## Discontinuous Galerkin time-domain computations of metallic nanostructures

Optics Express, Vol. 17, Issue 17, pp. 14934-14947 (2009)

http://dx.doi.org/10.1364/OE.17.014934

Acrobat PDF (487 KB)

### Abstract

We apply the three-dimensional Discontinuous-Galerkin Time-Domain method to the investigation of the optical properties of bar- and V-shaped metallic nanostructures on dielectric substrates. A flexible finite element-like mesh together with an expansion into high-order basis functions allows for an accurate resolution of complex geometries and strong field gradients. In turn, this provides accurate results on the optical response of realistic structures. We study in detail the influence of particle size and shape on resonance frequencies as well as on scattering and absorption efficiencies. Beyond a critical size which determines the onset of the quasi-static limit we find significant deviations from the quasi-static theory. Furthermore, we investigate the influence of the excitation by comparing normal illumination and attenuated total internal reflection setups. Finally, we examine the possibility of coherently controlling the local field enhancement of V-structures via chirped pulses.

© 2009 Optical Society of America

## 1. Introduction

3. S. Kim, J. Jin, Y.-J. Kim, I.-Y. Park, Y. Kim, and S.-W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature **453**, 757–760 (2008).
[PubMed]

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

15. V. Myroshnychenko, J. Rodriguez-Fernandez, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Martin, and F. J. Garcia de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. **37**, 1792–1805 (2008).
[PubMed]

15. V. Myroshnychenko, J. Rodriguez-Fernandez, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Martin, and F. J. Garcia de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. **37**, 1792–1805 (2008).
[PubMed]

21. H. Fischer and O. J. F. Martin, “Engineering the optical response of plasmonic nanoantennas,” Opt. Express **16**, 9144–9154 (2008).
[PubMed]

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

## 2. The discontinuous Galerkin time-domain method

*p*. In contrast to the classical finite element method [16], this expansion is purely local and the fields are allowed to be discontinuous across element boundaries. In order to facilitate the coupling between neighboring elements, one introduces an additional penalty term, which weakly enforces the physical boundary conditions.

## 3. Scattering by a metallic sphere

*ω*=1.39 ·10

_{d}^{16}s

^{-1}and the collision rate

*γd*=3.23 ·10

^{13}s

^{-1}correspond to the values given by Johnson and Christy [10] for silver. However, it should be noted that these values underestimate the losses in the ultraviolet regime. We take the sphere to have a radius of 50nm and to be embedded in vacuum.

*p*=3, 4), different numbers of tetrahedra for the sphere (in five steps from 160 to 2750) and different sizes of the PML region (either one or two cells). Although we find that the interpolation order and the size of the PML do have a slight influence on the accuracy of the result, the error is strongly dominated by the tetrahedrization of the sphere. This result has been anticipated, since we are using straight-sided elements which essentially lead to a polygonal approximation of the sphere’s shape. In Fig. 1, we display the results for the scattering and absorption cross sections for a computation when the sphere is meshed with 2750 tetrahedra and where a third-order spatial discretization is used. A comparison with analytic Mie theory [29] shows that the relative pointwise error of the spectrum is below 2 % except in the region of the highest resonance, where the error is significant (up to 10%) even for the finest discretization. This indicates that for the reproduction of the exact position and strength of high-Q resonances, one should resort to curved elements or much finer discretizations. For lower-Q resonances as well as for the general structure of the spectrum the presented calculations yield well-converged results.

## 4. Optical properties of silver nano-bars

*r*

_{end}=

*w*/2 at the endings of the bar. In this context, we would like to point out that the finite-element discretization allows us to model the rounding without staircasing as would be the case within standard FDTD. While this might be of limited relevance in the case of a bar-shaped object, it certainly is of high relevance for a V-shaped object as discussed in section 5.

### 4.1. Local field enhancement

*f*

_{res}of the structure by recording the scattering and absorption cross sections as described in section 3. Then, in a second computation, we irradiate the structure by a monochromatic plane wave with the very resonance frequency, where the amplitude is slowly ramped up from zero to its final value. This time, we record the values of the fields in front of the tip. The simulation is stopped when the enhancement has saturated. For the systems described below, this takes approximately 30 optical cycles (cf. Fig. 3(a)). We have carried out these computations for bars of fixed widths and heights,

*w*=

*h*=20nm, and whose lengths range between

*l*=100-200nm. Figure 3(b) shows the extracted enhancements 10nm above the substrate at sites

*A*and

*B*that are located 1nm and 5nm away from the tip, respectively (c.f. Fig. 2(b)). To check whether our results are converged, we have performed all calculations with interpolation orders

*p*=2 and

*p*=3. Finally, for completeness we provide in Table 1 the wavelengths

*λ*

_{res}corresponding to the lowest resonance frequencies as well as the respective quality factors

*Q*for different bars.

