Abstract

We demonstrate that simulating plasmonic nanostructures by means of curved elements (CEs) significantly increases the accuracy and computation speed not only in the linear but also in the nonlinear regime. We implemented CEs within the discontinuous Galerkin (DG) method and, as an example of a nonlinear effect, investigated second-harmonic generation (SHG) at a silver nanoparticle. The second-harmonic response of the material is simulated by an extended Lorentz model (ELM). In the linear regime the CEs are ≈ 9 times faster than ordinary elements for the same accuracy, provide a much better convergence and show fewer unphysical field artifacts. For DG-SHG calculations CEs are almost indispensable to obtain physically reasonable results at all. Additionally, their boundary approximation has to be of the same order as their polynomial degree to achieve artifact-free field distributions. In return, the use of such CEs with the DG method pays off more than evidently, since the additional computation time is only 1%.

© 2011 OSA

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  1. V. Sandoghdar, E. Klotzsch, V. Jacobsen, A. Renn, U. Håkanson, M. Agio, I. Gerhardt, J. Seelig, and G. Wrigge, “Optical detection of very small nonfluorescent nanoparticles,” Chimia 60, 761–764 (2006).
    [CrossRef]
  2. M. T. Wenzel, T. Härtling, P. Olk, S. C. Kehr, S. Grafström, S. Winnerl, M. Helm, and L. M. Eng, “Gold nanoparticle tips for optical field confinement in infrared scattering near-field optical microscopy,” Opt. Express 16, 12302–12312 (2008).
    [CrossRef] [PubMed]
  3. V. Deckert, “Tip-enhanced Raman spectroscopy,” J. Raman Spectrosc. 40, 1336–1337 (2009).
    [CrossRef]
  4. P. Olk, J. Renger, T. Härtling, M. T. Wenzel, and L. M. Eng, “Two particle enhanced nano Raman microscopy and spectroscopy,” Nano Lett. 7, 1736–1740 (2007).
    [CrossRef] [PubMed]
  5. J. Renger, R. Quidant, N. V. Hulst, and L. Novotny, “Surface-enhanced nonlinear four-wave mixing,” Phys. Rev. Lett. 104, 046803 (2010).
    [CrossRef] [PubMed]
  6. T. Hanke, G. Krauss, D. Träutlein, B. Wild, R. Bratschitsch, and A. Leitenstorfer, “Efficient nonlinear light emission of single gold optical antennas driven by few-cycle near-infrared pulses,” Phys. Rev. Lett. 103, 257404 (2009).
    [CrossRef]
  7. M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100, 203002 (2008).
    [CrossRef] [PubMed]
  8. G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
    [CrossRef]
  9. A. Hille, R. Kullock, S. Grafstrom, and L. M. Eng, “Improving nano-optical simulations through curved elements implemented within the discontinuous Galerkin method,” J. Comput. Theor. Nanosci. 7, 1581–1586 (2010).
    [CrossRef]
  10. J. Niegemann, M. Konig, C. Prohm, R. Diehl, and K. Busch, “Using curved elements in the discontinuous Galerkin time-domain approach,” AIP Conf. Proc. 1291, 76–78 (2010).
    [CrossRef]
  11. J. Hesthaven and T. Warburton, “Nodal high-order methods on unstructured grids I. time-domain solution of Maxwell’s equations,” J. Comput. Phys. 181, 186–221 (2002).
    [CrossRef]
  12. T. Lu, P. Zhang, and W. Cai, “Discontinuous Galerkin methods for dispersive and lossy Maxwell’s equations and PML boundary conditions,” J. Comput. Phys. 200, 549–580 (2004).
    [CrossRef]
  13. K. Stannigel, M. König, J. Niegemann, and K. Busch, “Discontinuous Galerkin time-domain computations of metallic nanostructures,” Opt. Express 17, 14934–14947 (2009).
    [CrossRef] [PubMed]
  14. J. Niegemann, M. König, K. Stannigel, and K. Busch, “Higher-order time-domain methods for the analysis of nano-photonic systems,” Photonics Nanostruct. Fundam. Appl. 7, 2–11 (2009).
    [CrossRef]
  15. J. Niegemann, W. Pernice, and K. Busch, “Simulation of optical resonators using DGTD and FDTD,” J. Opt. A, Pure Appl. Opt. 11, 114015 (2009).
    [CrossRef]
  16. J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications (Springer, 2008).
    [CrossRef]
  17. R. Diehl, K. Busch, and J. Niegemann, “Comparison of low-storage Runge-Kutta schemes for discontinuous-Galerkin time-domain simulations of Maxwell’s equations,” J. Comput. Theor. Nanosci. 7, 1572–1580 (2010).
    [CrossRef]
  18. J. Schöberl, “NETGEN an advancing front 2D/3D-mesh generator based on abstract rules,” Comput. Visual. Sci. 1, 41–52 (1997).
    [CrossRef]
  19. C. Hafner, Post-Modern Electromagnetics: Using Intelligent MaXwell Solvers (John Wiley & Sons, 1999).
  20. J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett. 83, 4045–4048 (1999).
    [CrossRef]
  21. G. Bachelier, J. Butet, I. Russier-Antoine, C. Jonin, E. Benichou, and P. Brevet, “Origin of optical second-harmonic generation in spherical gold nanoparticles: Local surface and nonlocal bulk contributions,” Phys. Rev. B 82, 235403 (2010).
    [CrossRef]
  22. J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second-harmonic generation at metal surfaces,” Phys. Rev. B 21, 4389–4402 (1980).
    [CrossRef]
  23. M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82, 043828 (2010).
    [CrossRef]
  24. M. C. Larciprete, A. Belardini, M. G. Cappeddu, D. de Ceglia, M. Centini, E. Fazio, C. Sibilia, M. J. Bloemer, and M. Scalora, “Second-harmonic generation from metallodielectric multilayer photonic-band-gap structures,” Phys. Rev. A 77, 013809 (2008).
    [CrossRef]
  25. J. Butet, G. Bachelier, I. Russier-Antoine, C. Jonin, E. Benichou, and P. Brevet, “Interference between selected dipoles and octupoles in the optical second-harmonic generation from spherical gold nanoparticles,” Phys. Rev. Lett. 105, 077401 (2010).
    [CrossRef] [PubMed]
  26. M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, F. J. G. de Abajo, W. Pfeiffer, M. Rohmer, C. Spindler, and F. Steeb, “Adaptive subwavelength control of nano-optical fields,” Nature 446, 301–304 (2007).
    [CrossRef] [PubMed]

