Abstract

Transient electromagnetic interactions on plasmonic nanostructures are analyzed by solving the Poggio-Miller-Chan-Harrington-Wu-Tsai (PMCHWT) surface integral equation (SIE). Equivalent (unknown) electric and magnetic current densities, which are introduced on the surfaces of the nanostructures, are expanded using Rao-Wilton-Glisson and polynomial basis functions in space and time, respectively. Inserting this expansion into the PMCHWT-SIE and Galerkin testing the resulting equation at discrete times yield a system of equations that is solved for the current expansion coefficients by a marching on-in-time (MOT) scheme. The resulting MOT-PMCHWT-SIE solver calls for computation of additional convolutions between the temporal basis function and the plasmonic medium’s permittivity and Green function. This computation is carried out with almost no additional cost and without changing the computational complexity of the solver. Time-domain samples of the permittivity and the Green function required by these convolutions are obtained from their frequency-domain samples using a fast relaxed vector fitting algorithm. Numerical results demonstrate the accuracy and applicability of the proposed MOT-PMCHWT solver.

© 2016 Optical Society of America

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2015 (2)

I. E. Uysal, H. A. Ulku, and H. Bagci, “MOT solution of the PMCHWT equation for analyzing transient scattering from conductive dielectrics,” IEEE Antennas Wireless Propag. Lett. 14, 507–510 (2015).
[Crossref]

D. J. de Aberasturi, A. B. Serrano-Montes, and L. M. Liz-Marzán, “Modern applications of plasmonic nanoparticles: from energy to health,” Adv. Opt. Mater. 3, 602–617 (2015).
[Crossref]

2012 (3)

2011 (5)

J. M. Taboada, J. Rivero, F. Obelleiro, M. G. Araújo, and L. Landesa, “Method-of-moments formulation for the analysis of plasmonic nano-optical antennas,” J. Opt. Soc. Am. A 28, 1341–1348 (2011).
[Crossref]

K. Sirenko, V. Pazynin, Y. K. Sirenko, and H. Bagci, “An FFT-accelerated FDTD scheme with exact absorbing conditions for characterizing axially symmetric resonant structures,” Prog. Electromagn. Res. 111, 331–364 (2011).
[Crossref]

H. A. Ulku and A. A. Ergin, “On the singularity of the closed-form expression of the magnetic field in time domain,” IEEE Trans. Antennas Propag. 59, 691–694 (2011).
[Crossref]

H. A. Ulku and A. A. Ergin, “Application of analytical retarded-time potential expressions to the solution of time domain integral equations,” IEEE Trans. Antennas Propag. 59, 4123–4131 (2011).
[Crossref]

X. Wang and D. S. Weile, “Implicit Runge-Kutta methods for the discretization of time domain integral equations,” IEEE Trans. Antennas Propag. 59, 4651–4663 (2011).
[Crossref]

2010 (5)

H. Bagci, A. Yilmaz, and E. Michielssen, “An FFT-accelerated time-domain multiconductor transmission line simulator,” IEEE Trans. Electromagn. Compat. 52, 199–214 (2010).
[Crossref]

Y. Hu, S. J. Noelck, and R. A. Drezek, “Symmetry breaking in gold-silica-gold multilayer nanoshells,” ACS Nano 4, 1521–1528 (2010).
[Crossref]

X. Wang and D. S. Weile, “Electromagnetic scattering from dispersive dielectric scatterers using the finite difference delay modeling method,” IEEE Trans. Antennas Propag. 58, 1720–1730 (2010).
[Crossref]

E. R. Encina and E. A. Coronado, “Plasmon coupling in silver nanosphere pairs,” J. Phys. Chem. C 114, 3918–3923 (2010).
[Crossref]

B. Gallinet, A. M. Kern, and O. J. Martin, “Accurate and versatile modeling of electromagnetic scattering on periodic nanostructures with a surface integral approach,” J. Opt. Soc. Am. A 27, 2261–2271 (2010).
[Crossref]

2009 (2)

A. M. Kern and O. J. Martin, “Surface integral formulation for 3D simulations of plasmonic and high permittivity nanostructures,” J. Opt. Soc. Am. A 26, 732–740 (2009).
[Crossref]

B. Shanker, M. Lu, J. Yuan, and E. Michielssen, “Time domain integral equation analysis of scattering from composite bodies via exact evaluation of radiation fields,” IEEE Trans. Antennas Propag. 57, 1506–1520 (2009).
[Crossref]

2008 (2)

D. Deschrijver, M. Mrozowski, T. Dhaene, and D. De Zutter, “Macromodeling of multiport systems using a fast implementation of the vector fitting method,” IEEE Microwave Wireless Compon. Lett. 18, 383–385 (2008).
[Crossref]

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. V. Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7, 442–453 (2008).
[Crossref]

2007 (3)

F. Hao and P. Nordlander, “Efficient dielectric function for FDTD simulation of the optical properties of silver and gold nanoparticles,” Chem. Phys. Lett. 446, 115–118 (2007).
[Crossref]

H. A. Ulku and A. A. Ergin, “Analytical evaluation of transient magnetic fields due to RWG current bases,” IEEE Trans. Antennas Propag. 55, 3565–3575 (2007).
[Crossref]

H. Bagci, A. Yilmaz, J.-M. Jin, and E. Michielssen, “Fast and rigorous analysis of EMC/EMI phenomena on electrically large and complex cable-loaded structures,” IEEE Trans. Electromagn. Compat. 49, 361–381 (2007).
[Crossref]

2006 (3)

B. Gustavsen, “Improving the pole relocating properties of vector fitting,” IEEE Trans. Power Del. 21, 1587–1592 (2006).
[Crossref]

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440, 508–511 (2006).
[Crossref]

J. H. Greene and A. Taflove, “General vector auxiliary differential equation finite-difference time-domain method for nonlinear optics,” Opt. Express 14, 8305–8310 (2006).
[Crossref]

2005 (3)

