1. Introduction

Second harmonic generation (SHG) in metal nanostructures has been a target of extensive research in both theoretical and experimental aspects. The ability of metal nanostructures to concentrate optical fields offers a great potential towards the enhancement of nonlinear optical effects by many orders of magnitude [1,2], opening new perspectives in such applications as sensing [2,3] and nonlinear optical imaging [4–6]. The ongoing research devoted to SHG in metal nanostructures generally focuses either on optimizing the efficiency of nonlinear light conversion [2,7–10] or probing the spatial distribution of local optical fields at the nanoscale [10–19]. The first objective involves various strategies for maximizing the local field factors at fundamental and second harmonic frequency, such as the use of nanolens [2], double resonant geometries [7], nanogaps [8], or Fano resonance [9]. The second objective takes advantage of the extreme sensitivity of nonlinear optical phenomena to the local geometry of the material [10–14,17,18] and may be considered as one of the most promising avenues towards combining nano-plasmonics and nonlinear optics. Metal nanostructures offer the possibility of controlling these effects by exploiting the influence of the local electric field distribution on the symmetry of the resultant nonlinear susceptibility tensor. Extensive studies on the multipolar SHG in spherical [3,14,15] and L-shaped gold nanoparticles [16,17], giant nonlinear chiroptical effect [18], and a number of polarization-resolved studies on diverse geometries such as nanocups [10], nanoparticle dimers [19,20], nanorods [21], and nanoapertures [22–26] constitute only a fraction of recent work in this direction.

A general feature of nonlinear optical phenomena is their ability to probe symmetry features that are not accessible to linear optics and spectroscopy. For example, a linear scheme will not be able to distinguish between two structures abiding respectively to four-fold (square-like) and three-fold (equilateral triangle-like) symmetry. Both will be recognized at the linear level as isotropic structures in the absence of further discriminating features. In contrast, higher order nonlinear processes accounted for by tensors of ranks higher than 2 (rank 3 for SHG and rank 4 for THG) allow to go beyond linear optics in their ability to discriminate between different symmetry types such as the above-mentioned ones, by means of their different polarization responses. Indeed, the higher the order of the nonlinear process (and therefore the rank of the attached tensor), the more powerful will be its symmetry recognition potential. Anisotropies of higher order than the real or imaginary part of the birefringence can thus be reached by nonlinear polarization analysis [27], in which the intensity of the x and y polarization components of the emitted nonlinear signal is measured as a function of the angle of the fundamental light polarization. Each type of nonlinear anisotropy possesses its own signature in the polarization analysis.

In this work we propose to analyze the anisotropy of SHG in metal nanostructures using terminology of the multipolar description of organic and organometallic molecules [28–30], where multipoles do not refer here to different modes (dipoles, quadrupoles…) of the optical field oscillations [14–17] but to irreducible components of the second order susceptibility tensor with respect to rotation point groups. We limit our considerations to electric dipoles, leaving aside other contributions such as electric quadrupoles [14,15] and magnetic dipoles [16]. While this approach may be considered as a large oversimplification, we will demonstrate that it is capable of giving an accurate description of the main phenomena investigated in our experiments, that are vectorial perturbations of the local field caused by SPPs. We will assume that higher order multipolar contributions are less pronounced and appear as a non-symmetrical distortions of the polarization response, resulting from random defects of the metal surface [17].

In the electric dipole approximation, SHG emitters may exhibit two types of anisotropy: dipolar and octupolar, the first one corresponding to vectorial breaking of centrosymmetry, and the second one being associated with the presence of three-fold rotational symmetry in the case of planar structures and of tetrahedral symmetries for 3-D ones [28–30]. Each second order susceptibility tensor χ⁽²⁾ can thus be decomposed into a dipolar (χ⁽²⁾_{J = 1}) and an octupolar term (χ⁽²⁾_{J = 3}). An interesting fact is that purely dipolar objects do not exist in nature, i.e. the octupolar term is always non-zero, if only |χ⁽²⁾| ≠ 0. On the other hand, purely octupolar objects can be simply realized in the equilateral triangular geometry, which possesses threefold rotational symmetry and is consequently non-centrosymmetric. Polarization analysis allows to distinguish these two components: the dipolar term produces two maxima in the polarization response, whereas the octupolar term produces four maxima (additional two at perpendicular polarizations). This peculiar feature originates from the relation between tensor elements imposed by three-fold rotational symmetry, which can be written as:

