1. Introduction

Scattering-type scanning near-field optical microscopy (s-SNOM) provides optical information on the 10-nm length scale over a broad spectral range from visible to far-infrared and THz wavelengths [1–6]. However, up to date, s-SNOM suffers from its dominant out-of-plane polarization response [7–9], restricting its application particularly to study biological and nano-materials with polarization-sensitive responses such as molecular vibrations [10–12], intermolecular interactions [13–15] as well as phonons and bond orientations within a sample [16–21]. Utilizing local vector fields at resonant excitation of materials as explored here, may guide future nanoscopic material analysis towards next generation near-field studies including polarization-sensitive 3D probing.

Published SNOM reports mostly focus on quantifying the out-of-plane (z) component of the local electric field [10,22,23]. Only few approaches utilize polarization effects for exciting and probing nanoscopic structures [24–29], while sample-resonant excitations so far were not considered. For such non-resonant excitations at visible wavelengths, e.g. tilted tips have been utilized to realize both in-plane and out-of-plane probing of local fields, hence breaking the axial symmetry [30,31]. On the other hand, s-SNOM is frequently applied at mid-infrared (mid-IR) to THz and GHz wavelengths [4,6,20,32] enabling to explore spectral fingerprint regions of molecules [12,33,34], lattice vibrations [35,36], and plasmonic responses in semiconductors [37,38]. In all these cases, the typical length scales of the near-field coupled system under consideration (including tip, tip-sample distance, lateral and vertical field confinement) measure a few nanometer, only, and thus are tiny fractions of the wavelength in use, well justifying the dipole approximations. Impressively, resolution at these wavelengths overcomes the diffraction limit by several orders of magnitude reaching values of up to λ/4600 at THz wavelength [6] and even λ/1.000.000 in the GHz regime [32]. Moreover, applying s-SNOM at those wavelengths enables resonant sample material excitations, resulting in both enhanced signal intensities and improved sensitivity [20,38–40], as well as indicating the ability to probe different polarization states by s-SNOM [21,41–43]. Nevertheless, a thorough and detailed theoretical discussion is needed in order to understand the corresponding scattering signatures in s-SNOM, and to identify the distribution of local fields for resonant excitation.

In this article, we theoretically investigate the existence of local in-plane vector field components at mid-IR wavelengths by applying 3-dimensional (3D) parametric simulations based on finite element modelling (FEM). We fundamentally analyze the influence of resonant excitation by comparing four cases of near-field coupled systems that represent all combinations of resonant-/non-resonant tip and/or sample excitations. As typical material representatives for non-resonant (metallic) and phonon-based resonant excitation at λ = 15 µm, we chose gold (Au) and strontium titanate (STO), respectively, the latter being a typical material utilized in infrared-optical applications and a widely used substrate for epitaxial thin-film growth. Note that our general findings can directly be applied to many other material systems by spectrally shifting the excitation to the corresponding material resonances, e.g. phononic or plasmonic modes at their corresponding wavelengths.

The four different cases studied here are:

 

Fig. 1. Comparison of the field distributions at the sample surface for the four configurations of near-field coupled systems (rows) including a spherical tip (radius a = 5 nm) at a tip-sample separation of h₀=1 nm and an extended sample for p- and s-polarized excitation (columns). For each case 1 - 4, the vector field distribution is shown along with the extracted values for every field component E_||,x, E_||,y, and E_⊥,z. Notably, the fields on the sample surface have very different characteristics: For p-polarization, the field distribution is dominated by the z-component with a maximum located at x = 0 (underneath the tip). On the other hand, for s-polarization, maxima for E_⊥,z are observed at y ≠ 0, while for resonant excitation a dominant in-plane contribution is found at y = 0.

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2. Setup configuration and analytical description

As starting point for our simulations here, we first analytically calculate the near-field enhancement at the scattering probe in close proximity to a sample, based on the dipole model approach. This model is widely used to estimate the scattering cross section in s-SNOM [44] and is adopted here to represent local electric fields. Note, that the dipole model is known to show several limitations, resulting e.g. in inaccuracy of the near-field's decay behavior as well as in restrictions to field orientations perpendicular to the sample surface [45,46]. In order to overcome these limitations, we carry out parametric 3D simulation of the near-field-coupled systems.

