Abstract

The analysis of fields in periodic dielectric structures arise in numerous applications of recent interest, ranging from photonic bandgap structures and plasmonically active nanostructures to metamaterials. To achieve an accurate representation of the fields in these structures using numerical methods, dense spatial discretization is required. This, in turn, affects the cost of analysis, particularly for integral-equation-based methods, for which traditional iterative methods require O(N2) operations, N being the number of spatial degrees of freedom. In this paper, we introduce a method for the rapid solution of volumetric electric field integral equations used in the analysis of doubly periodic dielectric structures. The crux of our method is the accelerated Cartesian expansion algorithm, which is used to evaluate the requisite potentials in O(N) cost. Results are provided that corroborate our claims of acceleration without compromising accuracy, as well as the application of our method to a number of compelling photonics applications.

© 2012 Optical Society of America

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    [CrossRef]
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2011 (1)

2010 (4)

M. Vikram, A. Baczewski, B. Shanker, and L. Kempel, “Accelerated Cartesian expansion (ACE) based framework for the rapid evaluation of diffusion, lossy wave, and Klein–Gordon potentials,” J. Comput. Phys. 229, 9119–9134 (2010).
[CrossRef]

S. Xiao, V. Drachev, A. Kildishev, X. Ni, U. Chettiar, H.-K. Yuan, and V. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[CrossRef]

S. Li, D. Orden Van, and V. Lomakin, “Fast periodic interpolation method for periodic unit cell problems,” IEEE Trans. Antennas Propag. 58, 4005–4014 (2010).
[CrossRef]

S. Li, B. Livshitz, and V. Lomakin, “Fast evaluation of Helmholtz potential on graphics processing units (GPUs),” J. Comput. Phys. 229, 8463–8483 (2010).
[CrossRef]

2009 (2)

M. Vikram, H. Huang, B. Shanker, and T. Van, “A novel wideband FMM for fast integral equation solution of multiscale problems in electromagnetics,” IEEE Trans. Antennas Propag. 57, 2094–2104 (2009).
[CrossRef]

S. Xiao, U. Chettiar, A. Kildishev, V. Drachev, and V. Shalaev, “Yellow-light negative-index metamaterials,” Opt. Lett. 34, 3478–3480 (2009).
[CrossRef]

2008 (1)

Y. Otani and N. Nishimura, “A periodic FMM for Maxwell’s equations in 3D and its applications to problems related to photonic crystals,” J. Comput. Phys. 227, 4630–4652(2008).
[CrossRef]

2007 (4)

A. Parker and H. Townley, “Biomimetics of photonic nanostructures,” Nat. Nanotechnol. 2, 347–353 (2007).
[CrossRef]

B. Shanker and H. Huang, “Accelerated Cartesian expansions—a fast method for computing of potential of the form r−ν for all real ν,” J. Comput. Phys. 226, 732–753 (2007).
[CrossRef]

M. Vikram and B. Shanker, “Fast evaluation of time domain fields in sub-wavelength source/observer distributions using accelerated Cartesian expansions (ACE),” J. Comput. Phys. 227, 1007–1023 (2007).
[CrossRef]

V. Shalaev, “Optical negative-index metamaterials,” Nat. Photon. 1, 41–48 (2007).
[CrossRef]

2006 (2)

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[CrossRef]

J. Huang, X. Wang, and Z. Wang, “Controlled replication of butterfly wings for achieving tunable photonic properties,” Nano Lett. 6, 2325–2331 (2006).
[CrossRef]

2005 (2)

L. Jiang and W. Chew, “A mixed-form fast multipole algorithm,” IEEE Trans. Antennas Propag. 53, 4145–4156 (2005).
[CrossRef]

G. Kobidze, B. Shanker, and D. Nyquist, “Efficient integral-equation-based method for accurate analysis of scattering from periodically arranged nanostructures,” Phys. Rev. E 72, 056702 (2005).
[CrossRef]

2004 (1)

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. Gibbs, G. Rupper, C. Ell, O. Shchekin, and D. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[CrossRef]

2003 (2)

W. Barnes, A. Dereuk, and T. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef]

P. Vukusic and J. Sambles, “Photonic structures in biology,” Nature 424, 852–855 (2003).
[CrossRef]

1999 (3)

M. Srinivasarao, “Nano-optics in the biological world: Beetles, butterflies, birds, and moths,” Chem. Rev. 99, 1935–1962 (1999).
[CrossRef]

T. Eibert, J. Volakis, D. Wilton, and D. Jackson, “Hybrid FE/BI modeling of 3-D doubly periodic structures utilizing triangular prismatic elements and an MPIE formulation accelerated by the Ewald transformation,” IEEE Trans. Antennas Propag. 47, 843–850 (1999).
[CrossRef]

