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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 27 — Dec. 19, 2011
  • pp: 26752–26767
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Second harmonic generation from 3D nanoantennas: on the surface and bulk contributions by far-field pattern analysis

Alessio Benedetti, Marco Centini, Mario Bertolotti, and Concita Sibilia  »View Author Affiliations


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

During the last decade a large number of works has been devoted to the investigation of the optical properties of metallic nanostructures and nanoparticles [1

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

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]

]. These kind of structures allow to locally enhance the incoming electromagnetic (e.m.) field by several orders of magnitude thanks to the high field localization across the metal-dielectric interfaces corresponding to the excitation of localized surface plasmon polariton (LSPP) modes [7

7. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles, Wiley, (1983).

9

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

]. The frequency of a LSPP can be tuned by varying the geometry of the structure, the metal material adopted and the environment morphology and composition. For these reasons metal nanostructures have been proposed for several interesting applications in chemistry, optical sensing and signal processing at the nanoscale [10

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

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]

].

Recently the combination of antenna devices with the LSPP excitation in nanoscale systems attracted growing interest, thanks to the possibility to manipulate the e.m. information at optical frequencies. Dipolar [13

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]

] and bow-tie [14

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,” Opt. Express 16(11), 7756-7766 (2008).

,15

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]

] nanoantennas have been deeply investigated; also metallic nanostructures with different shapes, and nano Yagi-Uda antennas [16

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

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]

] have been considered in plasmonics research.

Metal nanostructures show other important features when the intrinsic nonlinear optical properties of metals come into play. Second order effects such as second harmonic generation (SHG) [18

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

20

20. A. Liebsch, Electronic Excitations at Metal Surfaces,” Plenum, New York, (1997). chap 5.

] have been studied and experimentally observed. In particular, these phenomena have been widely investigated both theoretically and experimentally for the case of flat metal surfaces and gratings [21

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

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]

]; SHG enhancement from nanoparticles, nanoantennas and nanodimers has been observed both in near and far fields [23

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

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]

]. Interest for SHG in plasmonic nanostructures is growing rapidly, the aim is the creation of a new class of artificial nonlinear devices with specifically tailored properties. For example, generation of SH was observed from systems composed by split ring resonators [31

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]

] and chiral nanostructures [32

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]

] revealing peculiar properties. Of particular interest is the fact that the SHG is enhanced by the presence of “hot spots”, namely regions where the pump field is strongly localized. Recently, experimental techniques have been developed to map the hot spots and to study the connection between field localization and SHG properties in optical metamaterials [33

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

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]

]. Since the local field distribution in nanoplasmonic structures is highly inhomogeneous it is not obvious to establish the role of surface and bulk nonlinearities as well as their relative weight in the overall second harmonic (SH) signal. This subject has been investigated both theoretically and experimentally [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]

,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]

]. 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]

] it was analytically shown that for thin gold films and thin membranes the SH field is generated by the entire volume, provided that higher order multipole effects are negligible. On the other hand, 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]

] surface and bulk contributions to the second order nonlinear response of a 150 nm thick film of gold have been separately measured by using a two-beam SHG setup; the analysis of the experimental data suggest that surface nonlinearities dominate. Nevertheless if resonant structures are considered, the high field localization at the hot spots could enhance the SH signal from the bulk relaxation induced nonlinear response. Unfortunately, due to the amount of geometrical parameters, which affect both the linear and nonlinear response of three-dimensional (3D) disposition of scatterers to the impinging e.m. field, the calculations must be performed numerically [37

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

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]

] with the exception of single spheres [43

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]

]. Thus every case must be considered and analyzed by itself and the question concerning the relevance of bulk and surface nonlinearities in nanoplasmonic structures is still under debate.

In this paper we numerically investigate the SHG properties of 3D gold nanoantennas made of two coupled gold rods with square cross section. The separation between the two elements is kept constant at 30 nm. Due to their high selectivity in polarization, we consider the effects of SHG under two distinct illumination modes, such as the 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

Here a general outline of the adopted method will be presented. Although in the developed model it is possible to consider an arbitrary incident field profile with arbitrary polarization and angle of incidence, we limit our study to the case of plane wave incidence along the x-axis (see Fig. 1a
Fig. 1 (a) Schematic of the system under examination. (b) Simplified representation: when the kvector lays on the x-z plane, a mirror-like perfect electric conductor (perfect magnetic conductor) can be placed in the middle of the antenna for TE (TM) plane wave illumination mode. kdenotes the wave vector and indicates the direction of propagation of the pump field. Aand Bvectors represent the directions of oscillation of the electric field for the TE (s) and TM (p) mode respectively.
). In particular, the study of SHG will be performed when a TM (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]

]. We take advantage of the intrinsic symmetry of our system by adopting a modified Green's tensor which groups the two elements composing the entire antenna (see Fig. 1b and Appendix A). This procedure allows us to reduce the number of performed numerical integrations, the computation time and the amount of memory required to store the data. The results will be the same if we insert an infinite sheet laying in the 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.