In the case of noble metals, which are centrosymmetric, the only electric dipole-allowed sources of SHG originate from the breaking of centrosymmetry at the surface, resulting in three independent contributions to the second order susceptibility:${\chi }_{\perp \perp \perp }^{(2)}$, ${\chi }_{\perp \left|\right|\left|\right|}^{(2)}$ and ${\chi }_{\left|\right|\left|\right|\perp }^{(2)}$ ($\perp$ standing for the surface normal, and || for the surface tangential direction). An additional contribution arises from coupling to electric field gradients in the bulk of the metal ($P_{bulk}^{(2\omega )}$$\propto$γ_bulk$\nabla$E.E) [15,18]. There are several conditions required to build up a non-zero electric dipole moment at the second harmonic frequency from the above microscopic sources. If a nanostructure is small enough to allow for the validity of the quasi-static approximation, it must be able to modify the distribution of the fundamental electric field. This can be achieved either by plasmonic phenomena or by a non-resonant lightning-rod effect [11] resulting in surface charge concentration at sharp cusps and edges. Furthermore, in case of small nanostructures a non-centrosymmetric shape is required [14], otherwise all second order polarization vectors cancel out. Larger objects may generate a measurable SHG signal despite their centrosymmetric shape, for instance due to retardation effects in the propagation direction, leading to higher order multipolar modes [14,15]. However, it was shown that, in any case, breaking the centrosymmetry increases the SHG signal by orders of magnitude [17,19,23–26], making the symmetry-allowed electric dipole contribution dominant over the higher order modes [17]. In view of that, metallic nanostructures possessing triangular or tetrahedral shape are particularly interesting for plasmon-enhanced SHG experiments.

In this work, we present our results of polarization-resolved SHG microscopy experiments on precisely designed patterned silver films deposited on tetrahedral pyramid templates. We report a clear evidence of SPP-mediated interactions leading to a significant modification of the local field symmetry around the pyramids, which, in turn, changes the symmetry of the effective second order susceptibility tensor probed by polarization analysis of the SHG emission. Our observations are explained by two different coupling mechanisms, respectively at the fundamental and at second-harmonic frequency. The first model assumes octupolar second order susceptibility of the pyramids and a perturbation of the local electric field by Hankel-type surface waves generated by neighboring nanostructures. The second model takes into account the change of the overall symmetry of the system resulting from the presence of similar Hankel-type waves at the second harmonic frequency.

2. Sample fabrication

For fabrication of the studied metallic nanostructures, we use a semi-insulating (111)B GaAs substrate covered with a PMMA layer, in which a pattern of triangular openings is generated by e-beam lithography. Subsequent anisotropic chemical etching exposes the (111)A GaAs facets, which form perfectly tetrahedral pyramidal recesses with sharp tips, located exactly at each mask opening [31]. A scanning electron microscope (SEM) image of such tetrahedral cavity etched in the GaAs substrate is shown in Fig. 1(a), with (111)A and (111)B surfaces indicated by arrows. The size of the etched pyramids is determined by the size of the initial mask openings and is 270 ± 20 nm. After removing the PMMA mask, a germanium wetting layer of ~5 nm thickness and an optically thick (~100 nm) silver film are deposited on the patterned GaAs substrate by e-beam evaporation. As a result, we obtain a metal surface with site-controlled “inverted pyramids”, as shown in Fig. 1(b) (top view) and Fig. 1(c) (side view, in cross-section). Several arrangements of such pyramids, shown in Fig. 1(d)-1(h), were fabricated with varying inter-pyramid distance D indicated in Fig. 1(d) and 1(f). Finally, an additional PMMA layer is spin-coated on top of the sample to protect the silver surface from oxidation.

 

Fig. 1 SEM images of the investigated nanostructures: (a-b) single inverted pyramid before (a) and after (b) Ag deposition, (c) side view of the pyramid in cross-section showing GaAs substrate and silver film on top, (d-h) arrangements of pyramids studied in this work: (d) horizontal pair, (e) horizontal triplet, (f) vertical pair, (g) vertical triplet, (h) triangular arrangement. Scale bar in the lower right corner of each image indicates 100 nm. Incident polarization angle α is indicated in (d) together with the axes of the laboratory frame (x and y).

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3. Experimental setup

In our optical experiments we use an inverted two-photon microscopy setup, with an excitation wavelength of 950 nm from a femtosecond Ti:Sapphire laser (Mai-Tai from Spectra-Physics, 100 fs, 80 MHz) and an average power of 5 mW, providing sufficiently strong SHG signal from our nanostructures without any additional broad band emission [22] while offering good temporal stability. After reflection from a dichroic mirror, the incident beam is focused on a sample by a microscope objective (Nikon CFI Plan, 20x, NA = 0.4). The sample is placed on a piezoelectric stage (Piezosystem Jena) and can be scanned in the xy plane to obtain nonlinear emission images. The signal emitted by the sample is collected by the same objective, and after being transmitted through the dichroic mirror and a set of appropriate filters, is decomposed into x and y polarizations by a polarizing beam splitter. The two polarizations are separately detected by avalanche photodiodes working in single photon counting regime (COUNT®blue from Laser Components). The total signal emitted by the sample can be directed to a spectrometer (Andor), in order to monitor the spectral features of the emission. The polarization of the incident beam (α) is controlled by a half-wave plate mounted on a motorized rotation stage. In addition, we use a liquid crystal phase retarder (Thorlabs) in order to compensate the ellipticity acquired by the incident light due to reflection onto the dichroic mirror, ensuring that the sample is excited by a linearly polarized fundamental electric field at any angle addressed during the polarization analysis.