A. Vial, A.-S. Grimault, D. Macas, D. Barchiesi, and M. L. de La Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71, 085416 (2005).
[Crossref]

P.-L. Jiang and E. Michielssen, “Multilevel plane wave time domain-enhanced MOT solver for analyzing electromagnetic scattering from objects residing in lossy media,” Proc. IEEE 3B, 447–450 (2005).
[Crossref]

H. Bagci, A. E. Yilmaz, V. Lomakin, and E. Michielssen, “Fast solution of mixed-potential time-domain integral equations for half-space environments,” IEEE Trans. Geosci. Remote Sens. 43, 269–279 (2005).
[Crossref]

2004 (2)

A. E. Yilmaz, J.-M. Jin, and E. Michielssen, “Time domain adaptive integral method for surface integral equations,” IEEE Trans. Antennas Propag. 52, 2692–2708 (2004).
[Crossref]

C. Oubre and P. Nordlander, “Optical properties of metallodielectric nanostructures calculated using the finite difference time domain method,” J. Phys. Chem. B 108, 17740–17747 (2004).
[Crossref]

2003 (3)

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
[Crossref]

B. Shanker, A. A. Ergin, M. Lu, and E. Michielssen, “Fast analysis of transient electromagnetic scattering phenomena using the multilevel plane wave time domain algorithm,” IEEE Trans. Antennas Propag. 51, 628–641 (2003).
[Crossref]

K. Donepudi, J.-M. Jin, and W. C. Chew, “A higher order multilevel fast multipole algorithm for scattering from mixed conducting/dielectric bodies,” IEEE Trans. Antennas Propag. 51, 2814–2821 (2003).
[Crossref]

2002 (1)

A. Yilmaz, D. S. Weile, B. Shanker, J.-M. Jin, and E. Michielssen, “Fast analysis of transient scattering in lossy media,” IEEE Antennas Wireless Propag. Lett. 1, 14–17 (2002).
[Crossref]

1999 (2)

A. A. Ergin, B. Shanker, and E. Michielssen, “The plane-wave time-domain algorithm for the fast analysis of transient wave phenomena,” IEEE Trans. Antennas Propag. 41, 39–52 (1999).
[Crossref]

B. Gustavsen and A. Semlyen, “Rational approximation of frequency domain responses by vector fitting,” IEEE Trans. Power Del. 14, 1052–1061 (1999).
[Crossref]

1998 (1)

1997 (1)

G. Manara, A. Monorchio, and R. Reggiannini, “A space-time discretization criterion for a stable time-marching solution of the electric field integral equation,” IEEE Trans. Antennas Propag. 45, 527–532 (1997).
[Crossref]

1995 (1)

C. Lubich and A. Ostermann, “Linearly implicit time discretization of non-linear parabolic equations,” IMA J. Numer. Anal. 15, 555–583 (1995).
[Crossref]

1994 (2)

B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A 11, 1491–1499 (1994).
[Crossref]

P. G. Petropoulos, “Stability and phase error analysis of FD-TD in dispersive dielectrics,” IEEE Trans. Antennas Propag. 42, 62–69 (1994).
[Crossref]

1992 (1)

D. Vechinski and S. M. Rao, “Transient scattering from two-dimensional dielectric cylinders of arbitrary shape,” IEEE Trans. Antennas Propag. 40, 1054–1060 (1992).
[Crossref]

1991 (1)

J. Putnam and L. Medgyesi-Mitschang, “Combined field integral equation formulation for inhomogeneous two and three-dimensional bodies: the junction problem,” IEEE Trans. Antennas Propag. 39, 667–672 (1991).
[Crossref]

1989 (1)

R. F. Harrington, “Boundary integral formulations for homogeneous material bodies,” J. Electromagn. Waves Appl. 3, 1–15 (1989).
[Crossref]

1988 (1)

C. Lubich, “Convolution quadrature and discretized operational calculus. I,” Numer. Math. 52, 129–145 (1988).
[Crossref]

1984 (1)

D. Wilton, S. Rao, A. Glisson, D. Schaubert, O. Al-Bundak, and C. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. 32, 276–281 (1984).
[Crossref]

1982 (1)

S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
[Crossref]

1972 (1)

P. B. Johnson and R.-W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6, 4370–4379 (1972).
[Crossref]

Al-Bundak, O.

D. Wilton, S. Rao, A. Glisson, D. Schaubert, O. Al-Bundak, and C. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. 32, 276–281 (1984).
[Crossref]

Amin, M.

M. Amin and H. Bagci, “Scattering properties of vein induced localized surface plasmon resonances on a gold disk,” in Proceedings of High Capacity Optical Networks and Enabling Technologies (HONET), Riyadh, Saudi Arabia (2011), pp. 237–240.

Anker, J. N.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. V. Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7, 442–453 (2008).
[Crossref]

Araújo, M.

Araújo, M. G.

Bagci, H.

I. E. Uysal, H. A. Ulku, and H. Bagci, “MOT solution of the PMCHWT equation for analyzing transient scattering from conductive dielectrics,” IEEE Antennas Wireless Propag. Lett. 14, 507–510 (2015).
[Crossref]

K. Sirenko, V. Pazynin, Y. K. Sirenko, and H. Bagci, “An FFT-accelerated FDTD scheme with exact absorbing conditions for characterizing axially symmetric resonant structures,” Prog. Electromagn. Res. 111, 331–364 (2011).
[Crossref]

H. Bagci, A. Yilmaz, and E. Michielssen, “An FFT-accelerated time-domain multiconductor transmission line simulator,” IEEE Trans. Electromagn. Compat. 52, 199–214 (2010).
[Crossref]

H. Bagci, A. Yilmaz, J.-M. Jin, and E. Michielssen, “Fast and rigorous analysis of EMC/EMI phenomena on electrically large and complex cable-loaded structures,” IEEE Trans. Electromagn. Compat. 49, 361–381 (2007).
[Crossref]

H. Bagci, A. E. Yilmaz, V. Lomakin, and E. Michielssen, “Fast solution of mixed-potential time-domain integral equations for half-space environments,” IEEE Trans. Geosci. Remote Sens. 43, 269–279 (2005).
[Crossref]

M. Amin and H. Bagci, “Scattering properties of vein induced localized surface plasmon resonances on a gold disk,” in Proceedings of High Capacity Optical Networks and Enabling Technologies (HONET), Riyadh, Saudi Arabia (2011), pp. 237–240.

