## Second harmonic generation from 3D nanoantennas: on the surface and bulk contributions by far-field pattern analysis |

Optics Express, Vol. 19, Issue 27, pp. 26752-26767 (2011)

http://dx.doi.org/10.1364/OE.19.026752

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### Abstract

We numerically study second harmonic generation from dipole gold nanoantennas by analyzing the different contributions of bulk and surface nonlinear terms. We focus our attention to the properties of the emitted field related to the different functional expressions of the two terms. The second harmonic field exhibits different far and near field patterns if both nonlinear contributions are taken into account or if only one of them is considered. This effect persists despite of the model used to estimate the parameters of the nonlinear sources and it is strictly related to the resonant behavior of the plasmonic nanostructure at the fundamental frequency field and to its linear properties at the second harmonic frequency. We show that the excitation of localized surface plasmon polaritons in these structures can remarkably modify the nonlinear response of the system by enhancing surface and/or bulk contributions, creating regimes where bulk nonlinear terms dominate over surface linear terms and vice versa. Finally, the results of our calculations suggest a method that could be implemented to experimentally extract information on the relevance of bulk and surface contributions by measuring and analyzing the generated far field second harmonic patterns in metal nanoantennas and, more in general, in plasmonic nanostructures.

© 2011 OSA

## 1. Introduction

1. S. J. Oldenburg, R. D. Averitt, S. L. Westcott, and N. J. Halas, “*Nanoengineering of optical resonances*,” Chem. Phys. Lett. **288**(2-4), 243–247 (1998). [CrossRef]

6. H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “*Nanorice: a hybrid plasmonic nanostructure*,” Nano Lett. **6**(4), 827–832 (2006). [CrossRef] [PubMed]

9. H. Fischer and O. J. F. Martin, “*Engineering the optical response of plasmonic nanoantennas*,” Opt. Express **16**(12), 9144–9154 (2008). [CrossRef] [PubMed]

10. A. Rasmussen and V. Deckert, “*Surface– and tip–enhanced Raman scattering of DNA components*,” J. Raman Spectrosc. **37**(1-3), 311–317 (2006). [CrossRef]

12. Y. H. Joo, S. H. Song, R. Magnusson, and R. Magnusson, “*Long-range surface plasmon-polariton waveguide sensors with a Bragg grating in the asymmetric double-electrode structure*,” Opt. Express **17**(13), 10606–10611 (2009). [CrossRef]

13. W. Zhang, H. Fischer, T. Schmid, R. Zenobi, and O. J. F. Martin, “*Mode-Selective Surface-Enhanced Raman Spectroscopy Using Nanofabricated Plasmonic Dipole Antennas*,” J. Phys. Chem. C **113**(33), 14672–14675 (2009). [CrossRef]

15. A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Müllen, and W. E. Moerner, “*Large single-molecule fluorescence enhancements produced by a Bowtie nanoantenna*,” Nat. Photonics **3**(11), 654–657 (2009). [CrossRef]

16. J. Li, A. Salandrino, and N. Engheta, “*Optical spectrometer at the nanoscale using optical Yagi-Uda nanoantennas*,” Phys. Rev. B **79**(19), 195104 (2009). [CrossRef]

17. A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “*Unidirectional emission of a quantum dot coupled to a nanoantenna*,” Science **329**(5994), 930–933 (2010). [CrossRef] [PubMed]

18. S. Jha, “*Theory of optical harmonic generation at a metal surface*,” Phys. Rev. **140**(6A), A2020–A2030 (1965). [CrossRef]

21. J. C. Quail and H. J. Simon, “*Second harmonic generation from silver and aluminium films in total internal reflection*,” Phys. Rev. B **31**(8), 4900–4905 (1985). [CrossRef]

22. G. A. Farias and A. A. Maradudin, “*Second harmonic generation in reflection from a metallic grating*,” Phys. Rev. B **30**(6), 3002–3015 (1984). [CrossRef]

23. K. Li, M. I. Stockman, and D. J. Bergman, “*Enhanced second harmonic generation in a self-similar chain of metal nanospheres*,” Phys. Rev. B **72**(15), 153401 (2005). [CrossRef]