4. Theoretical model

The most common strategy in theoretical description of SHG in plasmonic nanostructures involves the solution of Maxwell equations giving the electric field distribution at fundamental frequency, which is then used to calculate the sources of scattered SH radiation. Unfortunately, analytical solution does not exist for nanostructures of arbitrary shape, requiring the use of numerical methods, for instance finite elements [15]. These methods are quite effective in theoretical analysis of SHG from isolated nanoparticles embedded in a dielectric medium - in such configuration the scattering problem can be solved employing perfectly matched layers that mimic the decay of scattered radiation at infinity. Numerical simulations of periodic nanostructures, such as SPP crystals, can be performed easily as well, using periodic boundary conditions. Compared to the above, a finite ensemble of cavities in a continuous metallic film is an intermediate case, that turns out to be very challenging, first due to strong reflection of SPPs from the simulation boundaries, and second due to large area of the metal surface that needs to be meshed with spatial resolution high enough to resolve the skin depth. This makes the above approach impractical in our specific case. Therefore, we decided to develop a simplified theoretical model, that employs existing analytical solutions modified by SPP-mediated interactions. Due to simple vectorial nature of these interactions, only electric dipoles are included in our analysis, assuming that their contribution is dominant over higher order modes, which is justified also by the non-centrosymmetric shape of our nanostructures.

Excitation of SPP waves was analytically described by Nerkararyan et al. [32] for a circular nanohole under normally incident linearly polarized plane wave irradiation. In such case, the E_z component of the field can be described by integer order Hankel functions H_n^(1,2) = J_n ± iY_n which are solutions of the wave equation in cylindrical symmetry (J_n and Y_n are Bessel functions). Taking into account only the first mode (n = 1) which has been shown to be the only mode excited by normally incident linearly polarized light, the electric field of SPP waves can be expressed by:

In the next step we assume that a single pyramid is polarized by the electric field of the incident light E^inc, acquiring a dipole moment p^dip = q^dipd = ε₀χ⁽¹⁾E^inc. We do not refer to this dipole as a localized plasmon resonance, because our pyramidal cavities may not necessarily exhibit such resonance. Charges that appear at opposite edges of the pyramid ( + q^dip and −q^dip) distanced by d = |d|, create a local field E^dip = − ξ p^dip that is proportional and parallel to the incident field vector E^inc, but of opposite sign. If we define the factor ξ by ξ = 1/4πε₀R³, then E^dip is the average electric field inside a sphere of radius R containing the q^dip charges. Expression for the local field vector can be written as E^loc = E^dip + E^inc = L^(ω)E^inc, where L^(ω) is the local field factor at fundamental frequency. Assumption that L^(ω) is a scalar is justified by the fact, that the linear optical susceptibility χ⁽¹⁾ of a material possessing 3-fold rotational symmetry is a monopole-like scalar, i.e. such material (or object) is optically isotropic.

The coupling mechanism between nanoholes was proposed by Alaverdyan et al. [33] It assumes that two neighboring holes exchange their charges via long-range SPP waves, thus mutually affecting their local field distributions. We apply this model to our pyramids; in particular we express the total charge at the i-th edge q_i^tot as a coherent sum of the charge q_i^dip due to the dipole moment induced by the incident light, and the charge q_ij^SPP delivered to the i-th edge by SPP generated by j-th neighboring pyramid: q_i^tot = q_i^dip + q_ij^SPP. The additional charge due to SPP can be expressed by: q_ij^SPP = b_ij σ_j(r_i, ϕ_i), where b_ij is a coupling constant accounting for the surface area of the i-th edge and the coupling efficiency of the j-th SPP to i-th edge, and σ_j(r_i, ϕ_i) = ε₀ E_z,j(r_i, ϕ_i, z = 0) is the surface charge density of the j-th SPP wave at the location (r_i, ϕ_i) of the i-th edge with respect to the origin of j-th SPP wave. The charge due to SPP creates an additional local electric field E^SPP = − ξ p^SPP = − ξ ∑_i,j q_ij^SPP ζ_i, where ζ_i is the location of i-th edge of the considered pyramid with respect to the center of this pyramid. The total local electric field E^loc becomes E^loc = E^inc − ξ ∑_i q_i^tot ζ_i = E^inc + E^dip + E^SPP. In this simple way, E^loc of the considered pyramid is depending on the local field of the neighboring pyramids E_j^loc via:

Let us now consider a horizontal pair of identical pyramids. In this simple case each pyramid has only one neighbor located respectively at ϕ_i = 0 and π. For simplicity we assume that the average location of the edges ζ_i receiving charges from SPP waves is such that |ζ_i| = d/2. Because pyramids are identical and their arrangement has mirror symmetry, we can assume b_ij = b and define a constant a = ξε₀bc₁d/2 to simplify notations. Additionally we set H = H₁^(1,2)(Dk_SPP), where D is an effective distance between the edge and the origin of a SPP wave. We will assume that it is equal to the distance between the nearest edges of neighboring pyramids. Now, after simple derivations we obtain a direct expression for the local electric field: E^loc = L^(ω)E^inc + a$\widehat{x}$E_x,j^locH, which is the same for both pyramids. As we can see, E_y^loc remains the same as for an individual and isolated pyramid, i.e. E_y^loc = L^(ω)E_y^inc, but E_x^loc is affected by SPP-mediated interactions: E_x^loc = ηL^(ω)E_x^inc, where η = 1/(1−aH).

We can use the local field form derived above to calculate the second order nonlinear polarization and the SHG intensity using the usual quadratic field-polarization dependence P_i^(2ω) = ε₀∑_i,j,k χ⁽²⁾_ijk E_j^locE_k^loc and I_i^SHG$\propto$|P_i^(2ω)|², with i,j,k = x,y and the incident field defined as E^inc $\propto$ ($\widehat{x}$cosα + $\widehat{y}$sinα)e^i(ωt−kz). We can then derive two general expressions for the SHG intensity under 2D approximation: one in terms of the local field:

5. Results and discussion

5.1 SHG from GaAs substrate

In order to better understand the main results of our experiments, we start with some reference SHG sources. We found that the (111)B GaAs substrate used for the fabrication of our nanostructures can serve as a perfect reference for the octupolar polarization response, an example of which is shown in Fig. 2(a). With its surface parallel to the (111) crystallographic plane, the crystal lattice of the wafer exhibits three-fold rotational symmetry with respect to the direction of incidence of the fundamental light, and the polarization response in Fig. 2(a) is a signature of the corresponding octupolar tensor symmetry described by Eq. (1).

 

Fig. 2 SHG polarization response: x-polarized signal in red, y-polarized in blue, measured from various objects on a (111)-GaAs substrate: (a) bare GaAs wafer showing a signature of three-fold rotational symmetry along an axis perpendicular to the (111)B plane; (b) flat silver surface constituting the “isotropic” background; (c-d) individual pyramid: (c) raw data and (d) after subtraction of background signal shown in (b). (e) Diffraction-limited image (scan 4 x 4 µm) of a single pyramid obtained by detecting the y-polarized SHG signal (indicated by a white arrow marked with E^2ω) under x-polarized excitation (marked with E^ω). (f) Spectrum of the total signal emitted by the pyramid proving that the SHG peak at 475 nm is the only nonlinear emission measured in the experiment. The angle of incident polarization α is indicated in (b) and in (e), in the latter together with the axes (x and y).

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5.2 SHG from flat metal surface

A flat metal surface without patterned pyramidal recesses is our next reference source for SHG. In theory, SHG can be detected in reflection from a perfectly smooth surface, only if the normally incident beam is tightly focused by microscope objective. In such a configuration, a non-zero normal component (E_z) of the fundamental electric field produces coherent nonlinear sources located at different points of the focal spot. SHG from these sources interferes constructively in the far-field at some particular angles, and some fraction of this coherent SHG can be collected by the objective. We try to minimize this contribution by using an objective with low numerical aperture (NA). It helps to reduce the magnitude of E_z, so that it is relatively small compared to the lateral (x and y) electric field components (obviously, it is not the case when SPPs are excited, which will be discussed in Section 5.4). However, surface roughness [12] may still constitute an additional source of background signal that is excited by lateral in-plane fundamental fields. In this case, the emission is incoherent due to phase randomization from multiple scattering events involving photons and plasmons at both fundamental and harmonic frequency. The total signal coming from the metal surface is thus a mixture of a coherent and incoherent emission. Both coherent and incoherent parts of this background are seen as an “isotropic” signal in the polarization analysis, with x and y components following cos²α and sin²α dependence (Fig. 2(b)). As a result, any octupolar source will show up as an anisotropic perturbation to the otherwise isotropic background, arising from its significant χ⁽²⁾_xxy tensor element, which accounts for the emission of y-polarized second-harmonic under an x-polarized fundamental excitation.