2010 (7)

J. Renger, R. Quidant, N. V. Hulst, and L. Novotny, “Surface-enhanced nonlinear four-wave mixing,” Phys. Rev. Lett. 104, 046803 (2010).
[CrossRef] [PubMed]

A. Hille, R. Kullock, S. Grafstrom, and L. M. Eng, “Improving nano-optical simulations through curved elements implemented within the discontinuous Galerkin method,” J. Comput. Theor. Nanosci. 7, 1581–1586 (2010).
[CrossRef]

J. Niegemann, M. Konig, C. Prohm, R. Diehl, and K. Busch, “Using curved elements in the discontinuous Galerkin time-domain approach,” AIP Conf. Proc. 1291, 76–78 (2010).
[CrossRef]

R. Diehl, K. Busch, and J. Niegemann, “Comparison of low-storage Runge-Kutta schemes for discontinuous-Galerkin time-domain simulations of Maxwell’s equations,” J. Comput. Theor. Nanosci. 7, 1572–1580 (2010).
[CrossRef]

G. Bachelier, J. Butet, I. Russier-Antoine, C. Jonin, E. Benichou, and P. Brevet, “Origin of optical second-harmonic generation in spherical gold nanoparticles: Local surface and nonlocal bulk contributions,” Phys. Rev. B 82, 235403 (2010).
[CrossRef]

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82, 043828 (2010).
[CrossRef]

J. Butet, G. Bachelier, I. Russier-Antoine, C. Jonin, E. Benichou, and P. Brevet, “Interference between selected dipoles and octupoles in the optical second-harmonic generation from spherical gold nanoparticles,” Phys. Rev. Lett. 105, 077401 (2010).
[CrossRef] [PubMed]

2009 (5)

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

J. Niegemann, M. König, K. Stannigel, and K. Busch, “Higher-order time-domain methods for the analysis of nano-photonic systems,” Photonics Nanostruct. Fundam. Appl. 7, 2–11 (2009).
[CrossRef]

J. Niegemann, W. Pernice, and K. Busch, “Simulation of optical resonators using DGTD and FDTD,” J. Opt. A, Pure Appl. Opt. 11, 114015 (2009).
[CrossRef]

V. Deckert, “Tip-enhanced Raman spectroscopy,” J. Raman Spectrosc. 40, 1336–1337 (2009).
[CrossRef]

T. Hanke, G. Krauss, D. Träutlein, B. Wild, R. Bratschitsch, and A. Leitenstorfer, “Efficient nonlinear light emission of single gold optical antennas driven by few-cycle near-infrared pulses,” Phys. Rev. Lett. 103, 257404 (2009).
[CrossRef]

2008 (4)

M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100, 203002 (2008).
[CrossRef] [PubMed]

J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications (Springer, 2008).
[CrossRef]

M. T. Wenzel, T. Härtling, P. Olk, S. C. Kehr, S. Grafström, S. Winnerl, M. Helm, and L. M. Eng, “Gold nanoparticle tips for optical field confinement in infrared scattering near-field optical microscopy,” Opt. Express 16, 12302–12312 (2008).
[CrossRef] [PubMed]

M. C. Larciprete, A. Belardini, M. G. Cappeddu, D. de Ceglia, M. Centini, E. Fazio, C. Sibilia, M. J. Bloemer, and M. Scalora, “Second-harmonic generation from metallodielectric multilayer photonic-band-gap structures,” Phys. Rev. A 77, 013809 (2008).
[CrossRef]

2007 (2)

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, F. J. G. de Abajo, W. Pfeiffer, M. Rohmer, C. Spindler, and F. Steeb, “Adaptive subwavelength control of nano-optical fields,” Nature 446, 301–304 (2007).
[CrossRef] [PubMed]

P. Olk, J. Renger, T. Härtling, M. T. Wenzel, and L. M. Eng, “Two particle enhanced nano Raman microscopy and spectroscopy,” Nano Lett. 7, 1736–1740 (2007).
[CrossRef] [PubMed]

2006 (1)

V. Sandoghdar, E. Klotzsch, V. Jacobsen, A. Renn, U. Håkanson, M. Agio, I. Gerhardt, J. Seelig, and G. Wrigge, “Optical detection of very small nonfluorescent nanoparticles,” Chimia 60, 761–764 (2006).
[CrossRef]

2004 (1)

T. Lu, P. Zhang, and W. Cai, “Discontinuous Galerkin methods for dispersive and lossy Maxwell’s equations and PML boundary conditions,” J. Comput. Phys. 200, 549–580 (2004).
[CrossRef]

2002 (1)

J. Hesthaven and T. Warburton, “Nodal high-order methods on unstructured grids I. time-domain solution of Maxwell’s equations,” J. Comput. Phys. 181, 186–221 (2002).
[CrossRef]

1999 (2)

C. Hafner, Post-Modern Electromagnetics: Using Intelligent MaXwell Solvers (John Wiley & Sons, 1999).

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett. 83, 4045–4048 (1999).
[CrossRef]

1997 (1)

J. Schöberl, “NETGEN an advancing front 2D/3D-mesh generator based on abstract rules,” Comput. Visual. Sci. 1, 41–52 (1997).
[CrossRef]

1980 (1)

J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second-harmonic generation at metal surfaces,” Phys. Rev. B 21, 4389–4402 (1980).
[CrossRef]

1908 (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Aeschlimann, M.