Barchiesi, D.

A. Vial, A.-S. Grimault, D. Macas, D. Barchiesi, and M. L. de La Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71, 085416 (2005).
[Crossref]

Bozhevolnyi, S. I.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440, 508–511 (2006).
[Crossref]

Butler, C.

D. Wilton, S. Rao, A. Glisson, D. Schaubert, O. Al-Bundak, and C. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. 32, 276–281 (1984).
[Crossref]

Chew, W. C.

K. Donepudi, J.-M. Jin, and W. C. Chew, “A higher order multilevel fast multipole algorithm for scattering from mixed conducting/dielectric bodies,” IEEE Trans. Antennas Propag. 51, 2814–2821 (2003).
[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]

Coronado, E.

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
[Crossref]

Coronado, E. A.

E. R. Encina and E. A. Coronado, “Plasmon coupling in silver nanosphere pairs,” J. Phys. Chem. C 114, 3918–3923 (2010).
[Crossref]

Dal Negro, L.

de Aberasturi, D. J.

D. J. de Aberasturi, A. B. Serrano-Montes, and L. M. Liz-Marzán, “Modern applications of plasmonic nanoparticles: from energy to health,” Adv. Opt. Mater. 3, 602–617 (2015).
[Crossref]

de La Chapelle, M. L.

A. Vial, A.-S. Grimault, D. Macas, D. Barchiesi, and M. L. de La Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71, 085416 (2005).
[Crossref]

De Zutter, D.

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D. Deschrijver, M. Mrozowski, T. Dhaene, and D. De Zutter, “Macromodeling of multiport systems using a fast implementation of the vector fitting method,” IEEE Microwave Wireless Compon. Lett. 18, 383–385 (2008).
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J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. V. Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7, 442–453 (2008).
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H. A. Ulku and A. A. Ergin, “Application of analytical retarded-time potential expressions to the solution of time domain integral equations,” IEEE Trans. Antennas Propag. 59, 4123–4131 (2011).
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H. A. Ulku and A. A. Ergin, “On the singularity of the closed-form expression of the magnetic field in time domain,” IEEE Trans. Antennas Propag. 59, 691–694 (2011).
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H. A. Ulku and A. A. Ergin, “Analytical evaluation of transient magnetic fields due to RWG current bases,” IEEE Trans. Antennas Propag. 55, 3565–3575 (2007).
[Crossref]

B. Shanker, A. A. Ergin, M. Lu, and E. Michielssen, “Fast analysis of transient electromagnetic scattering phenomena using the multilevel plane wave time domain algorithm,” IEEE Trans. Antennas Propag. 51, 628–641 (2003).
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A. A. Ergin, B. Shanker, and E. Michielssen, “The plane-wave time-domain algorithm for the fast analysis of transient wave phenomena,” IEEE Trans. Antennas Propag. 41, 39–52 (1999).
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D. Wilton, S. Rao, A. Glisson, D. Schaubert, O. Al-Bundak, and C. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. 32, 276–281 (1984).
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F. Hao and P. Nordlander, “Efficient dielectric function for FDTD simulation of the optical properties of silver and gold nanoparticles,” Chem. Phys. Lett. 446, 115–118 (2007).
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H. Bagci, A. Yilmaz, J.-M. Jin, and E. Michielssen, “Fast and rigorous analysis of EMC/EMI phenomena on electrically large and complex cable-loaded structures,” IEEE Trans. Electromagn. Compat. 49, 361–381 (2007).
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A. E. Yilmaz, J.-M. Jin, and E. Michielssen, “Time domain adaptive integral method for surface integral equations,” IEEE Trans. Antennas Propag. 52, 2692–2708 (2004).
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K. Donepudi, J.-M. Jin, and W. C. Chew, “A higher order multilevel fast multipole algorithm for scattering from mixed conducting/dielectric bodies,” IEEE Trans. Antennas Propag. 51, 2814–2821 (2003).
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A. Yilmaz, D. S. Weile, B. Shanker, J.-M. Jin, and E. Michielssen, “Fast analysis of transient scattering in lossy media,” IEEE Antennas Wireless Propag. Lett. 1, 14–17 (2002).
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S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440, 508–511 (2006).
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Liz-Marzán, L. M.

D. J. de Aberasturi, A. B. Serrano-Montes, and L. M. Liz-Marzán, “Modern applications of plasmonic nanoparticles: from energy to health,” Adv. Opt. Mater. 3, 602–617 (2015).
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H. Bagci, A. E. Yilmaz, V. Lomakin, and E. Michielssen, “Fast solution of mixed-potential time-domain integral equations for half-space environments,” IEEE Trans. Geosci. Remote Sens. 43, 269–279 (2005).
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B. Shanker, M. Lu, J. Yuan, and E. Michielssen, “Time domain integral equation analysis of scattering from composite bodies via exact evaluation of radiation fields,” IEEE Trans. Antennas Propag. 57, 1506–1520 (2009).
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B. Shanker, A. A. Ergin, M. Lu, and E. Michielssen, “Fast analysis of transient electromagnetic scattering phenomena using the multilevel plane wave time domain algorithm,” IEEE Trans. Antennas Propag. 51, 628–641 (2003).
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J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. V. Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7, 442–453 (2008).
[Crossref]

Macas, D.

A. Vial, A.-S. Grimault, D. Macas, D. Barchiesi, and M. L. de La Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71, 085416 (2005).
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Medgyesi-Mitschang, L.

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Michielssen, E.

