In this s-SNOM study, we investigate the formation of the near-field intensity distribution of SPPs on platelets at two wavelengths in the red (*λ*_{0,}_{r}=633 nm) and green (*λ*_{0,}_{g}=522 nm) part of the visible spectrum. The respective wavelength of the SPPs is given by

$${\lambda}_{\text{SPP}}={\lambda}_{0}\mathrm{/}\Re \text{{}\tilde{n}\text{}}\mathrm{,}$$(11)

where *λ*_{0} is the free-space wavelength of the incident light and $\tilde{n}=\tilde{k}\mathrm{/}{k}_{0}=\sqrt{\tilde{\epsilon}/\mathrm{(}1+\tilde{\epsilon}\mathrm{)}}$ is the complex-valued effective refractive index of the SPPs. We consider only SPPs propagating at the air-Au interface and neglect any hybridization effects with SPPs propagating at the SiO_{2}/Au interface. This is motivated by the sufficient thickness of our flakes (above 100 nm) [40] and validated by our calculations (see Supplementary Material C). By using the data of Johnson and Christie [41], the complex-valued dielectric function of gold $\mathrm{(}\tilde{\epsilon}\mathrm{)}$ at the two wavelengths yields the values $\tilde{\epsilon}\mathrm{(}{\lambda}_{\mathrm{0,}r}\mathrm{)}=-11.74+1.26i$ and $\tilde{\epsilon}\mathrm{(}{\lambda}_{\mathrm{0,}g}\mathrm{)}=-3.95+2.58i.$ Note that we are thus taking the non-neglectable effect of interband absorption into account, which is the limiting factor for both confinement and propagation length at our illumination wavelengths. We will use these analytically calculated values to compare our experimental results. The respective theoretical values for the SPP wavelengths are thus *λ*_{SPP}_{,}_{r}=606 nm and *λ*_{SPP}_{,}*g*=477 nm. The corresponding propagation lengths are given by *L*=1/(2*k*_{x}_{″}), where ${k}_{{x}^{\u2033}}=2\pi \Im m\text{{}\tilde{n}\text{}}\text{\hspace{0.05em}}\mathrm{/}\text{\hspace{0.05em}}{\lambda}_{0},$ which yields *L*_{r}=9.76 µm and *L*_{g}=9.76 µm.

Figure 4
shows (A) the AFM topography of a platelet and (B) the associated non-interferometric s-SNOM signal image taken at the third overtone of the tapping frequency of the s-SNOM (≈285 kHz) using the red HeNe laser at *λ*_{0}_{,}*r*. The topography shows that the height of the platelet is around 110 nm and that the surface of the sample is very smooth (RMS surface roughness from AFM measurement ≈210 pm) with almost no impurities. The red arrow indicates the angle of incidence of the laser beam. The sample has been oriented so that this angle is in the range of 75° (here *φ*=74°), where we got the clearest signal of SPPs due to the anisotropic excitation of SPPs at the tip apex [29]. To minimize the scan area, it has been aligned with this edge by rotating the scan area by 29° and the spatial resolution has been set to 25 nm.

Figure 3: Schematic illustrations of the interfering signal pathways in K- and real space.

(A) Schematic of wave vectors and angles for the derivation of the fringe spacings of edge-launched SPPs and (B) schematics of beam path of incident light and edge-launched SPPs.

Figure 4: Topographic and near-field optical signals at an illumination wavelength of 633 nm.

(A) AFM and (B) s-SNOM images of a monocrystalline gold platelet.

The image of the s-SNOM signal (Figure 4B) shows a pattern that is confined on the sample which proves that we are probing the near-field, also confirmed by the approach curves [37]. It consists of lines of maxima and minima that are parallel to the edges, at which strong modulations are apparent. As we have described in the previous section, several signal channels overlap at the edges. Especially, the excitation of edge-launched and tip-reflected edge-launched SPPs is strongly dependent on the position and orientation of the incident beam spot, which leads to the apparent anisotropic interference pattern at the edges, and it is challenging to explain quantitatively the intensity pattern. Therefore, we focus on the investigation of the periodicity of these modulations. We extracted line profiles binned over 128 pixels along the three arrows with Gwyddion [42]. Then, we performed a fast Fourier transform (FFT) to obtain the spatial frequency components (*K*) of theses profiles. However, we plot the spectra as a function of the fringe spacing Λ=2*π*/*K*, because this way the peaks for small fringe spacings become much more apparent.