 

Fig. 2. Schematic setup of the near-field-coupled tip-sample s-SNOM system, utilized for both analytical calculations and modelling. If not stated otherwise, we assume the following parameters: tip apex radius a = 5 nm, tip-sample distance ${h_0}$=1 nm, illumination with p- or s-polarized light that impinges at an angle of θ = 45°, as well as complex permittivities ${{\varepsilon }_{\textrm{tip}}}$,${\varepsilon _s}$, and ${{\varepsilon }_\textrm{m}}$ for tip, sample, and surrounding medium (air), respectively.

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For both the dipole model and our simulations, the following parameters were used: polarization direction of the incident light (p- or s-polarized light), tip-sample distance h₀ and complex material permittivities ɛ, hence including losses. For the simulations, a 3D sphere is defined with perfectly-matched layer (PML) boundary conditions, surrounding the near-field-coupled system. The layout consists of a spherical tip with radius a = 5 nm at a distance h₀ to a planar sample surface (see Fig. 2). The sample itself has a size of 150 µm x 150 µm x 100 µm along widths (x,y) and thickness (z), respectively, and is placed in the center of the 3D sphere used for FEM modeling. Note that for this fundamental study of local fields, we assume a symmetric (spherical) tip shape. In experiments, typically elongated tips of different shapes (ellipsoids, pyramids, etc.) are applied, which might be described by a weighting of the different components found here as long as the system is of cylindrical symmetry. However, it shall be noted that additional in-plane field components might be introduced, if this symmetry is broken, e.g. by tilting a non-spherical tip with respect to the sample surface as discussed in [30,31]. A finite element analysis (FEA) method is applied on the coupled system by building accurate mesh for each domain: For instance, the physical domain that includes tip, sample, and surrounding medium (air) is determined with a tetrahedral mesh type with maximum element size of λ/100. Linearly p- or s-polarized light is illuminating the tip-sample system at an incident angle of $\mathrm{\theta }\; = \; 45^\circ $. While for s-polarized light, this geometry results in purely in-plane-(y)-oriented fields, for p-polarized light, this represents the typically mixed polarization state assuming equal components parallel (x) and perpendicular (z) to the sample surface (note that other values for $\mathrm{\theta }$ are expected to principally show the same behavior, however, with different weighting of these two components). In the following p- and s-polarization always refers to the field orientation of the incident light, whereas ‖ and ⊥ as well as x/y and z are used to describe the local in-plane and out-of-plane field components in the sample coordinate system (see Fig. 2). In the simulation, Maxwell equations in frequency domain in the spectral range from 13.5 µm to 16 µm are solved for resonantly excited phonon modes, taking the real and the imaginary part of the material permittivity of tip (${{\varepsilon }_{\textrm{tip}}}$) and sample (${{\varepsilon }_\textrm{s}}$) into account. The proposed schematic setup of the near-field-coupled tip-sample system utilized for both the analytical calculation and FEM modelling is shown in Fig. 2.

Analytically, we apply the dipole model [44] that assumes a purely dipolar excitation of the near-field coupled system. This approximation is particularly well met for infrared wavelengths, since the nanoscopic tip is more than 3 orders of magnitude smaller than the wavelength [47]. In this model, the local vector field on the sample surface underneath the tip is given as:

