1. Introduction

The diffraction limit of classical optical microscopy has been overcome by apertureless optical near-field microscopy that employs light scattering from a scanning-probe tip (s-SNOM) [1]. This probe is usually a solid cone of high-index or metallic material with a sharp tip with curvature radius a of about 20 nm. The spatial resolution was found to approximately equal the radius a and not to be influenced by the wavelength λ ≫ a of the probing light[2]. This fact suggested in the past the use of the non-retarded electrostatic approximation for describing the near-field distribution, and thus encouraged to approximate the s-SNOM probe simply by a point dipole located at the tip’s center of curvature[3,4]. The corresponding point-dipole model of the near-field interaction between probe and sample has indeed explained the observed high spatial resolution in the s-SNOM images, and equally importantly, it has successfully predicted the s-SNOM relative near-field contrasts between different materials, in amplitude and in phase, and vs. frequency[5–13].

The absolute efficiency of the s-SNOM scattering response, on the other hand, is also of great practical interest but it has not yet been investigated. This quantity certainly could not be predicted by a model that reduces the conical probe to a small particle of size a, and certainly not by electrostatic theory, as it involves the coupling of focused far-field radiation with the fields on the probe structure which include the tip-confined and tip-enhanced near field. An exception is the scattering by a subwavelength-sized, plasmon-resonant metal particle supported on a dielectric shaft as in the original proposal of s-SNOM[14], experimentally realized in the visible[15], where a low-order Mie mode enhances both the scattering cross section and the local field[16]. The s-SNOM’s usual extended probe supports higher and propagating modes with the consequence that the scattering efficiency may not directly relate to the local field at the tip apex. In this Letter we present a simulation of a realistic probe structure and an experimental determination of the absolute scattering efficiency from a standard cantilevered probe. The experiment employs the recently introduced spectroscopic technique[17,18] which we have earlier applied to measure near-field spectra[19].

It is instructive to recall earlier studies of antenna-coupled infrared detectors. “Cat-whisker” wire antennas were introduced around 1970 to finely focus THz and mid-infrared beams into a sub-wavelength-size volume at the tip apex. There, nonlinear optical effects such as tunneling, mixing, rectification, or absorption were observed and exploited for detection and spectroscopy applications[20–29]. The efficiency of this antenna-based infrared nano-focusing can be approximately extracted from several of these reports, by interpreting the d.c. or a.c. voltages measured across the tip-sample gap or across an absorbing sample structure. If we define a “near-field-focusing efficiency” as η_N = P_N/P_I where P_I is the power of the incident coherent beam, and P^N is the power converted via the localized near field, the result comes out as varied as η_N = 0.0003 [21] and 0.000001 [25]. Optimal η_N requires orienting the polarization such that the electric field is along the wire, as observed also in the near infrared[29] and visible[28], and as is optimum for mid-infrared s-SNOM[30]. The applicability of long-wire antenna theory[21,23,24] was verified by the observation of antenna “lobes,” angular patterns of coupling efficiency that peak at certain angles between input and wire directions, observed in the far infrared[20] as well as in the mid infrared[22,25]. Shorter wires exhibit a smaller number of lobes[22]. Even in the near-infrared the antenna-typical performance was indicated when measuring a mixing response[27] that agreed with the extrapolation of quantitative mid-infrared measurements[26]. The latter experiment (with a Pt/Ir tip of 20° full-cone angle contacting Au) determined, in addition to a weak η_N = 0.000002, a “scattering efficiency” η_S = P_$/P_I = 0.03 which we define by P_S designating the power scattered out of an incident coherent beam of power P₁ which in ref [26] has 60 μm (FWHM) width. Another experiment at 633 nm wavelength determined a value η_N = 0.0001 for a Si tip contacting glass and a 4-μm wide input beam[31].

2. Experimental determination of back-scattering efficiency in s-SNOM

We investigate standard cantilevered Pt/Ir probes (MikroMasch CSC37/Ti-Pt) inclined 7° from the normal as used in our home-built s-SNOM[19]. They have a 25° full cone angle and 22 μm length (see Fig. 3 insets), and a ≈ 20-30 nm. The probe is approached to tap with 26 nmpp amplitude at ω = 20 KHz on a flat sample (the free amplitude is 29 nm_pp). For illumination we start with a 12 mm wide, collimated, broadband-infrared beam, containing wavelengths between 9 and 12 μm. We focus this beam to the tip to near the diffraction limit by a parabolic reflector that has a 10 mm focal length, a horizontal axis, and a rectangular shape to accomodate focal rays between 5° and 60° elevation (Fig. 1). Backward-scattered light is recollimated by the same mirror and measured using a ZnSe partial reflector and a photovoltaic HgCdTe detector (Kolmar KMPV11).