Figure 5A shows the intensity profile 1 illustrated in Figure 4B. As defined in Figure 3, the angle subtended by the incident light and the corresponding edge *φ* is −74° and +74°. The profile rises steeply on the left-hand side where the platelet starts; then the signal oscillates and decays rather smoothly until the center of the flake. After approximately 10 µm, a longer oscillation period appears which changes into a complex pattern of short fringe distances overlapped with a strong signal modulation. Finally, the signal decreases steeply at the other end of the sample. To analyze the spacing of the fringes we chose a range of the profile (green curve) and spans over the three colored regions in Figure 5A and performed an FFT over the entire range. The resulting spectrum plotted over Λ=2*π*/*K* is shown in Figure 5B (green curve). The range has been set approximately 0.5–1 µm away from the edges to avoid taking into account coupling effects between the tip and the edges [43]. We calculated the expected values of the fringe spacings due to the effects described in the previous section and obtained Λ_{tl}_{,}_{r}=0.303 µm, Λ_{el}_{1,}_{r} (60°, −74°)=0.347 µm and Λ_{trel}_{1,}_{r} (30°, +74°)=1.149 µm where we found the best agreement when substituting *ϑ*=60° with *ϑ*′=30°. The corresponding values are marked by the vertical lines in Figure 5B, and we find that the maxima of the three peaks (green curve) overlap well with the calculated values. The predicted value for Λ_{el}_{1,}_{r} (60°, +74°)=3.423 µm is masked by the DC offset so we limit the plotting range to Λ=2.0 µm. To identify which part of the signal profile contributes to these peaks, we divided the profile into three parts marked by the light blue region (blue curve), the light purple region (purple curve) and light red region (red curve) in Figure 5A and performed an FFT of the s-SNOM signal on each interval. Note that we performed the FFTs of the entire profile and the subintervals as they are shown without subtracting the background or using soft windows. The corresponding spectra are also plotted in Figure 5B. The spectrum of the first interval (blue curve) close to the edge, where *φ* is −74°, peaks around the expected value for Λ_{el}_{1,}_{r} (60°, −74°). The broadness of the peak reflects the fact that edge-launched SPPs can only be excited as long as the incident beam hits the edge; furthermore, tip-launched SPPs contribute to this signal. The spectrum of the second interval (purple curve) has one sharp peak centered almost exactly at the predicted value for tip-launched SPPs. We therefore conclude that this part of the profile is solely due to tip-launched SPPs. Impressively, this result shows that we are able to measure a signal of the SPPs after they have propagated a distance of at least 2×9 µm=18 µm because they are reflected at the edge, and we will use this signal, as we will show below, to determine the SPP wavelength. For the third subinterval we expected to obtain two peaks centered at Λ_{tl}_{,}_{r} and Λ_{trel}_{1,}_{r} (30°, +74°). The broad peak at Λ≈1.15 µm is in accordance with our estimation for tip-reflected edge-launched SPPs. We assign it to this signal channel because the oscillation with this period length is dominant between *x*=10 µm and *x*=15 µm. This range is far away from the edge, and it is unlikely that the incident beam is still illuminating the edge at these tip positions. Thus, other edge-related signal channels cannot be considered. Furthermore, we find a dominant peak which has its maximum close to the predicted value for edge-launched SPPs which cannot contribute to this signal because they can only be excited on the other side of the sample [Λ_{el}_{1,}_{r} (60°, −74°)]. We therefore believe that this signal is due to edge-launched SPPs interfering with the incident light scattered at the same edge [Λ_{el}_{2,}_{r} (60°, 74°)]. The slightly longer fringe spacing can be explained because the edge is not perpendicular with respect to the incident beam and the incident beam is not a plane wave but a light cone with the maximal half-angle of 23°. Therefore, it is likely that a part of the incident light will be reflected/scattered by the edge within the collection angle of the PM and contribute to the interference signal with a slightly different phase difference. Another feature of this peak is a shoulder on its left side which is presumably due to tip-launched SPPs.

Figure 5: Profiles and corresponding FFTs from the s-SNOM scan at 633 nm shown in Figure 4B.

(A) profile 1, (B) corresponding FFT, (C) profile 2, (D) corresponding FFT, (E) profile 3 and (F) corresponding FFT.