### 4.2. Quasi-static limit

*ε*(

*ω*) and unit permeability for which the efficiencies in the quasi-static limit can be determined analytically as [33]

*k*is the wave vector of the incident light and a the radius, i.e., the characteristic length, of the particle. The above expressions inform us that the spectral profile of the efficiencies—and in particular the position of the resonance—is independent of the particle

*size*, but rather depends on the particle

*shape*. The size merely influences the magnitudes of the efficiencies. In fact, this statement is true for scatterers of arbitrary shape. Thus, the quasi-static limit is reached if the resonance frequency of the particle is size independent and the efficiencies scale as

*Q*

_{sca}∝

*a*

^{4}and

*Q*

_{abs}∝

*a*with the particle’s characteristic length

*a*. Note that we are not considering the cross sections which would scale with an additional factor of

*a*

^{2}each.

*s*<1, and repeat the computations. The behavior of the efficiencies as a function of the scaling factor s can be used to detect the onset of the quasi-static limit. For simplicity, we reduce the shape of the nano-bar to a rectangular box with sharp corners. This reduces the computational effort since small tetrahedra in the rounded endings of the bars limit the simulation time-step. At the same time, while details of the individual structures slightly differ, the scaling behavior for rounded bars and bars with sharp corners can be assumed to be the same.

^{3}down to 5% of its original size. For simplicity, we omit the dielectric substrate in these calculations. From the overview plots Fig. 4(a) and Fig. 4(b), we observe the well-known fact that absorption in small particles dominates over scattering. With regard to the first criterion formulated above, we notice that the position of the resonance saturates for scaling factors

*s*≤

*s*

_{crit}≈0.15. Regarding the second criterion, we deduce the power law dependence of the cross sections from panels (c) and (d). Consistent with the first criterion, we find the same critical scaling factor scrit ≈0.15. Therefore, we conclude that for the setup consisting of silver nano-bars in vacuum, the quasi-static limit in the sense defined above sets in for structures with spatial extensions as small as

*l*≲15nm.

## 5. Optical properties of silver V-shaped nanostructures

### 5.1. Local field enhancement

*A*and

*B*that are, respectively, located 1nm and 5nm in front of the tip and

*h*/2 above the glass substrate. We list the resonance wavelengths and corresponding

*Q*-factors for the lowest frequency resonance of V-structures with different lengths, fixed

*w*=

*h*=20nm, and fixed apex angle

*α*=30° on a glass substrate in Table 2.

*A*and

*B*. It is obvious that the general tendency is the same as in the case of the nano-bars, namely that the enhancement increases notably with particle size. Remarkably, the local field enhancements attainable for V-structures under plane wave illumination are slightly below those of nano-bars of same lengths and tip radii (c.f. Fig. 3(b)).

*w*=

*h*=20nm and

*l*=150nm and tip radii of

*r*

_{tip}=8nm and

*r*

_{tip}=12nm. As expected, the enhancement increases with decreasing radius and the resulting field enhancements deviate from those with

*r*

_{tip}=

*w*/2=10nm by approximately ±5–8%.

*A*and

*B*in front of the tip we also extracted the complete field distributions on resonance. In Fig. 6, we display two corresponding intensity maps: Panel (a) depicts the intensity distribution |

*E*(

*x,y*)|

^{2}in the center plane of the structure, i.e., 10nm above the substrate, and panel (b) depicts the intensity distribution in a plane 30nm above the substrate. These computations reveal a total of three hotspots, two at the ends of the arms and one at the end of the tip of the nano-V.

*t*

^{2}/2

*σ*) and their spectra were centered at the resonance frequencies of the respective structures. The temporal widths

*σ*were chosen to be three optical cycles. However, according to the discussion of section 4, this does not suffice for the enhancement to saturate. As a result, the recorded peak enhancements have been about a factor of 2/3 smaller than those obtained with the monochromatic excitation.

### 5.2. ATIR- vs. NIT-spectroscopy

34. C. Sonnichsen, S. Geier, N. E. Hecker, G. von Plessen, J. Feldmann, H. Ditlbacher, B. Lamprecht, J. R. Krenn, F. R. Aussenegg, V. Z-H. Chan, J. P. Spatz, and M. Moller, “Spectroscopy of single metallic nanoparticles using total internal reflection microscopy,” Appl. Phys. Lett. **77**, 2949–2951 (2000).