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, F. J. G. de Abajo, W. Pfeiffer, M. Rohmer, C. Spindler, and F. Steeb, “Adaptive subwavelength control of nano-optical fields,” Nature 446, 301–304 (2007).
[CrossRef] [PubMed]

Agio, M.

V. Sandoghdar, E. Klotzsch, V. Jacobsen, A. Renn, U. Håkanson, M. Agio, I. Gerhardt, J. Seelig, and G. Wrigge, “Optical detection of very small nonfluorescent nanoparticles,” Chimia 60, 761–764 (2006).
[CrossRef]

Akozbek, N.

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82, 043828 (2010).
[CrossRef]

Bachelier, G.

J. Butet, G. Bachelier, I. Russier-Antoine, C. Jonin, E. Benichou, and P. Brevet, “Interference between selected dipoles and octupoles in the optical second-harmonic generation from spherical gold nanoparticles,” Phys. Rev. Lett. 105, 077401 (2010).
[CrossRef] [PubMed]

G. Bachelier, J. Butet, I. Russier-Antoine, C. Jonin, E. Benichou, and P. Brevet, “Origin of optical second-harmonic generation in spherical gold nanoparticles: Local surface and nonlocal bulk contributions,” Phys. Rev. B 82, 235403 (2010).
[CrossRef]

Bauer, M.

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, F. J. G. de Abajo, W. Pfeiffer, M. Rohmer, C. Spindler, and F. Steeb, “Adaptive subwavelength control of nano-optical fields,” Nature 446, 301–304 (2007).
[CrossRef] [PubMed]

Bayer, D.

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, F. J. G. de Abajo, W. Pfeiffer, M. Rohmer, C. Spindler, and F. Steeb, “Adaptive subwavelength control of nano-optical fields,” Nature 446, 301–304 (2007).
[CrossRef] [PubMed]

Belardini, A.

M. C. Larciprete, A. Belardini, M. G. Cappeddu, D. de Ceglia, M. Centini, E. Fazio, C. Sibilia, M. J. Bloemer, and M. Scalora, “Second-harmonic generation from metallodielectric multilayer photonic-band-gap structures,” Phys. Rev. A 77, 013809 (2008).
[CrossRef]

Benichou, E.

J. Butet, G. Bachelier, I. Russier-Antoine, C. Jonin, E. Benichou, and P. Brevet, “Interference between selected dipoles and octupoles in the optical second-harmonic generation from spherical gold nanoparticles,” Phys. Rev. Lett. 105, 077401 (2010).
[CrossRef] [PubMed]

G. Bachelier, J. Butet, I. Russier-Antoine, C. Jonin, E. Benichou, and P. Brevet, “Origin of optical second-harmonic generation in spherical gold nanoparticles: Local surface and nonlocal bulk contributions,” Phys. Rev. B 82, 235403 (2010).
[CrossRef]

Bloemer, M. J.

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82, 043828 (2010).
[CrossRef]

M. C. Larciprete, A. Belardini, M. G. Cappeddu, D. de Ceglia, M. Centini, E. Fazio, C. Sibilia, M. J. Bloemer, and M. Scalora, “Second-harmonic generation from metallodielectric multilayer photonic-band-gap structures,” Phys. Rev. A 77, 013809 (2008).
[CrossRef]

Bratschitsch, R.

T. Hanke, G. Krauss, D. Träutlein, B. Wild, R. Bratschitsch, and A. Leitenstorfer, “Efficient nonlinear light emission of single gold optical antennas driven by few-cycle near-infrared pulses,” Phys. Rev. Lett. 103, 257404 (2009).
[CrossRef]

Brevet, P.

G. Bachelier, J. Butet, I. Russier-Antoine, C. Jonin, E. Benichou, and P. Brevet, “Origin of optical second-harmonic generation in spherical gold nanoparticles: Local surface and nonlocal bulk contributions,” Phys. Rev. B 82, 235403 (2010).
[CrossRef]

J. Butet, G. Bachelier, I. Russier-Antoine, C. Jonin, E. Benichou, and P. Brevet, “Interference between selected dipoles and octupoles in the optical second-harmonic generation from spherical gold nanoparticles,” Phys. Rev. Lett. 105, 077401 (2010).
[CrossRef] [PubMed]

Brixner, T.

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, F. J. G. de Abajo, W. Pfeiffer, M. Rohmer, C. Spindler, and F. Steeb, “Adaptive subwavelength control of nano-optical fields,” Nature 446, 301–304 (2007).
[CrossRef] [PubMed]

Busch, K.

R. Diehl, K. Busch, and J. Niegemann, “Comparison of low-storage Runge-Kutta schemes for discontinuous-Galerkin time-domain simulations of Maxwell’s equations,” J. Comput. Theor. Nanosci. 7, 1572–1580 (2010).
[CrossRef]

J. Niegemann, M. Konig, C. Prohm, R. Diehl, and K. Busch, “Using curved elements in the discontinuous Galerkin time-domain approach,” AIP Conf. Proc. 1291, 76–78 (2010).
[CrossRef]

J. Niegemann, W. Pernice, and K. Busch, “Simulation of optical resonators using DGTD and FDTD,” J. Opt. A, Pure Appl. Opt. 11, 114015 (2009).
[CrossRef]

J. Niegemann, M. König, K. Stannigel, and K. Busch, “Higher-order time-domain methods for the analysis of nano-photonic systems,” Photonics Nanostruct. Fundam. Appl. 7, 2–11 (2009).
[CrossRef]

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

Butet, J.

