We report results of second harmonic generation calculations performed on Silver coupled 2D-nanoresonators. Coupling is responsible for the creation of resonant modes that can be localized on small portions of the structure or distributed over the whole structure. Different field profiles can be obtained by varying the parameters of the input field (i.e. the wavelength). The second harmonic generation nonlinear process is enhanced by the excitation of coupled surface plasmon polaritons. The emitted field is strongly affected by the linear properties of the structure behaving as a nano antenna. We note that different configurations of the pump field lead to different second harmonic far-field emission patterns. Also, we show that the angular emission of the second harmonic field contains information about the spatial location of the pump field hot spots at different frequencies. Applications to a new class of nano sources for single molecule fluorescence and sensors are proposed.
© 2011 OSA
Recently the development of nanotechnologies has made possible the creation of techniques for manipulation of matter at the nanoscale; consequently artificial materials exhibiting new properties compared with natural materials have been created. In particular, concerning the optical properties, it has been proved that some devices allow the confinement of light (more generally of the electromagnetic field), on sub wavelength spatial scales, beyond the limit imposed by the theory of diffraction. Nanofabrication processes are at the basis of the development of artificial materials, called meta materials (MM), operating in the range of wavelengths of visible or near infrared. They can be characterized macroscopically as homogeneous media with unique properties compared to natural materials [1–8]. The understanding of the macroscopic properties at a deeper level requires a knowledge of the properties of the single element that makes up the material, the “meta atom”. For this reason a part of research activities in the field of nano photonics is dedicated to the study of the optical response of nano particles or small clusters of particles with particular interest to the study of nano metallic structures also called plasmonic nanostructures (NS) . In the case of three-dimensional NS, if the appropriate resonance conditions are fulfilled, we observe the excitation of modes on the surface of the structure called localized surface plasmon polaritons (LSPP) . The confinement of the electromagnetic field on such a small size is very sensitive to small changes in morphology or surface imperfections which alter the conditions of resonance. These features are on the basis of the applications that have been proposed for the realization of highly sensitive sensors and biosensors  and for the improvement of the performances of solar cells by optimizing the amount of light energy converted into electrical energy . Reference  highlights the potential applications of plasmonic NS to enhance the fluorescence of single molecules and to produce integrated circuits able to perform logical operations on very small spaces . MM have recently been proposed to enhance the luminescence of a quantum dot by controlling the radiation spectrum . Similar mechanisms are used to monitor the far-field emission of nano sources by maximizing efficiency in terms of power transfer and directionality. For example, in  it was proved experimentally the control and the optimization of the emission of radiation by a quantum dot coupled to a nano-antenna designed according to the criteria of Yagi-Uda, widely used for radio frequency antennas.
Concerning nonlinear (NL) properties, interest in second and third order response of nanostructured media is growing rapidly [16–38]. Although centro-symmetric metal structures have a vanishing second order nonlinear electric susceptibility, second order effects arise due to contributions of magnetic dipole and electric quadrupole [20–23]. These mechanisms have been widely studied in the case of flat metal surfaces and gratings both theoretically and experimentally since the 60’s [39–45]. Other more recent works are devoted to the study of NL response of individual particles or percolated systems and to the description of experimental techniques of characterization [24–27]. Lately, the development of nanotechnologies and MM gave new life to the study of the properties of NL metal NS: recent results report on second harmonic (SH) generation from metal tips , nano antennas  and nano-dimers . It has also been shown that planar noncentrosymmetric geometries realized by metal NS produce macroscopically the effect of MM with a second-order nonlinearity. For example, generation of SH was observed in a MM consisting of split ring resonators  and in chiral nanostructures . Also, there is considerable interest for theoretical and experimental properties of SH generation in the near-field regime [22,31]. The generated field is located on very small spatial regions and can be used to selectively excite the fluorescence of single molecules and/or quantum dots. Concerning the theoretical activity, SH generation in the far field and near-field has been studied for two-dimensional cylindrical metal objects , also of arbitrary cross-section rods , for nanospheres and for arbitrary arrangements of cylinders [34,35]. Of particular interest is the fact that the emitted SH far-field profile depends very strongly on the regions where the pump field is localized but also by the geometry of the whole structure. It is possible to obtain different types of emission patterns by changing the geometry of the nano-scale emitters [33,35]. Given the complexity and variety of plasmonic NS, in most cases the theoretical study is carried out by resorting to numerical methods [36–38]. In a previous work  we numerically investigated the properties of far field and near field generated at the SH frequency from two coupled nanowires as a function of the shape of the sections and of the coupling between them.