### 5.3. Scaling analysis

*l*=150nm,

*w*=

*h*=25nm, and

*α*=30° by a common factor s down to

*s*=0.3. Again, the glass substrate was omitted in these studies. It is obvious that the structures are not yet in the quasi-static limit, since the resonance frequencies exhibit considerable shifts and the absolute values of the scattering cross section for intermediate scaling factors even increase when the structure is scaled down. As a scaling factor of

*s*=0.3 is equivalent to an arm width

*w*=7.5nm, the quasi-static limit seems to be well beyond the reach of current state-of-the-art fabrication techniques.

### 5.4. Coherent control via chirped pulses

*coherent control*of the optical near-fields, i.e., the interference of plasmon modes within the nanostructure may be tailored in a specific way to achieve a desired response [9

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

*σ*denotes the width of the Gaussian envelope and

*t*

_{0}the time of its maximum amplitude.

*ω*

_{0}is the central pulse frequency and

*β*represents the chirp parameter. The latter controls the extent to which the momentary frequency varies in time. For

*β*=0 we obviously recover a conventional Gaussian pulse. The above expression is valid for

*t*∈ [0,2

*t*

_{0}] and the incident field is taken to be zero otherwise. This of course requires a sufficiently small width σ, such that

*f*(

*t*=0,2

*t*

_{0})≈0.

*l*=150nm, identical width and height

*w*=

*h*=20nm, and apex angle

*α*=30° located on a glass substrate. Bearing in mind that the idea of coherent control is based on interference, the pulse width

*σ*should be of the order of the life time

*Q/ω*of the relevant particle plasmon resonance. We set the central frequency of the excitation to the fundamental resonance

_{res}*f*

_{res}of the V-structure and choose the following set of parameters:

*β*=±0.3. For reference, we also show the response to an unchirped pulse. As a first observation, we note that the magnitude of the response to chirped pulses is less than that of the response to the unchirped excitation. In addition, the envelope of the response is similar to that of the unchirped case until a sudden breakdown occurs just after the response has reached its maximum. We have performed similar computations for chirp parameters

*β*=-0.5…0.5 in steps of 0.1. In all cases, we observe the same qualitative behavior. However, the maximum attainable amplitude decreases with increasing magnitude of the chirp parameter and the breakdown happens more abruptly. Physically, the explanation for this behavior lies in the fact the excitation runs out of phase with the eigenmode that has already been (partly) established during the first half of the pulse. Consequently, at some point the further build-up of the particle-plasmon oscillation is stopped prematurely (as compared to the unchirped case) and an accelerated deexcitation (as compared to the unchirped case) based on destructive interference sets in.

## 6. Conclusion

*l*≲15nm. Outside the quasi-static limit, we have found that the attainable values for the field enhancement near nano-bars are about 50% higher than those of nano-Vs of comparable sizes. In this regime, the resonance frequencies and values for the field enhancement strongly depend on the bar or arm length. Furthermore, we have demonstrated that ATIR- and NIT-spectroscopy for these particles essentially provide the same spectral information, except that NIT-spectroscopy is capable of providing absolute values for cross sections. Finally, we have investigated coherent control schemes based on chirped pulse excitations of silver nano-Vs. While we find that a certain degree of control can be exerted, the full electromagnetic analysis suggests that this control is limited and more sophisticated control schemes and/or more complex geometries have to be employed in order to realize strong spatio-temporal localization of radiation in conjunction with large field enhancements.

## Acknowledgments

## References and links

1. | T. Soller, M. Ringler, M. Wunderlich, T. A. Klar, J. Feldmann, H.-P. Josel, Y. Markert, A. Nichl, and K. Kürzinger, “Radiative and nonradiative rates of phosphors attached to gold nanoparticles,” Nano Lett. |

2. | J. Steidtner and B. Pettinger, “Tip-enhanced Raman spectroscopy and microscopy on single dye molecules with 15 nm resolution,” Phys. Rev. Lett. |

3. | S. Kim, J. Jin, Y.-J. Kim, I.-Y. Park, Y. Kim, and S.-W. Kim, “High-harmonic generation by resonant plasmon field enhancement,” Nature |

4. | D. J. Bergman and M. I. Stockman, “Surface plasmon amplification by stimulated emission of radiation: Quantum generation of coherent surface plasmons in nanosystems,” Phys. Rev. Lett. |