T. Eibert and J. Volakis, “Adaptive integral method for hybrid FE/BI modelling of 3-D doubly periodic structures,” IEE Proc. H Microw. Antennas Propag. 146, 17–22 (1999).
[CrossRef]

1998 (3)

L. Greengard, J. Huang, V. Rokhlin, and S. Wandzura, “Accelerating fast multipole methods for the Helmholtz equation at low frequencies,” IEEE Comput. Sci. Eng. 5, 32–38 (1998).
[CrossRef]

T. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

S.-Y. Lin, E. Chow, V. Hietala, P. Villeneuve, and J. Joannopoulos, “Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal,” Science 282, 274–276 (1998).
[CrossRef]

1996 (1)

E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, “AIM: adaptive integral method for solving large-scale electromagnetic scattering and radiation problems,” Radio Sci. 31, 1225–1251 (1996).
[CrossRef]

1995 (1)

J. Song and W. Chew, “Multilevel fast-multipole algorithm for solving combined field integral equations of electromagnetic scattering,” Microw. Opt. Technol. Lett. 10, 14–19 (1995).
[CrossRef]

1987 (2)

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

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef]

1984 (2)

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]

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)

M. Inoue, K. Ohtaka, and S. Yanagawa, “Light scattering from macroscopic spherical bodies. II. Reflectivity of light and electromagnetic localized state in a periodic monolayer of dielectric spheres.” Phys. Rev. B 25, 689–699 (1982).
[CrossRef]

1972 (1)

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

1968 (1)

A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,” Z. Phys. 216, 398–410 (1968).
[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]

Baczewski, A.

M. Vikram, A. Baczewski, B. Shanker, and L. Kempel, “Accelerated Cartesian expansion (ACE) based framework for the rapid evaluation of diffusion, lossy wave, and Klein–Gordon potentials,” J. Comput. Phys. 229, 9119–9134 (2010).
[CrossRef]

A. Baczewski and B. Shanker, “An O(N) method for rapidly computing periodic potentials using accelerated Cartesian expansions,” submitted for publication to Journal of Computational Physics. Preprint available at http://arxiv.org/abs/1107.3069v1 .

Barnes, W.

W. Barnes, A. Dereuk, and T. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef]

Bleszynski, E.

E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, “AIM: adaptive integral method for solving large-scale electromagnetic scattering and radiation problems,” Radio Sci. 31, 1225–1251 (1996).
[CrossRef]

E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, “Rigorous modeling of electromagnetic wave interactions with large dense systems of discrete scatterers,” in Ultra-Wideband, Short Pulse Electromagnetics 9 (Springer, 2010), pp. 65–77.

Bleszynski, M.

E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, “AIM: adaptive integral method for solving large-scale electromagnetic scattering and radiation problems,” Radio Sci. 31, 1225–1251 (1996).
[CrossRef]

E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, “Rigorous modeling of electromagnetic wave interactions with large dense systems of discrete scatterers,” in Ultra-Wideband, Short Pulse Electromagnetics 9 (Springer, 2010), pp. 65–77.

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]

Chettiar, U.

S. Xiao, V. Drachev, A. Kildishev, X. Ni, U. Chettiar, H.-K. Yuan, and V. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[CrossRef]

S. Xiao, U. Chettiar, A. Kildishev, V. Drachev, and V. Shalaev, “Yellow-light negative-index metamaterials,” Opt. Lett. 34, 3478–3480 (2009).
[CrossRef]

Chew, W.

L. Jiang and W. Chew, “A mixed-form fast multipole algorithm,” IEEE Trans. Antennas Propag. 53, 4145–4156 (2005).
[CrossRef]

J. Song and W. Chew, “Multilevel fast-multipole algorithm for solving combined field integral equations of electromagnetic scattering,” Microw. Opt. Technol. Lett. 10, 14–19 (1995).
[CrossRef]

Chow, E.

S.-Y. Lin, E. Chow, V. Hietala, P. Villeneuve, and J. Joannopoulos, “Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal,” Science 282, 274–276 (1998).
[CrossRef]

Christy, R.

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

Deppe, D.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. Gibbs, G. Rupper, C. Ell, O. Shchekin, and D. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[CrossRef]

Dereuk, A.

W. Barnes, A. Dereuk, and T. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef]

Drachev, V.

S. Xiao, V. Drachev, A. Kildishev, X. Ni, U. Chettiar, H.-K. Yuan, and V. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[CrossRef]

S. Xiao, U. Chettiar, A. Kildishev, V. Drachev, and V. Shalaev, “Yellow-light negative-index metamaterials,” Opt. Lett. 34, 3478–3480 (2009).
[CrossRef]

Ebbesen, T.

W. Barnes, A. Dereuk, and T. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef]

T. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Eibert, T.