In this paper we investigate the nonlinear properties related to the SHG of a finite number of 2D coupled plasmon resonators. It has been shown that the electromagnetic field can be confined and guided on the nanoscale by using a chain of metallic blocks properly designed in order to be considered as a set of coupled resonators [46,47]. Our aim is to take advantage of this phenomenon in order to improve and maximize the nonlinear response, by maximizing the value of the SH nonlinear scattering cross section (NLSCS). Moreover we will show that it could be possible to spatially resolve the excitation of different LSPP modes at the fundamental frequency (FF), or to detect the presence of external agents by detecting the generated SH signal in the far field and/or in the near field. This property is strictly related to the effect of coupling between resonators. Indeed, coupling is responsible for the formation of resonant modes that can be localized on small portions of the structure or distributed over the whole structure. Different field profiles can be obtained by varying the parameters of the input field (for example the frequency or the spatial phase profile). Zones of high field confinement, called hot spots are characterized by local field intensity values 3 or 4 orders of magnitude higher than the incident field’s intensity. Thus, considering the process of SH generation the nonlinear emission comes from the hot spots with an enhancement up to 8 orders of magnitude with respect to the portions of the structure where the field is not localized. Then the structure behaves as a nanoantenna for the generated SH field and determines the radiation transfer between the localized sources and the free-radiation field pattern. In other words we are considering a regime where the nanostructure can be in good approximation considered as a “meta atom” for the field at the FF but its size becomes comparable to the wavelength at the SH field, thus photonics effect needs to be taken into account. As a results we will show that different FF localization patterns produce very different SH far field patterns. This feature might have interesting application for selective sensing and secure information encoding.
2. Linear properties
For the sake of simplicity and in order to improve the efficiency and speed of numerical calculations we consider 2D systems embedded in a homogeneous medium. Thus the nanoresonators are supposed to be infinitely long in the z-direction (see Fig. 2 ) and we consider air as the surrounding medium. Our aim here is to show how the coupling between nanoresonators can affect the properties of the FF and of the generated SH fields. More complex and realistic structures can be modeled by using and adapting the same numerical technique. The building block of the considered system is composed by two Silver rods of rectangular section (L = 212nm, T = 38nm) separated by a 28 nm gap. These values of the geometrical parameters have been chosen in order to have a resonant behavior in the wavelength range around 1 μm. Shorter (longer) resonant wavelengths can be tuned (for example) by decreasing (increasing) the length L of the rods. Concerning the linear response of Ag we considered a Drude-Lorentz model for the expression of the complex dielectric constant as a function of the angular frequency . The model provides an accurate description of the behavior of the electric permittivity in agreement with experimental data by considering both contribution of the free electrons and bound electrons. The behavior of bound electrons is described by using five Lorentz oscillators while the free electrons contribution are considered in the Drude model:49] based on a numerical integration of the wave equation in the frequency domain by using the Green’s dyadic approach. A study of the linear properties of the two block structure shows that in this configuration the system behaves as a nanoresonator. Indeed, if an incoming field at the proper wavelength and polarization is considered, the system response exhibits a strong resonant behavior and the field is tightly localized inside the nanocavity. As an example we show in Fig. 1(a) the modulus of the y component of the electric field at the FF normalized with respect to the modulus of the incident field when the NR is illuminated by a plane wave, polarized along the x axis and propagating upwards in the vertical (y) direction with a wavelength of 950 nm. We note that the y component of the electric field modulus inside the cavity reaches a value 20 times higher than the modulus of the incoming field and it resonates inside the cavity forming a quasi-standing wave. We also report in Fig. 1(b) the absorption coefficient (also called absorption efficiency factor) as a function of the incident field wavelength.