5. | N. I. Zheludev, S. I. Prosvirnin, N. Papasimakis, and V. A. Fedotov, “Lasing spaser,” Nature Photon. |

6. | M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, “Parallel and selective trapping in a patterned plasmonic landscape,” Nature Phys. |

7. | M. Righini, V. Giovani, C. Girard, D. Petrov, and R. Quidant, “Surface plasmon optical tweezers: Tunable optical manipulation in the femtonewton range,” Phys. Rev. Lett. |

8. | A. N. Grigorenko, N. W. Roberts, M. R. Dickinson, and Y. Zhang, “Nanometric optical tweezers based on nanostructured substrates,” Nature Photon. |

9. | M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, F. J. Garcia de Abajo, W. Pfeiffer, M. Rohmer, C. Spindler, and F. Steeb, “Adaptive subwavelength control of nano-optical fields,” Nature |

10. | P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B |

11. | C. M. Aikens, S. Li, and G. C. Schatz, “From discrete electronic states to plasmons: TDDFT optical absorption properties of Ag |

12. | M. I. Stockman, S. V. Faleev, and D. J. Bergman, “Coherent control of femtosecond energy localization in nanosystems,” Phys. Rev. Lett. |

13. | M. I. Stockman, D. J. Bergmann, and T. Kobayashi, “Coherent control of nanoscale localization of ultrafast optical excitation in nanosystems,” Phys. Rev. B |

14. | X. Li and M. I. Stockman, “Highly efficient spatiotemporal control in nanoplasmonics on a nanometer-femtosecond scale by time reversal,” Phys. Rev. B |

15. | V. Myroshnychenko, J. Rodriguez-Fernandez, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Martin, and F. J. Garcia de Abajo, “Modelling the optical response of gold nanoparticles,” Chem. Soc. Rev. |

16. | J. Jin, |

17. | C. Hafner, |

18. | N. Calander and M. Willander, “Theory of surface-plasmon resonance optical-field enhancement at prolate spheroids,” J. Appl. Phys. |

19. | R. Kappeler, D. Erni, C. Xudong, and L. Novotny, “Field computations of optical antennas,” J. Comput. Theor. Nanosci. |

20. | X. Cui, W. Zhang, B.-S. Yeo, R. Zenobi, Ch. Hafner, and D. Erni, “Tuning the resonance frequency of Ag-coated dielectric tips,” Opt. Express |

21. | H. Fischer and O. J. F. Martin, “Engineering the optical response of plasmonic nanoantennas,” Opt. Express |

22. | M. I. Stockman, “Ultrafast nanoplasmonics under coherent control,” New J. Phys. |

23. | A. Taflove and S. C. Hagness, |

24. | J. Niegemann, M. König, K. Stannigel, and K. Busch, “Higher-order time-domain methods for the analysis of nano-photonic systems,” Photon. Nanostruct. Fundam. Appl. |

25. | J. S. Hesthaven and T. Warburton “Nodal high-order methods on unstructured grids - I. Time-domain solution of Maxwell’s equations,” J. Comput. Phys. |

26. | T. Lu, P. Zhang, and W. Cai, “Discontinuous Galerkin methods for dispersive and lossy Maxwell’s equations and PML boundary conditions,” J. Comput. Phys. |

27. | M. H. Carpenter and C. A. Kennedy, “Fourth-Order 2N-Storage Runge-Kutta Schemes,” Technical Report NASA-TM-109112, NASA Langley Research Center, VA (1994). |

28. | K. Busch, J. Niegemann, M. Pototschnig, and L. Tkeshelashvili, “A Krylov-subspace based solver for the linear and nonlinear Maxwell equations,” phys. stat. sol. (b) |

29. | H. C. van de Hulst, |

30. | M. Liu, P. Guyot-Sionnest, T.-W. Lee, and S. Gray, “Optical properties of rodlike and bipyramidal gold nanoparticles from three-dimensional computations,” Phys. Rev. B |

31. | E. Hao and G. C. Schatz, “Electromagnetic fields around silver nanoparticles and dimers,” J. Chem. Phys. |

32. | H. Kuwata, H. Tamaru, K. Esumi, and K. Miyano, “Resonant light scattering from metal nanoparticles: Practical analysis beyond Rayleigh approximation,” Appl. Phys. Lett. |