 

Fig. 1. Optical layout of coupling a collimated infrared beam to the probe chip, using a parabolic mirror for both illumination and collection of back-scattering.

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We adjust the focus to the scattering probe by maximizing the 2ω = 40 kHz modulated part of the back-scattering detector signal, thus assuring the optimum near-field interaction[1]. This signal is found to reduce to one half at ≈ 7 μm of lateral misalignment. We conclude that the focus diameter is about λ, and furthermore, that the scattering structure responsible for the near-field interaction is overall not larger than a few μm. A related observation has been reported for the near infrared[27]. By measuring the power P_B of the back-scattered beam when the tip contacts a flat Au sample we determine the “back-scattering efficiency” defined as η_B = P_B/P_I to amount to 0.02. While in this letter we do not deal with the efficiency of the near-field interaction we note in passing that usually only a small fraction f_N of η_B originates from the near-field interaction between the probe and the sample. This fraction can be estimated as f_N ≈ 0.003 for our experimental situation, as determined from the signal levels already reported in Fig. 5 of ref. [19]; hence, the present probe geometry and optical setup, like previous ones[21,25,27], are still far from providing an efficient coupling between far field and near field.

The measured back-scattering efficiency η_B = 0.02 could be thought to serve for estimating the total scattering efficiency η_S. In order to compare with previous work, let us for a moment assume that the scattering occurs isotropically over the upper half space, which would mean that 12% of the total scattered power P_S reaches our receiver aperture; from this we would find η_S = P_S/P_I = η_B/0.12 = 0.17, a value considerably higher than η_S = 0.03 of ref.[26] where, however, the focusing was less tight. But while the question of angular distribution has not yet been posed in the context of infrared SNOM, the assumption of isotropic scattering is likely to be invalid in the light of the earlier work on whisker diodes which did show the existence of angular lobes of sensitivity[20–29].

Further insight into antenna effects is expected from changing the wavelength, since any geometry-related antenna property will be changed. To assess spectra of back-scattering experimentally, we use the fact that our broadband beam carries about 70,000 modes with equidistant frequencies in the range of 800 to 1100 cm^-1, i.e., a coherent frequency-comb. As shown before[18], a multi-heterodyne interferometric detection can then assess both the amplitude and phase spectra by superimposing a second, slightly detuned broadband frequency-comb beam as a reference; each frequency-comb element of the sample beam has just one element in the reference beam with which it produces a unique radio-frequency beat; the radio-frequency beat spectrum of the detector signal therefore replicates the mid-infrared spectrum. This coherent-comb infrared spectrometer (“comb-FTIR”) has been successfully coupled with s-SNOM[19]. We use here the identical, interferometric setup to measure direct back-scattering spectra with the tip contacting different sample materials (Fig. 2A). The amplitude spectra of Au and Si turn out to be similar to the input spectrum (Fig. 2b in ref. [19]) which means that η_B is approximately constant over the probed spectral range so antenna lobes have no grossly distorting effect.

 

Fig. 2. Infrared spectra of (A) measured back-scattering from s-SNOM tip in contact with a flat sample material, Au-red, Si-black, SiC-green; (B) calculated Fresnel reflection of SiC, normalized to a p-polarized plane wave incident at 45°.

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Interestingly, the scattering amplitude of Si is about half that of Au. This ratio becomes plausible when compared with the computed Fresnel reflection amplitude of 0.42 for Si, for 45°, p-polarized incidence. With SiC, however, a distinct spectral signature appears as a step-wise reduction of the back-scattering amplitude at frequencies above 975 cm^-1. This again resembles the theoretical Fresnel reflection amplitude plotted in Fig. 2B. The phase spectra have a basic, parabolic shape due to chirp generated by a NaCl plate in the sample beam[19].

The phase spectra do not seem influenced by the presence of Au and Si samples, whereas SiC evokes a distinct modification. Also in this respect the overall resemblance to plane-wave Fresnel reflection is striking. While no theory is presently available that treats the back-scattering spectrum from a tip-contacted sample, our result suggests that the basic interaction mechanism includes the step of just one sample reflection.

