Abstract

A combination of the multilevel fast multipole method (MLFMM) and boundary element method (BEM) can solve large scale photonics problems of arbitrary geometry. Here, MLFMM-BEM algorithm based on a scalar and vector potential formulation, instead of the more conventional electric and magnetic field formulations, is described. The method can deal with multiple lossy or lossless dielectric objects of arbitrary geometry, be they nested, in contact, or dispersed. Several examples are used to demonstrate that this method is able to efficiently handle 3D photonic scatterers involving large numbers of unknowns. Absorption, scattering, and extinction efficiencies of gold nanoparticle spheres, calculated by the MLFMM, are compared with Mie’s theory. MLFMM calculations of the bistatic radar cross section (RCS) of a gold sphere near the plasmon resonance and of a silica coated gold sphere are also compared with Mie theory predictions. Finally, the bistatic RCS of a nanoparticle gold–silver heterodimer calculated with MLFMM is compared with unmodified BEM calculations.

© 2013 Optical Society of America

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2012 (6)

Ö. Ergül, “Fast and accurate solutions of electromagnetics problems involving lossy dielectric objects with the multilevel fast multipole algorithm,” Eng. Anal. Bound. Elem. 36, 423–432 (2012).

Ö. Ergül, “Analysis of composite nanoparticles with surface integral equations and the multilevel fast multipole algorithm,” J. Opt. 14, 062701 (2012).
[CrossRef]

U. Hohenester and A. Trügler, “MNPBEM—A Matlab toolbox for the simulation of plasmonic nanoparticles,” Comput. Phys. Commun. 183, 370–381 (2012).
[CrossRef]

M. G. Araújo, J. M. Taboada, J. Rivero, D. M. Solís, and F. Obelleiro, “Solution of large-scale plasmonic problems with the multilevel fast multipole algorithm,” Opt. Lett. 37, 416 (2012).
[CrossRef]

M. G. Araújo, J. M. Taboada, D. M. Solís, J. Rivero, L. Landesa, and F. Obelleiro, “Comparison of surface integral equation formulations for electromagnetic analysis of plasmonic nanoscatterers,” Opt. Express 20, 9161–9171 (2012).
[CrossRef]

L. Landesa, M. G. Araújo, J. M. Taboada, L. Bote, and F. Obelleiro, “Improving condition number and convergence of the surface integral-equation method of moments for penetrable bodies,” Opt. Express 20, 17237–17249 (2012).
[CrossRef]

2011 (1)

2010 (1)

S. Mousavi and N. Sukumar, “Generalized Duffy transformation for integrating vertex singularities,” J. Comput. Mech. 45, 127–140 (2010).

2009 (2)

O. Ergul and L. Gurel, “Comparison of integral-equation formulations for the fast and accurate solution of scattering problems involving dielectric objects with the multilevel fast multipole algorithm,” IEEE Trans. Antennas Propag. 57, 176–187 (2009).
[CrossRef]

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

2008 (3)

Y. Shi, H. G. Wang, L. Li, and C. H. Chan, “Multilevel Green’s function interpolation method for scattering from composite metallic and dielectric objects,” J. Opt. Soc. Am. A 25, 2535–2548 (2008).
[CrossRef]

U. Hohenester and A. Trugler, “Interaction of single molecules with metallic nanoparticles,” IEEE J. Sel. Topics Quantum Electron. 14, 1430–1440 (2008).
[CrossRef]

J. Fostier and F. Olyslager, “An asynchronous parallel MLFMA for scattering at multiple dielectric objects,” IEEE Trans. Antennas Propag. 56, 2346–2355 (2008).
[CrossRef]

2007 (1)

L. Shen and Y. Liu, “An adaptive fast multipole boundary element method for three-dimensional acoustic wave problems based on the Burton–Miller formulation,” J. Comput. Mech. 40, 461–472 (2007).

2006 (1)

O. Ergul and L. Gurel, “Optimal interpolation of translation operator in multilevel fast multipole algorithm,” IEEE Trans. Antennas Propag. 54, 3822–3826 (2006).
[CrossRef]

2005 (2)

U. Hohenester and J. Krenn, “Surface plasmon resonances of single and coupled metallic nanoparticles: a boundary integral method approach,” Phys. Rev. B 72, 195429 (2005).
[CrossRef]

P. Yla-Oijala and M. Taskinen, “Application of combined field integral equation for electromagnetic scattering by dielectric and composite objects,” IEEE Trans. Antennas Propag. 53, 1168–1173 (2005).
[CrossRef]

2004 (2)

W.-B. Ewe, L.-W. Li, and M.-S. Leong, “Solving mixed dielectric/conducting scattering problem using adaptive method,” Prog. Electromagn. Res. 46, 143–163 (2004).
[CrossRef]

K. Sertel and J. L. Volakis, “Multilevel fast multipole method solution of volume integral equations using parametric geometry modeling,” IEEE Trans. Antennas Propag. 52, 1686–1692 (2004).
[CrossRef]

2003 (2)

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

S. Velamparambil, W. C. Chew, and J. Song, “10 million unknowns: is it that big? [computational electromagnetics],” IEEE Antennas Propag. Mag. 45(2), 43–58 (2003).
[CrossRef]

2002 (1)

F. Garcia de Abajo and A. Howie, “Retarded field calculation of electron energy loss in inhomogeneous dielectrics,” Phys. Rev. B 65, 115418 (2002).
[CrossRef]

2001 (1)

N. Geng, A. Sullivan, and L. Carin, “Fast multipole method for scattering from an arbitrary PEC target above or buried in a lossy half-space,” IEEE Trans. Antennas Propag. 49, 740–748 (2001).
[CrossRef]

2000 (3)

K. Sertel and J. L. Volakis, “Incomplete LU preconditioner for FMM implementation,” Microw. Opt. Technol. Lett. 26, 265–267 (2000).
[CrossRef]