We note the strong resonant behavior at the absorption peak corresponding to maximum field localization inside the nanocavity and at the metal/air interface. The absorption coefficient, defined for 2D geometries is:Fig. 2(a). If we consider coupled resonators by adding Ag rods of the same section and keeping gap between rods equal to 28 nm we notice that the overall system resonance frequency changes with respect to the case of the single resonator. For example we report calculations of the absorption coefficient in the case of a 4-rod structure in Fig. 2. We note that there is a main peak at 940 nm and a less pronounced secondary peak around 990 nm. These wavelengths correspond to different field profiles inside the system of resonators as shown in Fig. 3(a) and 3(b). A similar behavior is found if complex structure are considered by increasing the number of coupled resonators. As a first step it is straightforward to link the absorption coefficient to the second harmonic generation efficiency because higher absorption is achieved when the field penetrates in the metal and when the energy of the pump is coupled to the localized surface plasmon polariton mode. Nevertheless in what follows we will show that control and optimization of the second harmonic generation process in nanostructures depends on the region where the pump field is localized and how the generated field is irradiated by the nanostructure.
3. Second harmonic generation
Calculation of the generated second harmonic field has been performed by extending the Green tensor method used for linear calculation . The nonlinear response of metal has been modeled by considering a modified free electron gas model  and taking into account the contribution of free electrons only. For the linear response both free and bound electrons contributions are taken into account. Indeed, neglecting the effect of the bound electrons in the linear response would produce a shift of the resonant frequencies of the LSPP and underestimation of the losses at optical frequencies. Since the nonlinear frequency conversion enhancement is driven by the FF field localization an accurate description of the linear response of the metal structure is required. On the other hand, although previous calculations  performed by numerical integration in the time domain, show that second order contribution of bound electrons can play a significant role in the quantitative evaluation of the overall generated signal the main features of the generation process can be qualitatively described by neglecting their contributions. Both surface and bulk nonlinear terms are considered as effective surface and bulk current densities at the SH frequency, namely :41]. Nevertheless we used it in the development of our numerical integration algorithm because it allows a simple separate description for bulk and surface nonlinear terms for arbitrary shaped interfaces without the requirement of a flat, infinite surface [40–42] and it is suitable to the case of nanostructured metamaterials. Further calculations could be performed by implementing in our numerical integration method surface nonlinear terms obtained from different models and/or by taking into account for empirical correction parameters related to the features/defects of the surfaces. Indeed, in order to perform an accurate calculation on a real sample a detailed knowledge of the surface properties of the considered nanostructure is required and the parameters used to best fit experimental data might be very different with respect to the parameters of a flat infinite metal surface. For the 2D systems considered the main contribution to the SH generated signal is mostly given by the surface terms, indeed the SH generated by considering bulk contributions only is about 2 orders of magnitude lower than the total generated SH. Nevertheless preliminary calculation performed using the same method on 3D systems reveal that the relative efficiencies of surface and bulk contributions can be very different depending on the considered geometry. This topic goes beyond the scope of this paper and will be treated in detail in a future work. In what follows we focus on the possibility to tailor far field and near field emission of generated SH field in systems composed by 2D coupled Ag nanoresonators by taking advantage of FF field localization related to excitation of LSPPs.
The generated field pattern can be calculated by inserting Eqs. (3) into the equation for the generated SH electric field:Equation (5) is numerically solved by adopting the same numerical method used for the linear case . We calculated both the near field and far field features. Details on the far field emission are investigated by evaluation of the NLSCS  defined as:Fig. 4 . We note that the maximum enhancement is achieved when the FF field is tuned close to the maximum peak of absorption coefficient reported in Fig. 1(b).
We also show in Fig. 5 the Log10 of the modulus of the y component of the electric field at the SH frequency divided by the amplitude of the input field (a) and the differential nonlinear scattering cross section corresponding to the case of maximum efficiency (b) - pump field wavelength is 950 nm. In order to improve the appearance and readability of the depicted near field SH patterns we chose to plot the SH field in log scale. Indeed the near field exhibits highly evanescent components and sharp variations on a very short scale (few nanometers). In particular we note SH peaks at the corners of the rods. These peaks, due to the pump hot spots, depends on the sharpness of the corners. In our calculations this feature is limited by the size of the unit cell used for the discretization of the rods (1.5x1.5 nm2). However, being our numerical integration method valid for arbitrary geometries, for practical applications and/or for comparisons with real structures we can easily consider round corners according to the fabrication tolerances. We also note that the SH field is localized inside the nanocavity. This is possible because at the SH field frequency the relative permittivity of Ag is still negative. The doubly resonant behavior both at the FF and at the SH frequency is responsible for a higher enhancement of the efficiency of the generation process with respect to the case of gold nanorods (see for example results of ref. ).