It is important to mention that this approach is valid only for illumination functions with proper symmetry, which in the case of plane wave illumination means to have the wave vector klying in the x-z plane (see Fig. 1). For simplicity we consider only the case θ = 0.

The Green's tensor corresponding to an infinite homogeneous background medium is defined as:
G¯¯m,B(r,r')=[I¯¯+km,B2]gm,B(r,r')
(1)
with gm,B being the 3D Green's scalar function in the homogeneous background. The indices m = 1,2 refers to the fundamental frequency (FF) and SH cases. Because of the above mentioned approach, taking advantage of the symmetry of the system, for every couple of points (r,r'), Gm,B(r,r') changes into:
G¯¯m,BM,TM/TE[r,r']=G¯¯m,B[r,r']++(1)αG¯¯m,B[r,(x',2y0y',z')]·S¯¯,S¯¯=(1000+10001);r'=(x',y',z');
(2)
where α = 0 or 1 for impinging TE or TM modes, respectively, and y0 refers to the elevation of the plane of symmetry, which will be 0 in our following examples. The modified Green dyadic takes into account the contributions of both the point source located at r' and the source corresponding to its mirror image with respect to plane of symmetry. Assuming undepletion of the pump field, the solution of the FF field is obtained by considering only linear contributions. According to the theory and to the new rules coming from the symmetry of our system, the total electric field E1,i of the i-th scattering point at the FF (ω) depends on E0,i(the impinging field with a wavevector k1,B in the background medium, which in our examples is assumed to be the vacuum space), on the electric fields scattered by the other points, (with the exception of the mirror point placed on the opposite arm of the nanoantenna, which is represented separately in the numerical equation), and finally on the self patch contribution represented by the M¯¯1,i tensor [33

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]

] multiplied by the E1,i field itself (see Eq. (3).a)). The numerical procedure for the second harmonic (SH) field shares a similar approach with the FF one, with the exception that there is no input field, instead there are bulk and surface nonlinear sources represented by nonlinear current densities JNLBulk and JNLSurface (see Eq. (3).b)). Due to the symmetry properties of the nonlinear sources, the modified Green tensor for the SH field contributions fulfills the TM mode symmetry regardless of the polarization of the impinging field. This feature finds its explanation on the particular expressions in Eqs. (5-6) of the nonlinear sources and it is discussed in detail in Appendix A. The discrete equations representing both the FF and SH fields are:
E1,i=E0,i+j=1,jiN/2G¯¯1,BM,TE/TM(ri,rj)·[Δε1,jk1,B2E1,j]Δτj++(1)αG¯¯1,B[ri,(xi,2y0yi,zi)]·S¯¯·[Δε1,ik1,B2E1,i]Δτi+M¯¯1,i·E1,i; (3.a)
E2,l=j=1,jlN/2G¯¯2,BM,TM(rl,rj)·[Δε2,jk2,B2E2,j+i2ωμ0JNL,jBulk]Δτj+G¯¯2,B[rl,(xl,2y0yl,zl)]·S¯¯·[Δε2,lk2,B2E2,l+i2ωμ0JNL,lBulk]Δτl++M¯¯2,l·[E2,l+i2ωε0Δε2,lJNL,lBulk]++k=1NB/2G¯¯2,BM,TM(rl,rk)·[i2ωμ0JNL,kSurface]Δτk; (3.b)
where the x-z plane has been set as the plane of symmetry; N and NB refers to the total number of scattering and boundary points, respectively. Δτi is the volume of the i-th element of the discretized scatterer. ε0 and μ0 are the vacuum electric permittivity and magnetic permeability; thus in our case εB, the relative electric permittivity, is equal to 1. We adopted the following definitions:
M¯¯m,i=23Δεm,i[(1ikm,BRieff)eikm,BRieff12εB1]I¯¯,Δεm,i=εm,iεB,Rieff=(34πΔτi)13;
(4)
for the expression of the self patch term labeled as M¯¯m,i, which is meant to numerically resolve the singularity added by the dyadic Green's function G¯¯m,B when the emitting and observing points coincide. This numerical procedure allows the evaluation of the e.m. field inside the scattering points placed in the spatial region occupied by the nanoantenna; Once this calculation has been performed, the evaluation of the e.m. field in any region outside the scatterers is straightforward. The numerical algorithm shows good convergence, and it can be applied to metal structures of arbitrary shape even presenting inhomogeneities and anisotropic properties. In particular, we analyzed the SH pattern both in the near field and in a surrounding sphere in the far field region.