5.3 SHG from a single pyramid

A typical as-measured polarization response of a single pyramid is shown in Fig. 2(c). It can be interpreted as a mixture of “isotropic” (Fig. 2(b)) and octupolar (Fig. 2(a)) contribution. To some extent, this can be evidenced by subtraction of data in Fig. 2(b) from data in Fig. 2(c), which leads to the “corrected” polarization response of a single pyramid, as shown in Fig. 2(d). Despite significant noise, the result of this “correction” clearly features an octupolar character. However, subtraction of the two signals formally is not allowed here, because it requires that the “isotropic” contribution is totally incoherent and does not interfere with the octupolar radiation from the pyramid. In fact, the subtraction is not needed, because we are able to recognize the contribution from the pyramid. The presence of pyramid is clearly visible not only in polarization analysis, but also in polarization-resolved SHG images such as in Fig. 2(e). By measuring y-polarized SHG under x-polarized excitation we can recognize the pyramids as diffraction-limited spots with high contrast with respect to the background signal from the flat metal surface. The emission spectrum (Fig. 2(f)) confirms that the measured signal is purely originating from SHG (the same verification has been performed in all other measurements presented in this work).

There are two possible sources of octupolar character for the polarization response of a single pyramid. A natural suggestion is that is arises from the three-fold symmetry of the pyramid. Obviously, the incident field has both parallel and perpendicular component with respect to the metal surface inside the pyramid. However, thanks to the use of low-NA objective, only the projections of the nonlinear polarization vectors onto the xy-plane contribute to the measured SHG. As a result, the pyramid is seen as an object of second order susceptibility of the form in Eq. (1). On the other hand, one may suspect that the crystal lattice of the GaAs substrate may also contribute to the octupolar SHG emission due to the smaller thickness of the metal inside the cavity (see Fig. 1(c)) as opposed to the relatively high second order susceptibility coefficient of GaAs. However, taking into account the very low penetration depth of the optical fields in silver at both fundamental and second harmonic frequency (skin depth 22 and 26 nm, respectively) we can safely assume that the influence of the GaAs substrate is efficiently screened by the metal. We would like to emphasize that actually it is not a critical issue to determine what is the main source of SHG from pyramids. It is because the three-fold symmetry axis of the GaAs substrate perfectly coincides with the three-fold axis of the pyramids providing the same polarization dependence. In other words, as long as our pyramids exhibit octupolar symmetry of the observed χ⁽²⁾ described by Eq. (1), the exact origin of this character is not so important.

By analysis of SEM images (Fig. 1) one can find that the metal surface inside the pyramids is significantly rougher than elsewhere, forming terrace-like steps on the steep walls of the cavity. These surface defects, which certainly enhance the SHG signal via the lightning-rod effect [11], are also responsible for higher-order multipolar contributions [17] causing non-symmetrical distortions of the polarization response that are not repeatable when comparing different pyramids. Fortunately, these non-controllable fluctuations are much weaker than the electric dipole-allowed SHG determined by χ⁽²⁾, making pyramids relevant “building blocks” for more complex geometries, also in view of their ability to act as efficient sources of SHG without breaking the continuity of the embedding metal film, hence allowing for long-range communication via SPPs.

5.4 SHG from an ensemble of pyramids

The SPP-mediated interactions lead to modification of the local electric field of the pyramids, and the symmetry of these perturbations depends on the pyramids arrangement. In this work we investigate groups of two and three pyramids arranged horizontally, vertically and in equilateral triangle. In our patterns the distance between neighboring pyramids increases by steps of 50 nm, up to the vicinity of the diffraction limit, starting from edge-to-edge distance around 50 nm. To excite and detect SHG in these systems, we use a low-NA objective, ensuring that the diffraction limited spot size is large enough to excite coherently all pyramids in a given group by a fundamental electric field of similar magnitude. We investigate only groups in which the distance between pyramids is below our instrumental resolution limit, leading to roughly 9 elementary intervals, that is up to ~450 nm between neighboring pyramids. Obviously, the use of low-NA objective does not prevent the appearance of large E_z component associated with SPPs, however, as it was pointed out in Section 5.3, it allows to collect mostly the contribution of lateral components of nonlinear polarization vectors, induced by lateral components of the local electric field.

Let us start with the simplest case of horizontal pairs. An example of SHG image of such pairs (D = 200 nm) is shown in Fig. 3(a). Here we take again advantage of the fact that pyramids have different polarization properties than that of flat metal surface and we detect y-polarized SHG under x-polarized fundamental light, obtaining a good contrast with respect to the background. As we can see in Fig. 3(b), increasing the distance between pyramids causes significant changes in the measured SHG, which reflects the changes of an effective χ⁽²⁾_xxy tensor element. Modification of the effective tensor symmetry is clearly visible in the polarization dependence plots presented in Fig. 3(c). Of particular interest is the non-monotonous character of the observed dependence in intensity, which exhibits a global maximum at a distance D = 200 nm together with a second weaker maximum at D = 400 nm.

 

Fig. 3 (a) Diffraction-limited image of a horizontal arrangement of two pyramids distanced by 200 nm, obtained by detecting y-polarized SHG signal (E^2ω) under x-polarized excitation (E^ω). (b) Surface plots of SHG images such as in (a), showing the variation of the y-polarized SHG intensity excited by a x-polarized fundamental light in a series of horizontal pairs of pyramids with distances increasing from 50 to 450 nm. (c) Full SHG polarization analysis of horizontal pairs, in the same sequence as in (b).