E. Darve, “The fast multipole method: numerical implementation,” J. Comput. Phys. 160, 195–240 (2000).
[CrossRef]

N. Geng, A. Sullivan, and L. Carin, “Multilevel fast-multipole algorithm for scattering from conducting targets above or embedded in a lossy half-space,” IEEE Trans. Geosci. Remote Sens. 38, 1561–1573 (2000).
[CrossRef]

1999 (1)

H. Cheng, L. Greengard, and V. Rokhlin, “A fast adaptive multipole algorithm in three dimensions,” J. Comput. Phys. 155, 468–498 (1999).
[CrossRef]

1998 (1)

F. Garcia de Abajo and A. Howie, “Relativistic electron energy loss and electron-induced photon emission in inhomogeneous dielectrics,” Phys. Rev. Lett. 80, 5180–5183 (1998).
[CrossRef]

1997 (2)

L. Greengard and V. Rokhlin, “A new version of the fast multipole method for the Laplace equation in three dimensions,” ANU 6, 229–269 (1997).

J. Song, C. C. Lu, and W. C. Chew, “Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects,” IEEE Trans. Antennas Propag. 45, 1488–1493 (1997).
[CrossRef]

1994 (3)

R. L. Wagner and W. C. Chew, “A ray-propagation fast multipole algorithm,” Microw. Opt. Technol. Lett. 7, 435–438 (1994).
[CrossRef]

C. C. Lu and W. C. Chew, “A multilevel algorithm for solving a boundary integral equation of wave scattering,” Microw. Opt. Technol. Lett. 7, 466–470 (1994).
[CrossRef]

L. Medgyesi-Mitschang, J. Putnam, and M. Gedera, “Generalized method of moments for three-dimensional penetrable scatterers,” J. Opt. Sci. Am. A 11, 1383–1398 (1994).

1993 (1)

R. Coifman, V. Rokhlin, and S. Wandzura, “The fast multipole method for the wave equation: a pedestrian prescription,” IEEE Antennas Propag. Mag. 35, 7–12 (1993).
[CrossRef]

1992 (1)

N. Engheta, W. D. Murphy, V. Rokhlin, and M. S. Vassiliou, “The fast multipole method (FMM) for electromagnetic scattering problems,” IEEE Trans. Antennas Propag. 40, 634–641 (1992).
[CrossRef]

1988 (1)

A. Taflove, “Review of the formulation and applications of the finite-difference time-domain method for numerical modeling of electromagnetic wave interactions with arbitrary structures,” Wave Motion 10, 547–582 (1988).
[CrossRef]

1987 (1)

L. Greengard and V. Rokhlin, “A fast algorithm for particle simulations,” J. Comput. Phys. 73, 325–348 (1987).
[CrossRef]

1986 (2)

K. Umashankar, A. Taflove, and S. Rao, “Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric objects,” IEEE Trans. Antennas Propag. 34, 758–766 (1986).
[CrossRef]

Y. Saad and M. H. Schultz, “GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Statist. Comput. 7, 856–869 (1986).
[CrossRef]

1985 (1)

V. Rokhlin, “Rapid solution of integral equations of classical potential theory,” J. Comput. Phys. 60, 187–207 (1985).
[CrossRef]

1984 (1)

D. Schaubert, D. Wilton, and A. Glisson, “A tetrahedral modeling method for electromagnetic scattering by arbitrarily shaped inhomogeneous dielectric bodies,” IEEE Trans. Antennas Propag. 32, 77–85 (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]

Araújo, M. G.

Bote, L.

Carin, L.

N. Geng, A. Sullivan, and L. Carin, “Fast multipole method for scattering from an arbitrary PEC target above or buried in a lossy half-space,” IEEE Trans. Antennas Propag. 49, 740–748 (2001).
[CrossRef]

N. Geng, A. Sullivan, and L. Carin, “Multilevel fast-multipole algorithm for scattering from conducting targets above or embedded in a lossy half-space,” IEEE Trans. Geosci. Remote Sens. 38, 1561–1573 (2000).
[CrossRef]

Cha, C. C.

C. C. Cha and D. Wilkes, “Method of moments formulation for an arbitrary material configuration,” in Antennas and Propagation Society International Symposium, 1991. AP-S. Digest (IEEE, 1991), pp. 1508–1511.

Chan, C. H.

Cheng, H.

H. Cheng, L. Greengard, and V. Rokhlin, “A fast adaptive multipole algorithm in three dimensions,” J. Comput. Phys. 155, 468–498 (1999).
[CrossRef]

Chew, W.

C. Lu and W. Chew, “Fast algorithm for solving hybrid integral equations [EM wave scattering],” in Microwaves, Antennas and Propagation, IEE Proceedings H (IEE, 1993), Vol. 140, pp. 455–460.

Chew, W. C.

S. Velamparambil, W. C. Chew, and J. Song, “10 million unknowns: is it that big? [computational electromagnetics],” IEEE Antennas Propag. Mag. 45(2), 43–58 (2003).
[CrossRef]

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

J. Song, C. C. Lu, and W. C. Chew, “Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects,” IEEE Trans. Antennas Propag. 45, 1488–1493 (1997).
[CrossRef]

R. L. Wagner and W. C. Chew, “A ray-propagation fast multipole algorithm,” Microw. Opt. Technol. Lett. 7, 435–438 (1994).
[CrossRef]

C. C. Lu and W. C. Chew, “A multilevel algorithm for solving a boundary integral equation of wave scattering,” Microw. Opt. Technol. Lett. 7, 466–470 (1994).
[CrossRef]

J. M. Song and W. C. Chew, “Fast multipole method solution of combined field integral equation,” in 11th Annual Review of Progress in Applied Computational Electromagnetics 1 (Naval Postgraduate School, 1995), pp. 629–636.

Coifman, R.