33. | C. F. Bohren and D. R. Huffman, |

34. | C. Sonnichsen, S. Geier, N. E. Hecker, G. von Plessen, J. Feldmann, H. Ditlbacher, B. Lamprecht, J. R. Krenn, F. R. Aussenegg, V. Z-H. Chan, J. P. Spatz, and M. Moller, “Spectroscopy of single metallic nanoparticles using total internal reflection microscopy,” Appl. Phys. Lett. |

35. | A. Arbouet, D. Christofilos, N. Del Fatti, F. Vallée, J.R. Huntzinger, L. Arnaud, P. Billaud, and M. Broyer, “Direct Measurement of the Single-Metal-Cluster Optical Absorption,” Phys. Rev. Lett. |

36. | M. Husnik, M. W. Klein, N. Feth, M. König, J. Niegemann, K. Busch, S. Linden, and M. Wegener, “Absolute extinction cross-section of individual magnetic split-ring resonators,” Nature Photon. |

**OCIS Codes**

(000.3860) General : Mathematical methods in physics

(000.4430) General : Numerical approximation and analysis

(240.6680) Optics at surfaces : Surface plasmons

(260.3910) Physical optics : Metal optics

(290.5850) Scattering : Scattering, particles

(310.6628) Thin films : Subwavelength structures, nanostructures

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: April 27, 2009

Revised Manuscript: June 12, 2009

Manuscript Accepted: June 26, 2009

Published: August 7, 2009

**Citation**

Kai Stannigel, Michael König, Jens Niegemann, and Kurt Busch, "Discontinuous Galerkin time-domain computations of metallic nanostructures," Opt. Express **17**, 14934-14947 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-17-14934

