## 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 Optical Society of America

## 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 [1–6]. 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 [7–9]. 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 [10–12].

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] and bow-tie [14,15] nanoantennas have been deeply investigated; also metallic nanostructures with different shapes, and nano Yagi-Uda antennas [16,17] 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) [18–20] 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 [21,22]; SHG enhancement from nanoparticles, nanoantennas and nanodimers has been observed both in near and far fields [23–30]. 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] and chiral nanostructures [32] 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 [33,34]. 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 [35,36]. In [35] 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] 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 [37–42] with the exception of single spheres [43]. 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.

Our numerical results show that, despite what happens for thick flat surfaces where SHG is mostly driven by surface effects, for nanoantennas the bulk nonlinear contributions can dominate over the surface contributions if the input field parameters (frequency and polarization) are chosen far from the condition for excitation of LSPP. This result is in agreement with [35] and it is related to the fact that, for small objects with respect to the wavelength, the perturbation induced by the structure to the input field is very weak (i.e. scattering and absorption cross section are small compared to the resonant case) and retardation effects are negligible. On the other hand, approaching resonant conditions, the linear interaction between the input field and the nanoantenna is enhanced, as a consequence bulk and surface nonlinear terms are also both enhanced and their relative weight can be of the same order of magnitude. The physical interpretation of this phenomenon is suggested in [36]: the enhancement of absorption cross section due to high field localization in the gap region between the two rods corresponds to a deeper penetration of the field inside the metal and bulk nonlinear contributions can be enhanced more than surface terms. This result is also in agreement with [35] where it was shown that, for metal nano-membranes of thickness smaller than 30 nm, the SH field is generated by the entire volume of the structure and bulk and surface nonlinear contributions are comparable. Finally, our calculations reveal a more complex scenario. Indeed we report cases where the surface nonlinear contributions dominate with respect to bulk’s terms by one order of magnitude in the SH generated signal. Thus, according to [36], both surface and bulk contributions should always be taken into account and the decision to neglect either one or the other should be numerically verified case by case.

## 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
). 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]. 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 $\overrightarrow{k}$lying 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*being the 3D Green's scalar function in the homogeneous background. The indices

_{m,B}*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 $\left(\overrightarrow{r},\overrightarrow{r}\text{'}\right)$,

*G*changes into:

_{m,B$\left(\overrightarrow{r},\overrightarrow{r}\text{'}\right)$}*α*= 0 or 1 for impinging TE or TM modes, respectively, and

*y*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 $\overrightarrow{r}\text{'}$ 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 ${\overrightarrow{E}}_{1,i}$ of the

_{0}*i*-th scattering point at the FF (ω) depends on ${\overrightarrow{E}}_{0,i}$(the impinging field with a wavevector $\overrightarrow{k}$

*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 ${\overline{\overline{M}}}_{1,i}$ tensor [33] multiplied by the ${\overrightarrow{E}}_{1,i}$ 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 ${\overrightarrow{J}}_{NL}^{Bulk}{}_{}$ and ${\overrightarrow{J}}_{NL}^{Surface}{}_{}$ (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:*

_{1,B}*x-z*plane has been set as the plane of symmetry;

*N*and

*N*refers to the total number of scattering and boundary points, respectively. Δτ

_{B}*is the volume of the*

_{i}*i*-th element of the discretized scatterer.

*ε*

_{0}and

*μ*

_{0}are the vacuum electric permittivity and magnetic permeability; thus in our case

*ε*, the relative electric permittivity, is equal to 1. We adopted the following definitions:

_{B}The effective SH source, described as a current density of homogeneous, isotropic, centro-symmetric bulk materials is of the form [36]:

*γ*,

*d*and

*δ’*are frequency dependent material parameters and can be calculated using a microscopic response theory [21]. Concerning the surface terms, we adopted the parameterization introduced by Rudnick and Stern [21]. Thus the surface current densities have the expressions:

*a*and

*b*are related to the properties of the surface and are frequency dependent. With the signs + and – we refer to the choice of evaluating the discontinuous component of the field at the boundaries by taking its value either outside or inside the metal domain, respectively. Finally,

*δ*is the Dirac’s delta-function. Using the representation of Eqs. (5-6) we assume for the

*γ*parameter the value it assumes in the case of homogeneous electron gases $\gamma =\frac{e{\epsilon}_{0}{\omega}_{0}^{2}}{8m{\omega}^{4}}$ where