R. Coifman, V. Rokhlin, and S. Wandzura, “The fast multipole method for the wave equation: a pedestrian prescription,” IEEE Antennas Propag. Mag. 35, 7–12 (1993).
[CrossRef]

Darve, E.

E. Darve, “The fast multipole method: numerical implementation,” J. Comput. Phys. 160, 195–240 (2000).
[CrossRef]

E. Darve and O. Pironneau, “Méthodes multipôles rapides: résolution des équations de Maxwell par formulations intégrales,” Ph.D. dissertation (Université de Paris, 1999).

Donepudi, K. C.

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

Engheta, N.

N. Engheta, W. D. Murphy, V. Rokhlin, and M. S. Vassiliou, “The fast multipole method (FMM) for electromagnetic scattering problems,” IEEE Trans. Antennas Propag. 40, 634–641 (1992).
[CrossRef]

Ergul, O.

O. Ergul and L. Gurel, “Comparison of integral-equation formulations for the fast and accurate solution of scattering problems involving dielectric objects with the multilevel fast multipole algorithm,” IEEE Trans. Antennas Propag. 57, 176–187 (2009).
[CrossRef]

O. Ergul and L. Gurel, “Optimal interpolation of translation operator in multilevel fast multipole algorithm,” IEEE Trans. Antennas Propag. 54, 3822–3826 (2006).
[CrossRef]

Ergül, Ö.

Ö. Ergül, “Fast and accurate solutions of electromagnetics problems involving lossy dielectric objects with the multilevel fast multipole algorithm,” Eng. Anal. Bound. Elem. 36, 423–432 (2012).

Ö. Ergül, “Analysis of composite nanoparticles with surface integral equations and the multilevel fast multipole algorithm,” J. Opt. 14, 062701 (2012).
[CrossRef]

Ewe, W.-B.

W.-B. Ewe, L.-W. Li, and M.-S. Leong, “Solving mixed dielectric/conducting scattering problem using adaptive method,” Prog. Electromagn. Res. 46, 143–163 (2004).
[CrossRef]

Fostier, J.

J. Fostier and F. Olyslager, “An asynchronous parallel MLFMA for scattering at multiple dielectric objects,” IEEE Trans. Antennas Propag. 56, 2346–2355 (2008).
[CrossRef]

Garcia de Abajo, F.

F. Garcia de Abajo and A. Howie, “Retarded field calculation of electron energy loss in inhomogeneous dielectrics,” Phys. Rev. B 65, 115418 (2002).
[CrossRef]

F. Garcia de Abajo and A. Howie, “Relativistic electron energy loss and electron-induced photon emission in inhomogeneous dielectrics,” Phys. Rev. Lett. 80, 5180–5183 (1998).
[CrossRef]

Gedera, M.

L. Medgyesi-Mitschang, J. Putnam, and M. Gedera, “Generalized method of moments for three-dimensional penetrable scatterers,” J. Opt. Sci. Am. A 11, 1383–1398 (1994).

Geng, N.

N. Geng, A. Sullivan, and L. Carin, “Fast multipole method for scattering from an arbitrary PEC target above or buried in a lossy half-space,” IEEE Trans. Antennas Propag. 49, 740–748 (2001).
[CrossRef]

N. Geng, A. Sullivan, and L. Carin, “Multilevel fast-multipole algorithm for scattering from conducting targets above or embedded in a lossy half-space,” IEEE Trans. Geosci. Remote Sens. 38, 1561–1573 (2000).
[CrossRef]

Glisson, A.

D. Schaubert, D. Wilton, and A. Glisson, “A tetrahedral modeling method for electromagnetic scattering by arbitrarily shaped inhomogeneous dielectric bodies,” IEEE Trans. Antennas Propag. 32, 77–85 (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]

Greengard, L.

H. Cheng, L. Greengard, and V. Rokhlin, “A fast adaptive multipole algorithm in three dimensions,” J. Comput. Phys. 155, 468–498 (1999).
[CrossRef]

L. Greengard and V. Rokhlin, “A new version of the fast multipole method for the Laplace equation in three dimensions,” ANU 6, 229–269 (1997).

L. Greengard and V. Rokhlin, “A fast algorithm for particle simulations,” J. Comput. Phys. 73, 325–348 (1987).
[CrossRef]

Gurel, L.

O. Ergul and L. Gurel, “Comparison of integral-equation formulations for the fast and accurate solution of scattering problems involving dielectric objects with the multilevel fast multipole algorithm,” IEEE Trans. Antennas Propag. 57, 176–187 (2009).
[CrossRef]

O. Ergul and L. Gurel, “Optimal interpolation of translation operator in multilevel fast multipole algorithm,” IEEE Trans. Antennas Propag. 54, 3822–3826 (2006).
[CrossRef]

Hohenester, U.

U. Hohenester and A. Trügler, “MNPBEM—A Matlab toolbox for the simulation of plasmonic nanoparticles,” Comput. Phys. Commun. 183, 370–381 (2012).
[CrossRef]

U. Hohenester and A. Trugler, “Interaction of single molecules with metallic nanoparticles,” IEEE J. Sel. Topics Quantum Electron. 14, 1430–1440 (2008).
[CrossRef]

U. Hohenester and J. Krenn, “Surface plasmon resonances of single and coupled metallic nanoparticles: a boundary integral method approach,” Phys. Rev. B 72, 195429 (2005).
[CrossRef]

Howie, A.

F. Garcia de Abajo and A. Howie, “Retarded field calculation of electron energy loss in inhomogeneous dielectrics,” Phys. Rev. B 65, 115418 (2002).
[CrossRef]

F. Garcia de Abajo and A. Howie, “Relativistic electron energy loss and electron-induced photon emission in inhomogeneous dielectrics,” Phys. Rev. Lett. 80, 5180–5183 (1998).
[CrossRef]

Jin, J. M.