J. Butet, G. Bachelier, I. Russier-Antoine, C. Jonin, E. Benichou, and P. Brevet, “Interference between selected dipoles and octupoles in the optical second-harmonic generation from spherical gold nanoparticles,” Phys. Rev. Lett. 105, 077401 (2010).
[CrossRef] [PubMed]

G. Bachelier, J. Butet, I. Russier-Antoine, C. Jonin, E. Benichou, and P. Brevet, “Origin of optical second-harmonic generation in spherical gold nanoparticles: Local surface and nonlocal bulk contributions,” Phys. Rev. B 82, 235403 (2010).
[CrossRef]

Cai, W.

T. Lu, P. Zhang, and W. Cai, “Discontinuous Galerkin methods for dispersive and lossy Maxwell’s equations and PML boundary conditions,” J. Comput. Phys. 200, 549–580 (2004).
[CrossRef]

Cappeddu, M. G.

M. C. Larciprete, A. Belardini, M. G. Cappeddu, D. de Ceglia, M. Centini, E. Fazio, C. Sibilia, M. J. Bloemer, and M. Scalora, “Second-harmonic generation from metallodielectric multilayer photonic-band-gap structures,” Phys. Rev. A 77, 013809 (2008).
[CrossRef]

Centini, M.

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82, 043828 (2010).
[CrossRef]

M. C. Larciprete, A. Belardini, M. G. Cappeddu, D. de Ceglia, M. Centini, E. Fazio, C. Sibilia, M. J. Bloemer, and M. Scalora, “Second-harmonic generation from metallodielectric multilayer photonic-band-gap structures,” Phys. Rev. A 77, 013809 (2008).
[CrossRef]

Dadap, J. I.

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett. 83, 4045–4048 (1999).
[CrossRef]

de Abajo, F. J. G.

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, F. J. G. de Abajo, W. Pfeiffer, M. Rohmer, C. Spindler, and F. Steeb, “Adaptive subwavelength control of nano-optical fields,” Nature 446, 301–304 (2007).
[CrossRef] [PubMed]

de Ceglia, D.

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82, 043828 (2010).
[CrossRef]

M. C. Larciprete, A. Belardini, M. G. Cappeddu, D. de Ceglia, M. Centini, E. Fazio, C. Sibilia, M. J. Bloemer, and M. Scalora, “Second-harmonic generation from metallodielectric multilayer photonic-band-gap structures,” Phys. Rev. A 77, 013809 (2008).
[CrossRef]

Deckert, V.

V. Deckert, “Tip-enhanced Raman spectroscopy,” J. Raman Spectrosc. 40, 1336–1337 (2009).
[CrossRef]

Diehl, R.

J. Niegemann, M. Konig, C. Prohm, R. Diehl, and K. Busch, “Using curved elements in the discontinuous Galerkin time-domain approach,” AIP Conf. Proc. 1291, 76–78 (2010).
[CrossRef]

R. Diehl, K. Busch, and J. Niegemann, “Comparison of low-storage Runge-Kutta schemes for discontinuous-Galerkin time-domain simulations of Maxwell’s equations,” J. Comput. Theor. Nanosci. 7, 1572–1580 (2010).
[CrossRef]

Eisenthal, K. B.

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett. 83, 4045–4048 (1999).
[CrossRef]

Eng, L. M.

A. Hille, R. Kullock, S. Grafstrom, and L. M. Eng, “Improving nano-optical simulations through curved elements implemented within the discontinuous Galerkin method,” J. Comput. Theor. Nanosci. 7, 1581–1586 (2010).
[CrossRef]

M. T. Wenzel, T. Härtling, P. Olk, S. C. Kehr, S. Grafström, S. Winnerl, M. Helm, and L. M. Eng, “Gold nanoparticle tips for optical field confinement in infrared scattering near-field optical microscopy,” Opt. Express 16, 12302–12312 (2008).
[CrossRef] [PubMed]

P. Olk, J. Renger, T. Härtling, M. T. Wenzel, and L. M. Eng, “Two particle enhanced nano Raman microscopy and spectroscopy,” Nano Lett. 7, 1736–1740 (2007).
[CrossRef] [PubMed]

Fazio, E.

M. C. Larciprete, A. Belardini, M. G. Cappeddu, D. de Ceglia, M. Centini, E. Fazio, C. Sibilia, M. J. Bloemer, and M. Scalora, “Second-harmonic generation from metallodielectric multilayer photonic-band-gap structures,” Phys. Rev. A 77, 013809 (2008).
[CrossRef]

Feldmann, J.

M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100, 203002 (2008).
[CrossRef] [PubMed]

Fukui, M.

J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second-harmonic generation at metal surfaces,” Phys. Rev. B 21, 4389–4402 (1980).
[CrossRef]

Gerhardt, I.

V. Sandoghdar, E. Klotzsch, V. Jacobsen, A. Renn, U. Håkanson, M. Agio, I. Gerhardt, J. Seelig, and G. Wrigge, “Optical detection of very small nonfluorescent nanoparticles,” Chimia 60, 761–764 (2006).
[CrossRef]

Grafstrom, S.

A. Hille, R. Kullock, S. Grafstrom, and L. M. Eng, “Improving nano-optical simulations through curved elements implemented within the discontinuous Galerkin method,” J. Comput. Theor. Nanosci. 7, 1581–1586 (2010).
[CrossRef]

Grafström, S.

Hafner, C.

C. Hafner, Post-Modern Electromagnetics: Using Intelligent MaXwell Solvers (John Wiley & Sons, 1999).

Håkanson, U.

V. Sandoghdar, E. Klotzsch, V. Jacobsen, A. Renn, U. Håkanson, M. Agio, I. Gerhardt, J. Seelig, and G. Wrigge, “Optical detection of very small nonfluorescent nanoparticles,” Chimia 60, 761–764 (2006).
[CrossRef]

Hanke, T.