*e*is the electron charge modulus, ω

_{0}is the plasma frequency and m is the electron mass. All the corrections related to lattice defects and interband transitions among bound electrons, (responsible for modification of the effective mass and effective plasma frequency) are taken into account by the parameters

*d, δ’, a*and

*b*. We remind that the

*a*and

*b*parameters also take into account the surface features and conditions. For a lossless Drude metal

*d =*1,

*b =*−1,

*δ’ =*0 for all the frequencies while

*a*is strongly dependent on the frequency and can be evaluated for flat infinite surfaces by using a method detailed in [21]. For real samples,

*a*is very sensitive to surface features and has a relevant effect in the determination of the SHG from metal surfaces. We finally note that according to the definitions and symmetries of both boundary and bulk nonlinear contributions, it is easy to assign the TM-like symmetry to the nonlinear SH field vectorial equation, regardless of the pump's initial polarization (see appendix A for details).

In [38] we developed a semiclassical model of the nonlinear second order response of metals by starting from the assumption that linear response in the visible and near infrared range is strongly affected by both the free and bound electrons. Linear response is thus modeled by a Drude–Lorentz method taking into account also ohmic losses of free electrons. This is a crucial point because the field localization properties of the pump are determined by the linear response of the structure, thus an accurate model for the linear response of the metal is needed. Concerning the nonlinear response, the metal/dielectric interface is considered as an abrupt interface and the model does not take into account of several physical phenomena which play an important role in the nonlinear response of metal surfaces as done in [21]. Comparing results of [38] to the Eqs. (5-6) we find that the values of the parameters are:

*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] the nonlinear term proportional to

*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

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

*S*is a close surface surrounding the gold structure, ${\overrightarrow{E}}_{0,\omega}$and ${\overrightarrow{H}}_{0,\omega}$are the electric and magnetic field amplitudes of the FF monochromatic plane wave impinging on the structure.

Enhancement of the ACS is related to high field confinement across the metal/dielectric interface. This is a necessary condition for the enhancement of SHG. Due to the high number of free parameters, we choose to set the length of each element of the nanoantenna to 100 nm, while the spatial gap between each other is set to 30 nm. Finally, we allow the square cross section thickness to vary from a minimum of (10 x 10) nm^{2} to a maximum of (36 x 36) nm^{2}. Our calculations show that there is a resonant behavior if the thickness of the elements of the antenna is set at a value of (24 x 24) nm^{2} for a TE polarized FF wavelength tuned at 800 nm (Fig. 2
). This corresponds to the excitation of a LSPP mode, the FF field is localized in the air gap between the two rods and at the antenna’s tips [9].

Then we performed further calculations at a fixed wavelength of 800 nm varying the thickness of the antenna. Depicted results (Fig. 3
) show that maximum ACS and SCS is obtained for (24 x 24) nm^{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.

Finally we evaluated the nonlinear scattering cross section (NLSCS) as a function of the thickness of the antenna.

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

Figure 4
depicts the behavior of the NLSCS as a function of the thickness of the antenna for TE (Fig. 4a) and TM (Fig. 4b) polarized FF fields for a fixed value of 30 nm for the gap between the two rods. We note a certain correlation between the linear ACS and the NLSCS suggesting that the nonlinear properties are strongly related to the linear response of the nanoantenna. In particular we note that, for the case of TE polarization, there is an enhancement of the NLSCS by 4-5 orders of magnitude with respect to the case of TM polarized pump. Indeed the resonant excitation of the LSPP with consequent high field localization at air/metal interfaces (achieved when the thickness of the nanoantenna is (24x24) nm^{2}) is responsible for the enhancement of both the ACS and the NLSCS.

On the contrary, for TM polarized impinging pump, the size of the antenna along the direction of oscillation of the electric field is small compared to the wavelength (~λ/40), the perturbation to the input field related to the presence of the antenna is weak with respect to the TE case. We only note a monotonic increase of the ACS, SCS and NLSCS as the antenna becomes thicker. The enhancement of the SHG process is driven by the linear response of the metal at the pump frequency and by the considered geometry, in other words by the linear properties of the structure. We also calculate the NLSC when surface or bulk contributions are neglected. Results are depicted in Fig. 4. We note that the overall NLSCS is not equal to the sum of the NLSCS calculated by considering only bulk or surface nonlinear contributions. Indeed the calculation, performed by selectively switching off either one or the other kind of nonlinear source, is only an artificial tool to investigate their relative weight. In the real case they always contribute together to the overall nonlinear current density and they interfere constructively or destructively depending on the relative phase in different spatial portions of the antenna.