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

Jin, J.-M.

J.-M. Jin, The Finite Element Method in Electromagnetics (Wiley, 2002), pp. 165–202.

Kern, A. M.

Krenn, J.

U. Hohenester and J. Krenn, “Surface plasmon resonances of single and coupled metallic nanoparticles: a boundary integral method approach,” Phys. Rev. B 72, 195429 (2005).
[CrossRef]

Landesa, L.

Leong, M.-S.

W.-B. Ewe, L.-W. Li, and M.-S. Leong, “Solving mixed dielectric/conducting scattering problem using adaptive method,” Prog. Electromagn. Res. 46, 143–163 (2004).
[CrossRef]

Li, L.

Li, L.-W.

W.-B. Ewe, L.-W. Li, and M.-S. Leong, “Solving mixed dielectric/conducting scattering problem using adaptive method,” Prog. Electromagn. Res. 46, 143–163 (2004).
[CrossRef]

Liu, Y.

L. Shen and Y. Liu, “An adaptive fast multipole boundary element method for three-dimensional acoustic wave problems based on the Burton–Miller formulation,” J. Comput. Mech. 40, 461–472 (2007).

Lu, C.

C. Lu and W. Chew, “Fast algorithm for solving hybrid integral equations [EM wave scattering],” in Microwaves, Antennas and Propagation, IEE Proceedings H (IEE, 1993), Vol. 140, pp. 455–460.

Lu, C. C.

J. Song, C. C. Lu, and W. C. Chew, “Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects,” IEEE Trans. Antennas Propag. 45, 1488–1493 (1997).
[CrossRef]

C. C. Lu and W. C. Chew, “A multilevel algorithm for solving a boundary integral equation of wave scattering,” Microw. Opt. Technol. Lett. 7, 466–470 (1994).
[CrossRef]

Martin, O. J. F.

Medgyesi-Mitschang, L.

L. Medgyesi-Mitschang, J. Putnam, and M. Gedera, “Generalized method of moments for three-dimensional penetrable scatterers,” J. Opt. Sci. Am. A 11, 1383–1398 (1994).

Moshchalkov, V. V.

G. A. E. Vandenbosch, V. Volski, N. Verellen, and V. V. Moshchalkov, “On the use of the method of moments in plasmonic applications,” Radio Sci.46(5) (2011).
[CrossRef]

Mousavi, S.

S. Mousavi and N. Sukumar, “Generalized Duffy transformation for integrating vertex singularities,” J. Comput. Mech. 45, 127–140 (2010).

Murphy, W. D.

N. Engheta, W. D. Murphy, V. Rokhlin, and M. S. Vassiliou, “The fast multipole method (FMM) for electromagnetic scattering problems,” IEEE Trans. Antennas Propag. 40, 634–641 (1992).
[CrossRef]

Obelleiro, F.

Olyslager, F.

J. Fostier and F. Olyslager, “An asynchronous parallel MLFMA for scattering at multiple dielectric objects,” IEEE Trans. Antennas Propag. 56, 2346–2355 (2008).
[CrossRef]

Palik, E. D.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1998), pp. 294, 356.

Pironneau, O.

E. Darve and O. Pironneau, “Méthodes multipôles rapides: résolution des équations de Maxwell par formulations intégrales,” Ph.D. dissertation (Université de Paris, 1999).

Pozrikidis, C.

C. Pozrikidis, A Practical Guide to Boundary Element Methods with the Software Library Bemlib, illus. ed. (CRC Press, 2002), pp. 117–120.

Putnam, J.

L. Medgyesi-Mitschang, J. Putnam, and M. Gedera, “Generalized method of moments for three-dimensional penetrable scatterers,” J. Opt. Sci. Am. A 11, 1383–1398 (1994).

Rao, S.

K. Umashankar, A. Taflove, and S. Rao, “Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric objects,” IEEE Trans. Antennas Propag. 34, 758–766 (1986).
[CrossRef]

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

Rivero, J.

Rokhlin, V.

H. Cheng, L. Greengard, and V. Rokhlin, “A fast adaptive multipole algorithm in three dimensions,” J. Comput. Phys. 155, 468–498 (1999).
[CrossRef]

L. Greengard and V. Rokhlin, “A new version of the fast multipole method for the Laplace equation in three dimensions,” ANU 6, 229–269 (1997).

R. Coifman, V. Rokhlin, and S. Wandzura, “The fast multipole method for the wave equation: a pedestrian prescription,” IEEE Antennas Propag. Mag. 35, 7–12 (1993).
[CrossRef]

N. Engheta, W. D. Murphy, V. Rokhlin, and M. S. Vassiliou, “The fast multipole method (FMM) for electromagnetic scattering problems,” IEEE Trans. Antennas Propag. 40, 634–641 (1992).
[CrossRef]

L. Greengard and V. Rokhlin, “A fast algorithm for particle simulations,” J. Comput. Phys. 73, 325–348 (1987).
[CrossRef]

V. Rokhlin, “Rapid solution of integral equations of classical potential theory,” J. Comput. Phys. 60, 187–207 (1985).
[CrossRef]

Saad, Y.

Y. Saad and M. H. Schultz, “GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Statist. Comput. 7, 856–869 (1986).
[CrossRef]

Schaubert, D.

D. Schaubert, D. Wilton, and A. Glisson, “A tetrahedral modeling method for electromagnetic scattering by arbitrarily shaped inhomogeneous dielectric bodies,” IEEE Trans. Antennas Propag. 32, 77–85 (1984).
[CrossRef]

Schultz, M. H.

Y. Saad and M. H. Schultz, “GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Statist. Comput. 7, 856–869 (1986).
[CrossRef]

Sertel, K.