The absolute value of the local electric fields are plotted in Fig. 3 as functions of the real part of the tip and the sample permittivities, with imaginary parts kept fixed at 0.1. Strongly enhanced fields underneath the tip are obtained for both perpendicular ($\bot $) and parallel (||) components of the incident field ${\vec{\textrm{E}}_0}$ with respect to the sample surface by setting $ {\vec{\textrm{E}}_{\textrm{0,} \bot }}\textrm{ = }\left( {\begin{array}{c} \textrm{0}\\ {\begin{array}{c} \textrm{0}\\ \textrm{1} \end{array}} \end{array}} \right)$ for $\bot $-field orientation [Fig. 3(a)] and ${\vec{\textrm{E}}_{\textrm{0,||}}}\textrm{ = }\left( {\begin{array}{c} \textrm{1}\\ {\begin{array}{c} \textrm{0}\\ \textrm{0} \end{array}} \end{array}} \right)$ for $\textrm{||}$-field orientation [Fig. 3(b)]. Note that both, fixed imaginary part and strictly $\bot $/$\textrm{||}$ field orientation are in general not reflecting realistic experimental systems, but are applied here to discuss the characteristic behavior of the single components separately. Two branches of enhanced local fields are found for both field components corresponding to sample- and tip-mediated resonances around Re (${{\mathrm{\varepsilon} }_\textrm{s}}) ={-} 1$ and Re (${{\mathrm{\varepsilon} }_{\textrm{tip}}}) ={-} 2$ (Fröhlich resonance) [48], respectively. These branches show a splitting for both permittivities, taking on slightly negative real part values corresponding to the near-field coupling. The strength of this splitting depends on the distance h₀ that is set here to 1 nm. Enhancement factors of up to 315 and 52 are found for $\bot $ and || field orientation, respectively, when using resonant excitation of both tip and sample [area 4 marked in Fig. 3(a)]. Whenever only one component, either tip (area 2) or sample (area 3), is excited resonantly, the enhancement factors are reduced to 150 for $\bot $-field components and 15 for ||-field components. Note that for the latter, the enhancement is particularly strong for sample-resonant excitation.

 

Fig. 3. Local electric field strength plotted as a function of the real parts of tip (x-axes) and sample permittivity (y-axes), as calculated via the analytic dipole model within the ± 15 interval: (a) perpendicular (out-of-plane) field component; (b) parallel (in-plane) field orientation. Note that imaginary parts are fixed at 0.1, and the tip with radius a = 5 nm is approached to h_o = 1 nm to the sample surface. Notably, two branches of near-field enhancement (brighter colors) are observed for both field components used, corresponding to tip- (area 2) and sample- (area 3) related resonances. Strongest enhancement of the local fields is found for double resonant excitation (area 4) reaching values of up to 315 and 52 for (a) perpendicular ($\bot )$ and (b) parallel (||) field orientation, respectively.

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Even though the dipole model indicates areas of interest for enhanced local fields, it is limited in determining the exact local field distribution near the tip and the sample. Hence, in the following section we apply numerical simulations to model the four near-field-coupled systems, as corresponding to the circled areas 1 - 4 in Fig. 3(a). In the following, we explore the spectral regime around λ=13.5-16.0 μm at which Au shows strongly negative Re(${\mathrm{\varepsilon} })$-values corresponding to non-resonant material excitations in cases 1, 2, and 3, whereas STO is excited resonantly around Re(${\mathrm{\varepsilon} })$ =-1 to -2 within its phononic Reststrahlen band, hence realizing cases 2, 3, and 4 (exact ${\mathrm{\varepsilon} }$ -values are given below).

