## 4 Comparison of Wavefields of \"Perfect Beams\" and SPPs
### Wavefields of Perfect Beams Obtained with Binary Axicons and Overlap Integral  
In waveguide systems, the efficiency of the transformation of the exciting wavefield into a waveguide mode is usually estimated by the value of the overlap integral of the incident wavefield and the waveguide mode field (see, for example, [38]). The closer the distribution of the incident wave amplitude and phase to the expected distribution of these quantities in the waveguide mode, the greater the efficiency of excitation of the latter. Let us write the overlap integral for our case. It is known that the electric field of a plasmon on a cylinder has both radial and longitudinal components, while the incident wave at the lens focus has a plane wavefront; therefore, it is more convenient to write the overlap integral through the vectors of the magnetic field, which, in both cases, has only azimuthal components:  
\[\dfrac{d\eta(\psi)}{d\psi}\approx\dfrac{\left[\int_{a_{w}}^{\infty}\mathbf{H} _{\psi}^{(SPP)}(r,\psi)\cdot\mathbf{H}_{\psi}^{(QBB)*}(r,\psi)rdr\right]^{2}}{ \left[\int_{a_{w}}^{\infty}\mathbf{H}_{\psi}^{(SPP)*}(r,\psi)rdr\right]\left[ \int_{a_{w}}^{\infty}\mathbf{H}_{\psi}^{(QBB)}(r,\psi)\cdot\mathbf{H}_{\psi}^ {(QBB)*}(r,\psi)rdr\right]}. \tag{26}\]Unlike the notation conventional for the bulk waves, as, for example, in [39], we write here not the total efficiency of the transformation of a free wave into a plasmon mode, but its derivative as a function of the azimuthal angle. This notation takes into account the difference between the excitation of a waveguide mode, for example, in a resonator, and the excitation of a plasmon on a surface. In the first case, the excited mode is spatial, and its distribution across the entire volume is known in advance. All the energy absorbed transmits into this mode. In the case of plasmons, the situation is different. A plasmon is excited at the conductor edge and moves in a certain direction. Simultaneously, another plasmon moving in parallel can be excited nearby with the same field distribution, but, generally speaking, with a different intensity, set by the local pumping intensity. In other words, in the first approximation, the excitation of plasmons in different regions can be considered independent. This is evidenced by the observation of parallel non-expanding tracks of plasmons created by the diffraction of a Bessel beam at the edge of a convex surface [40]. To calculate expression (27), one should substitute the expected distribution of the plasmon field in the air, and the field of the vortex beam illuminating the end of the cylinder should be substituted into it. The plasmon field is described by the expression \(F(f,r,\psi)=e^{-\beta(r-a_{\mathrm{w}})}e^{i\phi t}\), where the \(\beta\) value can be taken from the experimental data for the Au-ZnS-air surface [34], and the incident wavefield can be calculated according to (22). Figure 5 shows the intensity and phase distributions in the Bessel beams generated by binary axicons and in perfect beams in the focal plane of the lens. The numerical calculations were carried out using the diffraction integral [41], whereas the analytical calculations were done using Equation (22). The results of the calculations are very similar. The parameters of the axicons were selected in a way such that the diameter of the beams at the focus of the lens was approximately 7 mm, which is close to the diameter of the cylindrical conductors prepared for experiments. One can see that the width of the brightest part of the ring is about 0.6 mm, which is in good agreement with the plasmon decay length estimated above. The wavefields of perfect beams produced by binary axicons of different orders (see Figure 5) have variations of brightness over the azimuthal angle, which has a number equal to \(2\ell\). Thus, we can expect the generation of \(2\ell\) parallel SPPs on a cylinder if the topological charge of a perfect beam is equal to \(\ell\).