K. Sertel and J. L. Volakis, “Multilevel fast multipole method solution of volume integral equations using parametric geometry modeling,” IEEE Trans. Antennas Propag. 52, 1686–1692 (2004).
[CrossRef]

K. Sertel and J. L. Volakis, “Incomplete LU preconditioner for FMM implementation,” Microw. Opt. Technol. Lett. 26, 265–267 (2000).
[CrossRef]

Shen, L.

L. Shen and Y. Liu, “An adaptive fast multipole boundary element method for three-dimensional acoustic wave problems based on the Burton–Miller formulation,” J. Comput. Mech. 40, 461–472 (2007).

Shi, Y.

Solís, D. M.

Song, J.

S. Velamparambil, W. C. Chew, and J. Song, “10 million unknowns: is it that big? [computational electromagnetics],” IEEE Antennas Propag. Mag. 45(2), 43–58 (2003).
[CrossRef]

J. Song, C. C. Lu, and W. C. Chew, “Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects,” IEEE Trans. Antennas Propag. 45, 1488–1493 (1997).
[CrossRef]

Song, J. M.

J. M. Song and W. C. Chew, “Fast multipole method solution of combined field integral equation,” in 11th Annual Review of Progress in Applied Computational Electromagnetics 1 (Naval Postgraduate School, 1995), pp. 629–636.

Sukumar, N.

S. Mousavi and N. Sukumar, “Generalized Duffy transformation for integrating vertex singularities,” J. Comput. Mech. 45, 127–140 (2010).

Sullivan, A.

N. Geng, A. Sullivan, and L. Carin, “Fast multipole method for scattering from an arbitrary PEC target above or buried in a lossy half-space,” IEEE Trans. Antennas Propag. 49, 740–748 (2001).
[CrossRef]

N. Geng, A. Sullivan, and L. Carin, “Multilevel fast-multipole algorithm for scattering from conducting targets above or embedded in a lossy half-space,” IEEE Trans. Geosci. Remote Sens. 38, 1561–1573 (2000).
[CrossRef]

Taboada, J. M.

Taflove, A.

A. Taflove, “Review of the formulation and applications of the finite-difference time-domain method for numerical modeling of electromagnetic wave interactions with arbitrary structures,” Wave Motion 10, 547–582 (1988).
[CrossRef]

K. Umashankar, A. Taflove, and S. Rao, “Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric objects,” IEEE Trans. Antennas Propag. 34, 758–766 (1986).
[CrossRef]

Taskinen, M.

P. Yla-Oijala and M. Taskinen, “Application of combined field integral equation for electromagnetic scattering by dielectric and composite objects,” IEEE Trans. Antennas Propag. 53, 1168–1173 (2005).
[CrossRef]

Trugler, A.

U. Hohenester and A. Trugler, “Interaction of single molecules with metallic nanoparticles,” IEEE J. Sel. Topics Quantum Electron. 14, 1430–1440 (2008).
[CrossRef]

Trügler, A.

U. Hohenester and A. Trügler, “MNPBEM—A Matlab toolbox for the simulation of plasmonic nanoparticles,” Comput. Phys. Commun. 183, 370–381 (2012).
[CrossRef]

Umashankar, K.

K. Umashankar, A. Taflove, and S. Rao, “Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric objects,” IEEE Trans. Antennas Propag. 34, 758–766 (1986).
[CrossRef]

Vandenbosch, G. A. E.

G. A. E. Vandenbosch, V. Volski, N. Verellen, and V. V. Moshchalkov, “On the use of the method of moments in plasmonic applications,” Radio Sci.46(5) (2011).
[CrossRef]

Vassiliou, M. S.

N. Engheta, W. D. Murphy, V. Rokhlin, and M. S. Vassiliou, “The fast multipole method (FMM) for electromagnetic scattering problems,” IEEE Trans. Antennas Propag. 40, 634–641 (1992).
[CrossRef]

Velamparambil, S.

S. Velamparambil, W. C. Chew, and J. Song, “10 million unknowns: is it that big? [computational electromagnetics],” IEEE Antennas Propag. Mag. 45(2), 43–58 (2003).
[CrossRef]

Verellen, N.

G. A. E. Vandenbosch, V. Volski, N. Verellen, and V. V. Moshchalkov, “On the use of the method of moments in plasmonic applications,” Radio Sci.46(5) (2011).
[CrossRef]

Volakis, J. L.

K. Sertel and J. L. Volakis, “Multilevel fast multipole method solution of volume integral equations using parametric geometry modeling,” IEEE Trans. Antennas Propag. 52, 1686–1692 (2004).
[CrossRef]

K. Sertel and J. L. Volakis, “Incomplete LU preconditioner for FMM implementation,” Microw. Opt. Technol. Lett. 26, 265–267 (2000).
[CrossRef]

Volski, V.

G. A. E. Vandenbosch, V. Volski, N. Verellen, and V. V. Moshchalkov, “On the use of the method of moments in plasmonic applications,” Radio Sci.46(5) (2011).
[CrossRef]

Wagner, R. L.

R. L. Wagner and W. C. Chew, “A ray-propagation fast multipole algorithm,” Microw. Opt. Technol. Lett. 7, 435–438 (1994).
[CrossRef]

Wandzura, S.

R. Coifman, V. Rokhlin, and S. Wandzura, “The fast multipole method for the wave equation: a pedestrian prescription,” IEEE Antennas Propag. Mag. 35, 7–12 (1993).
[CrossRef]

Wang, H. G.

Wilkes, D.

C. C. Cha and D. Wilkes, “Method of moments formulation for an arbitrary material configuration,” in Antennas and Propagation Society International Symposium, 1991. AP-S. Digest (IEEE, 1991), pp. 1508–1511.

Wilton, D.

D. Schaubert, D. Wilton, and A. Glisson, “A tetrahedral modeling method for electromagnetic scattering by arbitrarily shaped inhomogeneous dielectric bodies,” IEEE Trans. Antennas Propag. 32, 77–85 (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]

Yla-Oijala, P.