T. Hanke, G. Krauss, D. Träutlein, B. Wild, R. Bratschitsch, and A. Leitenstorfer, “Efficient nonlinear light emission of single gold optical antennas driven by few-cycle near-infrared pulses,” Phys. Rev. Lett. 103, 257404 (2009).
[CrossRef]

Härtling, T.

Heinz, T. F.

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett. 83, 4045–4048 (1999).
[CrossRef]

Helm, M.

Hesthaven, J.

J. Hesthaven and T. Warburton, “Nodal high-order methods on unstructured grids I. time-domain solution of Maxwell’s equations,” J. Comput. Phys. 181, 186–221 (2002).
[CrossRef]

Hesthaven, J. S.

J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications (Springer, 2008).
[CrossRef]

Hille, A.

A. Hille, R. Kullock, S. Grafstrom, and L. M. Eng, “Improving nano-optical simulations through curved elements implemented within the discontinuous Galerkin method,” J. Comput. Theor. Nanosci. 7, 1581–1586 (2010).
[CrossRef]

Hulst, N. V.

J. Renger, R. Quidant, N. V. Hulst, and L. Novotny, “Surface-enhanced nonlinear four-wave mixing,” Phys. Rev. Lett. 104, 046803 (2010).
[CrossRef] [PubMed]

Jacobsen, V.

V. Sandoghdar, E. Klotzsch, V. Jacobsen, A. Renn, U. Håkanson, M. Agio, I. Gerhardt, J. Seelig, and G. Wrigge, “Optical detection of very small nonfluorescent nanoparticles,” Chimia 60, 761–764 (2006).
[CrossRef]

Jonin, C.

J. Butet, G. Bachelier, I. Russier-Antoine, C. Jonin, E. Benichou, and P. Brevet, “Interference between selected dipoles and octupoles in the optical second-harmonic generation from spherical gold nanoparticles,” Phys. Rev. Lett. 105, 077401 (2010).
[CrossRef] [PubMed]

G. Bachelier, J. Butet, I. Russier-Antoine, C. Jonin, E. Benichou, and P. Brevet, “Origin of optical second-harmonic generation in spherical gold nanoparticles: Local surface and nonlocal bulk contributions,” Phys. Rev. B 82, 235403 (2010).
[CrossRef]

Kehr, S. C.

Klar, T. A.

M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100, 203002 (2008).
[CrossRef] [PubMed]

Klotzsch, E.

V. Sandoghdar, E. Klotzsch, V. Jacobsen, A. Renn, U. Håkanson, M. Agio, I. Gerhardt, J. Seelig, and G. Wrigge, “Optical detection of very small nonfluorescent nanoparticles,” Chimia 60, 761–764 (2006).
[CrossRef]

Konig, M.

J. Niegemann, M. Konig, C. Prohm, R. Diehl, and K. Busch, “Using curved elements in the discontinuous Galerkin time-domain approach,” AIP Conf. Proc. 1291, 76–78 (2010).
[CrossRef]

König, M.

J. Niegemann, M. König, K. Stannigel, and K. Busch, “Higher-order time-domain methods for the analysis of nano-photonic systems,” Photonics Nanostruct. Fundam. Appl. 7, 2–11 (2009).
[CrossRef]

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

Krauss, G.

T. Hanke, G. Krauss, D. Träutlein, B. Wild, R. Bratschitsch, and A. Leitenstorfer, “Efficient nonlinear light emission of single gold optical antennas driven by few-cycle near-infrared pulses,” Phys. Rev. Lett. 103, 257404 (2009).
[CrossRef]

Kullock, R.

A. Hille, R. Kullock, S. Grafstrom, and L. M. Eng, “Improving nano-optical simulations through curved elements implemented within the discontinuous Galerkin method,” J. Comput. Theor. Nanosci. 7, 1581–1586 (2010).
[CrossRef]

Kürzinger, K.

M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100, 203002 (2008).
[CrossRef] [PubMed]

Larciprete, M. C.

M. C. Larciprete, A. Belardini, M. G. Cappeddu, D. de Ceglia, M. Centini, E. Fazio, C. Sibilia, M. J. Bloemer, and M. Scalora, “Second-harmonic generation from metallodielectric multilayer photonic-band-gap structures,” Phys. Rev. A 77, 013809 (2008).
[CrossRef]

Leitenstorfer, A.

T. Hanke, G. Krauss, D. Träutlein, B. Wild, R. Bratschitsch, and A. Leitenstorfer, “Efficient nonlinear light emission of single gold optical antennas driven by few-cycle near-infrared pulses,” Phys. Rev. Lett. 103, 257404 (2009).
[CrossRef]

Lu, T.

T. Lu, P. Zhang, and W. Cai, “Discontinuous Galerkin methods for dispersive and lossy Maxwell’s equations and PML boundary conditions,” J. Comput. Phys. 200, 549–580 (2004).
[CrossRef]

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Nichtl, A.

M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100, 203002 (2008).
[CrossRef] [PubMed]

Niegemann, J.

J. Niegemann, M. Konig, C. Prohm, R. Diehl, and K. Busch, “Using curved elements in the discontinuous Galerkin time-domain approach,” AIP Conf. Proc. 1291, 76–78 (2010).
[CrossRef]

R. Diehl, K. Busch, and J. Niegemann, “Comparison of low-storage Runge-Kutta schemes for discontinuous-Galerkin time-domain simulations of Maxwell’s equations,” J. Comput. Theor. Nanosci. 7, 1572–1580 (2010).
[CrossRef]

J. Niegemann, M. König, K. Stannigel, and K. Busch, “Higher-order time-domain methods for the analysis of nano-photonic systems,” Photonics Nanostruct. Fundam. Appl. 7, 2–11 (2009).
[CrossRef]

J. Niegemann, W. Pernice, and K. Busch, “Simulation of optical resonators using DGTD and FDTD,” J. Opt. A, Pure Appl. Opt. 11, 114015 (2009).
[CrossRef]

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

Novotny, L.