T. Eibert, J. Volakis, D. Wilton, and D. Jackson, “Hybrid FE/BI modeling of 3-D doubly periodic structures utilizing triangular prismatic elements and an MPIE formulation accelerated by the Ewald transformation,” IEEE Trans. Antennas Propag. 47, 843–850 (1999).
[CrossRef]

T. Eibert and J. Volakis, “Adaptive integral method for hybrid FE/BI modelling of 3-D doubly periodic structures,” IEE Proc. H Microw. Antennas Propag. 146, 17–22 (1999).
[CrossRef]

Ell, C.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. Gibbs, G. Rupper, C. Ell, O. Shchekin, and D. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[CrossRef]

Gervais, F.

F. Gervais, Handbook of Optical Constants of Solids (Academic, 1991), Vol. 2, pp. 761–775.

Ghaemi, H.

T. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Gibbs, H.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. Gibbs, G. Rupper, C. Ell, O. Shchekin, and D. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[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]

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]

Greengard, L.

L. Greengard, J. Huang, V. Rokhlin, and S. Wandzura, “Accelerating fast multipole methods for the Helmholtz equation at low frequencies,” IEEE Comput. Sci. Eng. 5, 32–38 (1998).
[CrossRef]

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

Harrington, R.

R. Harrington, Time-Harmonic Electromagnetic Fields (Wiley-IEEE, 2001).

Hendrickson, J.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. Gibbs, G. Rupper, C. Ell, O. Shchekin, and D. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[CrossRef]

Hietala, V.

S.-Y. Lin, E. Chow, V. Hietala, P. Villeneuve, and J. Joannopoulos, “Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal,” Science 282, 274–276 (1998).
[CrossRef]

Huang, H.

M. Vikram, H. Huang, B. Shanker, and T. Van, “A novel wideband FMM for fast integral equation solution of multiscale problems in electromagnetics,” IEEE Trans. Antennas Propag. 57, 2094–2104 (2009).
[CrossRef]

B. Shanker and H. Huang, “Accelerated Cartesian expansions—a fast method for computing of potential of the form r−ν for all real ν,” J. Comput. Phys. 226, 732–753 (2007).
[CrossRef]

Huang, J.

J. Huang, X. Wang, and Z. Wang, “Controlled replication of butterfly wings for achieving tunable photonic properties,” Nano Lett. 6, 2325–2331 (2006).
[CrossRef]

L. Greengard, J. Huang, V. Rokhlin, and S. Wandzura, “Accelerating fast multipole methods for the Helmholtz equation at low frequencies,” IEEE Comput. Sci. Eng. 5, 32–38 (1998).
[CrossRef]

Inoue, M.

M. Inoue, K. Ohtaka, and S. Yanagawa, “Light scattering from macroscopic spherical bodies. II. Reflectivity of light and electromagnetic localized state in a periodic monolayer of dielectric spheres.” Phys. Rev. B 25, 689–699 (1982).
[CrossRef]

Jackson, D.

T. Eibert, J. Volakis, D. Wilton, and D. Jackson, “Hybrid FE/BI modeling of 3-D doubly periodic structures utilizing triangular prismatic elements and an MPIE formulation accelerated by the Ewald transformation,” IEEE Trans. Antennas Propag. 47, 843–850 (1999).
[CrossRef]

Jaroszewicz, T.

E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, “AIM: adaptive integral method for solving large-scale electromagnetic scattering and radiation problems,” Radio Sci. 31, 1225–1251 (1996).
[CrossRef]

E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, “Rigorous modeling of electromagnetic wave interactions with large dense systems of discrete scatterers,” in Ultra-Wideband, Short Pulse Electromagnetics 9 (Springer, 2010), pp. 65–77.

Jiang, L.

L. Jiang and W. Chew, “A mixed-form fast multipole algorithm,” IEEE Trans. Antennas Propag. 53, 4145–4156 (2005).
[CrossRef]

Joannopoulos, J.

S.-Y. Lin, E. Chow, V. Hietala, P. Villeneuve, and J. Joannopoulos, “Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal,” Science 282, 274–276 (1998).
[CrossRef]

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef]

Johnson, P.

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

Kempel, L.

M. Vikram, A. Baczewski, B. Shanker, and L. Kempel, “Accelerated Cartesian expansion (ACE) based framework for the rapid evaluation of diffusion, lossy wave, and Klein–Gordon potentials,” J. Comput. Phys. 229, 9119–9134 (2010).
[CrossRef]

Khitrova, G.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. Gibbs, G. Rupper, C. Ell, O. Shchekin, and D. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[CrossRef]

Kildishev, A.

S. Xiao, V. Drachev, A. Kildishev, X. Ni, U. Chettiar, H.-K. Yuan, and V. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[CrossRef]

S. Xiao, U. Chettiar, A. Kildishev, V. Drachev, and V. Shalaev, “Yellow-light negative-index metamaterials,” Opt. Lett. 34, 3478–3480 (2009).
[CrossRef]

Kobidze, G.