H. Bagci, A. Yilmaz, and E. Michielssen, “An FFT-accelerated time-domain multiconductor transmission line simulator,” IEEE Trans. Electromagn. Compat. 52, 199–214 (2010).
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B. Shanker, M. Lu, J. Yuan, and E. Michielssen, “Time domain integral equation analysis of scattering from composite bodies via exact evaluation of radiation fields,” IEEE Trans. Antennas Propag. 57, 1506–1520 (2009).
[Crossref]

H. Bagci, A. Yilmaz, J.-M. Jin, and E. Michielssen, “Fast and rigorous analysis of EMC/EMI phenomena on electrically large and complex cable-loaded structures,” IEEE Trans. Electromagn. Compat. 49, 361–381 (2007).
[Crossref]

P.-L. Jiang and E. Michielssen, “Multilevel plane wave time domain-enhanced MOT solver for analyzing electromagnetic scattering from objects residing in lossy media,” Proc. IEEE 3B, 447–450 (2005).
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H. Bagci, A. E. Yilmaz, V. Lomakin, and E. Michielssen, “Fast solution of mixed-potential time-domain integral equations for half-space environments,” IEEE Trans. Geosci. Remote Sens. 43, 269–279 (2005).
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A. E. Yilmaz, J.-M. Jin, and E. Michielssen, “Time domain adaptive integral method for surface integral equations,” IEEE Trans. Antennas Propag. 52, 2692–2708 (2004).
[Crossref]

B. Shanker, A. A. Ergin, M. Lu, and E. Michielssen, “Fast analysis of transient electromagnetic scattering phenomena using the multilevel plane wave time domain algorithm,” IEEE Trans. Antennas Propag. 51, 628–641 (2003).
[Crossref]

A. Yilmaz, D. S. Weile, B. Shanker, J.-M. Jin, and E. Michielssen, “Fast analysis of transient scattering in lossy media,” IEEE Antennas Wireless Propag. Lett. 1, 14–17 (2002).
[Crossref]

A. A. Ergin, B. Shanker, and E. Michielssen, “The plane-wave time-domain algorithm for the fast analysis of transient wave phenomena,” IEEE Trans. Antennas Propag. 41, 39–52 (1999).
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M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University, 2002).

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G. Manara, A. Monorchio, and R. Reggiannini, “A space-time discretization criterion for a stable time-marching solution of the electric field integral equation,” IEEE Trans. Antennas Propag. 45, 527–532 (1997).
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P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, 1953).

Mrozowski, M.

D. Deschrijver, M. Mrozowski, T. Dhaene, and D. De Zutter, “Macromodeling of multiport systems using a fast implementation of the vector fitting method,” IEEE Microwave Wireless Compon. Lett. 18, 383–385 (2008).
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Y. Hu, S. J. Noelck, and R. A. Drezek, “Symmetry breaking in gold-silica-gold multilayer nanoshells,” ACS Nano 4, 1521–1528 (2010).
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F. Hao and P. Nordlander, “Efficient dielectric function for FDTD simulation of the optical properties of silver and gold nanoparticles,” Chem. Phys. Lett. 446, 115–118 (2007).
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C. Lubich and A. Ostermann, “Linearly implicit time discretization of non-linear parabolic equations,” IMA J. Numer. Anal. 15, 555–583 (1995).
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J. Putnam and L. Medgyesi-Mitschang, “Combined field integral equation formulation for inhomogeneous two and three-dimensional bodies: the junction problem,” IEEE Trans. Antennas Propag. 39, 667–672 (1991).
[Crossref]

Rao, S.

D. Wilton, S. Rao, A. Glisson, D. Schaubert, O. Al-Bundak, and C. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. 32, 276–281 (1984).
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S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
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G. Manara, A. Monorchio, and R. Reggiannini, “A space-time discretization criterion for a stable time-marching solution of the electric field integral equation,” IEEE Trans. Antennas Propag. 45, 527–532 (1997).
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Rivero, J.

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K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
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Schaubert, D.

D. Wilton, S. Rao, A. Glisson, D. Schaubert, O. Al-Bundak, and C. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. 32, 276–281 (1984).
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B. Gustavsen and A. Semlyen, “Rational approximation of frequency domain responses by vector fitting,” IEEE Trans. Power Del. 14, 1052–1061 (1999).
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D. J. de Aberasturi, A. B. Serrano-Montes, and L. M. Liz-Marzán, “Modern applications of plasmonic nanoparticles: from energy to health,” Adv. Opt. Mater. 3, 602–617 (2015).
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Shah, N. C.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. V. Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7, 442–453 (2008).
[Crossref]

Shanker, B.

B. Shanker, M. Lu, J. Yuan, and E. Michielssen, “Time domain integral equation analysis of scattering from composite bodies via exact evaluation of radiation fields,” IEEE Trans. Antennas Propag. 57, 1506–1520 (2009).
[Crossref]

B. Shanker, A. A. Ergin, M. Lu, and E. Michielssen, “Fast analysis of transient electromagnetic scattering phenomena using the multilevel plane wave time domain algorithm,” IEEE Trans. Antennas Propag. 51, 628–641 (2003).
[Crossref]

A. Yilmaz, D. S. Weile, B. Shanker, J.-M. Jin, and E. Michielssen, “Fast analysis of transient scattering in lossy media,” IEEE Antennas Wireless Propag. Lett. 1, 14–17 (2002).
[Crossref]

A. A. Ergin, B. Shanker, and E. Michielssen, “The plane-wave time-domain algorithm for the fast analysis of transient wave phenomena,” IEEE Trans. Antennas Propag. 41, 39–52 (1999).
[Crossref]

Sirenko, K.

K. Sirenko, V. Pazynin, Y. K. Sirenko, and H. Bagci, “An FFT-accelerated FDTD scheme with exact absorbing conditions for characterizing axially symmetric resonant structures,” Prog. Electromagn. Res. 111, 331–364 (2011).
[Crossref]

Sirenko, Y. K.

K. Sirenko, V. Pazynin, Y. K. Sirenko, and H. Bagci, “An FFT-accelerated FDTD scheme with exact absorbing conditions for characterizing axially symmetric resonant structures,” Prog. Electromagn. Res. 111, 331–364 (2011).
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Ulku, H. A.