3. Near-field-coupled system: simulations and results

In order to explore local vector fields beyond the limitations of the analytical dipole model [Eq. (1)], we simulate the four exemplary near-field-coupled systems for all possible combinations of tip and sample being either resonantly or non-resonantly excited (see areas 1 to 4 in Fig. 3). As model materials, we consider STO and Au in the spectral range of 13.5-16 μm [47]. Here, STO is excited within its phononic Reststrahlen band resulting in ${\mathrm{\varepsilon} }$-values from ${{\mathrm{\varepsilon} }_{\textrm{STO,13}.5\; \mathrm{\mu}\textrm{m}}} =- 0\textrm{.17 + i} \cdot 1.35$ to ${{\mathrm{\varepsilon} }_{\textrm{STO},16\; \mathrm{\mu}\textrm{m}}} =- 5.48 +\textrm{i} \cdot 2.31$ [49] covering possible resonant excitation when used as tip or sample material. On the other hand, Au shows a strongly metallic response for this wavelength region with ${{\mathrm{\varepsilon} }_{\textrm{Au,13}.5\; \mathrm{\mu}\textrm{m}}} \cong - 5746 + \textrm{i} \cdot 2340 $ to ${{\mathrm{\varepsilon} }_{\textrm{Au},16\; \mathrm{\mu}\textrm{m}}} \cong - 7511 + \textrm{i} \cdot 3287$ [50], representing a typically non-resonant material for both tip and sample. In the following subsections, the tip with an apex radius equal to a = 5 nm is illuminated with linearly p- or s-polarized incident light at an angle of θ = 45° (see Fig. 2) and the tip-sample distance is determined for h₀ = 1 nm if not stated otherwise, for all configurations. Note that here, unlike in the dipole model discussed above, p-polarized incident light results in mixed local excitation with $\bot $ (z) and || (x)-field components, whereas s-polarized incident light corresponds to strict || (y)-excitation.

3.1 Case 1: Au-tip and Au-sample (non-resonant tip and non-resonant sample)

In this configuration, Au is used as the non-resonant material for both tip and sample. The field enhancement in the vicinity of the tip is plotted in Fig. 4. There exists no field that penetrates either tip or sample, exactly fulfilling the electrostatic boundary conditions of a good metal. The local vector field orientations for both p- and s-polarized incident light are shown in Figs. 4(a) and 4(b), respectively, displaying a strict z-oriented field at the sample surface due to the metal’s boundary condition (see Fig. 1 for a detailed view of these fields at the sample surface, including extracted values for E_x, E_y, and E_z). The lobe shape of the field around the tip generally confirms the dipole model concept, however, indicates a slight asymmetry with stronger, more confined fields towards the sample as known from previous works [51].

 

Fig. 4. Local vector field distributions for case 1 when using non-resonant Au for both tip and sample in the mid-IR wavelength regime, plotted for (a) p- and (b) s-polarized incident light. (c,d) display the total electric fields underneath the tip as a function of tip-sample distance h₀ (varied from 1 - 5 nm) and wavelength λ (spectral range from 13.5–16 µm), when using (c) p- and (d) s-polarized incident light. A spectrally flat near-field is found that characteristically decays vs. distance h₀. Largest field enhancements of 15.4 are found for p-polarized light at h₀=1nm, whereas for s-polarized light, the field at the sample surface is negligibly small (factor of 1 × 10⁻⁶). As expected from a good metal, the vector field on the sample surface has z-components, only.

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In Figs. 4(c) and 4(d), the absolute values for field enhancement at the sample surface underneath the tip is displayed as a function of tip-sample distance h₀ and wavelength λ for both p- and s-polarizations, respectively, showing the characteristic near-field decay with h₀ and a flat spectral response over the full wavelength regime of interest here. The highest enhancement factor reaches 15.4 for p-polarized light at a distance of h₀= 1 nm. For s-polarized incident light, though, the field is more than 6 orders of magnitudes smaller as compared to p-polarization, reaching factors of 2.6 × 10⁻⁶ only, one of the reasons, why s-polarization is usually neglected in near-field microscopy.

3.2 Case 2: STO-tip and Au-sample (resonant tip and non-resonant sample)

In the second scenario, we consider the tip to be excited resonantly by choosing STO as the tip material while the sample (Au) remains non-resonantly excited. Experimental realization of such a system includes special tip coatings or nanoparticles attached to non-resonant tips [52]. Compared to the double-non-resonant case discussed above (in section 3.1) the field enhancement for p- and s-polarization as illustrated in Fig. 5, shows several distinct differences:

 

Fig. 5. Local vector field distribution for resonant tip (STO) and non-resonant sample (Au), illustrated for (a) p- and (b) s-polarized incident light. (c,d) show the electric field strength at the sample surface underneath the tip as a function of tip-sample distance h₀ and wavelength λ for (c) p- and (d) s-polarization. The local field enhancement on the sample surface reaches 41.7 and 0.00041 for p- and s-polarization at h₀=1nm, respectively (maxima marked by red circles for selected h₀). The field penetrates the resonant tip with a strongly wavelength dependent behavior. The field inside the metallic sample is exactly zero, forming the boundary condition for strictly z-oriented field components at the sample surface.

