Figure displays a raster image of the scattered radiation signal, upon lateral translation of the sample under the illuminated tip, where the moderately focused, incident laser beam is indicated by the blue arrow $\overrightarrow{k}.$ We observe periodic spatial signal modulations at all four edges (labeled 1–4), with the intensity peaks and valleys running parallel to the edges. One observes an additional modulation – most clearly visible in regions 2 and 4 – which is parallel to the wave front of the incoming laser radiation. This is a measurement artifact discussed in Supplemental C in connection with Supplemental Figure 3 shown there. The modulations inside the Au window are modes of the nano-ring meta-material not addressed further here.

The signal profiles along the lines in Figure are displayed in Figure . Along line 1, the signal corresponds to a damped sine wave (superposed on a constant offset) with a spatial period of Λ_{1}=460 nm; similarly along line 2, only with a period of Λ_{2}=1200 nm. Along line 3, i.e. in the downstream direction with respect to the incoming laser beam, the signal is more complex, as discussed below in connection with Figure . Note that these features were reproduced with many other samples, both in the 2Ω- and 3Ω-demodulated signals (see examples in Supplemental Figure 2).

Figure 2: Numerical simulation of the s-SNOM signal along line 3 in Figure .

(A) Sketch of four coherent s-SNOM signal contributions: (1) incoming beam; (2) SPP generated at the metal edge; (3) SPP launched at the s-SNOM tip, propagating to the edge of the metal film, reflected there and returning to the tip; (4) incoming beam reflected from the tip shaft onto the metal edge, generating an SPP there that propagates to the tip. (B–E) Calculated s-SNOM signals (time-averaged absolute square of the sum of the fields) at the location of the tip a distance *y*_{3} from the metal edge for various combinations of the four signal contributions in A. In D and E, the tip-reflected edge-launched SPP contribution (4) is modeled with an envelope that initially grows from *y*_{3}=0 and then decays to zero for *y*_{3}>1.8 μm (dashed line in D), as discussed in the text.

In the following, we focus on the identification of the origin of the spatial modulations. They arise from the interference of waves – SPPs and free-space radiation – which need to be determined. Characteristic features help us narrow down the options and avoid consideration of all reflected/scattered waves reaching the photodiode (for theoretical aspects of a full-wave approach, especially also for the application of the reciprocity theory, see Refs. [18], [19], [20], [21]; for a consideration of the role of light that is scattered at the edges but does not convert to SPPs, see comments in Supplemental D). The modulations must arise from SPPs generated at or reflected from the edges to form intensity peaks/valleys parallel to them. They propagate to the tip and interfere there with the incoming radiation. As we will see, we need SPPs generated in different ways to fully explain the modulations that we have set out to identify.

A first scenario involves the directly edge-excited SPPs, schematically shown in Figure , sketch 2, which interfere at the probe tip with the incident light wave (see sketch 1 in Figure ). The light wave and the SPP arrive at the probe tip with wave vectors $\overrightarrow{k}$ and ${\overrightarrow{k}}_{\text{SPP}},$ respectively, resulting in a spatial modulation of the total s-SNOM intensity with wave vector $\overrightarrow{K}={\overrightarrow{k}}_{\text{SPP}}-{\overrightarrow{k}}_{\parallel}$ (where ${\overrightarrow{k}}_{\parallel}$ is the projection of $\overrightarrow{k}$ in the surface plane; see Figure ). Note that while the focused incoming beam is indeed composed of a distribution of wave vectors (with a full cone angle of Δ*θ*~12° for the beam used here), we assume that the total interaction averages essentially to that of a quasi-plane wave where $\overrightarrow{k}$ is along the beam axis. The SPP propagation direction ${\overrightarrow{k}}_{\text{SPP}}$ is dictated by the phase-matching condition along the edge, i.e. *k*_{SPP,x} =*k*_{x}, which results in interference fringes that are always parallel to the respective edge [2] (as *K*_{x}=0), and corresponds to a generalized Snell’s (Ibn Sahl’s) law:

$$\mathrm{cos}\mathrm{(}\varphi \mathrm{)}\cdot \mathrm{sin}\mathrm{(}\theta \mathrm{)}=n\cdot \mathrm{cos}\mathrm{(}\beta \mathrm{}\mathrm{)}\mathrm{,}$$(1)where the angle of incidence *θ* (0°≤*θ*≤90°) and azimuthal angles *ϕ* (−180°<*ϕ*≤180°), *β* (0°≤*β*≤180°) are depicted in Figure . Here, $n=\Re \mathrm{(}\sqrt{\u03f5/\mathrm{(}1+\u03f5\mathrm{)}}\mathrm{)}$ is the real part of the effective refractive index of the SPP, with *ϵ* being the complex-valued dielectric function of gold [9], [22]. Interpolating data from Ref. [23], we obtain $\sqrt{\u03f5}=\text{0}\text{.165}+\text{i5}\text{.32}$ for thin-film gold at the wavelength λ=850 nm (*ħω*=1.46 eV), hence *n*=1.018. Figure illustrates the situation for *θ*=45° and *ϕ*=20°, yielding *β*=49.3°.