J. Renger, R. Quidant, N. V. Hulst, and L. Novotny, “Surface-enhanced nonlinear four-wave mixing,” Phys. Rev. Lett. 104, 046803 (2010).
[CrossRef] [PubMed]

Olk, P.

Pernice, W.

J. Niegemann, W. Pernice, and K. Busch, “Simulation of optical resonators using DGTD and FDTD,” J. Opt. A, Pure Appl. Opt. 11, 114015 (2009).
[CrossRef]

Pfeiffer, W.

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, F. J. G. de Abajo, W. Pfeiffer, M. Rohmer, C. Spindler, and F. Steeb, “Adaptive subwavelength control of nano-optical fields,” Nature 446, 301–304 (2007).
[CrossRef] [PubMed]

Prohm, C.

J. Niegemann, M. Konig, C. Prohm, R. Diehl, and K. Busch, “Using curved elements in the discontinuous Galerkin time-domain approach,” AIP Conf. Proc. 1291, 76–78 (2010).
[CrossRef]

Quidant, R.

J. Renger, R. Quidant, N. V. Hulst, and L. Novotny, “Surface-enhanced nonlinear four-wave mixing,” Phys. Rev. Lett. 104, 046803 (2010).
[CrossRef] [PubMed]

Renger, J.

J. Renger, R. Quidant, N. V. Hulst, and L. Novotny, “Surface-enhanced nonlinear four-wave mixing,” Phys. Rev. Lett. 104, 046803 (2010).
[CrossRef] [PubMed]

P. Olk, J. Renger, T. Härtling, M. T. Wenzel, and L. M. Eng, “Two particle enhanced nano Raman microscopy and spectroscopy,” Nano Lett. 7, 1736–1740 (2007).
[CrossRef] [PubMed]

Renn, A.

V. Sandoghdar, E. Klotzsch, V. Jacobsen, A. Renn, U. Håkanson, M. Agio, I. Gerhardt, J. Seelig, and G. Wrigge, “Optical detection of very small nonfluorescent nanoparticles,” Chimia 60, 761–764 (2006).
[CrossRef]

Ringler, M.

M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100, 203002 (2008).
[CrossRef] [PubMed]

Rohmer, M.

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, F. J. G. de Abajo, W. Pfeiffer, M. Rohmer, C. Spindler, and F. Steeb, “Adaptive subwavelength control of nano-optical fields,” Nature 446, 301–304 (2007).
[CrossRef] [PubMed]

Roppo, V.

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82, 043828 (2010).
[CrossRef]

Russier-Antoine, I.

G. Bachelier, J. Butet, I. Russier-Antoine, C. Jonin, E. Benichou, and P. Brevet, “Origin of optical second-harmonic generation in spherical gold nanoparticles: Local surface and nonlocal bulk contributions,” Phys. Rev. B 82, 235403 (2010).
[CrossRef]

J. Butet, G. Bachelier, I. Russier-Antoine, C. Jonin, E. Benichou, and P. Brevet, “Interference between selected dipoles and octupoles in the optical second-harmonic generation from spherical gold nanoparticles,” Phys. Rev. Lett. 105, 077401 (2010).
[CrossRef] [PubMed]

Sandoghdar, V.

V. Sandoghdar, E. Klotzsch, V. Jacobsen, A. Renn, U. Håkanson, M. Agio, I. Gerhardt, J. Seelig, and G. Wrigge, “Optical detection of very small nonfluorescent nanoparticles,” Chimia 60, 761–764 (2006).
[CrossRef]

Scalora, M.

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82, 043828 (2010).
[CrossRef]

M. C. Larciprete, A. Belardini, M. G. Cappeddu, D. de Ceglia, M. Centini, E. Fazio, C. Sibilia, M. J. Bloemer, and M. Scalora, “Second-harmonic generation from metallodielectric multilayer photonic-band-gap structures,” Phys. Rev. A 77, 013809 (2008).
[CrossRef]

Schöberl, J.

J. Schöberl, “NETGEN an advancing front 2D/3D-mesh generator based on abstract rules,” Comput. Visual. Sci. 1, 41–52 (1997).
[CrossRef]

Schwemer, A.

M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100, 203002 (2008).
[CrossRef] [PubMed]

Seelig, J.

V. Sandoghdar, E. Klotzsch, V. Jacobsen, A. Renn, U. Håkanson, M. Agio, I. Gerhardt, J. Seelig, and G. Wrigge, “Optical detection of very small nonfluorescent nanoparticles,” Chimia 60, 761–764 (2006).
[CrossRef]

Shan, J.

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett. 83, 4045–4048 (1999).
[CrossRef]

Sibilia, C.

M. C. Larciprete, A. Belardini, M. G. Cappeddu, D. de Ceglia, M. Centini, E. Fazio, C. Sibilia, M. J. Bloemer, and M. Scalora, “Second-harmonic generation from metallodielectric multilayer photonic-band-gap structures,” Phys. Rev. A 77, 013809 (2008).
[CrossRef]

Sipe, J. E.

J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second-harmonic generation at metal surfaces,” Phys. Rev. B 21, 4389–4402 (1980).
[CrossRef]

So, V. C. Y.

J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second-harmonic generation at metal surfaces,” Phys. Rev. B 21, 4389–4402 (1980).
[CrossRef]

Spindler, C.

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, F. J. G. de Abajo, W. Pfeiffer, M. Rohmer, C. Spindler, and F. Steeb, “Adaptive subwavelength control of nano-optical fields,” Nature 446, 301–304 (2007).
[CrossRef] [PubMed]

Stannigel, K.