G. Kobidze, B. Shanker, and D. Nyquist, “Efficient integral-equation-based method for accurate analysis of scattering from periodically arranged nanostructures,” Phys. Rev. E 72, 056702 (2005).
[CrossRef]

Lezec, H.

T. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Li, S.

S. Li, B. Livshitz, and V. Lomakin, “Fast evaluation of Helmholtz potential on graphics processing units (GPUs),” J. Comput. Phys. 229, 8463–8483 (2010).
[CrossRef]

S. Li, D. Orden Van, and V. Lomakin, “Fast periodic interpolation method for periodic unit cell problems,” IEEE Trans. Antennas Propag. 58, 4005–4014 (2010).
[CrossRef]

Lin, S.-Y.

S.-Y. Lin, E. Chow, V. Hietala, P. Villeneuve, and J. Joannopoulos, “Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal,” Science 282, 274–276 (1998).
[CrossRef]

Livshitz, B.

S. Li, B. Livshitz, and V. Lomakin, “Fast evaluation of Helmholtz potential on graphics processing units (GPUs),” J. Comput. Phys. 229, 8463–8483 (2010).
[CrossRef]

Lomakin, V.

S. Li, B. Livshitz, and V. Lomakin, “Fast evaluation of Helmholtz potential on graphics processing units (GPUs),” J. Comput. Phys. 229, 8463–8483 (2010).
[CrossRef]

S. Li, D. Orden Van, and V. Lomakin, “Fast periodic interpolation method for periodic unit cell problems,” IEEE Trans. Antennas Propag. 58, 4005–4014 (2010).
[CrossRef]

Mittra, R.

A. Peterson, S. Ray, and R. Mittra, Computational Methods for Electromagnetics (Wiley-IEEE, 1998).

Nair, N.

Ni, X.

S. Xiao, V. Drachev, A. Kildishev, X. Ni, U. Chettiar, H.-K. Yuan, and V. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[CrossRef]

Nishimura, N.

Y. Otani and N. Nishimura, “A periodic FMM for Maxwell’s equations in 3D and its applications to problems related to photonic crystals,” J. Comput. Phys. 227, 4630–4652(2008).
[CrossRef]

Nyquist, D.

G. Kobidze, B. Shanker, and D. Nyquist, “Efficient integral-equation-based method for accurate analysis of scattering from periodically arranged nanostructures,” Phys. Rev. E 72, 056702 (2005).
[CrossRef]

Ohtaka, K.

M. Inoue, K. Ohtaka, and S. Yanagawa, “Light scattering from macroscopic spherical bodies. II. Reflectivity of light and electromagnetic localized state in a periodic monolayer of dielectric spheres.” Phys. Rev. B 25, 689–699 (1982).
[CrossRef]

Orden Van, D.

S. Li, D. Orden Van, and V. Lomakin, “Fast periodic interpolation method for periodic unit cell problems,” IEEE Trans. Antennas Propag. 58, 4005–4014 (2010).
[CrossRef]

Otani, Y.

Y. Otani and N. Nishimura, “A periodic FMM for Maxwell’s equations in 3D and its applications to problems related to photonic crystals,” J. Comput. Phys. 227, 4630–4652(2008).
[CrossRef]

Otto, A.

A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,” Z. Phys. 216, 398–410 (1968).
[CrossRef]

Ozbay, E.

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[CrossRef]

Parker, A.

A. Parker and H. Townley, “Biomimetics of photonic nanostructures,” Nat. Nanotechnol. 2, 347–353 (2007).
[CrossRef]

Peterson, A.

A. Peterson, S. Ray, and R. Mittra, Computational Methods for Electromagnetics (Wiley-IEEE, 1998).

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).
[CrossRef]

Ray, S.

A. Peterson, S. Ray, and R. Mittra, Computational Methods for Electromagnetics (Wiley-IEEE, 1998).

Rokhlin, V.

L. Greengard, J. Huang, V. Rokhlin, and S. Wandzura, “Accelerating fast multipole methods for the Helmholtz equation at low frequencies,” IEEE Comput. Sci. Eng. 5, 32–38 (1998).
[CrossRef]

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

V. Rokhlin and S. Wandzura, “The fast multipole method for periodic structures,” in Antennas and Propagation Society International Symposium (IEEE, 1994), pp. 424–426.

Rupper, G.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. Gibbs, G. Rupper, C. Ell, O. Shchekin, and D. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[CrossRef]

Saad, Y.

Y. Saad, Iterative Methods for Sparse Linear Systems (SIAM, 2003).

Sambles, J.

P. Vukusic and J. Sambles, “Photonic structures in biology,” Nature 424, 852–855 (2003).
[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]

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]

Scherer, A.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. Gibbs, G. Rupper, C. Ell, O. Shchekin, and D. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[CrossRef]

Shalaev, V.