3. Main results

As a first step we evaluated absorption cross section (ACS) and the scattering cross section (SCS) for the FF field as defined in [44

44. J. V.an Bladel, “Electromagnetic Fields (2nd Edition),” IEEE Press Wiley (2007).

]:
SCS=SRe(ESC,ω×HSC,ω*)n^dSRe(E0,ω×H0,ω*)
(9)
ACS=SRe(Eω×Hω*)n^dSRe(E0,ω×H0,ω*)
(10)
where S is a close surface surrounding the gold structure, E0,ωand H0,ωare the electric and magnetic field amplitudes of the FF monochromatic plane wave impinging on the structure.

Then we performed further calculations at a fixed wavelength of 800 nm varying the thickness of the antenna. Depicted results (Fig. 3
Fig. 3 Linear absorption cross section (ACS- red curve) and scattering cross section (SCS-blue curve) values for TE (solid) and TM (dashed) polarization.
) show that maximum ACS and SCS is obtained for (24 x 24) nm2 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.

Finally we evaluated the nonlinear scattering cross section (NLSCS) as a function of the thickness of the antenna.
NLSCS=Sσ(θ,φ)dΩ;σ(θ,φ)=2SincRe(ESC,2ω×HSC,2ω*)·n^R2P0,ω2;
(11)
where σ denotes the SH differential nonlinear scattering cross section.

We note that for TE polarized pump and 25-35 nm thick structures, bulk nonlinear contributions are dominant and the NLSCS evaluated by considering only surface terms is about one order of magnitude lower than the one obtained by taking into account both contributions (Fig. 4a). The two contributions are comparable for nanoantennas with square section around (20 x 20) nm2. 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]

]. If thinner structures are considered, (around 12-15 nm) the NLSCS evaluated by considering only the surface contributions is about one order of magnitude higher than the one obtained by considering only bulk terms (Fig. 4a).

We can give a physical interpretation to this phenomenon by considering the results presented 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]

]. For TE polarized pump the linear response of the system is stronger with respect to the TM case. The size of the antenna along the direction of oscillation of the electric field is approximately λ/4 and LSPP modes are excited. Indeed, SCS and ACS are almost 3 orders of magnitude higher. This is the signature of high localization of the field at metal/air interfaces, higher absorption means higher field penetration inside the metal. At first (for 10-18 nm thick antennas) surface contributions are stronger, as expected for a film. Increasing the thickness of the antenna, the e.m. field becomes more localized (close to the air gap between the two rods and at the antenna’s tips) reducing the amount of surface effectively contributing to the process and enhancing the bulk nonlinear response by increasing the number of plane wave expansion coefficients used to describe the hot spot, as discussed 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]

]. For the geometry under consideration our results show that bulk contributions become dominant if the antenna is thicker than 20 nm. Different structures might reveal different behaviors, both surface and bulk nonlinear contributions should be considered and analyzed case by case.

The first concern is to find out whether the rough model used to estimate the gold nonlinear response might affect the accuracy of our results quantitatively and/or qualitatively. Usually, for flat metal layers, the surface effects dominate because the a coefficient is sensibly higher than the others and accurate evaluation is required. For example, following Eq. (7) we have a value of a = a0 = 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

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]

]), thus, varying the a parameter does not significantly affect the results.

4.Conclusions

Appendix A

It is possible to take advantage of the intrinsic symmetry of the system under investigation to reduce the number of points by considering only one half of the antenna. This procedure is performed by the adoption of a particular Green's tensor for the FF whose configuration depends on the illumination mode, either TE or TM, according to the direction of the electric pump field. If a generic point P of coordinates (xP,yP,zP) of the nanoantenna in y>0 domain is considered, its mirror point with respect to the plane of symmetry y=0 is S of coordinates (xP,-yP,zP). If the FF input field is TE (TM) polarized, the scattered electric field E(xP,yP,zP) is connected to the scattered electric field in S E(xP,-yP,zP) by the equation:

{ES,x=EP,xES,y=+EP,yES,z=EP,zTEpump;{ES,x=+EP,xES,y=EP,yES,z=+EP,zTMpump;
[A1]

In what follows we show that the generated SH field fulfills the same symmetry rules used for the TM polarized FF field for both TE and TM polarization of the pump. This results is obtained by evaluating the nonlinear polarization source terms.