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Direct comparison between the distance dependence of the experimentally measured ratio I_y^SHG(α = 0)/ I_y^SHG(α = π/2) and the factor |η|⁴ derived in the theoretical section is presented in Fig. 4(a). To shorten the notation, we will refer to this ratio as “xxy/yyy”. Experimental values are obtained by averaging the measurements of several copies of the investigated pattern, and the error is estimated from statistics. The “isotropic” background is not subtracted – this results in an overestimation of I_y^SHG(α = π/2) by a constant factor, however it does not affect the overall trend of the dependence on distance. We should point out that in a given measurement the distance dependencies of xxy/yyy and I_y^SHG(α = 0) are nearly identical. The only difference appears at larger distances, when the pyramids cannot completely fit inside the exact center of the focal spot, which results in a decreased SHG signal. Despite the similarity of the distance dependence, it is more adequate to analyze the values of the xxy/yyy ratio, as the absolute values of the SHG intensity are more sensitive to the specific experimental conditions pertaining to a particular measurement, such as proper focusing of the incident beam.

 

Fig. 4 Experimental values of the ratio xxy/yyy for horizontal pairs (a) and horizontal triplets (b), and experimental values of yyy/xxy for vertical pairs (c) and vertical triplets (d) as a function of the distance between pyramids (D), superimposed with theoretical curves representing the influence of SPP-mediated coupling at fundamental (solid lines) and second-harmonic frequency (dashed lines). Insets show the corresponding pyramid arrangements. Diffraction limited SHG images of the nanostructures for all investigated distances are shown beneath each plot (I_y^SHG(α = 0) for horizontal arrangements and I_y^SHG(α = π/2) for vertical arrangements).

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As we can see, experimental points at short distances (50-200 nm) are relatively well fitted by our model that assumes η = 1/(1−aH), with a as a fitting parameter. The fitted value is a = 0.3, and the theoretical curve is normalized to experimental data. To calculate k_SSP we used the complex permittivity of silver at 950 nm, namely ε_m = −45.7 + 0.54i [34], and the dielectric constant of the PMMA layer ε_d = n² with the PMMA refractive index n = 1.48 [35]. This gives the physical wavelength of the SPPs (λ_SPP = 2π/k_SPP) equal 625 nm. No second maximum is predicted by this model, and apparently the SPP-mediated interactions are weaker at D > 200 nm compared to the theoretical curve. The observed discrepancy may be caused by an interaction mediated by SPPs at the second-harmonic frequency (SH SPPs), which is neglected in the above model. The wavelength of these SPPs: ${\lambda }_{SPP}^{(2\omega )}$ = 271 nm (calculated with the same manner as λ_SPP, taking the optical constants of silver and PMMA at second-harmonic frequency: ε_m = −8.41 + 0.29i [34] and n = 1.50 [35]) is comparable with the size of pyramids, which may result in much higher probability of excitation of higher order (n > 1) Hankel waves that emanate from the pyramids with angular distribution described by the cos[n(ϕ − β)] function. It is enough that the second mode (n = 2) is excited, and the second-harmonic dipoles oriented along the y axis may communicate in the horizontal direction, producing a constructive and destructive contribution to the observed dependence. This contribution can be described by an SPP-mediated coupling mechanism similar to that for the fundamental frequency. Due to coupling at second-harmonic frequency, the xxy/yyy ratio changes by a factor ρ = |η^(2ω)|², where η^(2ω) = 1/(1−aH^(2ω)) and H^(2ω) = H_n^(1,2)($D{k}_{SPP}^{(2\omega )}$). This factor is plotted in Fig. 4(a) (dashed line), assuming n = 2 and a = 0.5. As we can see, the relative drop of xxy/yyy ratio at distances around 300 and the increase at 400 nm, respectively, corresponds exactly with the negative and positive influence of the SH SPPs.

Our theoretical model can be used to describe SPP-mediated coupling between any number of octupolar sources arranged in any pattern. Besides horizontal pairs, we shall consider three other simple arrangements: horizontal triplets, vertical pairs and vertical triplets. It should be pointed out that in the case of three pyramids arranged in a line, the changes of the local field due to SPP waves at the lateral pyramids will be different from that at the central site (the central pyramid receives SPP waves from two neighbors). In the horizontal arrangement, the parameter η at the lateral pyramids will be equal to (1 + aH)/(1−2a²H²) and at the central one: (1 + 2aH)/(1−2a²H²). However, both expressions produce almost exactly the same line shape in the distance dependence, therefore only one (corresponding to the central pyramid) is plotted in Fig. 4(b), corresponding to coupling at fundamental (solid line) and second-harmonic frequency (dashed line) with a = 0.2 for both theoretical curves. In this case the influence of second-harmonic SPPs is less pronounced. The experimental dependence has only one clear maximum, located close to the maximum of coupling at the fundamental frequency. A small shift between experimental and theoretical maximum originates probably from the increased influence of the periodicity - an effect associated with formation of a SPP crystal (which period is obviously larger than D).