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Notably, electric fields clearly penetrate into the dielectric tip, which needs to be fully accounted for when utilizing coated tips. Moreover, when considering resonant tips, the system is expected to be highly sensitive to the tip shape, also introducing geometrical resonances for the small scatterer used here [53]. As for the vector fields on the metallic sample surface, the surface field is fully z-oriented and the field within the sample is zero.

3.3 Case 3: Au-tip and STO-sample (non-resonant tip and resonant sample)

We consider now a near-field-coupled system consisting of a resonantly excited sample (STO) next to a non-resonant Au tip. The local fields deviate clearly from the pure dipolar shape due to a stronger distortion by the sample surface (Fig. 6). The highest fields at the sample surface underneath the tip at h₀= 1 nm for p- and s-polarized incident light, respectively, reach values of 25.9 and 1.74, being 1.7- and 663000-times larger as compared to the fully non-resonant case 1. Notably, here, not only p-polarized light yields strongly localized electric fields, but equally the (often neglected) s-polarized case shows appreciable in-plane field components reaching significant enhancement factors, a fact that is characteristic to any sample-resonant system. Spectrally, compared to the tip-resonant configuration (case 2), the resonance is slightly red-shifted with maximum fields obtained at λ = 15.5 μm for h₀= 1 nm. The spectral positions of these maxima depend strongly on the tip-sample separation h₀ [see red circles in Figs. 6(c) and 6(d)], being most distinct for p-polarized light with shifts of Δλ_p= 1500 nm and Δλ_s= 200 nm. This lobe-like behavior is well known for sample-resonant excitation of near-field coupled system, both when applying the dipole model or from experimental studies [21,39,43,54]. For the latter, a variety of sample materials has been studied, showing a quantitative dependence of spectral distribution, field strength and distance dependence on the damping and oscillator strength of the corresponding phonon mode [20,21,39,40,55]. However, the qualitative signature of the field distribution is expected to be similar to STO. While the electric field does not penetrate the metallic tip, appreciable field strengths inside the resonant-sample are observed, enabling the examination of buried structures and deducing sub-surface sample properties. Notably, in-plane local vector fields components are distinguishable both at the sample surface and within the sample, particularly in s-polarized configuration. In fact, the in-plane-field reaches much larger values as compared to cases 1 and 2, reaching values of up to 1.74 for s-polarization.

 

Fig. 6. Local vector field distributions for non-resonant tip (Au) and resonant sample (STO), shown for (a) p- and (b) s-polarized incident light, respectively. (c,d) display field plots underneath the tip, as a function of tip-sample distance h₀ and wavelength λ for (c) p- and (d) s-polarization (spectral maxima marked by red circles). Field enhancements of up to 25.9 and 1.74 are found for p- and s-polarization, respectively. The local fields penetrate the STO sample and show significant in-plane components both at the sample surface and within the sample, particularly for s-polarized excitation.

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3.4 Case 4: STO-tip and STO-sample (resonant tip and resonant sample)

