Dispersion and dissipation error in high-order Runge–Kutta discontinuous Galerkin discretisations of the Maxwell equations,” J. Sci. Comput. 33(1), 47–74 (2007).
[Crossref]

van Exter, M. P.

C. Genet, M. P. van Exter, and J. P. Woerdman, “Fano-type interpretation of red shifts and red tails in hole array transmission spectra,” Opt. Express Commun. 225(4-6), 331–336 (2003).
[Crossref]

Veysoglu, M. E.

M. E. Veysoglu, R. T. Shin, and J. A. Kong, “A finite-difference time domain analysis of wave scattering from periodic structures: Oblique incidence case,” J. Electromagn. Waves Appl. 7(12), 1595–1607 (1993).
[Crossref]

Vial, A.

A. Vial and T. Laroche, “Comparison of gold and silver dispersion laws suitable for FDTD simulations,” Appl. Phys. B 93(1), 139–143 (2008).
[Crossref]

Wang, D.

N. K. Emani, D. Wang, T.-F. Chung, L. J. Prokopeva, A. V. Kildishev, V. M. Shalaev, Y. P. Chen, and A. Boltasseva, “Plasmon resonance in multilayer graphene nanoribbons,” Laser Photonics Rev. 9(6), 650–655 (2015).
[Crossref]

Werner, D. H.

Q. Ren, H. Bao, S. D. Campbell, L. J. Prokopeva, A. V. Kildishev, and D. H. Werner, “Continuous-discontinuous Galerkin time domain (CDGTD) method with generalized dispersive material (GDM) model for computational photonics,” Opt. Express 26(22), 29005–29016 (2018).
[Crossref] [PubMed]

Q. Ren, Y. Bian, L. Kang, P. L. Werner, and D. H. Werner, “Leap-frog continuous–discontinuous Galerkin time domain method for nanoarchitectures with the Drude model,” J. Lit. Technol. 35(22), 4888–4896 (2017).
[Crossref]

H. Bao, L. Kang, S. D. Campbell, and D. H. Werner, “PML implementation in a non-conforming mixed-element DGTD method for periodic structure analysis,” (unpublished).

Werner, P. L.

Q. Ren, Y. Bian, L. Kang, P. L. Werner, and D. H. Werner, “Leap-frog continuous–discontinuous Galerkin time domain method for nanoarchitectures with the Drude model,” J. Lit. Technol. 35(22), 4888–4896 (2017).
[Crossref]

Woerdman, J. P.

C. Genet, M. P. van Exter, and J. P. Woerdman, “Fano-type interpretation of red shifts and red tails in hole array transmission spectra,” Opt. Express Commun. 225(4-6), 331–336 (2003).
[Crossref]

Xu, H.

H. Xu, D. Ding, J. Bi, and R. S. Chen, “A 3-D continuous-discontinuous Galerkin finite-element time-domain method for Maxwell’s equations,” IEEE Antennas Wirel. Propag. Lett. 16, 908–911 (2017).
[Crossref]

Yan, S.

S. Yan, A. D. Greenwood, and J.-M. Jin, “Simulation of high-power microwave air breakdown modeled by a coupled Maxwell–Euler system with a non-Maxwellian EEDF,” IEEE. Trans. Antennas Propagat. 66(4), 1882–1893 (2018).
[Crossref]

S. Yan, C.-P. Lin, R. R. Arslanbekov, V. I. Kolobov, and J.-M. Jin, “A discontinuous Galerkin time-domain method with dynamically adaptive Cartesian mesh for computational electromagnetics,” IEEE. Trans. Antennas Propagat. 65(6), 3122–3133 (2017).
[Crossref]

Yang, S.

S. Jeon, J.-U. Park, R. Cirelli, S. Yang, C. E. Heitzman, P. V. Braun, P. J. Kenis, and J. A. Rogers, “Fabricating complex three-dimensional nanostructures with high-resolution conformable phase masks,” Proc. Natl. Acad. Sci. U.S.A. 101(34), 12428–12433 (2004).
[Crossref] [PubMed]

Yu, N.

H. T. Chen, A. J. Taylor, and N. Yu, “A review of metasurfaces: Physics and applications,” Rep. Prog. Phys. 79(7), 076401 (2016).
[Crossref] [PubMed]

Zhang, B.

Zhao, B.

S. Dosopoulos, B. Zhao, and J. F. Lee, “Non-conformal and parallel discontinuous Galerkin time domain method for Maxwell’s equations: EM analysis of IC packages,” J. Comput. Phys. 238, 48–70 (2013).
[Crossref]

Zhao, Y.