Sort: Year | Journal | Reset

### References

- T. Soller,M. Ringler,M. Wunderlich, T. A. Klar, J. Feldmann, H.-P. Josel, Y. Markert, A. Nichl, and K. K¨urzinger, "Radiative and nonradiative rates of phosphors attached to gold nanoparticles," Nano Lett. 7, 1941-1946 (2008).
- J. Steidtner and B. Pettinger, "Tip-enhanced Raman spectroscopy and microscopy on single dye molecules with 15 nm resolution," Phys. Rev. Lett. 100, 236101-1-4 (2008).
- S. Kim, J. Jin., Y.-J. Kim, I.-Y. Park, Y. Kim, and S.-W. Kim, "High-harmonic generation by resonant Plasmon field enhancement," Nature 453, 757-760 (2008). [PubMed]
- D. J. Bergman and M. I. Stockman, "Surface plasmon amplification by stimulated emission of radiation: Quantum generation of coherent surface plasmons in nanosystems," Phys. Rev. Lett. 90, 027402-1-4 (2003).
- N. I. Zheludev, S. I. Prosvirnin, N. Papasimakis, and V. A. Fedotov, "Lasing spaser," Nature Photon. 2, 351-354 (2008).
- M. Righini, A. S. Zelenina, C. Girard, and R. Quidant, "Parallel and selective trapping in a patterned plasmonic landscape," Nature Phys. 3, 477-480 (2008).
- M. Righini, V. Giovani, C. Girard, D. Petrov, and R. Quidant, "Surface plasmon optical tweezers: Tunable optical manipulation in the femtonewton range," Phys. Rev. Lett. 100, 186804-1-4 (2008).
- A. N. Grigorenko, N. W. Roberts, M. R. Dickinson, and Y. Zhang, "Nanometric optical tweezers based on nanostructured substrates," Nature Photon. 2, 365-370 (2008).
- M. Aeschlimann, M. Bauer, D. Bayer, T. Brixner, F. J. Garcia de Abajo,W. Pfeiffer, M. Rohmer, C. Spindler, and F. Steeb, "Adaptive subwavelength control of nano-optical fields," Nature 446, 301-304 (2007). [PubMed]
- P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
- C. M. Aikens, S. Li, and G. C. Schatz, "From discrete electronic states to plasmons: TDDFT optical absorption properties of Agn(n=10,20,35,56,84,120) tetrahedral clusters," J. Phys. Chem. C 112, 11272-11279 (2008).
- M. I. Stockman, S. V. Faleev, and D. J. Bergman, "Coherent control of femtosecond energy localization in nanosystems," Phys. Rev. Lett. 88, 067402-1-4 (2002).
- M. I. Stockman, D. J. Bergmann, and T. Kobayashi, "Coherent control of nanoscale localization of ultrafast optical excitation in nanosystems," Phys. Rev. B 69, 054202-054211 (2004).
- X. Li and M. I. Stockman, "Highly efficient spatiotemporal control in nanoplasmonics on a nanometerfemtosecond scale by time reversal," Phys. Rev. B 77, 195109 (2008).
- V. Myroshnychenko, J. Rodriguez-Fernandez, I. Pastoriza-Santos, A. M. Funston, C. Novo, P. Mulvaney, L. M. Liz-Martin, and F. J. Garcia de Abajo, "Modelling the optical response of gold nanoparticles," Chem. Soc. Rev. 37, 1792-1805 (2008). [PubMed]
- J. Jin, Computational Electrodynamics: The Finite Element Method in Electromagnetics (2nd edition, John Wiley & Sons, New York, 2002).
- C. Hafner, Post-modern Electromagnetics (John Wiley & Sons, New York, 1999).
- N. Calander and M. Willander, "Theory of surface-plasmon resonance optical-field enhancement at prolate spheroids," J. Appl. Phys. 92, 4878-4884, (2002).
- R. Kappeler, D. Erni, C. Xudong, and L. Novotny, "Field computations of optical antennas," J. Comput. Theor. Nanosci. 4, 686-691 (2007).
- X. Cui,W. Zhang, B.-S. Yeo, R. Zenobi, Ch. Hafner, and D. Erni, "Tuning the resonance frequency of Ag-coated dielectric tips," Opt. Express 15, 8309-8316 (2007). [PubMed]
- H. Fischer and O. J. F. Martin, "Engineering the optical response of plasmonic nanoantennas," Opt. Express 16, 9144-9154 (2008). [PubMed]
- M. I. Stockman, "Ultrafast nanoplasmonics under coherent control," New J. Phys. 10, 025031 (2008).
- A. Taflove and S. C. Hagness, Computational electrodynamics (3rd edition, Artech House, Boston, 2005).
- J. Niegemann, M. K¨onig, K. Stannigel, and K. Busch, "Higher-order time-domain methods for the analysis of nano-photonic systems," Photon. Nanostruct. Fundam. Appl. 7, 2-11 (2008).
- J. S. 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).
- T. Lu, P. Zhang, and W. Cai, "Discontinuous Galerkin methods for dispersive and lossy Maxwell’s equations and ML boundary conditions," J. Comput. Phys. 200, 549-580 (2004).
- M. H. Carpenter and C. A. Kennedy, "Fourth-Order 2N-Storage Runge-Kutta Schemes," Technical Report NASA-TM-109112, NASA Langley Research Center, VA (1994).
- K. Busch, J. Niegemann, M. Pototschnig, and L. Tkeshelashvili, "A Krylov-subspace based solver for the linear and nonlinear Maxwell equations," phys. stat. sol. (b) 244, 3479-2496 (2007).
- H. C. van de Hulst, Light scattering by small particles (Dover Publ., New York, 1981).
- M. Liu, P. Guyot-Sionnest, T.-W. Lee, and S. K. Gray, "Optical properties of rodlike and bipyramidal gold nanoparticles from three-dimensional computations," Phys. Rev. B 76, 235428 (2007).
- E. Hao and G. C. Schatz, "Electromagnetic fields around silver nanoparticles and dimers," J. Chem. Phys. 120, 357-366 (2004). [PubMed]
- H. Kuwata, H. Tamaru, K. Esumi, and K. Miyano, "Resonant light scattering from metal nanoparticles: Practical analysis beyond Rayleigh approximation," Appl. Phys. Lett. 83, 4625-4627 (2003).
- C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (JohnWiley & Sons, New York, 1983).
- C. Sonnichsen, S. Geier, N. E. Hecker, G. von Plessen, J. Feldmann, H. Ditlbacher, B. Lamprecht, J. R. Krenn, F. R. Aussenegg, V. Z-H. Chan, J. P. Spatz, and M. Moller, "Spectroscopy of single metallic nanoparticles using total internal reflection microscopy," Appl. Phys. Lett. 77, 2949-2951 (2000).
- A. Arbouet, D. Christofilos, N. Del Fatti, F. Vall´ee, J.R. Huntzinger, L. Arnaud, P. Billaud, and M. Broyer, "Direct Measurement of the Single-Metal-Cluster Optical Absorption," Phys. Rev. Lett. 93, 127401-1-4 (2004).
- M. Husnik, M. W. Klein, N. Feth, M. K¨onig, J. Niegemann, K. Busch, S. Linden, and M. Wegener, "Absolute extinction cross-section of individual magnetic split-ring resonators," Nature Photon. 2, 614-617 (2008).

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.