In Figure 3 we show that the back-scattered spectrum does not significantly change with systematic misalignment of the focus position on the tip. It is noteworthy that the amplitude reduces in about the same way as the near-field scattering 2ω signal mentioned above. This suggests that the back-scattering process and the near-field scattering process share at least one common step which could be that the incident beam with power P_I excites the modes of the extended tip structure at power > P_S (the > sign accounts for absorption[24] of these modes). In a separate step then some of these modes would in turn excite the localized near field at the apex, at power P_N.

 

Fig. 3. Amplitude spectra of mid-infrared radiation back-scattered from tip in contact with SiC, for eight different focus positions separated by 5 μm, vertically (upper panel) and horizontally (lower panel). The colours correspond to the focus positions (upper panel) and the incident beam positions (lower panel), respectively, marked by 5 μm wide coloured symbols on the inserted SEM images showing two different views of the actually used tip.

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A closer look at the back-scattering spectrum of Au normalized to the input spectrum (Fig. 4) reveals, indeed, that even with dispersionless samples a significant spectral variation does exist. The amplitude varies by 20% around the average value of 0.14 (this value confirms the frequency-averaged efficiency η_B = 0.02 (= 0.14²) determined above). When the sample recedes from the tip, the efficiency η_B diminishes strongly, and the broad spectral feature shifts to lower frequency. The latter effect can be explained, in principle, by the interference of some partial waves whose phase difference increases with the tip-sample distance z. One possibility of such interference could be that the incident wave illuminates the shaft partly directly, and partly also indirectly via sample reflection (a similar consideration applies to the outgoing, scattered waves). Indeed, a simple calculation of such two-wave interference can approximately recover the measured spectra up to z = 3 μm, but only if the waves’ origin (scattering point) was assumed to be remote from the apex, at a height of 6 μm on the shaft; this appeared, however, not plausible[32].

 

Fig. 4. Normalized amplitude spectra √η_B of tip back-scattering, at varied distance z between tip and Au sample, as designated in μm in the colour table.

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The alternative interference mechanism would involve modes of the extended tip structure, viz surface waves which propagate along the shaft and reflect from the cantilever at one end, and from the sample at the other. In the theory of travelling-wave antennas, such resonances generate the angular antenna lobes mentioned above and observed in the infrared before[20,22,26]. Theoretically, a cylindrical wire of length 3λ on a metal ground plane should exhibit three lobes with maxima at elevation angles of 10°, 30° and 60°, as were in fact experimentally observed at 10.6 μm wavelength for conical W tips of length 3λ [22]. In our experiment the tip is conical and has a length of 22 μm ≈ 2λ, so the antenna efficiency should have two lobes with the maxima at about 20° and 50° elevation. A modification to the theory and the mentioned experiment seems required from the facts that (i) our focus spot is smaller than the shaft length, and (ii) the optical system integrates over a large angular range from 5° to 60° elevation. Still, antenna resonance should leave a spectral signature. This leads us to conclude that the modulations in Fig. 4 likely indicate classical antenna behaviour because, qualitatively, the spectral feature shifts down in frequency with the increasing tip-to-sample distance z. Quantitatively, at z = 2.2 μm the antenna length (wire plus gap) should be effectively increased by 10%. This should red-shift any spectral feature by also 10% as is indeed in agreement with Fig. 4.

Altogether, we find that our mid-infrared s-SNOM contacting Au returns η_B = 2% of the input power to the detector, and that spectroscopic analysis reveals antenna effects of the probe structure.

3. Observation of a configurational resonance

When the tip probes near the edge of an extended, 20 nm thick Au film on SiC we find a surprisingly strong, resonant response in the direct back-scattering (Fig. 5). The three η_B spectra taken near the boundary show a pronounced feature not seen in the two other spectra taken far (> 100 μm) from the boundary on either Au or SiC. The narrow resonance is centered at about 955 cm^-1 and features a width of ≈ 25 cm^-1 (FWHM) and a variable spectral shape depending on the tip’s position. The effect remains observable when the probe is several μm away from the edge, in horizontal as vertical directions. No signal is observable with the probe removed, so the resonance is a property of the geometric/dielectric configuration comprising both sample and probe. In this configurational resonance the direct back-scattering efficiency is enhanced six-fold over that of Au, amounting to η_B = 13%.

 

Fig. 5. Normalized power efficiency spectra η_B of tip back-scattering when the tip is in contact with SiC (green) or Au (red). Strong, resonant enhancement around 955 cm^-1 is observed when the tip probes near the edge of a 20 nm thick Au film on SiC, at three different positions (blue, light bue, purple)).