S. Xiao, V. Drachev, A. Kildishev, X. Ni, U. Chettiar, H.-K. Yuan, and V. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[CrossRef]

S. Xiao, U. Chettiar, A. Kildishev, V. Drachev, and V. Shalaev, “Yellow-light negative-index metamaterials,” Opt. Lett. 34, 3478–3480 (2009).
[CrossRef]

V. Shalaev, “Optical negative-index metamaterials,” Nat. Photon. 1, 41–48 (2007).
[CrossRef]

Shanker, B.

N. Nair and B. Shanker, “Generalized method of moments: a framework for analyzing scattering from homogeneous dielectric bodies,” J. Opt. Soc. Am. A 28, 328–340 (2011).
[CrossRef]

M. Vikram, A. Baczewski, B. Shanker, and L. Kempel, “Accelerated Cartesian expansion (ACE) based framework for the rapid evaluation of diffusion, lossy wave, and Klein–Gordon potentials,” J. Comput. Phys. 229, 9119–9134 (2010).
[CrossRef]

M. Vikram, H. Huang, B. Shanker, and T. Van, “A novel wideband FMM for fast integral equation solution of multiscale problems in electromagnetics,” IEEE Trans. Antennas Propag. 57, 2094–2104 (2009).
[CrossRef]

B. Shanker and H. Huang, “Accelerated Cartesian expansions—a fast method for computing of potential of the form r−ν for all real ν,” J. Comput. Phys. 226, 732–753 (2007).
[CrossRef]

M. Vikram and B. Shanker, “Fast evaluation of time domain fields in sub-wavelength source/observer distributions using accelerated Cartesian expansions (ACE),” J. Comput. Phys. 227, 1007–1023 (2007).
[CrossRef]

G. Kobidze, B. Shanker, and D. Nyquist, “Efficient integral-equation-based method for accurate analysis of scattering from periodically arranged nanostructures,” Phys. Rev. E 72, 056702 (2005).
[CrossRef]

A. Baczewski and B. Shanker, “An O(N) method for rapidly computing periodic potentials using accelerated Cartesian expansions,” submitted for publication to Journal of Computational Physics. Preprint available at http://arxiv.org/abs/1107.3069v1 .

Shchekin, O.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. Gibbs, G. Rupper, C. Ell, O. Shchekin, and D. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[CrossRef]

Song, J.

J. Song and W. Chew, “Multilevel fast-multipole algorithm for solving combined field integral equations of electromagnetic scattering,” Microw. Opt. Technol. Lett. 10, 14–19 (1995).
[CrossRef]

Srinivasarao, M.

M. Srinivasarao, “Nano-optics in the biological world: Beetles, butterflies, birds, and moths,” Chem. Rev. 99, 1935–1962 (1999).
[CrossRef]

Thio, T.

T. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Townley, H.

A. Parker and H. Townley, “Biomimetics of photonic nanostructures,” Nat. Nanotechnol. 2, 347–353 (2007).
[CrossRef]

Van, T.

M. Vikram, H. Huang, B. Shanker, and T. Van, “A novel wideband FMM for fast integral equation solution of multiscale problems in electromagnetics,” IEEE Trans. Antennas Propag. 57, 2094–2104 (2009).
[CrossRef]

Vikram, M.

M. Vikram, A. Baczewski, B. Shanker, and L. Kempel, “Accelerated Cartesian expansion (ACE) based framework for the rapid evaluation of diffusion, lossy wave, and Klein–Gordon potentials,” J. Comput. Phys. 229, 9119–9134 (2010).
[CrossRef]

M. Vikram, H. Huang, B. Shanker, and T. Van, “A novel wideband FMM for fast integral equation solution of multiscale problems in electromagnetics,” IEEE Trans. Antennas Propag. 57, 2094–2104 (2009).
[CrossRef]

M. Vikram and B. Shanker, “Fast evaluation of time domain fields in sub-wavelength source/observer distributions using accelerated Cartesian expansions (ACE),” J. Comput. Phys. 227, 1007–1023 (2007).
[CrossRef]

Villeneuve, P.

S.-Y. Lin, E. Chow, V. Hietala, P. Villeneuve, and J. Joannopoulos, “Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal,” Science 282, 274–276 (1998).
[CrossRef]

Volakis, J.

T. Eibert and J. Volakis, “Adaptive integral method for hybrid FE/BI modelling of 3-D doubly periodic structures,” IEE Proc. H Microw. Antennas Propag. 146, 17–22 (1999).
[CrossRef]

T. Eibert, J. Volakis, D. Wilton, and D. Jackson, “Hybrid FE/BI modeling of 3-D doubly periodic structures utilizing triangular prismatic elements and an MPIE formulation accelerated by the Ewald transformation,” IEEE Trans. Antennas Propag. 47, 843–850 (1999).
[CrossRef]

Vukusic, P.