For vertical arrangements there may be two different coupling constants due to the asymmetric character of the interactions. Pyramids are turned apex-to-base, which may result in a different receiving efficiency of SPP waves at the upper or lower edge of the cavity. To take this into account one needs to introduce another fitting parameter (a second coupling constant). Instead, we decided to neglect the asymmetry and use only one coupling constant a for both upward and downward interactions. It is easy to predict that all the expressions for ρ will be the same as in horizontal arrangements except that in vertical arrangements the η factor will affect only the y component of the local field instead of the x component: E_y^loc = ηL^(ω)E_y^inc. Following the derivations of I^SHG we can directly compare the theoretical $|{\chi }_{yyy}^{(2),eff}/{\chi }_{xxy}^{(2),eff}{|}^2$ value with the experimental I_y^SHG(α = π/2)/ I_y^SHG(α = 0) ratio (yyy/xxy). However, as we can see from Fig. 4(c) (pairs) and Fig. 4(d) (triplets), the fit is very poor (solid lines). Indeed, in vertical arrangements the second-harmonic dipoles oriented along the y axis can interact strongly by SH SPPs excited in the dominant n = 1 mode. Therefore the coupling at second-harmonic frequency will be much stronger than in the case of horizontal arrangements. This effect will change the yyy/xxy ratio by a factor ρ = |1/(1−aH^(2ω))|² with H^(2ω) = H₁^(1,2)($D{k}_{SPP}^{(2\omega )}$), which is plotted in Fig. 4(c) and Fig. 4(d), assuming a = 0.5 for pairs and 0.7 for triplets. As we can see, the theoretical curves (dashed lines) almost perfectly fit the experimental points. Such agreement, which is not found for a theoretical coupling at the fundamental frequency, provides convincing evidence that the SHG polarization properties of vertical arrangements are dominantly affected by the SPP-mediated interactions at the second-harmonic frequency leading to two maxima for yyy/xxy within the distance range in our investigation. In horizontal arrangements the coupling between second-harmonic dipoles is much weaker, because communication via the n = 1 mode is forbidden, and therefore the changes in SHG polarization properties are caused mainly by the SPPs at fundamental frequency.

There is an essential difference between coupling at fundamental and second-harmonic frequency. The change of the ratio yyy/xxy cannot be explained by the same analysis as for SPP waves at the fundamental frequency, assuming that the y-component of the nonlinear polarization vector is affected by SH SPPs from neighboring pyramids proportional to the y-component of their own nonlinear polarization. This would lead to P'_y^(2ω) = ηP_y^(2ω) with P_y^(2ω) calculated using tensor elements in Eq. (1) and a fundamental electric field E^inc. One can easily predict that in such situation the polarization analysis of the SHG intensity I_y^SHG = |P'_y^(2ω)|² would exhibit the same symmetry as an individual octupole, with ratio yyy/xxy = 1. In fact, the presence of SH SPPs modifies the symmetry of the entire system, i.e. the symmetry of the eigenmode of the electromagnetic field at second harmonic frequency and the coupling of E^loc to this eigenmode. We can account for this effect analytically only by introduction of a phenomenological relation between tensor elements:

In addition to the simple nonlinear anisotropy parameters xxy/yyy and yyy/xxy we should compare experimental and theoretical results of a full polarization analysis. If the coupling occurs at fundamental frequency, the polarization response is described in terms of the local field at fundamental frequency, like in Eq. (6). An example is shown in Fig. 5(a), corresponding to the case of three pyramids arranged horizontally and distanced by D = 300 nm, with a coupling constant a = 0.35. It can be compared with the experimental data of a horizontal triplet of the same D, plotted in Fig. 5(b). As we can see, the SPP-mediated interaction results in a partial transition from octupolar to effective dipolar anisotropy in the horizontal direction. Similarly, a dipolar anisotropy is induced in vertical arrangements along the vertical direction, which is demonstrated theoretically in Fig. 5(c) and experimentally in Fig. 5(d). Presented data correspond to a vertical triplet of D = 250 nm (minimum of coupling by SH SPPs) and coupling constant a = 0.35.

 

Fig. 5 (a) Theoretical SHG polarization response assuming coupling at fundamental frequency in a horizontal triplet of D = 300 nm, a = 0.35, and (b) corresponding experimental data; (c) theoretical response for coupling at fundamental frequency in a vertical triplet of D = 250 nm, a = 0.35, and (d) corresponding experimental data; (e) theoretical response for coupling at second-harmonic frequency in a vertical triplet of D = 90 nm, a = 0.8, and (f) experimental data corresponding to a vertical triplet of D = 150 nm.