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From its spectral location, the resonance’s physical origin likely is in the surface phonon polariton dispersion of SiC. This dispersion manifests itself as a Reststrahlen edge in Figs. 2 and 3, and is known to cause a spectral concentration of surface phonon polariton states[33][34]. These surface waves represent additional channels for the scattering interaction. The coupling to these waves seems weak in our experiment when the tip probes an extended SiC area, since no resonant spectral feature around 955 cm^-1 is discernible in the back-scattering spectra of Figs. 2A and 3. But strong coupling obviously occurs when the probe is near a SiC/Au edge which is known for its ability to launch surface waves[35] as recently investigated[36]. Future angle-resolved measurements might map highly anisotropic scattering in this configuration, such as a lobe pointing towards the input beam (see simulations below).

Generally any sample inhomogeneity in the illuminated area contributes to the tip illumination and thus modifies the back-scattering. For small-scale inhomogeneities («λ) this influence could be treated by an extended effective-medium theory[37,38]. In contrast, any large inhomogeneity acts as an individual, localized antenna that illuminates the tip by its near field or far field. Accordingly we could show before that such localized fields can perfectly well be mapped by s-SNOM for the cases of plasmon-resonant particles[39], particle-gaps[40], and metal edges[36]. In the earlier phase of the present experiment[19] we have already reported how the direct back-scattering spectra change smoothly with distance from the edge, up to 20 μm, and also depend on the orientation of the edge relative to the incident beam direction. Very remarkably, this edge influence seemed not to show up in the simultaneously measured near-field interaction spectra which change abruptly at the material edge and stay constant away from the edge (Fig. 5 of ref. [19]).

4. Simulation of the field distribution in the scattering zone

The complicated scattering of a real s-SNOM probe cannot be treated under the electrostatics approximation. Rather, electromagnetic analysis with numerical simulation is essential for the understanding of the complex interactions[41–43]. Accordingly we calculated field distributions in the scattering zone as well as angle-dependent scattering efficiencies in the far field, with an emphasis on direct back-scattering. The tip was positioned close to a flat sample made up of two materials.

For this simulation we use the commercial package HFSS^TM. It employs the finite element method[44] (FEM) with adaptive mesh refinement. FEM is a well established and powerful tool, arguably the most robust and reliable due to the built-in adaptivity and error control. We have used it in the past for various problems in plasmonics and nanophotonics[45–47]. While the use of high-performance computers could certainly enhance the simulation capabilities if necessary, for practical convenience our simulations were performed on a conventional 3.4 GHz PC with 4 GB RAM. A typical electrodynamic simulation of the real tip scattering involves about 200,000 tetrahedral elements and takes several hours. To ensure reliability of the numerical results, we monitored a posteriori error estimates provided by HFSS and verified consistency of these results for different FE meshes.

Here we consider the realistic, three-dimensional structure of a cantilevered, Pt coated AFM tip which rests above a flat surface in close proximity to it. As a sample we consider a SiC substrate partly covered with a 30 nm thick Au film. Our special emphasis is on simulating the situation where the tip is near the edge of the Au film. The simulated space includes, as depicted in Fig. 6, the whole conical Pt tip of 22 μm height, with its axis tilted at 7° from the normal, characterized by a 25° full-cone angle and a 50 nm radius of curvature at the apex. The cantilever is represented by a 1 μm thick Pt disk with 12.5 μm radius. The tip apex is at a fixed height of 100 nm above the SiC surface. The computational domain is a cuboid of 30 × 13.5 × 32 μm³, with the solution extended by symmetry to 30 × 27 × 32 μm³. For illumination we consider a p-polarized monochromatic Gaussian beam of 15 μm width (FWHM) incident from an elevation angle of 30°. The dielectric values of Pt, Au and SiC were obtained from the literature as -1431+868i, -5538+1259i, and -0.757+0.148i, respectively, for 950 cm^-1, and -1560+964i, -6136+1473i, and -4.265 + 0.301i, respectively, for 900 cm^-1.[48–50] It can be assumed that the dimensions of the system are sufficiently large for bulk material parameters to be applicable with good accuracy. Full electrodynamic wave analysis was performed.

While from the mathematical point of view the problem is considered in the unbounded space, the numerical treatment calls for boundary conditions imposed on the walls of the computational domain. We used the second order radiation boundary condition built into HFSS (by Ansoft Corp.); this condition, according to the HFSS manual included with the software, has the form

where E _tan is the component of the E-field that is tangential to the surface and ◻_tan is the gradient operator on that surface. (Note that the exp(+jωt) convention common in electrical engineering is used in HFSS for phasors; one can switch to the exp(-jωt) convention of physics and optics by formally substituting -j for j.)