P. Vukusic and J. Sambles, “Photonic structures in biology,” Nature 424, 852–855 (2003).
[CrossRef]

Wandzura, S.

L. Greengard, J. Huang, V. Rokhlin, and S. Wandzura, “Accelerating fast multipole methods for the Helmholtz equation at low frequencies,” IEEE Comput. Sci. Eng. 5, 32–38 (1998).
[CrossRef]

V. Rokhlin and S. Wandzura, “The fast multipole method for periodic structures,” in Antennas and Propagation Society International Symposium (IEEE, 1994), pp. 424–426.

Wang, X.

J. Huang, X. Wang, and Z. Wang, “Controlled replication of butterfly wings for achieving tunable photonic properties,” Nano Lett. 6, 2325–2331 (2006).
[CrossRef]

Wang, Z.

J. Huang, X. Wang, and Z. Wang, “Controlled replication of butterfly wings for achieving tunable photonic properties,” Nano Lett. 6, 2325–2331 (2006).
[CrossRef]

Wilton, D.

T. Eibert, J. Volakis, D. Wilton, and D. Jackson, “Hybrid FE/BI modeling of 3-D doubly periodic structures utilizing triangular prismatic elements and an MPIE formulation accelerated by the Ewald transformation,” IEEE Trans. Antennas Propag. 47, 843–850 (1999).
[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]

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]

Wolff, P.

T. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

Xiao, S.

S. Xiao, V. Drachev, A. Kildishev, X. Ni, U. Chettiar, H.-K. Yuan, and V. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[CrossRef]

S. Xiao, U. Chettiar, A. Kildishev, V. Drachev, and V. Shalaev, “Yellow-light negative-index metamaterials,” Opt. Lett. 34, 3478–3480 (2009).
[CrossRef]

Yanagawa, S.

M. Inoue, K. Ohtaka, and S. Yanagawa, “Light scattering from macroscopic spherical bodies. II. Reflectivity of light and electromagnetic localized state in a periodic monolayer of dielectric spheres.” Phys. Rev. B 25, 689–699 (1982).
[CrossRef]

Yoshie, T.

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. Gibbs, G. Rupper, C. Ell, O. Shchekin, and D. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[CrossRef]

Yuan, H.-K.

S. Xiao, V. Drachev, A. Kildishev, X. Ni, U. Chettiar, H.-K. Yuan, and V. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[CrossRef]

Chem. Rev. (1)

M. Srinivasarao, “Nano-optics in the biological world: Beetles, butterflies, birds, and moths,” Chem. Rev. 99, 1935–1962 (1999).
[CrossRef]

IEE Proc. H Microw. Antennas Propag. (1)

T. Eibert and J. Volakis, “Adaptive integral method for hybrid FE/BI modelling of 3-D doubly periodic structures,” IEE Proc. H Microw. Antennas Propag. 146, 17–22 (1999).
[CrossRef]

IEEE Comput. Sci. Eng. (1)

L. Greengard, J. Huang, V. Rokhlin, and S. Wandzura, “Accelerating fast multipole methods for the Helmholtz equation at low frequencies,” IEEE Comput. Sci. Eng. 5, 32–38 (1998).
[CrossRef]

IEEE Trans. Antennas Propag. (6)

L. Jiang and W. Chew, “A mixed-form fast multipole algorithm,” IEEE Trans. Antennas Propag. 53, 4145–4156 (2005).
[CrossRef]

M. Vikram, H. Huang, B. Shanker, and T. Van, “A novel wideband FMM for fast integral equation solution of multiscale problems in electromagnetics,” IEEE Trans. Antennas Propag. 57, 2094–2104 (2009).
[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]

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]

T. Eibert, J. Volakis, D. Wilton, and D. Jackson, “Hybrid FE/BI modeling of 3-D doubly periodic structures utilizing triangular prismatic elements and an MPIE formulation accelerated by the Ewald transformation,” IEEE Trans. Antennas Propag. 47, 843–850 (1999).
[CrossRef]

S. Li, D. Orden Van, and V. Lomakin, “Fast periodic interpolation method for periodic unit cell problems,” IEEE Trans. Antennas Propag. 58, 4005–4014 (2010).
[CrossRef]

J. Comput. Phys. (6)

S. Li, B. Livshitz, and V. Lomakin, “Fast evaluation of Helmholtz potential on graphics processing units (GPUs),” J. Comput. Phys. 229, 8463–8483 (2010).
[CrossRef]

B. Shanker and H. Huang, “Accelerated Cartesian expansions—a fast method for computing of potential of the form r−ν for all real ν,” J. Comput. Phys. 226, 732–753 (2007).
[CrossRef]

M. Vikram and B. Shanker, “Fast evaluation of time domain fields in sub-wavelength source/observer distributions using accelerated Cartesian expansions (ACE),” J. Comput. Phys. 227, 1007–1023 (2007).
[CrossRef]