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Full polarization response reveals the difference between coupling at fundamental and second harmonic frequency. In the first case both x and y polarization components of the emitted SHG are affected by SPPs, which leads to comparable enhancement of both I_x^SHG and I_y^SHG. This is not the case, when the interaction occurs at second-harmonic frequency in the vertical arrangements. In such situation only the y polarization component will be influenced, leading to a large difference between the magnitudes of I_x^SHG and I_y^SHG. In order to demonstrate this effect, we can calculate an example theoretical polarization response using Eq. (8) with ρ = |(1 + 2aH^(2ω))/[1−2(aH^(2ω))²]|² and H^(2ω) = H₁^(1,2)($D{k}_{SPP}^{(2\omega )}$), assuming E^loc = E^inc, D = 90 nm (theoretical maximum) and a = 0.8. It is shown in Fig. 5(e), above the experimental data for a vertical triplet at D = 150 nm plotted in Fig. 5(f) for comparison. Obviously, in general case we should account also for the local field at fundamental frequency (by assuming E_y^loc = ηL^(ω)E_y^inc instead of E^loc = E^inc), however it is clearly visible that the effect of just the symmetry breaking due to SH SPPs has a dominant influence on the measured polarization response, causing much stronger dipolar anisotropy along the vertical direction.

The last pattern of our investigation is an equilateral triangular arrangement of three pyramids, shown in Fig. 1(h). Seven different pyramid-to-pyramid distances of this pattern were studied, ranging from 50 up to 350 nm. It is easy to predict that in such arrangement the SPP-mediated interaction enhances equally both x and y component of the local field, either at fundamental or at second-harmonic frequency, following a theoretical distance dependence defined by factor η = 1/(1−aH). This results in enhancement of the effective octupolar character of the system, which only improves the polarization contrast with respect to the “isotropic” background, but it does not influence the overall symmetry. This analysis is in agreement with our experimental results: for most distances, the SHG polarization response is similar to the “enhanced octupole” presented in Fig. 6(a), where the experimental data correspond to D = 200 nm. Only at the shortest and at the longest distance D the polarization analysis gives results similar to that of an uncoupled nanopyramid (Fig. 2(c)). Therefore, instead of the xxy/yyy ratio (which is constant) we analyze the absolute magnitude of y-polarized SHG signal under x-polarized excitation (I_y^SHG(α = 0)). The experimental distance dependence is shown in Fig. 6(b), together with a theoretical fit assuming interactions at fundamental frequency with a = 0.55. As we can see, the theory predicts that maximal enhancement occurs around D = 200 nm, which is in excellent agreement with experimental data. This dependence is clearly visible in a diffraction limited SHG image (Fig. 6(c)), which is compared with an SEM image of the same nanostructures (Fig. 6(d)).

 

Fig. 6 (a) Polarization response of triangular assembly of pyramids with edge-to-edge distance D = 200 nm, (b) distance dependence of normalized I_y^SHG(α = 0) superimposed with a theoretical curve (solid line) assuming coupling at fundamental frequency with a = 0.55; inset shows an SEM image with indicated distance D, (c) I_y^SHG(α = 0) scanning two-photon microscopy image of the entire investigated pattern and (d) corresponding SEM image.

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At the end we would like to point out that an investigation similar to the work presented here could have been performed by tuning the excitation wavelength. However, there are two very important advantages for keeping the wavelength fixed while changing the distance. Firstly, one does not need to worry about the spectral dependence of transmittance and reflectance of the optical elements used in the experiment (filters, objective, dichroic mirror, beam-splitter) or the quantum efficiency of avalanche photodiodes, as well as the spectroscopic properties of the nanostructures themselves. Secondly, varying the distance allows to explore plasmonic standing waves in a much broader range than when tuning the excitation wavelength from ~800 to ~1000 nm, which are the limits of our experimental setup.

7. Conclusion

We have applied two-photon microscopy to study the polarization properties of second harmonic generation in horizontal, vertical and triangular arrangements of inverted silver-coated pyramids with varying distances. It has been experimentally demonstrated that surface plasmon polaritons can strongly affect both the efficiency and the polarization properties of second harmonic generation. Furthermore, we can precisely control these effects using octupolar SHG sources, such as tetrahedral or triangular nanocavities connected by continuous metal film. In particular, by tuning the distance between octupolar emitters or by changing their arrangement we can select the SPP-mediated coupling mechanism, i.e. we can determine whether the interactions occur either at fundamental or at second-harmonic frequency. Coupling at fundamental frequency can be accounted for in terms of local field modifications caused by additional surface charges generated by SPPs from neighboring nanostructures. In contrast, coupling at second-harmonic frequency requires to consider the change of the overall symmetry of the system. The possibility of controlling these effects opens new perspectives for tailoring the nonlinear optical properties of plasmonic nanostructures and towards possible applications of SPPs in a future nanoplasmonic circuitry, possibly involving plasmonic coupling to semiconductor quantum dots and nanowires grown in inverted pyramids [36].