P. Yla-Oijala and M. Taskinen, “Application of combined field integral equation for electromagnetic scattering by dielectric and composite objects,” IEEE Trans. Antennas Propag. 53, 1168–1173 (2005).
[CrossRef]

ANU (1)

L. Greengard and V. Rokhlin, “A new version of the fast multipole method for the Laplace equation in three dimensions,” ANU 6, 229–269 (1997).

Comput. Phys. Commun. (1)

U. Hohenester and A. Trügler, “MNPBEM—A Matlab toolbox for the simulation of plasmonic nanoparticles,” Comput. Phys. Commun. 183, 370–381 (2012).
[CrossRef]

Eng. Anal. Bound. Elem. (1)

Ö. Ergül, “Fast and accurate solutions of electromagnetics problems involving lossy dielectric objects with the multilevel fast multipole algorithm,” Eng. Anal. Bound. Elem. 36, 423–432 (2012).

IEEE Antennas Propag. Mag. (2)

S. Velamparambil, W. C. Chew, and J. Song, “10 million unknowns: is it that big? [computational electromagnetics],” IEEE Antennas Propag. Mag. 45(2), 43–58 (2003).
[CrossRef]

R. Coifman, V. Rokhlin, and S. Wandzura, “The fast multipole method for the wave equation: a pedestrian prescription,” IEEE Antennas Propag. Mag. 35, 7–12 (1993).
[CrossRef]

IEEE J. Sel. Topics Quantum Electron. (1)

U. Hohenester and A. Trugler, “Interaction of single molecules with metallic nanoparticles,” IEEE J. Sel. Topics Quantum Electron. 14, 1430–1440 (2008).
[CrossRef]

IEEE Trans. Antenna Propag. (1)

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

IEEE Trans. Antennas Propag. (11)

J. Fostier and F. Olyslager, “An asynchronous parallel MLFMA for scattering at multiple dielectric objects,” IEEE Trans. Antennas Propag. 56, 2346–2355 (2008).
[CrossRef]

K. Sertel and J. L. Volakis, “Multilevel fast multipole method solution of volume integral equations using parametric geometry modeling,” IEEE Trans. Antennas Propag. 52, 1686–1692 (2004).
[CrossRef]

O. Ergul and L. Gurel, “Comparison of integral-equation formulations for the fast and accurate solution of scattering problems involving dielectric objects with the multilevel fast multipole algorithm,” IEEE Trans. Antennas Propag. 57, 176–187 (2009).
[CrossRef]

P. Yla-Oijala and M. Taskinen, “Application of combined field integral equation for electromagnetic scattering by dielectric and composite objects,” IEEE Trans. Antennas Propag. 53, 1168–1173 (2005).
[CrossRef]

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

K. Umashankar, A. Taflove, and S. Rao, “Electromagnetic scattering by arbitrary shaped three-dimensional homogeneous lossy dielectric objects,” IEEE Trans. Antennas Propag. 34, 758–766 (1986).
[CrossRef]

D. Schaubert, D. Wilton, and A. Glisson, “A tetrahedral modeling method for electromagnetic scattering by arbitrarily shaped inhomogeneous dielectric bodies,” IEEE Trans. Antennas Propag. 32, 77–85 (1984).
[CrossRef]

O. Ergul and L. Gurel, “Optimal interpolation of translation operator in multilevel fast multipole algorithm,” IEEE Trans. Antennas Propag. 54, 3822–3826 (2006).
[CrossRef]

N. Engheta, W. D. Murphy, V. Rokhlin, and M. S. Vassiliou, “The fast multipole method (FMM) for electromagnetic scattering problems,” IEEE Trans. Antennas Propag. 40, 634–641 (1992).
[CrossRef]

J. Song, C. C. Lu, and W. C. Chew, “Multilevel fast multipole algorithm for electromagnetic scattering by large complex objects,” IEEE Trans. Antennas Propag. 45, 1488–1493 (1997).
[CrossRef]

N. Geng, A. Sullivan, and L. Carin, “Fast multipole method for scattering from an arbitrary PEC target above or buried in a lossy half-space,” IEEE Trans. Antennas Propag. 49, 740–748 (2001).
[CrossRef]

IEEE Trans. Geosci. Remote Sens. (1)

N. Geng, A. Sullivan, and L. Carin, “Multilevel fast-multipole algorithm for scattering from conducting targets above or embedded in a lossy half-space,” IEEE Trans. Geosci. Remote Sens. 38, 1561–1573 (2000).
[CrossRef]

J. Comput. Mech. (2)

L. Shen and Y. Liu, “An adaptive fast multipole boundary element method for three-dimensional acoustic wave problems based on the Burton–Miller formulation,” J. Comput. Mech. 40, 461–472 (2007).

S. Mousavi and N. Sukumar, “Generalized Duffy transformation for integrating vertex singularities,” J. Comput. Mech. 45, 127–140 (2010).

J. Comput. Phys. (4)

H. Cheng, L. Greengard, and V. Rokhlin, “A fast adaptive multipole algorithm in three dimensions,” J. Comput. Phys. 155, 468–498 (1999).
[CrossRef]

E. Darve, “The fast multipole method: numerical implementation,” J. Comput. Phys. 160, 195–240 (2000).
[CrossRef]

V. Rokhlin, “Rapid solution of integral equations of classical potential theory,” J. Comput. Phys. 60, 187–207 (1985).
[CrossRef]

L. Greengard and V. Rokhlin, “A fast algorithm for particle simulations,” J. Comput. Phys. 73, 325–348 (1987).
[CrossRef]

J. Opt. (1)

Ö. Ergül, “Analysis of composite nanoparticles with surface integral equations and the multilevel fast multipole algorithm,” J. Opt. 14, 062701 (2012).
[CrossRef]

J. Opt. Sci. Am. A (1)

L. Medgyesi-Mitschang, J. Putnam, and M. Gedera, “Generalized method of moments for three-dimensional penetrable scatterers,” J. Opt. Sci. Am. A 11, 1383–1398 (1994).