M. Vikram, A. Baczewski, B. Shanker, and L. Kempel, “Accelerated Cartesian expansion (ACE) based framework for the rapid evaluation of diffusion, lossy wave, and Klein–Gordon potentials,” J. Comput. Phys. 229, 9119–9134 (2010).
[CrossRef]

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

Y. Otani and N. Nishimura, “A periodic FMM for Maxwell’s equations in 3D and its applications to problems related to photonic crystals,” J. Comput. Phys. 227, 4630–4652(2008).
[CrossRef]

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

Microw. Opt. Technol. Lett. (1)

J. Song and W. Chew, “Multilevel fast-multipole algorithm for solving combined field integral equations of electromagnetic scattering,” Microw. Opt. Technol. Lett. 10, 14–19 (1995).
[CrossRef]

Nano Lett. (1)

J. Huang, X. Wang, and Z. Wang, “Controlled replication of butterfly wings for achieving tunable photonic properties,” Nano Lett. 6, 2325–2331 (2006).
[CrossRef]

Nat. Nanotechnol. (1)

A. Parker and H. Townley, “Biomimetics of photonic nanostructures,” Nat. Nanotechnol. 2, 347–353 (2007).
[CrossRef]

Nat. Photon. (1)

V. Shalaev, “Optical negative-index metamaterials,” Nat. Photon. 1, 41–48 (2007).
[CrossRef]

Nature (5)

S. Xiao, V. Drachev, A. Kildishev, X. Ni, U. Chettiar, H.-K. Yuan, and V. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735–738 (2010).
[CrossRef]

W. Barnes, A. Dereuk, and T. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424, 824–830 (2003).
[CrossRef]

T. Ebbesen, H. Lezec, H. Ghaemi, T. Thio, and P. Wolff, “Extraordinary optical transmission through sub-wavelength hole arrays,” Nature 391, 667–669 (1998).
[CrossRef]

T. Yoshie, A. Scherer, J. Hendrickson, G. Khitrova, H. Gibbs, G. Rupper, C. Ell, O. Shchekin, and D. Deppe, “Vacuum Rabi splitting with a single quantum dot in a photonic crystal nanocavity,” Nature 432, 200–203 (2004).
[CrossRef]

P. Vukusic and J. Sambles, “Photonic structures in biology,” Nature 424, 852–855 (2003).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. B (2)

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

M. Inoue, K. Ohtaka, and S. Yanagawa, “Light scattering from macroscopic spherical bodies. II. Reflectivity of light and electromagnetic localized state in a periodic monolayer of dielectric spheres.” Phys. Rev. B 25, 689–699 (1982).
[CrossRef]

Phys. Rev. E (1)

G. Kobidze, B. Shanker, and D. Nyquist, “Efficient integral-equation-based method for accurate analysis of scattering from periodically arranged nanostructures,” Phys. Rev. E 72, 056702 (2005).
[CrossRef]

Phys. Rev. Lett. (1)

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef]

Radio Sci. (1)

E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, “AIM: adaptive integral method for solving large-scale electromagnetic scattering and radiation problems,” Radio Sci. 31, 1225–1251 (1996).
[CrossRef]

Science (2)

S.-Y. Lin, E. Chow, V. Hietala, P. Villeneuve, and J. Joannopoulos, “Experimental demonstration of guiding and bending of electromagnetic waves in a photonic crystal,” Science 282, 274–276 (1998).
[CrossRef]

E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science 311, 189–193 (2006).
[CrossRef]

Z. Phys. (1)

A. Otto, “Excitation of nonradiative surface plasma waves in silver by the method of frustrated total reflection,” Z. Phys. 216, 398–410 (1968).
[CrossRef]

Other (7)

V. Rokhlin and S. Wandzura, “The fast multipole method for periodic structures,” in Antennas and Propagation Society International Symposium (IEEE, 1994), pp. 424–426.

A. Baczewski and B. Shanker, “An O(N) method for rapidly computing periodic potentials using accelerated Cartesian expansions,” submitted for publication to Journal of Computational Physics. Preprint available at http://arxiv.org/abs/1107.3069v1 .

F. Gervais, Handbook of Optical Constants of Solids (Academic, 1991), Vol. 2, pp. 761–775.

E. Bleszynski, M. Bleszynski, and T. Jaroszewicz, “Rigorous modeling of electromagnetic wave interactions with large dense systems of discrete scatterers,” in Ultra-Wideband, Short Pulse Electromagnetics 9 (Springer, 2010), pp. 65–77.

R. Harrington, Time-Harmonic Electromagnetic Fields (Wiley-IEEE, 2001).

Y. Saad, Iterative Methods for Sparse Linear Systems (SIAM, 2003).

A. Peterson, S. Ray, and R. Mittra, Computational Methods for Electromagnetics (Wiley-IEEE, 1998).

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Figures (9)

Fig. 1.
Fig. 1.