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

A completely different scenario is presented if we consider the case of TM pump. We note that the overall NLSCS is mainly due to bulk nonlinear terms (Fig. 4b). As stated before, for TM polarization the size of the antenna along the direction of oscillation of the field is small compared to the wavelength, retardation effects are negligible and the results of our calculations are in agreement with the case discussed in [35]. The increasing behavior of the NLSCS as a function of the antenna’s thickness is only related to the amount of metal.

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* = *a _{0}* = 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]), thus, varying the

*a*parameter does not significantly affect the results.

As an example we performed calculations for different values of the *a* parameter obtained by multiplying *a _{0}* for a variable coefficient ξ that can vary from −1 to 1. This way we performed a set of calculations for

*a*varying from

*–a*to

_{0}*+ a*. We considered the structure which exhibits stronger surface contributions with respect to bulk, namely the 13nmx13nm thick structure, with a TE polarized pump. We depict in Fig. 5 the overall NLSCS as a function of the value of ξ. We note that there is only a 30% difference between the NLSC evaluated when minimum or the maximum value of

_{0}*a*is considered, revealing that most of the surviving surface nonlinear contribution is due to the tangential polarization terms (corresponding to the case

*a*= 0). Our results show that for nanoantennas the nonlinear response related to surface contributions appears to be weaker with respect to the case of flat thick surfaces. Nevertheless the geometry can be responsible of significant modifications and there might be cases where surface terms are stronger and other cases where bulk terms dominate. This behavior cannot be predicted a priori and each single case must be evaluated by itself.

Finally we evaluated the SH far field features by calculating the differential NLSC as defined in Eq. (11). Differences on the far field patterns can be easily evidenced by comparing the SH field generated by TE (Fig. 6a ) and TM (Fig. 7a ) pump polarization for the 13nmx13nm section antenna. We also performed the same calculation by neglecting the surface nonlinear contribution. In this case there are two different features for the TE and TM pump polarization. Indeed when a TM polarized pump is considered we have shown that bulk terms dominate over surface terms by over one order of magnitude. Thus neglecting the surface terms there is not appreciable difference in the far field pattern (compare Fig. 7a and 7b). With a good level of accuracy the signal generated can be considered as related only to the bulk terms. On the other hand, when surface terms are comparable and/or bigger than bulk contribution (i.e. for TE polarization in the considered scheme) the calculated far field pattern can be very different if both terms are considered with respect to the case where surface terms are neglected. As shown in Fig. 6, the difference is not only in the value of the differential NLSCS but also in the angular distribution of the generated SH field. By comparing the two plots we can state that the forward generated SH with respect to the direction of the pump is related only to the surface terms. The analysis of the differential NLSCS (or the far field intensity angular spectrum) suggests a way to perform an indirect measure of the nonlinear coefficient for the bulk and for the surface terms separately by collecting the SH signal generated by either one or the other kind of nonlinear source at different detection angles. Moreover, performing experiments with different polarization of the pump field, the regime where surface or bulk contribution are dominant could be addressed separately.

## 4.Conclusions

We numerically studied the second order nonlinear properties of 3D gold dipole nanoantennas of the shape shown in Fig. 1 by the adoption of an integral method based on a modified Green's tensor to take into account for the symmetry of the structure under plane wave illumination. To simplify the study we choose a constant value for the gap of 30 nm, arm's length of 100 nm and a square section varying from (10 x 10) nm^{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

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 (*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}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:

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

For the nonlinear surface contributions in Eq. (6) we need to write the unitary local tangent and normal vectors in cartesian coordinates for each point of the surface. We have no ambiguities for the orientation of the normal vector, but the two local tangent vectors cannot be univocally defined. Anyway, by geometrical construction it can be shown that the cartesian coordinates of local tangent and normal vectors in the point P and S are related by the following equations:

The results in (A5) show a TM mode symmetry. The same procedure can be applied also to the third term in Eq. (6) related to the local unitary tangent vector *Z.* Finally, for the normal component of the nonlinear surface term we have:

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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