The scenario is completely different if we consider systems composed by multiple coupled nanoresonators. Adding more rods we allow coupling among each nano-cavity and two kind of effects are observed: Absorption coefficient increases and the field can be localized in different cavities. SH generation maxima do not necessarily correspond to the maximum of the absorption coefficient. Indeed the absorption is related to the localization of the field in the metal but the overlap between FF and SH field becomes important. For example we report results of calculations for the 4-rod structure depicted in the previous section. In Fig. 6 we show the NLSCS as a function of the pump wavelength. We note the presence of two peaks, one at 940 nm (corresponding to SH of 470 nm) an the other at 990 nm (SH at 495 nm). Then we calculate the differential NL scattering cross section for the two cases (Fig. 7 ). We note that the SH field is emitted at different angles and the emission properties are very sensitive to the tuning of FF. Speaking in terms of antennas, the localization of the FF field produce spots of high nonlinear polarization: by selecting the resonances of the FF field we are selecting the position of the feeder of our antenna. Then the emitted SH field pattern and efficiency depends on how the geometry of the antenna couples to the SH field. Complex structures provide a richer variety of cases. For example we considered a structure composed by 8 rods. In this case we have 7 coupled resonators. We report the NLSCS in Fig. 8 . As expected the overall conversion efficiency is of the same order of magnitude of the single resonator but we note a broader line composed by four peaks. The amount of pump power per unit length coupled to the plasmonic resonances is similar but, varying the wavelength the energy is spatially distributed in different patterns. Different peaks of nonlinear scattering cross section corresponds to different FF localization profiles and consequently different SH far field emission patterns. We also calculate the differential nonlinear scattering cross section corresponding to the peaks at 920 nm, 960 nm (Fig. 9(b) , 9(d)) and 980nm, 1030 nm (Fig. 10(b) , 10(d)). The respective modulus of the y component of the FF field normalized with respect to the input field amplitude is provided in Fig. 9(a), 9(c) and in Fig. 10(a), 10(c). We note that the SH field emitted corresponding to FF pump wavelengths of 920 nm and 960 is characterized by a narrower angular distribution with respect to the other cases. Also, there is a difference in the main direction of emission between the two cases of approximately 15 degrees. On the other hand, the emission of the SH field corresponding to pump wavelengths equal to 980 nm and 1030 nm is characterized by a broad angle (FF at 980 nm) or multiple directions (FF at 1030 mm) arising by multiple multipoles interference.
Finally we depict the generated SH field near the 8-rod nanostructure at the considered wavelengths (Figs. 11 ). We note that different SH near field patterns are obtained by tuning the FF wavelengths in a range from 920nm to −1030 nm. Thus, as expected, different FF localization patterns (Fig. 9(a), 9(c) and Fig. 10(a), 10(c)) produce different SH far field (Fig. 9(b), 9(d) and Fig. 10(b), 10(d)) and near field patterns (Fig. 11(a), 11(b), 11(c), 11(d)).
Our numerical results have shown that plasmonic NS can be realized to optimize and tailor second harmonic generation by taking advantage of multiple coupled nanoresonators. Enhancements of 2-3 orders of magnitude with respect to the case of single metal nanoparticles have been predicted. Moreover we have shown that both FF field localization and spatial distribution on the nanoscale can drastically affect the nonlinear response of the system. Maximum efficiency of the generation process is expected in the range where the absorption cross section is maximum but a more detailed investigation requires the calculation of the near field and far field SH generation. Indeed, by tuning the pump wavelength it is possible to modify the arrangement of the feeder (being the nonlinear polarization the source of the SH field) and then the generated field pattern. This property might be used to address a subwavelength size portion of the structure where the field is highly localized. Also the FF field pattern could be tailored by coherent control using two counterpropagating pump beams or with spatial phase-shaped beams as discussed in . In this way a nano source of SH field could be addressed and it could be possible to use it to study the fluorescence of a single molecule located in the hot spot by far field detection at the proper angle of emission, for example. Optimization of the structures in order to obtain high directionality or high selectivity can be performed by changing size and shapes of the nanoresonators. Also, an array of these kind of “meta atoms” displaying high selectivity in SH directionality could be used to improve the superprism effect discussed in  for photonic crystals or to bring it on a more compact scale.
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