The second profile marked in Figure 4B is shown in Figure 5C. For this profile the incident angle with respect to the two edges is *φ*=±46°. We apply the same procedure and use the same color coding for the curves and backgrounds as before. The predicted values for the angle-dependent fringe spacings are Λ_{el}_{1,}_{r} (60°, −46°)=0.428 µm and Λ_{trel}_{,}_{r} (30°, +46°)=1.018 µm for which we again substituted *ϑ*=60° with *ϑ*′=30°. Also here the period for edge-launched SPPs from the right-hand side Λ_{el}_{1,}_{r} (60°, +46°)=2.735 µm cannot be resolved by the FFT and thus is not shown. The spectrum of the entire range (green curve) consists of two broad peaks (Figure 5D). One peak is centered around the predicted value for edge-launched SPPs and shows indications for some contribution due to tip-launched SPPs on its left-hand side. The second peak is almost centered at the predicted value for tip-reflected edge-launched SPPs. The analysis of the subintervals reveals that the fringes close to the left edge, where *φ*=−46°, can be attributed to edge-launched SPPs. The second subinterval has one clear peak centered at Λ_{tl}_{,}_{r}. Thus, also in this direction tip-launched SPPs can propagate over the entire platelet. The third subinterval can be related to tip-reflected edge-launched SPPs. In contrast to the third subinterval of profile 1, we do not obtain a strong peak close to Λ_{el}_{1,}_{r} (60°, −46°), which is reasonable because it is unlikely that the incident light is reflected by the edge in the direction of the PM.

The results for the third profile in Figure 4B are presented in Figure 5E and F. The incident angle *φ* is with ±14° almost parallel to the respective edges. For this situation, the estimated values of the fringe spacings are Λ_{el}_{1,}_{r} (60°, −14°)=0.736 µm, Λ_{el}_{1,}_{r} (60°, +14°)=1.592 µm and Λ_{trel}_{1,}_{r} (30°, +14°)=0.797 µm. The spectrum of the entire range marked by the green curve (Figure 5F) peaks sharply at Λ_{tl}_{,}_{r} and has a dominant and broad maximum at approximately 0.95 µm, which is larger than the predicted values for both edge-launched SPPs from the left-hand side and tip-reflected edge-launched SPPs from the right-hand side. The amplitude rises again close to Λ_{el}_{1,}_{r} (60°, +14°), but stays up because of the DC offset. The profile in the first selected subinterval (blue curve) shows a highly complex oscillation which quickly drops. The spectrum of this region starts to rise around the predicted value of tip-launched SPPs but does not show further features. The profile in the light purple window (purple curve) features a small but clear oscillation. Accordingly, the spectrum features a single peak at the estimated value for Λ_{tl}_{,}_{r}. Thus, also for this orientation we can clearly identify tip-launched SPPs which must have propagated for approximately 20 µm. The oscillation in the third region (red curve) corresponds mainly to the peak around 0.95 µm. It is actually surprising that also at this angle tip-reflected SPPs contribute to the signal because of the momentum conservation of the parallel component of the SPP wave vector, which means that the SPP should propagate away from the tip. However, it is likely that the incident beam will be scattered at the edge of the tip in a variety of directions, thus causing SPPs to be excited at the edges of the sample that eventually propagate toward the tip and contribute to the signal. The width of the peak, as well as its dislocation, is an indication that the process is not well defined because of the steep angle of incident. The increase of the amplitude close to Λ_{el}_{1} (60°, +14°) is an indication for the contribution of edge-launched SPPs, but the resolution is too coarse for an exact assignment. Furthermore, we can see that in this region no signs for tip-launched SPPs can be identified. This is probably due to an inefficient generation of SPPs propagating in this direction. In all spectra shown in Figure 5, additional peaks and shoulders are apparent which do not overlap with the predicted values of the homodyne amplified signals [second term in Eq. (4)]. It is likely that these features are due to additional signal channels, especially due to the interference of different SPP-related scattering pathways [last term in Eq. (4)]. However, these effects are relatively small compared to the homodyne amplified pathways, which justifies our approximation to neglect the higher order scattering pathways in this study.

We performed the same experiment with a different gold platelet with slightly smaller dimensions, using 521 nm excitation wavelength. The spacial resolution of this scan was set to 20 nm. The laser beam was coupled into the near-field setup from the left-hand side, as indicated by the green arrow in Figure 6A. The surface of this sample is also of high quality and the height is constant at about 100 nm (RMS surface roughness ≈230 pm). The s-SNOM signal shown in Figure 6B is taken at the fourth harmonic of the tapping frequency. Also here, the signal is confined on the sample and features interference fringes parallel to the edges whose spacing varies from edge to edge. In contrast to the result obtained at 633 nm, no interference fringes can be seen in the center of the sample which is due to the much shorter propagation length of SPPs excited by light at 521 nm (*L*_{g}=540 nm). We extracted three profiles along the arrows indicated in Figure 6B to analyze the contributions of the different SPP signal channels.