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

J. Niegemann, M. König, K. Stannigel, and K. Busch, “Higher-order time-domain methods for the analysis of nano-photonic systems,” Photonics Nanostruct. Fundam. Appl. 7, 2–11 (2009).
[CrossRef]

Steeb, F.

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, F. J. G. de Abajo, W. Pfeiffer, M. Rohmer, C. Spindler, and F. Steeb, “Adaptive subwavelength control of nano-optical fields,” Nature 446, 301–304 (2007).
[CrossRef] [PubMed]

Stegeman, G. I.

J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second-harmonic generation at metal surfaces,” Phys. Rev. B 21, 4389–4402 (1980).
[CrossRef]

Träutlein, D.

T. Hanke, G. Krauss, D. Träutlein, B. Wild, R. Bratschitsch, and A. Leitenstorfer, “Efficient nonlinear light emission of single gold optical antennas driven by few-cycle near-infrared pulses,” Phys. Rev. Lett. 103, 257404 (2009).
[CrossRef]

Vincenti, M. A.

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82, 043828 (2010).
[CrossRef]

Warburton, T.

J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications (Springer, 2008).
[CrossRef]

J. Hesthaven and T. Warburton, “Nodal high-order methods on unstructured grids I. time-domain solution of Maxwell’s equations,” J. Comput. Phys. 181, 186–221 (2002).
[CrossRef]

Wenzel, M. T.

Wild, B.

T. Hanke, G. Krauss, D. Träutlein, B. Wild, R. Bratschitsch, and A. Leitenstorfer, “Efficient nonlinear light emission of single gold optical antennas driven by few-cycle near-infrared pulses,” Phys. Rev. Lett. 103, 257404 (2009).
[CrossRef]

Winnerl, S.

Wrigge, G.

V. Sandoghdar, E. Klotzsch, V. Jacobsen, A. Renn, U. Håkanson, M. Agio, I. Gerhardt, J. Seelig, and G. Wrigge, “Optical detection of very small nonfluorescent nanoparticles,” Chimia 60, 761–764 (2006).
[CrossRef]

Wunderlich, M.

M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100, 203002 (2008).
[CrossRef] [PubMed]

Zhang, P.

T. Lu, P. Zhang, and W. Cai, “Discontinuous Galerkin methods for dispersive and lossy Maxwell’s equations and PML boundary conditions,” J. Comput. Phys. 200, 549–580 (2004).
[CrossRef]

AIP Conf. Proc. (1)

J. Niegemann, M. Konig, C. Prohm, R. Diehl, and K. Busch, “Using curved elements in the discontinuous Galerkin time-domain approach,” AIP Conf. Proc. 1291, 76–78 (2010).
[CrossRef]

Ann. Phys. (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908).
[CrossRef]

Chimia (1)

V. Sandoghdar, E. Klotzsch, V. Jacobsen, A. Renn, U. Håkanson, M. Agio, I. Gerhardt, J. Seelig, and G. Wrigge, “Optical detection of very small nonfluorescent nanoparticles,” Chimia 60, 761–764 (2006).
[CrossRef]

Comput. Visual. Sci. (1)

J. Schöberl, “NETGEN an advancing front 2D/3D-mesh generator based on abstract rules,” Comput. Visual. Sci. 1, 41–52 (1997).
[CrossRef]

J. Comput. Phys. (2)

J. Hesthaven and T. Warburton, “Nodal high-order methods on unstructured grids I. time-domain solution of Maxwell’s equations,” J. Comput. Phys. 181, 186–221 (2002).
[CrossRef]

T. Lu, P. Zhang, and W. Cai, “Discontinuous Galerkin methods for dispersive and lossy Maxwell’s equations and PML boundary conditions,” J. Comput. Phys. 200, 549–580 (2004).
[CrossRef]

J. Comput. Theor. Nanosci. (2)

A. Hille, R. Kullock, S. Grafstrom, and L. M. Eng, “Improving nano-optical simulations through curved elements implemented within the discontinuous Galerkin method,” J. Comput. Theor. Nanosci. 7, 1581–1586 (2010).
[CrossRef]

R. Diehl, K. Busch, and J. Niegemann, “Comparison of low-storage Runge-Kutta schemes for discontinuous-Galerkin time-domain simulations of Maxwell’s equations,” J. Comput. Theor. Nanosci. 7, 1572–1580 (2010).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

J. Niegemann, W. Pernice, and K. Busch, “Simulation of optical resonators using DGTD and FDTD,” J. Opt. A, Pure Appl. Opt. 11, 114015 (2009).
[CrossRef]

J. Raman Spectrosc. (1)

V. Deckert, “Tip-enhanced Raman spectroscopy,” J. Raman Spectrosc. 40, 1336–1337 (2009).
[CrossRef]

Nano Lett. (1)

P. Olk, J. Renger, T. Härtling, M. T. Wenzel, and L. M. Eng, “Two particle enhanced nano Raman microscopy and spectroscopy,” Nano Lett. 7, 1736–1740 (2007).
[CrossRef] [PubMed]

Nature (1)

M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, F. J. G. de Abajo, W. Pfeiffer, M. Rohmer, C. Spindler, and F. Steeb, “Adaptive subwavelength control of nano-optical fields,” Nature 446, 301–304 (2007).
[CrossRef] [PubMed]

Opt. Express (2)

Photonics Nanostruct. Fundam. Appl. (1)

J. Niegemann, M. König, K. Stannigel, and K. Busch, “Higher-order time-domain methods for the analysis of nano-photonic systems,” Photonics Nanostruct. Fundam. Appl. 7, 2–11 (2009).
[CrossRef]

Phys. Rev. A (2)

M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “Second- and third-harmonic generation in metal-based structures,” Phys. Rev. A 82, 043828 (2010).
[CrossRef]