In this final configuration, we investigate the behavior of a double-resonant system using resonant STO for both tip and sample material. Clearly, the local fields around the tip are strongly distorted and markedly deviate from the dipolar shape, resulting in a highly polarization- and wavelength-sensitive response (see Fig. 7). The local vector field orientation is highly polarization-sensitive and the field enhancement values for h₀= 1 nm reach 79 and 3.4 for p- and s-polarized incident light, respectively, representing the highest numbers for all the 4 configurations investigated here. Note again, that local fields are significantly strong even for s-polarized incident light in this setup. The spectral positions of the field maxima [red circles in Figs. 7(c), 7(d)] are slightly red-shifted compared to the cases of resonant tip or sample discussed above, and show a distance dependent shift of up to Δλ_p= 100 nm and Δλ_s= 500 nm, in agreement with the distance-dependent splitting of the two near-field coupled branches shown in Fig. 3(a), area (4). Significant field penetration into the tip is also observed, being less homogeneous as compared to the previous case 2. Moreover, non-zero fields are present in the sample as well, however, being small as compared to the fields that surround the tip and the sample as discussed in the previous cases. In-plane vector fields are present for both polarizations, but being most distinct in the s-polarized configuration. Markedly for this scenario, two confined field maxima at the sample surface are observed with slightly off-centered positions, electric fields that might be utilized to excite e.g. nano-rod-shaped samples.

 

Fig. 7. Local vector field distribution for resonant tip and sample made up of STO, for both (a) p- and (b) s-polarized incident light. (c,d) display the local field underneath the tip as a function of tip-sample distance h₀ and wavelength λ, for (c) p- and (d) s-polarization (spectral maxima marked by red circles). This double-resonant configuration enables the strongest local field enhancements of up to 79 and 3.4 for p- and s-polarized excitation, respectively, demonstrating clearly the significance of s-polarized components for resonant excitation.

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4. Comparison and discussion

Comparing the four different setups (see Table 1) of near-field coupled resonant and/or non-resonant tip/sample systems in terms of absolute values and relative changes with reference to the doubly non-resonant case 1 (Au-tip plus Au-sample), we conclude several findings:

Table 1. Comparison of the four configurations of near-field coupled systems at a h₀=1nm tip-sample separation. We calculate here ratios of field enhancements underneath the tip for both p- and s-polarized incident light, taking the non-resonant tip/sample excitation case 1 as the reference (i.e. E_p/E_p,non=1; E_s/E_s,non-res=1). Particularly for resonant samples (cases 3 and 4), the local fields for s-polarized excitation are strongly enhanced, reaching significant values on the same order of magnitude as for non-resonant p-polarized excitation, but more than 5 orders of magnitude stronger as compared to the non-resonant s-polarized case.

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In general, the local fields on the sample surface show non-symmetric but characteristic distributions for the four different setups (see Fig. 1): For non-resonant (metallic) samples, the fields at the sample surface are oriented strictly perpendicular to the interface having a maximum underneath the tip for p-polarized excitation, whereas the comparably small fields for s-polarized incident light are zero underneath the tip. On the other hand, for resonant samples, in-plane components are present at the sample surface for both p- and s-polarized excitation. More specifically, for p-polarization, out-of-plane components are dominant, and underneath the tip (at x = 0), in-plane field contributions of 5.4% and 2.4% of the total fields are observed in the Au-STO [case 3] and STO-STO [case 4] scenario, respectively. Depending on x ≷ 0, the in-plane component has a different sign, resulting in a characteristic fan-like distributions for p-polarized excitation of sample-resonant systems. On the other hand, for all configurations, s-polarized excitation results in field distributions with two maxima at y ≠ 0 with opposite sign in the out-of-plane component. The shapes of the local fields show different characteristics for the different cases: Whereas for non-resonant, metallic samples, all fields are oriented strictly perpendicular to the sample surface with zero fields at the tip position, the field vectors for resonant samples spatially rotate from one maxima to the other, resulting in maxima of in-plane components at y = 0. Consequently, at this position, even purely in-plane-oriented fields can be observed for cases 3 and 4, respectively. Notable, the sample-resonant case 3 includes a constant, positive field offset for the in-plane component, whereas for the double-resonant case 4, the in-plane field changes sign at the position of the total-field maxima at y ≠ 0.

In conclusion of the simulated cases, we find that all vector components at the sample surface are available for resonant excitation of the sample in a near-field coupled system when considering both, p- and s-polarized light. As such, in-plane fields are present both slightly above the sample surface as well as within the sample; they particularly enable the excitation of molecules and nanoscopic specimen at sample surfaces as well as the investigation of sub-surface sample properties and buried structures.