Figure 6: Topographic and near-field optical signals at an illumination wavelength of 521 nm.

(A) AFM and (B) s-SNOM images of a monocrystalline gold platelet.

The first profile is presented in Figure 7A. The incident angle at the right- and left-hand side is *φ*=±78°, respectively. We obtain the following predicted values for the fringe spacings: Λ_{tl}_{,}_{g}=0.238 µm, Λ_{el}_{1,}_{g} (60°, −78°)=0.270 µm, Λ_{trel}_{,}_{g} (30°, 78°)=0.869 µm for which we have also exchanged $\vartheta {=60}^{\circ}$ with ${\vartheta}^{\prime}{=30}^{\circ}$ and Λ_{el}_{1,}_{g} (60°, +78°)=2.243 µm, which could not be resolved by the FFT and therefore is outside the plotted range. The most left and right vertical lines in Figure 7A mark the entire range we analyzed (green curve). The resulting FFT (Figure 7B, green curve) shows a small peak at Λ_{tl}_{,}_{g}; then it also peaks at a slightly larger fringe spacing than the predicted value for edge-launched SPPs at *φ*=−78° and features a broad peak at the estimated value for tip-reflected SPPs. Similar to profile 1 at 633 nm, the light blue subinterval which is at the edge facing the incident beam (blue curve, *φ*=+78°) yields two peaks: one as expected for tip-reflected edge-launched SPPs Λ_{trel}_{,}_{g} (30°, +74°) and a strong and narrower peak which is centered at a slightly larger fringe spacing than the predicted value for edge-launched SPPs excited at the opposite side of the sample where *φ*=−78°. Therefore, we assign the latter peak to SPPs which are launched by the edge and propagate toward the tip, where they are scattered and interfere with the backscattered incident light hitting the edge (see Figure 2E). In addition, this peak has a small plateau on the left-hand side which fits to tip-launched SPPs that are reflected at the edge. In the center of profile 1 we cannot find any signature of SPPs which is in accordance with the short propagation length. The FFT of the light red subinterval close to the edge where *φ*=−78° (red curve) has the expected peak centered at Λ_{el}_{1,}_{g} (60°, −78°).

Figure 7: Profiles and corresponding FFTs from the s-SNOM scan at 521 nm shown in Figure 6B.

(A) profile 1, (B) corresponding FFT, (C) profile 2, (D) corresponding FFT, (E) profile 3 and (F) corresponding FFT.

For the second profile shown in Figure 7C the incident angle at the two edges is *φ*=+42°, respectively. The FFT of the entire range (green curve) has maxima at Λ_{tl}_{,}_{g}, Λ_{el}_{1,}_{g} (60°, −42°)=256 µm and Λ_{trel}_{,}_{g} (30°, +42°)=0.750 µm, and the amplitude rises toward Λ_{el}_{1,}_{g} (60°, +42°)=1.706 µm. The analysis of the signal at the left-hand side of the profile (light blue region) shows, however, no clear features that could be assigned to the expected fringe spacings. In contrast, the signal on the other side of the profile (light red region) can clearly be appointed to edge-launched SPPs where *φ*=−42° (Figure 7D).

The results for the third profile, where *φ*=±18°, are shown in Figure 7E and F. For these incident azimuth angles we estimate the following fringe spacings: Λ_{el}_{1,}_{g} (60°, −18°)=0.528 µm, Λ_{trel}_{,}_{g} (30°, +18°)=0.627 µm and Λ_{el}_{1,}_{g} (60°, +18°)=1.151 µm. Also here, the FFT of the entire range (green curve) and the two subintervals at the respective edges yield peaks which can be assigned to the calculated fringe spacings of the respective signal channels. As for the illumination at 633 nm, we find no clear contribution of tip-launched SPPs at the edge where *φ*=±18° (blue curve); however, the FFT yields a peak which can be assigned to tip-reflected SPPs, while the result for the other side of the profile (*φ*=−18°) yields two peaks which fit to both tip-launched and edge-launched SPPs. Also at this illumination wavelength we can find small peaks (especially in profiles 1 and 3 around 0.4 µm^{−1}) which are not overlapping with our predicted fringe spacings. These peaks are probably due to additional signal channels like interfering SPP-related scattering pathways.

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