M. C. Larciprete, A. Belardini, M. G. Cappeddu, D. de Ceglia, M. Centini, E. Fazio, C. Sibilia, M. J. Bloemer, and M. Scalora, “Second-harmonic generation from metallodielectric multilayer photonic-band-gap structures,” Phys. Rev. A 77, 013809 (2008).
[CrossRef]

Phys. Rev. B (2)

G. Bachelier, J. Butet, I. Russier-Antoine, C. Jonin, E. Benichou, and P. Brevet, “Origin of optical second-harmonic generation in spherical gold nanoparticles: Local surface and nonlocal bulk contributions,” Phys. Rev. B 82, 235403 (2010).
[CrossRef]

J. E. Sipe, V. C. Y. So, M. Fukui, and G. I. Stegeman, “Analysis of second-harmonic generation at metal surfaces,” Phys. Rev. B 21, 4389–4402 (1980).
[CrossRef]

Phys. Rev. Lett. (5)

J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “Second-harmonic Rayleigh scattering from a sphere of centrosymmetric material,” Phys. Rev. Lett. 83, 4045–4048 (1999).
[CrossRef]

J. Butet, G. Bachelier, I. Russier-Antoine, C. Jonin, E. Benichou, and P. Brevet, “Interference between selected dipoles and octupoles in the optical second-harmonic generation from spherical gold nanoparticles,” Phys. Rev. Lett. 105, 077401 (2010).
[CrossRef] [PubMed]

J. Renger, R. Quidant, N. V. Hulst, and L. Novotny, “Surface-enhanced nonlinear four-wave mixing,” Phys. Rev. Lett. 104, 046803 (2010).
[CrossRef] [PubMed]

T. Hanke, G. Krauss, D. Träutlein, B. Wild, R. Bratschitsch, and A. Leitenstorfer, “Efficient nonlinear light emission of single gold optical antennas driven by few-cycle near-infrared pulses,” Phys. Rev. Lett. 103, 257404 (2009).
[CrossRef]

M. Ringler, A. Schwemer, M. Wunderlich, A. Nichtl, K. Kürzinger, T. A. Klar, and J. Feldmann, “Shaping emission spectra of fluorescent molecules with single plasmonic nanoresonators,” Phys. Rev. Lett. 100, 203002 (2008).
[CrossRef] [PubMed]

Other (2)

J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications (Springer, 2008).
[CrossRef]

C. Hafner, Post-Modern Electromagnetics: Using Intelligent MaXwell Solvers (John Wiley & Sons, 1999).

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Figures (11)

Fig. 1
Fig. 1

Schematic drawing of the numerical setup used for calculating the SHG response from a 3D sphere: The scattered field / total field (SF/TF) contour is used to inject a short laser pulse from the left exciting the 80-nm Ag nanoparticle. The resulting scattered light is then monitored and absorbed by a perfectly matched layer (PML, parameters as in [9]).

Fig. 2
Fig. 2

Polynomial approximation of the field representation for a 3D element. As a visualization of the polynomials the distribution of their nodal points is sketched for the 1st, 2nd and 4th orders (P).

Fig. 3
Fig. 3

Schematics of the nodal distribution for linear and curved elements: (a) 2D linear, (b) 2D curved, (c) 3D linear and (d) 3D curved elements. The blue dots represent the (surface) nodes.

Fig. 4
Fig. 4

Geometric description of a 3D sphere using a 4th-order icosahedron and a 1st-, 2nd-and 4th-order boundary approximation (B). Edges of the elements are visible for both B=1 (linear elements) and for B=2 (CEs), but not for CEs with a 4th-order approximation (B=4).

Fig. 5
Fig. 5

Visualization of the meshes used (M=1...4) by means of the grid on the sphere. The four meshes consist of 6675, 7803, 10154 and 33226 elements, respectively, having 176, 385, 780 and 2278 boundary elements.

Fig. 6
Fig. 6

Maximum field enhancement and scattering cross section (normalized to the geometrical cross section of the particle) for the different meshes (M=1...4) as well as for linear and curved elements (B=1,2, P=2).

Fig. 7
Fig. 7

Maximum field enhancement and normalized scattering cross section for the two coarsest meshes, 4th-order polynomial degree (P=4) and 2nd- and 4th-order of boundary approximation. In the lower two plots also the error of the MMP reference is shown, (left) averaged relative error and (right) maximal relative error. Note, for (M=2,B=4) or (M=2, B=4) the geometrical description is rather good and other effects such as reflections from the PML or the detailed discretization of the mesh become dominant. They cause the spectral variations of the error profile.

Fig. 8
Fig. 8

Linear near-field distribution of the Ag sphere after excitation by a plane wave from the left. The averaged E field at a wavelength of 251 nm is plotted for three different configurations as indicated. All plots are scaled equally and the color range is: white/yellow – high field strength; black – low field strength.

Fig. 9
Fig. 9

Influence of the parameter set (mesh M, boundary approximation B, polynomial degree P) on the accuracy of the maximal near-field enhancement. The errors for the fundamental and second-harmonic frequency are shown for CEs and the different sets.

Fig. 10
Fig. 10

SHG field of a Ag sphere illuminated with a femtosecond pulse from the left. The parameters M, B, P are the same as in Fig. 8 and all plots are scaled equally.

Fig. 11
Fig. 11

Visualization of the geometrical mismatch for different boundary approximations (B) and polynomial orders (P). Black line: boundary to be described; blue line: shape of the CE; red dots: nodes of the polynomial representation. Note, B < P leads to a wrong boundary description.

Equations (4)

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

t 2 P = N d q 2 m ( E + μ N d q t P × H ) - γ t P ω 0 2 P ,
E E 0 + 1 N d q ( ( P ) E ) 0
t P × H t P × H 0 .
t 2 P ω p 2 ɛ 0 E 0 + q m ( ( P ) E ) 0 + μ q m t P × H 0 - γ t P - ω 0 2 P ,

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