## SHG simulations of plasmonic nanoparticles using curved elements |

Optics Express, Vol. 19, Issue 15, pp. 14426-14436 (2011)

http://dx.doi.org/10.1364/OE.19.014426

Acrobat PDF (1049 KB)

### 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

## 1. Introduction

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]

*al.*further supported this statement by investigations on a 3D spherical cavity [10

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]

## 2. Discontinuous Galerkin method and curved elements

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]

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]

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

## 3. Linear calculations

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

*ω*= 1.30 × 10

_{p}^{16}

*s*

^{−1},

*γ*= 3.23 × 10

^{13}

*s*

^{−1}), embedded in air (cf. Fig. 1) and excited with a differentiated Gaussian pulse (see [14

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]

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]

## 4. SHG calculation

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]

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]

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]

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]

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]

**E**being the electric field,

**H**the magnetic field,

*N*the dipole density,

_{d}*q*the charge of an electron,

*m*the effective electron mass,

*μ*the permeability,

*γ*the damping coefficient and

*ω*

_{0}the dipole eigenfrequency. Now,

**E**and

**P**×

**H**can be approximated with a 2nd-order Taylor expansion around

**P**= 0 (note that the standard Drude-Lorentz model only uses a 1st-order expansion): By inserting these expressions in Eq. (1) we obtain the ELM: with

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]

*E*

_{0}; the second term provides the nonlinear response and is proportional to the gradient of

*E*

_{0}times the polarization. So, in contrast to the linear case, the SHG calculations are not only dependent on the field in the previous time step but also on the gradient of that previous field. Therefore, an as-good-as-possible geometrical description gets even more important.

*al.*chose B=2 and up to P=6 in [10

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]

*linear*calculations.

## 5. Numerical effort

mesh | M1 | M2 | M3 | M4 |
---|---|---|---|---|

linear elements | 0.25 | 0.57 | 1.04 | 8.75 |

curved elements | 0.85 | 1 | 1.69 | 10.20 |

## 6. Conclusions and outlook

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]

*r*>5 nm [9

**7**, 1581–1586 (2010). [CrossRef]

## Acknowledgments

## References and links

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 |

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 |

3. | V. Deckert, “Tip-enhanced Raman spectroscopy,” J. Raman Spectrosc. |

4. | P. Olk, J. Renger, T. Härtling, M. T. Wenzel, and L. M. Eng, “Two particle enhanced nano Raman microscopy and spectroscopy,” Nano Lett. |

5. | J. Renger, R. Quidant, N. V. Hulst, and L. Novotny, “Surface-enhanced nonlinear four-wave mixing,” Phys. Rev. Lett. |

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. |

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. |

8. | G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. |

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. |

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. |

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

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. |

13. | K. Stannigel, M. König, J. Niegemann, and K. Busch, “Discontinuous Galerkin time-domain computations of metallic nanostructures,” Opt. Express |

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. |

15. | J. Niegemann, W. Pernice, and K. Busch, “Simulation of optical resonators using DGTD and FDTD,” J. Opt. A, Pure Appl. Opt. |

16. | J. S. Hesthaven and T. Warburton, |

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. |

18. | J. Schöberl, “NETGEN an advancing front 2D/3D-mesh generator based on abstract rules,” Comput. Visual. Sci. |

19. | C. Hafner, |

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. |

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 |

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 |

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 |

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 |

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. |

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 |

**OCIS Codes**

(190.2620) Nonlinear optics : Harmonic generation and mixing

(290.5850) Scattering : Scattering, particles

(250.5403) Optoelectronics : Plasmonics

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: March 25, 2011

Revised Manuscript: May 5, 2011

Manuscript Accepted: June 1, 2011

Published: July 13, 2011

**Citation**

René Kullock, Andreas Hille, Alexander Haußmann, Stefan Grafström, and Lukas M. Eng, "SHG simulations of plasmonic nanoparticles using curved elements," Opt. Express **19**, 14426-14436 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-15-14426

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### References

- 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]
- 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]
- V. Deckert, “Tip-enhanced Raman spectroscopy,” J. Raman Spectrosc. 40, 1336–1337 (2009). [CrossRef]
- 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]
- 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]
- G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 25, 377–445 (1908). [CrossRef]
- 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]
- 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]
- 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]
- J. S. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications (Springer, 2008). [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. Schöberl, “NETGEN an advancing front 2D/3D-mesh generator based on abstract rules,” Comput. Visual. Sci. 1, 41–52 (1997). [CrossRef]
- 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]
- 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]
- 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]
- 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]
- 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]

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