With the generalized Snell’s law from Eq. (1), one can show that the wavelength of the intensity interference fringes Λ=2*π*/*K*=2*π*/*K*_{y} is given by

$$\Lambda =\frac{\lambda}{-\mathrm{sin}\mathrm{(}\theta \mathrm{)}\cdot \mathrm{sin}\mathrm{(}\varphi \mathrm{)}+\sqrt{{\mathrm{sin}}^{2}\mathrm{(}\theta \mathrm{)}\cdot {\mathrm{sin}}^{2}\mathrm{(}\varphi \mathrm{)}-{\mathrm{sin}}^{2}\mathrm{(}\theta \mathrm{)}+{n}^{2}}}\mathrm{.}$$(2)We note that this equation contains a correction to that given in Ref. [2]. For co-propagating incoming wave and SPP [0°≤*ϕ*≤180°, hence sin(*ϕ*)≥0], Λ is always larger than for the case of counter-propagation (−180°<*ϕ*<0°, hence sin(*ϕ*)<0). Employing Eq. (2) with the observed wavelengths Λ_{1} and Λ_{2} allows one to solve for the angles of the incoming beam axis: *θ*=59°, *ϕ*_{1}=−80°. The wavelengths of the signal modulation in each region 1–4 (identified with edge-emitted SPPs) and the corresponding model curve from Eq. (2) are shown in Figure . With this value of *ϕ*_{1}, the wave vector $\overrightarrow{k}$ has only a small component along the *x*_{1}-axis and the SPP is emitted nearly in the opposite direction [from Eq. (1), *β*=82°]. In contrast, the SPP launched into region 2 has a value of *β*=34°. For region 4, Eq. (2) evaluates to Λ_{4}=2.0 μm, for which a modulation feature can be identified in the profile (Figure ).

Profile 3 exhibits a more complex behavior, being composed of three modulation patterns (see below), and it is also characterized by the highest s-SNOM signals. Near the edge, there are three maxima separated by ≈650 nm; further away, in the range *y*_{3}>2.5 μm, a fit of the data with two damped sine waves reproduces the data well, with periods of 430 nm and 5.4 μm, respectively. As shown in Figure , the longer of these two wavelengths is indeed consistent with the value Λ_{3} expected for the edge-launched SPP in this region (*ϕ*_{3}=*ϕ*_{1}+180°=100°).

We tentatively assign the 430-nm component to an SPP generated at the tip, and propagating to the edge where it is partly reflected back to the tip [24], [25], producing a distinct s-SNOM signal component. The expected period of this interference signal is ${\Lambda}_{\text{SPP,tip}}=\frac{1}{2}\lambda \mathrm{/}n=417.4\text{\hspace{0.17em}nm},$ which is indeed consistent with the fitted value. This mechanism is reminiscent of s-SNOM observations of SPPs on graphene, of phonon polaritons on boron nitride, both excited in the infrared [3], [5], [6], [26], and of SPPs on Ag excited at 532 nm [4], where in all cases the respective tip also simultaneously launched and detected the surface waves.

The full interference pattern in profile 3 (in particular, for *y*_{3}=2 μm) still remains to be explained. For this task, we employ a one-dimensional treatment of the various signal components, i.e. based on sinusoidal interference patterns with the predicted wavelengths, while the amplitudes, phases, and envelope functions are chosen to best reproduce the experimental data. The three mechanisms leading to the signals discussed so far are depicted schematically in Figure , (1–3). Figure show that these can account well for the experimental data for *y*_{3}>2.5 μm. In order to reproduce the full structure, we find that one additional contribution is required. From a series of simulations, we identified another channel for SPP generation that possesses the correct interference period to reconcile the data. This corresponds to a portion of the focused beam that first reflects from a portion of the tip, then encounters the metal edge and generates an additional SPP, as depicted in the fourth scheme (tip-reflected edge-launched SPP) in Figure .

The additional optical path of this beam gives rise to the additional optical path of this beam gives rise to interference fringes with a significantly shorter wavelength Λ_{SPP,refl} (which also depends sensitively on the tip apex angle *α*) than that of the directly edge-launched SPP (Λ_{3}). We find that a value of *α*=31° allows us to best fit the data with a resulting value of Λ_{SPP,refl}=874 nm. This contribution is shown in Figure (superimposed on the directly edge-launched SPP signal from Figure ). As the reflected ray is close to normal incidence (with *θ*′=*θ*−2*α*=−2.6°), due to the finite beam diameter, one expects that this component can only be present when the tip is close enough to the edge (i.e. for *y*_{3} not larger than the incoming beam radius of 2–2.5 μm). Also, as the tip-edge distance *y* increases from zero, the section of the illuminated tip that reflects down onto the edge shifts upwards from the apex. This corresponds to a larger reflecting area of the tip wall and hence total power on the edge. These two effects justify the use of the envelope function applied for this signal component (see Figure ). The total simulated signal from all four waves is shown in Figure , and is seen to reproduce the full structure of profile 3 in Figure well, providing support for this proposed additional SPP generation mechanism.