 

Fig. 6. Computational domain for simulating the field distribution near a realistic s-SNOM cantilevered tip probing near the edge of a 30 nm thick Au film on a SiC substrate.

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The HFSS package uses proprietary matrix solvers based on a variety of mathematical algorithms, including domain decomposition and adaptive multigrid/multilevel preconditioned conjugate gradient algorithms.

The main computational challenge has to do with the multiscale nature of the problem: the free space wavelength (10 μm) and the length of the tip (22 μm) are orders of magnitude larger than the tip radius at the apex, the Au film thickness, and the penetration depth in Pt and Au, all of which are below 50 nm. This imposed some restrictions due to computer memory and time constraints. The simulations are restricted for two frequencies below the Reststrahlen edge of SiC, especially including the interesting case of 950 cm^-1, where our experiment reveals extraordinarily strong back-scattering (Fig. 5). Further, a trade-off between convergence of adaptive mesh refinement and the domain size had to be found. The resultant computational domain truncates the incident beam slightly and contains ∼80% of the beam power. The final results, however, are normalized to the total beam power.

The distribution of the amplitude of the total field, for 950 cm^-1, is shown in Fig.7. The field in the SiC substrate is very small below the Au film, due to screening, except in the region extending about 2 μm from the film edge where it is enhanced ∼10-fold in respect to the incident field strength. We note in passing that the enhancement is as large as ∼150 at the tip apex and ∼50 in the “hot spot” of about 100 nm width below the tip apex, but its discussion would be beyond the scope of this letter.

 

Fig. 7. Calculated distribution of the total field amplitude /E_tot/ (normalized to the input field at beam center) in the central plane of the computational domain, for 950 cm^-1 with the tip on Au, at 2.4 μm distance of from the film edge. The colour in the upper graph is in logarithmic scale. The lower, zoomed graphs (linear colour scales) show enhanced near fields around the tip apex (left) and the film edge (right).

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Clearly the calculated distribution is far from that of the incoming beam which is sketched by its FWHM contours in Figs. 6&7. Furthermore, the distribution is not at all dominated by the interference pattern of the incident and a sample-reflected beam, which would consist of surface-parallel fringes. Instead, an intense field which in its maxima exceeds the incident field 4-8 times extends between cantilever and sample. Its strong modulation with 5-μm period reveals the interference of counterpropagating waves predominantly in the radial directions to/from the tip apex. These modes exhibit a non-zero, indeed even a maximum field strength right at the surfaces of the shaft and also of the Au film. This lets us assign these modes a predominantly surface-wave character (note that surface plasmon polaritons in the mid infrared propagate at nearly undistinguishable speed compared to free space waves). Inspection of the standing-wave pattern just under the cantilever reveals that the plasmon propagating along the shaft is strongly reflected at the cantilever, and with around 180° phase shift as can be recognized from the near-zero field minimum at the cantilever surface. Also, our simulation shows the distribution of energy flow near the tip-sample structure (Fig. 8). The flow exhibits vortices that are not uncommon in multiple-wave interference fields. The simplest example of such a vortice is the case of three intersecting beams polarized in the same direction, with the wave vectors at 120° to one another.

One again observes an enhanced energy flow along the metal surfaces of the shaft and the sample and in close proximity to them. This indicates that surface plasmons participate in the energy transport.

 

Fig. 8. Distribution of energy flow (normalized to the input value at beam center) for the same geometry and parameters as in Fig. 7. Logarithmic colour scale.

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5. Prediction of scattering lobes and of back-scattering efficiency

As the standing wave pattern marks a geometrical resonance between the cantilever and sample its signature should be manifest in the far-field scattering efficiency. From our discussion above we expect about two main lobes of enhanced scattering[20–29]. To render the expectation more realistic, we use the simulation to calculate the far-field scattering pattern, and in particular, its dependence on the position of the tip relative to the Au film edge. (Far-field calculation is a built-in feature of the HFSS package. It relies on the surface equivalence principle and integration of the electromagnetic fields over the radiation surface.)