J. Opt. Soc. Am. A (3)

Microw. Opt. Technol. Lett. (3)

K. Sertel and J. L. Volakis, “Incomplete LU preconditioner for FMM implementation,” Microw. Opt. Technol. Lett. 26, 265–267 (2000).
[CrossRef]

R. L. Wagner and W. C. Chew, “A ray-propagation fast multipole algorithm,” Microw. Opt. Technol. Lett. 7, 435–438 (1994).
[CrossRef]

C. C. Lu and W. C. Chew, “A multilevel algorithm for solving a boundary integral equation of wave scattering,” Microw. Opt. Technol. Lett. 7, 466–470 (1994).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. B (2)

F. Garcia de Abajo and A. Howie, “Retarded field calculation of electron energy loss in inhomogeneous dielectrics,” Phys. Rev. B 65, 115418 (2002).
[CrossRef]

U. Hohenester and J. Krenn, “Surface plasmon resonances of single and coupled metallic nanoparticles: a boundary integral method approach,” Phys. Rev. B 72, 195429 (2005).
[CrossRef]

Phys. Rev. Lett. (1)

F. Garcia de Abajo and A. Howie, “Relativistic electron energy loss and electron-induced photon emission in inhomogeneous dielectrics,” Phys. Rev. Lett. 80, 5180–5183 (1998).
[CrossRef]

Prog. Electromagn. Res. (1)

W.-B. Ewe, L.-W. Li, and M.-S. Leong, “Solving mixed dielectric/conducting scattering problem using adaptive method,” Prog. Electromagn. Res. 46, 143–163 (2004).
[CrossRef]

SIAM J. Sci. Statist. Comput. (1)

Y. Saad and M. H. Schultz, “GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems,” SIAM J. Sci. Statist. Comput. 7, 856–869 (1986).
[CrossRef]

Wave Motion (1)

A. Taflove, “Review of the formulation and applications of the finite-difference time-domain method for numerical modeling of electromagnetic wave interactions with arbitrary structures,” Wave Motion 10, 547–582 (1988).
[CrossRef]

Other (8)

C. C. Cha and D. Wilkes, “Method of moments formulation for an arbitrary material configuration,” in Antennas and Propagation Society International Symposium, 1991. AP-S. Digest (IEEE, 1991), pp. 1508–1511.

C. Pozrikidis, A Practical Guide to Boundary Element Methods with the Software Library Bemlib, illus. ed. (CRC Press, 2002), pp. 117–120.

G. A. E. Vandenbosch, V. Volski, N. Verellen, and V. V. Moshchalkov, “On the use of the method of moments in plasmonic applications,” Radio Sci.46(5) (2011).
[CrossRef]

J.-M. Jin, The Finite Element Method in Electromagnetics (Wiley, 2002), pp. 165–202.

E. Darve and O. Pironneau, “Méthodes multipôles rapides: résolution des équations de Maxwell par formulations intégrales,” Ph.D. dissertation (Université de Paris, 1999).

J. M. Song and W. C. Chew, “Fast multipole method solution of combined field integral equation,” in 11th Annual Review of Progress in Applied Computational Electromagnetics 1 (Naval Postgraduate School, 1995), pp. 629–636.

C. Lu and W. Chew, “Fast algorithm for solving hybrid integral equations [EM wave scattering],” in Microwaves, Antennas and Propagation, IEE Proceedings H (IEE, 1993), Vol. 140, pp. 455–460.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1998), pp. 294, 356.

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Figures (8)

Fig. 1.
Fig. 1.

Host medium is denoted by Ω1 and is characterized by a permittivity ε1 (to remain consistent with Section 2.B), while the volume inside the lossy dielectric body is denoted by Ω2and characterized by ε2. The interface between the two media is denoted by Γ and the normal vector nΓ is oriented toward Ω1.

Fig. 2.
Fig. 2.

Three objects and four distinct media denoted by Ωj, where j=14, with permittivity εj. The host medium is denoted by Ω1. Let us remark that Ω3 is delimited by the two interfaces Γ2 and Γ3 The orientations of normal vectors are defined only once regardless of the medium. The normal vectors are always directed toward the host medium.

Fig. 3.
Fig. 3.

Hierarchical division of all objects in an octree structure. The right figure is a zoom on the content of the grey box.

Fig. 4.
Fig. 4.

Absorption, scattering and extinction efficiencies for a gold sphere. Comparison between MLFMM and Mie’s theory.

Fig. 5.
Fig. 5.

Bistatic RCS for a near resonance gold sphere of radius 0.145λ0.

Fig. 6.
Fig. 6.

Normalized RMS error (erms) of the extinction efficiency (top) and average relative error BEM-MLFMM of gold spheres of radius R1=λ0/2 at λ0=548.6nm and εr=5.8.6+2.1i (bottom).

Fig. 7.
Fig. 7.

Bistatic RCS of a silica coated gold sphere.

Fig. 8.
Fig. 8.

MLFMM and BEM simulation of scattering on nanoparticle Au–Ag heterodimer. Norm of the electric field represented on the surface of particles (top). Comparison of the bistatic RCS MLFMM and BEM computation (bottom).