Zubrzycki, W.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz, and J. Bur, “A three-dimensional photonic crystal operating at infrared wavelengths,” Nature 394(6690), 251–253 (1998).
[Crossref]

Appl. Phys. B (1)

A. Vial and T. Laroche, “Comparison of gold and silver dispersion laws suitable for FDTD simulations,” Appl. Phys. B 93(1), 139–143 (2008).
[Crossref]

ESAIM: M2AN (1)

L. Fezoui, S. Lanteri, S. Lohrengel, and S. Piperno, “Convergence and stability of a discontinuous Galerkin time-domain method for the 3-D heterogeneous Maxwell equations on unstructured meshes,” ESAIM: M2AN 39(6), 1149–1176 (2005).
[Crossref]

Eur. Phys. J. B Cond. Matter Complex Syst. (1)

H. Chen and W. Z. Shen, “Perspectives in the characteristics and applications of Tauc-Lorentz dielectric function model,” Eur. Phys. J. B Cond. Matter Complex Syst. 43(4), 503–507 (2005).
[Crossref]

IEEE Antennas Propag. Mag. (1)

J. Alvarez, L. D. Angulo, A. R. Bretones, C. M. de Jong van Coevorden, and S. G. Garcia, “Efficient antenna modeling by DGTD: Leap-frog discontinuous Galerkin time-domain method,” IEEE Antennas Propag. Mag. 57(3), 95–106 (2015).
[Crossref]

IEEE Antennas Wirel. Propag. Lett. (1)

H. Xu, D. Ding, J. Bi, and R. S. Chen, “A 3-D continuous-discontinuous Galerkin finite-element time-domain method for Maxwell’s equations,” IEEE Antennas Wirel. Propag. Lett. 16, 908–911 (2017).
[Crossref]

IEEE J. Multiscale Multiphys. Comput. Techn. (1)

K. Sirenko, Y. Sirenko, and H. Bağcı, “Exact absorbing boundary conditions for periodic three-dimensional structures: Derivation and implementation in discontinuous Galerkin time-domain method,” IEEE J. Multiscale Multiphys. Comput. Techn. 3, 108–120 (2018).
[Crossref]

IEEE Trans. Antenn. Propag. (5)

A. Aminian and Y. Rahmat-Samii, “Spectral FDTD: A novel technique for the analysis of oblique incident plane wave on periodic structures,” IEEE Trans. Antenn. Propag. 54(6), 1818–1825 (2006).
[Crossref]

L. E. R. Petersson and J. M. Jin, “A three-dimensional time-domain finite-element formulation for periodic structures,” IEEE Trans. Antenn. Propag. 54(1), 12–19 (2006).
[Crossref]

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Figures (4)

Fig. 1
Fig. 1 Illustration of a doubly periodic structure under oblique illumination.
Fig. 2
Fig. 2 Gold film with D2CP model under plane wave illumination. (a) The simulation setup. (b) The transient reflected and transmitted E field at probes 1 and 2. (c,d,e) The reflection and transmission coefficients under plane wave illumination with different pitch angles (c) θi = 0°, (d) θi = 30°, and (e) θi = 60°.
Fig. 3
Fig. 3 Surface plasmon excitation enabled extraordinary transmission of a metallic hole array structure. (a) Unit cell geometry. (b) The reflection and transmission coefficients at normal incidence. (c) Electric field distribution (Ez) in a cut plane at y = –300 nm.
Fig. 4
Fig. 4 A dual-band infrared absorber. (a) Geometry of a unit cell. (b,c,d) Reflectance spectra under plane wave illuminations with different pitch angles (b) θi = 0°, (c) θi = 15°, and (d) θi = 30°. (e) Absorptivity as a function of both wavelength and pitch angle of incidence.

Tables (2)

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Table 1 GDM Parameters for Different Dispersion Terms

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Table 2 Comparison on Computing Resources

Equations (33)