30. A. Belardini, M. C. Larciprete, M. Centini, E. Fazio, C. Sibilia, M. Bertolotti, A. Toma, D. Chiappe, and F. Buatier de Mongeot, “*Tailored second harmonic generation from self-organized metal nano-wires arrays*,” Opt. Express **17**(5), 3603–3609 (2009). [CrossRef] [PubMed]

31. M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “*Second-harmonic generation from magnetic metamaterials*,” Science **313**(5786), 502–504 (2006). [CrossRef] [PubMed]

32. V. K. Valev, A. V. Silhanek, N. Verellen, W. Gillijns, P. Van Dorpe, O. A. Aktsipetrov, G. A. Vandenbosch, V. V. Moshchalkov, and T. Verbiest, “*Asymmetric Optical Second-Harmonic Generation from Chiral G-Shaped Gold Nanostructures*,” Phys. Rev. Lett. **104**(12), 127401 (2010). [CrossRef] [PubMed]

33. V. K. Valev, A. V. Silhanek, Y. Jeyaram, D. Denkova, B. De Clercq, V. Petkov, X. Zheng, V. Volskiy, W. Gillijns, G. A. E. Vandenbosch, O. A. Aktsipetrov, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “*Hotspot Decorations Map Plasmonic Patterns with the Resolution of Scanning Probe Techniques*,” Phys. Rev. Lett. **106**(22), 226803 (2011). [CrossRef] [PubMed]

34. V. K. Valev, X. Zheng, C. G. Biris, A. V. Silhanek, V. Volskiy, B. De Clercq, O. A. Aktsipetrov, M. Ameloot, N. C. Panoiu, G. A. E. Vandenbosch, and V. V. Moshchalkov, “*The Origin of Second Harmonic Generation Hotspots in Chiral Optical Metamaterials*,” Opt. Mater. Express **1**(1), 36–45 (2011). [CrossRef]

35. Y. Zeng and J. V. Moloney, “*Volume electric dipole origin of second-harmonic generation from metallic membrane with noncentrosymmetric patterns*,” Opt. Lett. **34**(18), 2844–2846 (2009). [CrossRef] [PubMed]

36. F. X. Wang, F. J. Rodríguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, “*Surface and bulk contributions to the second-order nonlinear optical response of a gold film*,” Phys. Rev. B **80**(23), 233402 (2009). [CrossRef]

35. Y. Zeng and J. V. Moloney, “*Volume electric dipole origin of second-harmonic generation from metallic membrane with noncentrosymmetric patterns*,” Opt. Lett. **34**(18), 2844–2846 (2009). [CrossRef] [PubMed]

36. F. X. Wang, F. J. Rodríguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, “*Surface and bulk contributions to the second-order nonlinear optical response of a gold film*,” Phys. Rev. B **80**(23), 233402 (2009). [CrossRef]

37. C. G. Biris and N. C. Panoiu, “*Second harmonic generation in metamaterials based on homogeneous centrosymmetric nanowires*,” Phys. Rev. B **81**(19), 195102 (2010). [CrossRef]

42. 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**(4), 043828 (2010). [CrossRef]

43. 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**(20), 4045–4048 (1999). [CrossRef]

*s*(also referred to as TE) and

*p*(TM) polarization modes, under which either the magnetic or the electric field are parallel to main axis of the two rods. Changing the input polarization and the thickness of the antenna we can investigate the different nonlinear behavior of the system when resonant excitation of LSPP conditions are fulfilled or not.

## 2. Numerical model

*p*) or TE (

*s*) polarized wave impinges across a gold, square cross-shaped nanoantenna, where with TE or TM we refer to the case of an impinging electric

*E*field parallel or orthogonal to the main nanoantenna axis, respectively. The solution of the leading differential equations is performed by using an integral method based on the dyadic Green's functions numerically implemented in [38

38. A. Benedetti, M. Centini, C. Sibilia, and M. Bertolotti, “*Engineering the Second Harmonic Generation Pattern from Coupled Gold Nanowires*,” J. Opt. Soc. Am. B **27**(3), 408 (2010). [CrossRef]

*y =*0 plane acting as a perfect electric conductor (PEC) for the TE input field and by a perfect magnetic conductor (PMC) for TM input field, respectively.