I. E. Uysal, H. A. Ulku, and H. Bagci, “MOT solution of the PMCHWT equation for analyzing transient scattering from conductive dielectrics,” IEEE Antennas Wireless Propag. Lett. 14, 507–510 (2015).
[Crossref]

H. A. Ulku and A. A. Ergin, “On the singularity of the closed-form expression of the magnetic field in time domain,” IEEE Trans. Antennas Propag. 59, 691–694 (2011).
[Crossref]

H. A. Ulku and A. A. Ergin, “Application of analytical retarded-time potential expressions to the solution of time domain integral equations,” IEEE Trans. Antennas Propag. 59, 4123–4131 (2011).
[Crossref]

H. A. Ulku and A. A. Ergin, “Analytical evaluation of transient magnetic fields due to RWG current bases,” IEEE Trans. Antennas Propag. 55, 3565–3575 (2007).
[Crossref]

Uysal, I. E.

I. E. Uysal, H. A. Ulku, and H. Bagci, “MOT solution of the PMCHWT equation for analyzing transient scattering from conductive dielectrics,” IEEE Antennas Wireless Propag. Lett. 14, 507–510 (2015).
[Crossref]

Vechinski, D.

D. Vechinski and S. M. Rao, “Transient scattering from two-dimensional dielectric cylinders of arbitrary shape,” IEEE Trans. Antennas Propag. 40, 1054–1060 (1992).
[Crossref]

Vial, A.

A. Vial, A.-S. Grimault, D. Macas, D. Barchiesi, and M. L. de La Chapelle, “Improved analytical fit of gold dispersion: application to the modeling of extinction spectra with a finite-difference time-domain method,” Phys. Rev. B 71, 085416 (2005).
[Crossref]

Volkov, V. S.

S. I. Bozhevolnyi, V. S. Volkov, E. Devaux, J.-Y. Laluet, and T. W. Ebbesen, “Channel plasmon subwavelength waveguide components including interferometers and ring resonators,” Nature 440, 508–511 (2006).
[Crossref]

Wang, X.

X. Wang and D. S. Weile, “Implicit Runge-Kutta methods for the discretization of time domain integral equations,” IEEE Trans. Antennas Propag. 59, 4651–4663 (2011).
[Crossref]

X. Wang and D. S. Weile, “Electromagnetic scattering from dispersive dielectric scatterers using the finite difference delay modeling method,” IEEE Trans. Antennas Propag. 58, 1720–1730 (2010).
[Crossref]

Weile, D. S.

X. Wang and D. S. Weile, “Implicit Runge-Kutta methods for the discretization of time domain integral equations,” IEEE Trans. Antennas Propag. 59, 4651–4663 (2011).
[Crossref]

X. Wang and D. S. Weile, “Electromagnetic scattering from dispersive dielectric scatterers using the finite difference delay modeling method,” IEEE Trans. Antennas Propag. 58, 1720–1730 (2010).
[Crossref]

A. Yilmaz, D. S. Weile, B. Shanker, J.-M. Jin, and E. Michielssen, “Fast analysis of transient scattering in lossy media,” IEEE Antennas Wireless Propag. Lett. 1, 14–17 (2002).
[Crossref]

Wilton, D.

D. Wilton, S. Rao, A. Glisson, D. Schaubert, O. Al-Bundak, and C. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. 32, 276–281 (1984).
[Crossref]

S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
[Crossref]

Yilmaz, A.

H. Bagci, A. Yilmaz, and E. Michielssen, “An FFT-accelerated time-domain multiconductor transmission line simulator,” IEEE Trans. Electromagn. Compat. 52, 199–214 (2010).
[Crossref]

H. Bagci, A. Yilmaz, J.-M. Jin, and E. Michielssen, “Fast and rigorous analysis of EMC/EMI phenomena on electrically large and complex cable-loaded structures,” IEEE Trans. Electromagn. Compat. 49, 361–381 (2007).
[Crossref]

A. Yilmaz, D. S. Weile, B. Shanker, J.-M. Jin, and E. Michielssen, “Fast analysis of transient scattering in lossy media,” IEEE Antennas Wireless Propag. Lett. 1, 14–17 (2002).
[Crossref]

Yilmaz, A. E.

H. Bagci, A. E. Yilmaz, V. Lomakin, and E. Michielssen, “Fast solution of mixed-potential time-domain integral equations for half-space environments,” IEEE Trans. Geosci. Remote Sens. 43, 269–279 (2005).
[Crossref]

A. E. Yilmaz, J.-M. Jin, and E. Michielssen, “Time domain adaptive integral method for surface integral equations,” IEEE Trans. Antennas Propag. 52, 2692–2708 (2004).
[Crossref]

Yuan, J.

B. Shanker, M. Lu, J. Yuan, and E. Michielssen, “Time domain integral equation analysis of scattering from composite bodies via exact evaluation of radiation fields,” IEEE Trans. Antennas Propag. 57, 1506–1520 (2009).
[Crossref]

Zhao, J.

J. N. Anker, W. P. Hall, O. Lyandres, N. C. Shah, J. Zhao, and R. P. V. Duyne, “Biosensing with plasmonic nanosensors,” Nat. Mater. 7, 442–453 (2008).
[Crossref]

Zhao, L. L.