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E (x,y,z,ω)={ E (x+ x p ,y,z,ω) e j k 0 x p sin θ i cos φ i E (x,y+ y p ,z,ω) e j k 0 y p sin θ i cos φ i
H (x,y,z,ω)={ H (x+ x p ,y,z,ω) e j k 0 x p sin θ i cos φ i H (x,y+ y p ,z,ω) e j k 0 y p sin θ i cos φ i
P e (x,y,z,ω)= E (x,y,z,ω) e j( k 0 xsin θ i cos φ i + k 0 ysin θ i sin φ i )
P h (x,y,z,ω)= H (x,y,z,ω) e j( k 0 xsin θ i cos φ i + k 0 ysin θ i sin φ i )
P e (x,y,z,ω)={ P e (x+ x p ,y,z,ω) P e (x,y+ y p ,z,ω)
P h (x,y,z,ω)={ P h (x+ x p ,y,z,ω) P h (x,y+ y p ,z,ω) .
jω ε 0 ε r P e =× P h jω 1 c k t × P h
jω μ 0 μ r P h =× P e +jω 1 c k t × P e
ε r (ω)= ε + m χ m (ω) = ε + m a 0,m +jω a 1,m b 0,m +jω b 1,m ω 2 .
Q m = ε 0 a 0,m +jω a 1,m b 0,m +jω b 1,m ω 2 P e ,
jω ε 0 ε P e + m jω Q m =× P h jω 1 c k t × P h
ε 0 ε P e t + m Q m t =× P h 1 c k t × P h t
μ 0 μ r P h t =× P e + 1 c k t × P e t
2 Q m t 2 + b 1,m Q m t + b 0,m Q m = ε 0 a 0,m P e + ε 0 a 1,m P e t
P e (x,y,z,t)={ P e (x+ x p ,y,z,t) P e (x,y+ y p ,z,t)
P h (x,y,z,t)={ P h (x+ x p ,y,z,t) P h (x,y+ y p ,z,t)
P e = p e Φ , P h = p h Φ , Q m = q m Φ
Φ ε 0 ε P e t dV + m Φ Q m t dV = × Φ P h dV Φ ( 1 c k t × P h t )dV + s Φ ( n × P h )dS
Φ μ 0 μ r P h t dV = × Φ P e dV + Φ ( 1 c k t × P e t )dV s Φ ( n × P e )dS
Φ 2 Q m t 2 dV + b 1,m Φ Q m t dV + b 0,m Φ Q m dV = ε 0 a 0,m Φ P e dV + ε 0 a 1,m Φ P e t dV .
{ n × P e = n × (Y P e n × P h )+( Y + P e + n × P h + ) Y+ Y + n × P h = n × (Z P h n × P e )+( Z + P h + n × P e + ) Z+ Z +
T ee ε 0 ε p e t + M eh p h t + T ee 1 q m t = S ee p e + S eh p h + F ee p e + + F eh p h +
T hh μ 0 μ r p h t M he p e t = S hh p h + S he p e + F hh p h + + F he p e +
2 q m t 2 + b 1,m q m t ε 0 a 1,m p e t = ε 0 a 0,m p e b 0,m q m
[ T rs ξ ] ij =ξ Φ ri Φ sj dV [ M rs ] ij = Φ ri ( 1 c k t × Φ sj )dV [ S ee ] ij = s Φ ei n × n × Φ ej Z+ Z + dS [ S eh ] ij = × Φ ei Φ hj dV + s Φ ei n ×Z Φ hj Z+ Z + dS [ F ee ] ij = s Φ ei n × n × Φ ej + Z+ Z + dS [ F eh ] ij = s Φ ei n × Z + Φ hj + Z+ Z + dS [ S hh ] ij = s Φ hi n × n × Φ hj Y+ Y + dS [ S he ] ij = × Φ hi Φ ej dV s Φ hi n ×Y Φ hj Y+ Y + dS [ F hh ] ij = s Φ hi n × n × Φ hj + Y+ Y + dS [ F he ] ij = s Φ hi n × Y + Φ ej + Y+ Y + dS
q m t = γ m .
M u t = L 1 u+ L 1 u +
M=[ T ee ε 0 ε M eh 0 0 M he T hh μ 0 μ r 0 0 ε 0 a 1,m I 0 0 I 0 0 I 0 ]
L 1 =[ S ee S eh 0 T eq 1 S he S hh 0 0 ε 0 a 0,m I 0 b 0,m I b 1,m I 0 0 0 I ]
L 2 =[ F ee F eh 0 0 F he F hh 0 0 0 0 0 0 0 0 0 0 ]
{ u(t+ c i Δt)=u(t)+Δt j=1 s a ij M 1 ( L 1 u(t+ c j Δt)+ L 1 u + (t+ c j Δt)),1is u(t+Δt)=u(t)+Δt i=1 s b i M 1 ( L 1 u(t+ c i Δt)+ L 1 u + (t+ c i Δt))
E inc (z,t)= e ( t cos θ i (z z s ) t w t s ) 2 sin[ 2π f 0 ( t cos θ i (z z s ) t w t s ) ]
Δt d min ε r, 5c

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