Tables (1)

Tables Icon

Table 1. Number of Unknowns, Number of Levels, Drop Tolerance Value, Number of Iterations, Memory Requirements and Computation Time for BEM and MLFMM, and Relative Error of Extinction (eext)

Equations (53)

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

×EikB=0,·B=0,×H+ikD=jc,·D=ρ,
E=ikAϕ,
μH=×A,
·A=ikεμϕ.
(+k2εμ)ϕ=(ρ/εσΓ),
(+k2εμ)A=1c(μj+hΓ).
ϕ(r)=ϕje(r)+ΓGj(|rs|)σj(s)ds,rΩj,j=1,2,
A(r)=Aje(r)+ΓGj(|rs|)hj(s)ds,rΩj,j=1,2,
ϕje(r)=1εj(ω)ΩjGj(|rr|)ρ(r)dr,
Aje(r)=μj(ω)cΩjGj(|rr|)j(r)dr,
Gj(|r|)=exp(ikj|r|)4π|r|,rΩj,
[Gint]σint[Gout]σout=ϕouteϕinte,
[Gint]hint[Gout]hout=AouteAinte,
[Hint]hint[Hout]houtiknT([Gint]εintσint[Gout]εoutσout)=α,
[Hint]εintσint[Hout]εoutσoutiknT·([Gint]εinthint[Gout]εouthout)=De,
α=(nΓ·Γ)(AouteAinte)+iknΓ(εintμintϕinteεoutμoutϕoute),De=nΓ·[εint(ikAinteΓϕinte)εout(ikAouteΓϕoute)].
[Gj]a,b=TbGj(|sas|)ds,saTa,
[Hj]a,b=TbnTa·TaGj(|sas|)ds,saTa,
je:{Discretization(a=2NΓa)}{1,,N},Taje(Ta)=n
[Ge]a,b=[Gje(Ta)]a,bδje(Ta)je(Tb),
[Gje(Ta)]a,b=TbGje(Ta)(|sas|)ds.
[Ge]aσe=b=1N[Gje(Ta)]a,bσe,bδje(Ta)je(Tb),
[He]aσe=b=1N[Hje(Ta)]a,bσe,bδje(Ta)je(Tb),
[Ge]aσe=b=1N[Gje(Ta)]a,b(σe,bδje(Ta)je(Tb)+σe,bδje(Ta)je(Tb)).
[Ge]a,b=[Gje(Tb)]a,b(δje(Ta)je(Tb)δje(Ta)je(Tb)),
[He]a,b=[Hje(Tb)]a,b(δje(Ta)je(Tb)δje(Ta)je(Tb)),
Cyl(n,·)Γ={TCylΓ|jint(T)=n},
Cyl(·,m)={TCylΓ|jout(T)=m}.
[Ge]σe=[Ge]nearσe+[Ge]farσe,
[He]σe=[He]nearσe+[He]farσe,
CxlΓ=nCxl(n,·)Γ=mCxl(·,m)Γ.
(Gefar·σ)=Cylsuburb(Cxl)Cy(l)ΓGje(Tb)(|xay|)(δje(Ta)je(Tb)δje(Ta)je(Tb))σ(y)dy,
(Hefar·σ)=Cylsuburb(Cxl)Cy(l)ΓHje(Tb)(|xay|)(δje(Ta)je(Tb)δje(Ta)je(Tb))σ(y)dy.
(Gintfar·σ)=ikn16π2S2eikns,xlxGCxl(n,·)(s)dsmjout{Cxl(n,·)}ikm16π2S2eikms,xlxGCxl(n,m)(s)ds,
(Hintfar·σ)=kn216π2S2eikns,xlxHCxl(n,·)(s)ds+mjout{Cxl(n,·)}km216π2S2eikms,xlxHCxl(n,m)(s)ds,
GCx(l)(n,·)(s)=Cylsuburb(Cxl)Πjint(Cy(l))(n)TrlL(s)FCy(l)(n,·)(s),
GCx(l)(n,m)(s)=Cylsuburb(Cxl)Πjint(Cy(l))(m)TrlL(s)FCy(l)(m,·)(s),
HCx(l)(n,·)(s)=Cylsuburb(Cxl)Πjint(Cy(l))(n)TrlL(s)F¯Cy(l)(n,·)(s),
HCx(l)(n,m)(s)=Cylsuburb(Cxl)Πjint(Cy(l))(m)TrlL(s)F¯Cy(l)(m,·)(s),
FCy(l)(p,·)(s)=Cy(p,·)Γeikps,yylσ(y)dy,
F¯Cy(l)(p,·)(s)=Cy(l)(p,·)Γ(s·nCy(l)(p,·)Γ)eikps,yylσ(y)dy,
(Gextfar·σ)=ikm16π2sS2eikms,xlxGCxl(·,m)(s)dsnjint{Cxl(·,m)}ikn16π2S2eikns,xlxG¯Cxl(n,m)(s)ds,
(Hextfar·σ)=km216π2S2eikms,xlxHCxl(·,m)(s)ds+mjint{Cxl(·,m)}kn216π2S2eikns,xlxH¯Cxl(n,m)(s)ds,
GCx(l)(·,m)(s)=Cylsuburb(Cxl)Πjext(Cy(l))(m)TrlL(s)FCy(l)(·,m)(s),
G¯Cx(l)(n,m)(s)=Cylsuburb(Cxl)Πjext(Cy(l))(n)TrlL(s)FCy(l)(·,n)(s),
HCx(l)(·,m)(s)=Cylsuburb(Cxl)Πjext(Cy(l))(m)TrlL(s)F¯Cy(l)(·,m)(s),
H¯Cx(l)(n,m)(s)=Cylsuburb(Cxl)Πjext(Cy(l))(n)TrlL(s)F¯Cy(l)(·,n)(s),
FCy(l)(·,p)(s)=Cy(l)(·,p)Γeikps,yylσ(y)dy,
F¯Cy(l)(·,p)(s)=Cy(l)(.,p)Γ(s·nCy(l)(·,p)Γ)eikp<s,yyl>σ(y)dy.
ΠC(x)={1sixC0sinon
TrlL(s,n)=l=0L(2l+1)i2hl(1)(knrl)Pl(cos(s,rl)),
[A]x=([A]near+[A]far)x,
[A]near1[A]x=([I]+[A]near1[A]far)x.

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