K. L. Kelly, E. Coronado, L. L. Zhao, and G. C. Schatz, “The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment,” J. Phys. Chem. B 107, 668–677 (2003).
[Crossref]

ACS Nano (1)

Y. Hu, S. J. Noelck, and R. A. Drezek, “Symmetry breaking in gold-silica-gold multilayer nanoshells,” ACS Nano 4, 1521–1528 (2010).
[Crossref]

Adv. Opt. Mater. (1)

D. J. de Aberasturi, A. B. Serrano-Montes, and L. M. Liz-Marzán, “Modern applications of plasmonic nanoparticles: from energy to health,” Adv. Opt. Mater. 3, 602–617 (2015).
[Crossref]

Chem. Phys. Lett. (1)

F. Hao and P. Nordlander, “Efficient dielectric function for FDTD simulation of the optical properties of silver and gold nanoparticles,” Chem. Phys. Lett. 446, 115–118 (2007).
[Crossref]

IEEE Antennas Wireless Propag. Lett. (2)

A. Yilmaz, D. S. Weile, B. Shanker, J.-M. Jin, and E. Michielssen, “Fast analysis of transient scattering in lossy media,” IEEE Antennas Wireless Propag. Lett. 1, 14–17 (2002).
[Crossref]

I. E. Uysal, H. A. Ulku, and H. Bagci, “MOT solution of the PMCHWT equation for analyzing transient scattering from conductive dielectrics,” IEEE Antennas Wireless Propag. Lett. 14, 507–510 (2015).
[Crossref]

IEEE Microwave Wireless Compon. Lett. (1)

D. Deschrijver, M. Mrozowski, T. Dhaene, and D. De Zutter, “Macromodeling of multiport systems using a fast implementation of the vector fitting method,” IEEE Microwave Wireless Compon. Lett. 18, 383–385 (2008).
[Crossref]

IEEE Trans. Antennas Propag. (16)

J. Putnam and L. Medgyesi-Mitschang, “Combined field integral equation formulation for inhomogeneous two and three-dimensional bodies: the junction problem,” IEEE Trans. Antennas Propag. 39, 667–672 (1991).
[Crossref]

K. Donepudi, J.-M. Jin, and W. C. Chew, “A higher order multilevel fast multipole algorithm for scattering from mixed conducting/dielectric bodies,” IEEE Trans. Antennas Propag. 51, 2814–2821 (2003).
[Crossref]

S. Rao, D. Wilton, and A. Glisson, “Electromagnetic scattering by surfaces of arbitrary shape,” IEEE Trans. Antennas Propag. 30, 409–418 (1982).
[Crossref]

D. Wilton, S. Rao, A. Glisson, D. Schaubert, O. Al-Bundak, and C. Butler, “Potential integrals for uniform and linear source distributions on polygonal and polyhedral domains,” IEEE Trans. Antennas Propag. 32, 276–281 (1984).
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Figures (10)

Fig. 1.
Fig. 1. (a) Real and (b) imaginary parts of ε 1 ( ω ) for gold.
Fig. 2.
Fig. 2. (a) Real and (b) imaginary parts of ε 1 ( ω ) for silver.
Fig. 3.
Fig. 3. Amplitude of (a)  γ 1 ( t ) and (b)  γ ¯ 1 ( t ) for gold and silver.
Fig. 4.
Fig. 4. Amplitude of J 1 ( t ) and M 1 ( t ) induced on (a) the gold sphere and (b) the silver sphere.
Fig. 5.
Fig. 5. Q ext ( ω ) computed for (a) the gold sphere and (b) the silver sphere.
Fig. 6.
Fig. 6. C ext ( ω ) computed for the gold rounded cube.
Fig. 7.
Fig. 7. C ext ( ω ) computed for the gold rounded triangular prism.
Fig. 8.
Fig. 8. Q ext ( ω ) computed for the gold shell with silica core.
Fig. 9.
Fig. 9. C ext ( ω ) computed for the silver dimer under two excitations with different polarizations.
Fig. 10.
Fig. 10. C sca ( ω ) computed for the gold disk with the nonconcentric cavity.

Equations (68)