*x-z*plane (see Fig. 1). For simplicity we consider only the case θ = 0.

38. A. Benedetti, M. Centini, C. Sibilia, and M. Bertolotti, “*Engineering the Second Harmonic Generation Pattern from Coupled Gold Nanowires*,” J. Opt. Soc. Am. B **27**(3), 408 (2010). [CrossRef]

21. J. C. Quail and H. J. Simon, “*Second harmonic generation from silver and aluminium films in total internal reflection*,” Phys. Rev. B **31**(8), 4900–4905 (1985). [CrossRef]

38. A. Benedetti, M. Centini, C. Sibilia, and M. Bertolotti, “*Engineering the Second Harmonic Generation Pattern from Coupled Gold Nanowires*,” J. Opt. Soc. Am. B **27**(3), 408 (2010). [CrossRef]

*κ*the damping coefficient in the Drude law accounting for losses and

_{0}*ε*the relative electric permittivity at the fundamental frequency field. We note that in the lossless case (

_{r,ω}*κ*= 0)

_{0}*α = β =*1, while

*d, δ’, b*assume the typical value for a free electron gas (

*d =*1

*,δ’ =*0

*,b = -*1). We also note that according to the definition given in [36

36. F. X. Wang, F. J. Rodríguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, “*Surface and bulk contributions to the second-order nonlinear optical response of a gold film*,” Phys. Rev. B **80**(23), 233402 (2009). [CrossRef]

*δ’*is relaxation induced, indeed

*δ’*vanishes if perfect lossless electron gas model is considered. In the next section we will show that for the considered nanostructures the generated SH signal is only weakly dependent on the value of

*a*in the considered wavelength range and it does not qualitatively affect the results. This feature could also give an explanation why results obtained with simple models based on lossless free electron gas response, not considering bulk and surface terms separately, in some cases are in agreement with experimental results in metal nanoresonators, split rings and nanostructures.

## 3. Main results

*S*is a close surface surrounding the gold structure,

^{2}thick antennas with TE polarized impinging fields. We note that ACS and SCS values for TM polarized fields are 3 orders of magnitude lower, thus at 800 nm wavelength the antenna is out of resonance for TM polarized fields.

*σ*denotes the SH differential nonlinear scattering cross section.

^{2}. This appears to be the region where the structure behaves according to the results shown in [35

35. Y. Zeng and J. V. Moloney, “*Volume electric dipole origin of second-harmonic generation from metallic membrane with noncentrosymmetric patterns*,” Opt. Lett. **34**(18), 2844–2846 (2009). [CrossRef] [PubMed]

*Surface and bulk contributions to the second-order nonlinear optical response of a gold film*,” Phys. Rev. B **80**(23), 233402 (2009). [CrossRef]

*Surface and bulk contributions to the second-order nonlinear optical response of a gold film*,” Phys. Rev. B **80**(23), 233402 (2009). [CrossRef]

*a*coefficient is sensibly higher than the others and accurate evaluation is required. For example, following Eq. (7) we have a value of

*a*=

*a*= 8.4923 - 1.7634i for a pump wavelength of 800 nm. Nevertheless in the studied geometries the contributions due to the nonlinear response in the normal direction with respect to the metal/dielectric interface tends to destructively interfere (as discussed in [35

_{0}**34**(18), 2844–2846 (2009). [CrossRef] [PubMed]

*a*parameter does not significantly affect the results.

## 4.Conclusions

^{2}to (36 x 36) nm

^{2}. We also set the pump wavelength to 800 nm, being the value at which the nanoantenna with an average thickness of (24 x 24) nm

^{2}resonates. Our numerical results show that for nanoscaled metal structures the nonlinear surface contributions strongly reduce their relative weight in the overall SHG with respect to bulk contributions. If the pump is tuned far from the condition of excitation of LSPP (TM pump) the nonlinear surface contributions are negligible (over one order of magnitude lower) with respect to bulk ones. On the other hand, considering a TE polarized pump, the nanoantenna's nonlinear response is affected by the excitation of LSPP and a variety of cases can be obtained. In particular, for thin structures (15 x 15) nm