At first we consider the nonlinear bulk contributions. According to Eq. (5) we need to evaluate the symmetry properties of two terms:
(ES·)ES=(ES,x,ES,y,ES,z)·(ES,xxES,yxES,zxES,xyES,yyES,zyES,xzES,yzES,zz)=={(EP,x,EP,y,EP,z)·(EP,xxEP,yxEP,zxEP,xyEP,yyEP,zyEP,xzEP,yzEP,zz)TEpumpmode(EP,x,EP,y,EP,z)·(E1,xxE1,yxE1,zxE1,xyE1,yyE1,zyE1,xzE1,yzE1,zz)TMpumpmode==([(EP·)EP]x,[(EP·)EP]y,[(EP·)EP]z)
[A2]
and:

(|ES|2)=(|EP|2x,|EP|2y,|EP|2z)=([(|EP|2)]x,[(|EP|2)]y,[(|EP|2)]z)
[A3]

Eqs. (A2) and (A3), clearly show that the bulk nonlinear terms fulfill the TM mode symmetry according to (A1).

ES,XS()ES,YS()={±[EP,XP()][EP,YP()]TEpumpmode±[EP,XP()][EP,YP()]TMpumpmode=EP,XP()EP,YP(){ES,XS()ES,YS()X^S,x=+EP,XP()EP,YP()X^P,xES,XS()ES,YS()X^S,y=EP,XP()EP,YP()X^P,yES,XS()ES,YS()X^S,z=+EP,XP()EP,YP()X^P,z
[A5]

[E2,Y()]2=[E1,Y()]2{[E2,Y()]2Y^2,x=+[E1,Y()]2Y^1,x[E2,Y()]2Y^2,y=[E1,Y()]2Y^1,y[E2,Y()]2Y^2,z=+[E1,Y()]2Y^1,z
[A6]

In conclusion both bulk and surface nonlinear terms fulfill the TM mode symmetry.

Acknowledgments

The authors wish to acknowledge M. Kauranen for interesting and stimulating discussions. This work was financially supported by PRIN programme.

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11.

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. B 109(43), 20522–20528 (2005). [CrossRef] [PubMed]

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]

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,” Opt. Express 16(11), 7756-7766 (2008).

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]

19.

N. Bloembergen, R. K. Chang, and C. H. Lee, “Second harmonic generation of light in reflection from media with inversion symmetry,” Phys. Rev. Lett. 16(22), 986–989 (1966). [CrossRef]

20.

A. Liebsch, Electronic Excitations at Metal Surfaces,” Plenum, New York, (1997). chap 5.

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]

24.

J. I. Dadap, H. B. de Aguiar, and S. Roke, “Nonlinear light scattering from clusters and single particles,” J. Chem. Phys. 130(21), 214710 (2009). [CrossRef] [PubMed]

25.

J. Butet, J. Duboisset, G. Bachelier, I. Russier-Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Optical second harmonic generation of single metallic nanoparticles embedded in a homogeneous medium,” Nano Lett. 10(5), 1717–1721 (2010). [CrossRef] [PubMed]

26.

J. Butet, G. Bachelier, I. Russier-Antoine, C. Jonin, E. Benichou, and P. F. Brevet, “Interference between selected dipoles and octupoles in the optical second-harmonic generation from spherical gold nanoparticles,” Phys. Rev. Lett. 105(7), 077401 (2010). [CrossRef] [PubMed]

27.

B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in noncentrosymmetric nanodimers,” Nano Lett. 7(5), 1251–1255 (2007). [CrossRef] [PubMed]

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, “Near-field second-harmonic generation in single gold nanoparticles,” Appl. Phys. Lett. 92(9), 093119 (2008). [CrossRef]

29.

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(25), 257404 (2009). [CrossRef] [PubMed]

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]

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]

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]

39.

Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B 79(23), 235109 (2009). [CrossRef]

40.

W. L. Schaich, “Second harmonic genaration by periodically-structured metal surfaces,” Phys. Rev. B 78(19), 195416 (2008). [CrossRef]

41.