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n ^ ( r ) × t E inc ( r , t ) = n ^ ( r ) × [ t E 0 sca ( r , t ) t E 1 sca ( r , t ) ] ,
n ^ ( r ) × t H inc ( r , t ) = n ^ ( r ) × [ t H 0 sca ( r , t ) t H 1 sca ( r , t ) ] .
t E 0 sca ( r , t ) = L 0 { μ 0 J ( r , t ) } + Q 0 { ε 0 1 J ( r , t ) } K 0 { M ( r , t ) } ,
t H 0 sca ( r , t ) = L 0 { ε 0 M ( r , t ) } + Q 0 { μ 0 1 M ( r , t ) } + K 0 { J ( r , t ) } ,
t E 1 sca ( r , t ) = L 1 { μ 1 J ( r , t ) } Q 1 { ε ¯ 1 ( t ) * J ( r , t ) } + K 1 { M ( r , t ) } ,
t H 1 sca ( r , t ) = L 1 { ε 1 ( t ) * M ( r , t ) } Q 1 { μ 1 1 M ( r , t ) } K 1 { J ( r , t ) } .
L p { X ( r , t ) } = S g p ( R , t ) * t 2 X ( r , t ) d s ,
Q p { X ( r , t ) } = S g p ( R , t ) * · X ( r , t ) d s ,
K p { X ( r , t ) } = × S g p ( R , t ) * t X ( r , t ) d s .
ε 1 ( t ) = F 1 { ε 1 ( ω ) } ,
ε ¯ 1 ( t ) = F 1 { ε ¯ 1 ( ω ) } = F 1 { 1 / ε 1 ( ω ) } .
ε 1 ( ω ) ε 1 m = 1 N p VF b m ϵ j ω + a m ϵ ,
ε ¯ 1 ( ω ) 1 ε 1 m = 1 N p VF b m ϵ ¯ j ω + a m ϵ ¯ ,
ε 1 ( t ) ε 1 δ ( t ) + m = 1 N p VF b m ϵ u ( t ) e a m ϵ t = ε 1 δ ( t ) + γ 1 ( t ) ,
ε ¯ 1 ( t ) 1 ε 1 δ ( t ) + m = 1 N p VF b m ϵ ¯ u ( t ) e a m ϵ ¯ t = 1 ε 1 δ ( t ) + γ ¯ 1 ( t ) ,
e j ω R / c 1 [ g 1 ( R , ω ) e j ω R / c 1 4 π R ] = e j ( k 1 R ω R / c 1 ) 1 4 π R d g ( R ) + m = 1 N p VF b m g ( R ) j ω + a m g ( R ) .
g 1 ( R , ω ) e j ω R / c 1 [ 1 4 π R + d g ( R ) + m = 1 N p VF b m g ( R ) j ω + a m g ( R ) ] .
g 1 ( R , t ) δ ( τ 1 ) 4 π R + d g ( R ) δ ( τ 1 ) + m = 1 N p VF b m g ( R ) u ( τ 1 ) e a m g ( R ) τ 1 ,
e j ω R / c 1 [ R g 1 ( R , ω ) + ( j ω c 1 R + 1 R 2 ) e j ω R / c 1 4 π ] = e j ( k 1 R ω R / c 1 ) 4 π ( j k 1 R + 1 R 2 ) + 1 4 π R 2 + j ω 4 π c 1 R d g R ( R ) + j ω f g R ( R ) + m = 1 N p VF b m g R ( R ) j ω + a m g R ( R ) .
R g 1 ( R , ω ) e j ω R / c 1 [ j ω ( f g R ( R ) 1 4 π c 1 R ) + d g R ( R ) 1 4 π R 2 + m = 1 N p VF b m g R ( R ) j ω + a m g R ( R ) ] .
R g 1 ( R , t ) δ ( τ 1 ) ( f g R ( R ) 1 4 π c 1 R ) + δ ( τ 1 ) ( d g R ( R ) 1 4 π R 2 ) + m = 1 N p VF b m g R ( R ) u ( τ 1 ) e a m g R ( R ) τ 1 .
J ( r , t ) = j = 1 N t n = 1 N s J n ( j Δ t ) f n ( r ) T j ( t ) = j = 1 N t n = 1 N s { I ¯ j J } n f n ( r ) T j ( t ) ,
M ( r , t ) = j = 1 N t n = 1 N s M n ( j Δ t ) f n ( r ) T j ( t ) = j = 1 N t n = 1 N s { I ¯ j M } n f n ( r ) T j ( t ) .
[ Z ¯ ¯ 0 JJ Z ¯ ¯ 0 JM Z ¯ ¯ 0 JM Z ¯ ¯ 0 MM ] [ I ¯ j J I ¯ j M ] = [ V ¯ j J V ¯ j M ] j = 1 j 1 [ Z ¯ ¯ j j JJ Z ¯ ¯ j j JM Z ¯ ¯ j j JM Z ¯ ¯ j j MM ] [ I ¯ j J I ¯ j M ] .
{ V ¯ j J } n = f n ( r ) , t E 0 inc ( r , t ) t = j Δ t ,
{ V ¯ j M } n = f n ( r ) , t H 0 inc ( r , t ) t = j Δ t .
{ Z ¯ ¯ j j JJ } n , n = { Z ¯ ¯ j j JJ , 0 } n , n + { Z ¯ ¯ j j JJ , 1 } n , n ,
{ Z ¯ ¯ j j JM } n , n = { Z ¯ ¯ j j JM , 0 } n , n + { Z ¯ ¯ j j JM , 1 } n , n ,
{ Z ¯ ¯ j j MM } n , n = { Z ¯ ¯ j j MM , 0 } n , n + { Z ¯ ¯ j j MM , 1 } n , n ,
{ Z ¯ ¯ j j JJ , 0 } n , n = μ 0 f n ( r ) , L 0 { f n ( r ) T j ( t ) } t = j Δ t + 1 ε 0 f n ( r ) , Q 0 { f n ( r ) T j ( t ) } t = j Δ t ,
{ Z ¯ ¯ j j JM , 0 } n , n = f n ( r ) , K 0 { f n ( r ) T j ( t ) } t = j Δ t ,
{ Z ¯ ¯ j j MM , 0 } n , n = ε 0 μ 0 { Z ¯ ¯ j j JJ , 0 } n , n ,
{ Z ¯ ¯ j j JJ , 1 } n , n = μ 1 f n ( r ) , L 1 { f n ( r ) T j ( t ) } t = j Δ t + 1 ε 1 f n ( r ) , Q 1 { f n ( r ) T j ( t ) } t = j Δ t + f n ( r ) , Q 1 { γ ¯ 1 ( t ) * f n ( r ) T j ( t ) } t = j Δ t ,
{ Z ¯ ¯ j j JM , 1 } n , n = f n ( r ) , K 1 { f n ( r ) T j ( t ) } t = j Δ t ,
{ Z ¯ ¯ j j MM , 1 } n , n = ε 1 f n ( r ) , L 1 { f n ( r ) T j ( t ) } t = j Δ t f n ( r ) , L 1 { γ 1 ( t ) * f n ( r ) T j ( t ) } t = j Δ t + 1 μ 1 f n ( r ) , Q 1 { f n ( r ) T j ( t ) } t = j Δ t .
{ Z ¯ ¯ j j MM , 0 } n , n = ε 0 μ 0 { Z ¯ ¯ j j JJ , 0 } n , n = ε 0 4 π S n f n ( r ) · S n f n ( r ) t 2 T j ( t ) | t = j Δ t R / c 0 R d s d s 1 4 π μ 0 S n · f n ( r ) S n · f n ( r ) T j ( j Δ t R / c 0 ) R d s d s ,