Figure 9 shows the calculated far-field-scattering intensity (power per unit solid angle). In the case of a homogeneous sample (Fig. 9(A)) the scattering occurs overwhelmingly in the forward direction. It is interesting that whereas the tip modifies the profile of the forward, sample-reflected beam very significantly, it does not produce strong scattering in other directions. This behaviour is reminiscent of Mie scattering of large scatterers[16]. Integrating the far-field patterns over the elevation angles 10°-50° (and azimuthal angles within ±20°), corresponding to the angular aperture of the receiver optics in our experiment, we find that the backward-scattered beam contains η_B = 0.004 and 0.019 of the incident power, respectively, for the cases of SiC and Au. These values agree in the order of magnitude with our experimental determination of the back-scattering efficiencies η_B = 0.02 and 0.025 for SiC and Au, respectively (Fig. 5). The respective forward-scattering efficiencies come out to be 0.63 and 0.81.

 

Fig. 9. Calculated far-field angular scattering intensity for 950 cm^-1. The angular range of the convergent input beam is indicated as a grey sector impinging from the left. (A) Two cases of homogeneous, flat samples, SiC (full) or Au (dashed); (B) a case of a partially Au covered SiC sample, with the tip positioned right on the edge (dashed), or at 2.4 μm distance from the edge on the Au film (full); the lines below the graph serve to visualize the Au film and its edge.

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Drastically different scattering patterns are calculated when the tip is near the edge of an Au film (Fig. 9(B)). The most prominent features are the dramatic reduction of the forward scattering, and the appearance of a strong scattering lobe in the backward direction as well as two lobes into high elevation angles around 60°, both on the forward and backward sides of the probe. Here the integrated back-scattering efficiency is as large as η_B = 15% for the case that the tip is 2.4 μm away from the edge (full curve). Further calculations (Fig. 10) predict that there exists a broad maximum of η_B = 0.15 at about 2.5 μm or λ/4 from the edge. This agrees fairly well with the experimental value of η_B = 0.13 (Fig. 5). For the 900 cm^-1 frequency case we compute a much reduced efficiency, as is also experimentally observed. Thus our simulation supports the experimental observation of the configurational resonance as displayed in Fig. 5.

It is noteworthy that the edge of the Au film should have an effect on the specular reflection in the absence of the tip. Since there is a considerable phase difference in Fresnel reflection from Au and SiC (see Fig. 2(B)), we expect that the beam reflected at the edge should be partially suppressed, and a multi-peaked angular modulation is impressed. The scattering mechanism must therefore, as is certainly implicit in the simulation, include the scattering at the edge into bulk waves as well as into surface waves.

 

Fig. 10. Calculated back-scattering efficiency η_B integrated over the receiver aperture, for 950 cm^-1 and 900 cm^-1, vs. tip position from the Au film edge.

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Further, we use our simulation to compute the total scattered power, summed over all scattering lobes which mostly lie in the plane of incidence and have 20° width. As a result, we calculate total scattering efficiencies η = 0.64 for SiC, 0.83 for Au, 0.77 for the tip on the film edge, and 0.80 for the tip at 2.4 μm from the edge, corresponding to the four cases in Fig. 9. The missing fraction of 0.17 to 0.36 can be thought of representing absorption in both tip and sample. Thus the simulated total scattering efficiency η_S = 0.83 for Au, however, greatly exceeds η_S = 0.17 tentatively deduced from our experiment in section 2 for hypothetical isotropic scattering. This discrepancy could significantly diminish if the anisotropic, forward scattering predicted in Fig. 9(A) was experimentally verified or at least taken into our experiment's analysis as this would significantly increase the deduced η_S. A second consideration is the extra absorption caused by surface roughness; if accounted for in the simulation, it would decrease the expected efficiency.

Conclusion

We find evidence from the simulation and experiment that the scattering in a mid-infrared s-SNOM is distinctly influenced by the antenna properties of the tip. The length of the shaft determines a geometrical resonance for surface waves that propagate along the shaft and form a standing-wave pattern due to reflection at both cantilever and sample. This standing wave induces an angular structure in the far-field scattering.

How this antenna-mediated scattering expresses itself in the near-field response of s-SNOM should be an important issue for future work. While one would anticipate a direct influence[24], our recent experiment pointed to the contrary[19]. Clearly the mastering of antenna effects in s-SNOM will be of enormous practical consequence, enabling the optimization of probe structures together with illumination/receiving geometries, in order to attain both high signal/noise and high signal/background ratio near-field imaging. Tools for this work have been outlined in this paper, namely the full 3D electrodynamic finite-element simulation of the tip-sample interaction on a conventional PC, and the quantitative measurement of complex-valued scattering spectra using c-FTIR coherent spectroscopy.