^{2}, surface terms can dominate over bulk terms while they become comparable for square cross sections larger than (20 x 20) nm

^{2}. We have also shown that because of this behavior the accurate evaluation of the

*a*parameter in the nonlinear response is not as crucial as for the case of thick flat surfaces. Nevertheless, from the data obtained by our simulations, we are lead to conclude that great attention must be paid when neglecting the nonlinear surface sources in numerical models of SHG, because, every case must be considered by itself, without performing a priori simplification. Finally, our calculations show that different spatial patterns of emission can be achieved by considering surface and/or bulk contributions. This feature suggests a simple but effective way to separate their actual role in the SH generated field by performing an angular map of the irradiation diagram. Performing experiments with different polarization of the pump field, the two regimes could be addressed in order to investigate dominant surface contributions and dominant bulk contributions separately.

## Appendix A

*P*of coordinates (

*x*,

_{P}*y*,

_{P}*z*) of the nanoantenna in

_{P}*y*>0 domain is considered, its mirror point with respect to the plane of symmetry

*y*=0 is S of coordinates (

*x*,-

_{P}*y*,

_{P}*z*). If the FF input field is TE (TM) polarized, the scattered electric field

_{P}*E*(

*x*,

_{P}*y*,

_{P}*z*) is connected to the scattered electric field in S

_{P}*E*(

*x*,-

_{P}*y*,

_{P}*z*) by the equation:

_{P}## Acknowledgments

## References and links

1. | S. J. Oldenburg, R. D. Averitt, S. L. Westcott, and N. J. Halas, “ |

2. | H. Tamaru, H. Kuwata, H. T. Miyazaki, and K. Miyano, “ |

3. | J. Aizpurua, P. Hanarp, D. S. Sutherland, M. Käll, G. W. Bryant, and F. J. García de Abajo, “ |

4. | C. L. Nehl, H. Liao, and J. H. Hafner, “ |

5. | L. J. Sherry, R. Jin, C. A. Mirkin, G. C. Schatz, and R. P. Van Duyne, “ |

6. | H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, “ |

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

8. | A. V. Zayats, I. I. Smolyaninov, and A. A. Maradudin, “ |

9. | H. Fischer and O. J. F. Martin, “ |

10. | A. Rasmussen and V. Deckert, “ |

11. | A. V. Whitney, J. W. Elam, S. L. Zou, A. V. Zinovev, P. C. Stair, G. C. Schatz, and R. P. Van Duyne, “ |

12. | Y. H. Joo, S. H. Song, R. Magnusson, and R. Magnusson, “ |

13. | W. Zhang, H. Fischer, T. Schmid, R. Zenobi, and O. J. F. Martin, “ |

14. | H. Guo, T. P. Meyrath, T. Zentgraf, N. Liu, L. Fu, H. Schweizer, and H. Giessen, “Optical resonances of bowtie slot antennas and their geometry and material dependence |

15. | A. Kinkhabwala, Z. Yu, S. Fan, Y. Avlasevich, K. Müllen, and W. E. Moerner, “ |

16. | J. Li, A. Salandrino, and N. Engheta, “ |

17. | A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “ |

18. | S. Jha, “ |

19. | N. Bloembergen, R. K. Chang, and C. H. Lee, “ |

20. | A. Liebsch, |

21. | J. C. Quail and H. J. Simon, “ |

22. | G. A. Farias and A. A. Maradudin, “ |

23. | K. Li, M. I. Stockman, and D. J. Bergman, “ |

24. | J. I. Dadap, H. B. de Aguiar, and S. Roke, “ |

25. | J. Butet, J. Duboisset, G. Bachelier, I. Russier-Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “ |

26. | J. Butet, G. Bachelier, I. Russier-Antoine, C. Jonin, E. Benichou, and P. F. Brevet, “ |

27. | B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “ |

28. | M. Zavelani-Rossi, M. Celebrano, P. Biagioni, D. Polli, M. Finazzi, L. Duò, G. Cerullo, M. Labardi, M. Allegrini, J. Grand, and P.-M. Adam, “ |