{ Z ¯ ¯ j j JM , 0 } n , n = 1 4 π S n f n ( r ) · S n R ^ × f n ( r ) R [ t 2 T j ( t ) c 0 + t T j ( t ) R ] t = j Δ t R / c 0 d s d s .
C T ( R , t ) = g 1 ( R , t ) * t 2 T j ( t ) ,
C ( R , t ) = g 1 ( R , t ) * T j ( t ) ,
C R ( R , t ) = R g 1 ( R , t ) * t T j ( t ) .
f n ( r ) , L 1 { f n ( r ) T j ( t ) } t = j Δ t = S n f n ( r ) · S n C T ( R , t ) f n ( r ) d s d s | t = j Δ t ,
f n ( r ) , Q 1 { f n ( r ) T j ( t ) } t = j Δ t = S n · f n ( r ) S n C ( R , t ) · f n ( r ) d s d s | t = j Δ t ,
f n ( r ) , K 1 { f n ( r ) T j ( t ) } t = j Δ t = S n f n ( r ) · S n R ^ × f n ( r ) C R ( R , t ) d s d s | t = j Δ t .
C T ( R , t ) = [ d g ( R ) + 1 4 π R ] t 2 T ( t ) | t = t j Δ t R / c 1 + m = 1 N p VF b m g ( R ) Δ t min ( t j Δ t R / c 1 , d Δ t ) u ( t t j Δ t R / c 1 ) e a m g ( R ) ( t t j Δ t R / c 1 ) t 2 T ( t ) d t ,
C ( R , t ) = [ d g ( R ) + 1 4 π R ] T ( t j Δ t R / c 1 ) + m = 1 N p VF b m g ( R ) Δ t min ( t j Δ t R / c 1 , d Δ t ) u ( t t j Δ t R / c 1 ) e a m g ( R ) ( t t j Δ t R / c 1 ) T ( t ) d t ,
C R ( R , t ) = [ f g R ( R ) 1 4 π R c 1 ] t 2 T ( t ) | t = t j Δ t R / c 1 + [ d g R ( R ) 1 4 π R 2 ] t T ( t ) | t = t j Δ t R / c 1 + m = 1 N p VF b m g R ( R ) Δ t min ( t j Δ t R / c 1 , d Δ t ) u ( t t j Δ t R / c 1 ) e a m g R ( R ) ( t t j Δ t R / c 1 ) t T ( t ) d t .
D ( R , t ) = g 1 ( R , t ) * γ ¯ 1 ( t ) * T j ( t ) ,
D T ( R , t ) = g 1 ( R , t ) * γ 1 ( t ) * t 2 T j ( t ) = t 2 g 1 ( R , t ) * γ 1 ( t ) * T j ( t ) .
f n ( r ) , Q 1 { γ ¯ 1 ( t ) * f n ( r ) T j ( t ) } t = j Δ t = S n · f n ( r ) S n D ( R , t ) · f n ( r ) d s d s | t = j Δ t ,
f n ( r ) , L 1 { γ 1 ( t ) * f n ( r ) T j ( t ) } t = j Δ t = S n f n ( r ) · S n D T ( R , t ) f n ( r ) d s d s | t = j Δ t .
F j ( t ) = γ ¯ 1 ( t ) * T j ( t ) l = 1 N t F l j T l ( t ) ,
F j T ( t ) = γ 1 ( t ) * T j ( t ) l = 1 N t F l j T T l ( t ) ,
F l j = F j ( l Δ t ) = γ ¯ 1 ( t ) * T j ( t ) | t = l Δ t = Δ t min ( l j , d ) Δ t γ ¯ 1 ( [ l j ] Δ t t ) T ( t ) d t = m = 1 N p VF b m ϵ ¯ Δ t min ( l j , d ) Δ t u ( [ l j ] Δ t t ) e a m ϵ ¯ ( [ l j ] Δ t t ) T ( t ) d t ,
F l j T = F j T ( l Δ t ) = γ 1 ( t ) * T j ( t ) | t = l Δ t = Δ t min ( l j , d ) Δ t γ 1 ( [ l j ] Δ t t ) T ( t ) d t = m = 1 N p VF b m ϵ Δ t min ( l j , d ) Δ t u ( [ l j ] Δ t t ) e a m ϵ ( [ l j ] Δ t t ) T ( t ) d t .
D ( R , t ) = g 1 ( R , t ) * F l j ( t ) l = 1 N t F l j g 1 ( R , t ) * T l ( t ) ,
D T ( R , t ) = t 2 g 1 ( R , t ) * F l j T ( t ) l = 1 N t F l j T t 2 g 1 ( R , t ) * T l ( t ) = l = 1 N t F l j T g 1 ( R , t ) * t 2 T l ( t ) .
f n ( r ) , Q 1 { γ ¯ 1 ( t ) * f n ( r ) T j ( t ) } t = j Δ t = S n · f n ( r ) S n [ l = 1 N t F l j g 1 ( R , t ) * T l ( t ) ] · f n ( r ) d s d s | t = j Δ t = l = 1 N t f n ( r ) , Q 1 { f n ( r ) T l ( t ) } t = j Δ t F l j ,
f n ( r ) , L 1 { γ 1 ( t ) * f n ( r ) T j ( t ) } t = j Δ t = S n f n ( r ) · S n [ l = 1 N t F l j T g 1 ( R , t ) * t 2 T l ( t ) ] f n ( r ) d s d s | t = j Δ t = l = 1 N t f n ( r ) , L 1 { f n ( r ) T l ( t ) } t = j Δ t F l j T .
E 0 inc ( r , t ) = p ^ E 0 inc G ( t k ^ · r / c 0 ) ,
C sca ( l Δ ω ) = 1 16 π 2 | E 0 inc | 2 Ω | F ( r ^ , l Δ ω ) | 2 d Ω ,
C ext ( l Δ ω ) = 1 k 0 | E 0 inc | 2 Im { E 0 inc p ^ · F ( k ^ , l Δ ω ) } .
F ( r ^ , l Δ ω ) = j ( l Δ ω ) μ 0 N ( r ^ , l Δ ω ) + j k 0 r ^ × L ( r ^ , l Δ ω ) ,
N ( r ^ , l Δ ω ) = n = 1 N s J n ( l Δ ω ) S n f n ( r ) e j k 0 r · r ^ d s ,
L ( r ^ , l Δ ω ) = n = 1 N s M n ( l Δ ω ) S n f n ( r ) e j k 0 r · r ^ d s .
F ( ω ) j ω f + d + m = 1 N b m j ω + a m .
S ( ω ) = 1 + m = 1 N b m S j ω + a ˜ m ,
S ( ω ) F ( ω ) j ω f SF + d SF + m = 1 N b m SF j ω + a ˜ m .
A ¯ ¯ X ¯ = Y ¯ .

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