M. Centini, A. Benedetti, C. Sibilia, and M. Bertolotti, “Coupled 2D Ag nano-resonator chains for enhanced and spatially tailored second harmonic generation,” Opt. Express 19(Issue 9), 8218–8232 (2011). [CrossRef] [PubMed]

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]

44.

J. V.an Bladel, “Electromagnetic Fields (2nd Edition),” IEEE Press Wiley (2007).

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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  9. H. Fischer and O. J. F. Martin, “Engineering the optical response of plasmonic nanoantennas,” Opt. Express16(12), 9144–9154 (2008). [CrossRef] [PubMed]
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  11. 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]
  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. Express17(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. C113(33), 14672–14675 (2009). [CrossRef]
  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,” Opt. Express16(11), 7756-7766 (2008).
  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. Photonics3(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. B79(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,” Science329(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]
  19. N. Bloembergen, R. K. Chang, and C. H. Lee, “Second harmonic generation of light in reflection from media with inversion symmetry,” Phys. Rev. Lett.16(22), 986–989 (1966). [CrossRef]
  20. A. Liebsch, “Electronic Excitations at Metal Surfaces,” Plenum, New York, (1997). chap 5.
  21. J. C. Quail and H. J. Simon, “Second harmonic generation from silver and aluminium films in total internal reflection,” Phys. Rev. B31(8), 4900–4905 (1985). [CrossRef]
  22. G. A. Farias and A. A. Maradudin, “Second harmonic generation in reflection from a metallic grating,” Phys. Rev. B30(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. B72(15), 153401 (2005). [CrossRef]
  24. J. I. Dadap, H. B. de Aguiar, and S. Roke, “Nonlinear light scattering from clusters and single particles,” J. Chem. Phys.130(21), 214710 (2009). [CrossRef] [PubMed]
  25. J. Butet, J. Duboisset, G. Bachelier, I. Russier-Antoine, E. Benichou, C. Jonin, and P. F. Brevet, “Optical second harmonic generation of single metallic nanoparticles embedded in a homogeneous medium,” Nano Lett.10(5), 1717–1721 (2010). [CrossRef] [PubMed]
  26. J. Butet, G. Bachelier, I. Russier-Antoine, C. Jonin, E. Benichou, and P. F. Brevet, “Interference between selected dipoles and octupoles in the optical second-harmonic generation from spherical gold nanoparticles,” Phys. Rev. Lett.105(7), 077401 (2010). [CrossRef] [PubMed]
  27. B. K. Canfield, H. Husu, J. Laukkanen, B. Bai, M. Kuittinen, J. Turunen, and M. Kauranen, “Local field asymmetry drives second-harmonic generation in noncentrosymmetric nanodimers,” Nano Lett.7(5), 1251–1255 (2007). [CrossRef] [PubMed]
  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, “Near-field second-harmonic generation in single gold nanoparticles,” Appl. Phys. Lett.92(9), 093119 (2008). [CrossRef]
  29. 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(25), 257404 (2009). [CrossRef] [PubMed]
  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. Express17(5), 3603–3609 (2009). [CrossRef] [PubMed]
  31. M. W. Klein, C. Enkrich, M. Wegener, and S. Linden, “Second-harmonic generation from magnetic metamaterials,” Science313(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. Express1(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. B80(23), 233402 (2009). [CrossRef]
  37. C. G. Biris and N. C. Panoiu, “Second harmonic generation in metamaterials based on homogeneous centrosymmetric nanowires,” Phys. Rev. B81(19), 195102 (2010). [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. B27(3), 408 (2010). [CrossRef]
  39. Y. Zeng, W. Hoyer, J. Liu, S. W. Koch, and J. V. Moloney, “Classical theory for second-harmonic generation from metallic nanoparticles,” Phys. Rev. B79(23), 235109 (2009). [CrossRef]
  40. W. L. Schaich, “Second harmonic genaration by periodically-structured metal surfaces,” Phys. Rev. B78(19), 195416 (2008). [CrossRef]
  41. M. Centini, A. Benedetti, C. Sibilia, and M. Bertolotti, “Coupled 2D Ag nano-resonator chains for enhanced and spatially tailored second harmonic generation,” Opt. Express19(Issue 9), 8218–8232 (2011). [CrossRef] [PubMed]
  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. A82(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]
  44. J. V.an Bladel, “Electromagnetic Fields (2nd Edition),” IEEE Press Wiley (2007).

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