In its qualitative behavior, the simulation results confirm the estimations of the analytical dipole model: We find distinct resonances of the near-field coupled system when either the tip- or sample-material shows slightly negative values for Re(ɛ). On the other hand compared to the simulation, quantitative values in the dipole model as discussed above (see Fig. 3) are always overestimated (by as much as 15-times) due to the assumed constant losses via Im(ɛ). When using realistic material parameters including losses and known corrections to the tip-dipole displacement [56,57], the dipole model matches much better, resulting in slightly overestimated (∼4-times higher) values only. In our simulation, we always consider the wavelength-dependent losses especially in STO. Here, the resonance is found within the materials’ Reststrahlen band and, consequently, the spectral width and losses are defined by the corresponding phonon mode(s). For such phonon-based resonances, typically, narrow resonances of a 1–2 µm spectral width are found accompanied with low losses [20,21,55]. When applying the concept of resonant excitation to other materials, particularly plasmonic resonances, enhanced losses have to be considered. Moreover, for resonant excitation at shorter wavelengths, e.g. in the visible and near-IR regime, the wavelength-to-tip-size ratio is reduced, a fact that significantly provokes multipole excitation and related field distortions [51]. At the mid-IR wavelengths, however, the 5-nm-sized tip is 3000-times smaller than the wavelength in use and the dipole approximation is well met. Also note, that the spatial confinement is on the order of the tip radius and, hence, the localization of the vector fields impressively exceeds the diffraction limit by more than 3 orders of magnitude. When increasing the tip radius slightly at mid-IR wavelengths, basically the same field distribution can be found with a direct scaling of distance dependence and lateral confinement with the tip radius. However, even though accompanied by a loss of resolution, in s-SNOM experiments larger tip radii are often chosen to increase the signal strength due to enhanced scattering efficiencies particularly at long wavelengths [58].

Finally, it shall be noted that, firstly, the present study does not consider scattering of the local fields into the far-field corresponding to the quantity that then is measured in a s-SNOM experiment. In order to describe the re-radiation of the local field into the far field, one needs to take the specific geometry and parameters of the experiment into account that particularly include the tensorial scattering efficiency of the tip as well as effects of antenna-resonances and angular dependence of scattering. In the recent study of McArdle et al. [40], these particular aspects of scattering signature are discussed for resonant excitation of an STO-sample, however, so far neglecting the corresponding effects of polarization. In this paper here, we explicitly focus on the local field distributions in order to fundamentally explore available vector fields and local sample excitations underneath the tip. Besides their utilization in classical s-SNOM, these local sample excitations might be of interest e.g. for local pump-probe configurations as well as for particle-enhanced Raman spectroscopy. Secondly, as shown in Figs. 4–7(c), 4–7(d), the spectral position of the resonances and the field strengths as well as the field distributions displayed in Fig. 1, strongly depend on the tip-sample distance h₀, offering the possibility to probe local contributions by modulating h₀, which might be applied to both in-plane and out-of-plane components as long as a field gradient in z-direction is present. Exploring these effects, however, is beyond the scope of the present paper and will be subject of future studies.

4. Conclusion

3D parametric simulations were carried out for four different configurations of non-resonant and resonant near-field coupled tip and/or sample for the example of mid-IR excitation. The field enhancement underneath the tip was studied in terms of strength, spatial distribution, and vector orientation reaching particularly strong values for material-resonant excitation of the sample and either non-resonant or resonant tip excitation. In these configurations, s-polarized incident light induces a significantly strong field enhancement, that also includes all vector components. Considering the drawback of strong dependences on the tip-shape and penetration through coating layers for resonant tips, we conclude that it is particularly the sample-resonant/tip-non-resonant configuration that is the most suitable option for realizing strongly enhanced local vector fields both at and below the sample surface. Based on these investigations, we envision the utilization of sample-resonant excitations in polarization-sensitive s-SNOM experiments, in order to unlock its high potential for nanoscopic material analysis via enhanced vector fields.