Illustration of the periodic scattering problem described in Section 2.

Fig. 2.
Fig. 2.

Top-down view of the geometry illustrated in Fig. 1 with a four-level octree structure superimposed. Interaction lists are indicated for the dark blue box. Boxes highlighted in blue are in the near field, whereas light blue boxes are in the far field. The interaction between sources in the dark blue box and boxes highlighted in red are effected at a higher level.

Fig. 3.
Fig. 3.

Error convergence for Φfar(r⃗) evaluated using Eq. (6).

Fig. 4.
Fig. 4.

Timing results for ACE kernel code.

Fig. 5.
Fig. 5.

Validation of our ACE-accelerated code against an analytic solution for scattering from a homogeneous dielectric slab of width 20 nm with εr=±4.

Fig. 6.
Fig. 6.

Validation against scattering from an electromagnetic bandgap structure solved using FE-BI in [32]. Discretization has N=6030 unknowns. Average time to solution (without acceleration): 813min. Average time to solution (ACE acceleration): 23min. Total speedup: 37×.

Fig. 7.
Fig. 7.

Validation against scattering from an array of polystyrene spheres (εr=2.56) solved analytically in [33]. Discretization has N=7328. Average time to solution (without acceleration): 2006min. Average time to solution (ACE acceleration): 43min. Total speedup: 46×.

Fig. 8.
Fig. 8.

(Top) Calculated reflectivity of a single model butterfly scale. (Bottom) Scale geometry. A single unit cell with |a⃗1|=|a⃗2|=320nm is outlined in red. The height of the structure out of plane is 280 nm, and the diameter of the larger cylinders is 130 nm, with a center–center spacing of 160 nm between nearest neighbors. The smaller cylinders of diameter 20 nm with axes in the plane of periodicity are oriented along the polarization vector of the incident field, with a center–center spacing of 70 nm out of the plane of periodicity. The resultant mesh has N=10,782. Average time to solution (without acceleration, extrapolated): 1430min. Average time to solution (ACE acceleration): 31min. Total speedup: 46×.

Fig. 9.
Fig. 9.

Demonstration of capability in solving a large scattering problem (N=147,374) inspired by a metamaterial design presented in [39]. An average of 188 iterations per frequency was required, with an average time per iteration of 3.18 min, and an average total solution time of 896 min.

Equations (19)

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

L2={t⃗m,n=ma⃗1+na⃗2|m,nZ}.
L2*={k⃗m,n=mb⃗1+nb⃗2|m,nZ},
J⃗V(r⃗)=jωκ(r⃗)D⃗(r⃗),whereκ(r⃗)=ε(r⃗,ω)ε0ε(r⃗,w).
E⃗inc(r⃗)=E⃗(r⃗)E⃗scat(r⃗),r⃗ΩD,
E⃗inc(r⃗)=D⃗(r⃗)/ε(r⃗,ω)iωμ0ΩDdr⃗g(r⃗,r⃗)κ(r⃗)D⃗(r⃗)iωε0ΩDdr⃗g(r⃗,r⃗)·(κ(r⃗)D⃗(r⃗)).
E⃗(r⃗+t⃗m,n)=eik⃗inc·t⃗m,nE⃗(r⃗).
E⃗inc(r⃗)=D⃗(r⃗,ω)/ε(r⃗,ω)ΩD*dr⃗G¯¯per(r⃗,r⃗)·[κ(r⃗)D⃗(r⃗)],ΩD*=supp(J⃗V)Ω0.
ZμνIν=Vμ,
Iν=cν,Vμ=f⃗μ(r⃗),E⃗inc(r⃗),
Zμν=f⃗μ(r⃗),f⃗ν(r⃗)/ε(r⃗,ω)f⃗μ(r⃗),dr⃗G¯¯per(r⃗,r⃗)·[κ(r⃗)f⃗ν(r⃗)],
D⃗(r⃗)=i=1Ncif⃗i(r⃗),r⃗ΩD*.
f(r⃗r⃗)=n=0(1)nn!r⃗(n)·n·(n)f(r⃗).
ZμνIνZμνnearIν+TACE(Iμ).
Mμ(n)=i=1Ns(1)nn!wμ,i(r⃗ir⃗sc)(n),0nP.
Lμ(n)=m=nP1n!Mμ(mn)·(mn)·(m)gper(|r⃗ocr⃗sc|),0nP.
E⃗μso(r⃗)=(I(2)+(2)k2)·1·n=0PLν(n)·n·(r⃗r⃗oc)(n),r⃗Ωo.
Φ(r⃗)=Ω0dr⃗gper(r⃗,r⃗)f(r⃗),
f(r⃗)=i=1Nwiδ(r⃗r⃗i).
ϵfar=i=1N||ΦfarACE(r⃗i)Φfardirect(r⃗i)||2i=1N||Φfardirect(r⃗i)||2.

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