29. | T. Hanke, G. Krauss, D. Träutlein, B. Wild, R. Bratschitsch, and A. Leitenstorfer, “ |

30. | A. Belardini, M. C. Larciprete, M. Centini, E. Fazio, C. Sibilia, M. Bertolotti, A. Toma, D. Chiappe, and F. Buatier de Mongeot, “ |

31. | M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “ |

32. | V. K. Valev, A. V. Silhanek, N. Verellen, W. Gillijns, P. Van Dorpe, O. A. Aktsipetrov, G. A. Vandenbosch, V. V. Moshchalkov, and T. Verbiest, “ |

33. | V. K. Valev, A. V. Silhanek, Y. Jeyaram, D. Denkova, B. De Clercq, V. Petkov, X. Zheng, V. Volskiy, W. Gillijns, G. A. E. Vandenbosch, O. A. Aktsipetrov, M. Ameloot, V. V. Moshchalkov, and T. Verbiest, “ |

34. | V. K. Valev, X. Zheng, C. G. Biris, A. V. Silhanek, V. Volskiy, B. De Clercq, O. A. Aktsipetrov, M. Ameloot, N. C. Panoiu, G. A. E. Vandenbosch, and V. V. Moshchalkov, “ |

35. | Y. Zeng and J. V. Moloney, “ |

36. | F. X. Wang, F. J. Rodríguez, W. M. Albers, R. Ahorinta, J. E. Sipe, and M. Kauranen, “ |

37. | C. G. Biris and N. C. Panoiu, “ |

38. | A. Benedetti, M. Centini, C. Sibilia, and M. Bertolotti, “ |

39. | Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “ |

40. | W. L. Schaich, “ |

41. | M. Centini, A. Benedetti, C. Sibilia, and M. Bertolotti, “ |

42. | M. Scalora, M. A. Vincenti, D. de Ceglia, V. Roppo, M. Centini, N. Akozbek, and M. J. Bloemer, “ |

43. | J. I. Dadap, J. Shan, K. B. Eisenthal, and T. F. Heinz, “ |

44. | J. V.an Bladel, “ |

**OCIS Codes**

(160.4330) Materials : Nonlinear optical materials

(190.3970) Nonlinear optics : Microparticle nonlinear optics

(350.4238) Other areas of optics : Nanophotonics and photonic crystals

(250.5403) Optoelectronics : Plasmonics

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: July 26, 2011

Revised Manuscript: September 10, 2011

Manuscript Accepted: September 12, 2011

Published: December 14, 2011

**Citation**

Alessio Benedetti, Marco Centini, Mario Bertolotti, and Concita Sibilia, "Second harmonic generation from 3D nanoantennas: on the surface and bulk contributions by far-field pattern analysis," Opt. Express **19**, 26752-26767 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-27-26752

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### References

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- H. Tamaru, H. Kuwata, H. T. Miyazaki, and K. Miyano, “Resonant light scattering from individual Ag nanoparticles and particle pairs,” Appl. Phys. Lett.80(10), 1826–1828 (2002). [CrossRef]
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- A. Rasmussen and V. Deckert, “Surface– and tip–enhanced Raman scattering of DNA components,” J. Raman Spectrosc.37(1-3), 311–317 (2006). [CrossRef]
- A. V. Whitney, J. W. Elam, S. L. Zou, A. V. Zinovev, P. C. Stair, G. C. Schatz, and R. P. Van Duyne, “Localized surface plasmon resonance nanosensor: a high-resolution distance-dependence study using atomic layer deposition,” J. Phys. Chem. B109(43), 20522–20528 (2005). [CrossRef] [PubMed]
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- W. Zhang, H. Fischer, T. Schmid, R. Zenobi, and O. J. F. Martin, “Mode-Selective Surface-Enhanced Raman Spectroscopy Using Nanofabricated Plasmonic Dipole Antennas,” J. Phys. Chem. C113(33), 14672–14675 (2009). [CrossRef]
- H. Guo, T. P. Meyrath, T. Zentgraf, N. Liu, L. Fu, H. Schweizer, and H. Giessen, “Optical resonances of bowtie slot antennas and their geometry and material dependence,” Opt. Express16(11), 7756-7766 (2008).
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