## 4 Comparison of Wavefields of \"Perfect Beams\" and SPPs  
### Characteristics of SPPs on Axisymmetric Conductors  
In this section, we will discuss the features of plasmon excitation by the end-fire coupling technique [13], which we must take into account when evaluating the applicability of one or another axicon for this. Surface plasmon polaritons [34] are essentially coupled oscillations of electrons in the surface layer of a conductor and a p-polarized electromagnetic field in an adjacent dielectric, which propagate along the surface. The field strength goes down on both sides of the interface, and the wave energy decreases due to ohmic losses in  
Figure 4: Distribution of intensity \(I(r,\psi,z)=E(r,\psi,z)\cdot E^{*}(r,\psi,z)\) of vortex beams with \(\ell=0,-1,-2\) (left, middle, and right columns, respectively) created by binary axions with periods \(p=3.1\) mm in the space behind the lens with focal length \(f=100\) mm. Dimensions are in millimeters. (**a,b**) Distribution of \(I(r,\psi,f)\) in the focal plane of lens, calculated by Formula (22). (**c**) Longitudinal section of the same beam \(I(r,0,z)\) for \(80\leq z\leq 120\) mm, calculated using Formula (21). The radiation wavelength is \(141\) μm.  
the conductor. In the case of cylindrical geometry, the normal component of the plasmon field value \(F(r)\) decreases according to the law (see, e.g., [35])  
\[F(r)=\left\{\begin{array}{ll}\dfrac{I_{1}(p_{m}r)}{I_{1}(p_{m}a_{w})}&r<a_{w}\\ \dfrac{K_{1}(p_{d}r)}{K_{1}(p_{d}a_{w})}&r>a_{w}\end{array}\right., \tag{23}\]  
where \(I_{1}(\bullet)\) and \(K_{1}(\bullet)\) are modified first-order Bessel functions of the first and second kind, respectively, \(p_{m,d}=\sqrt{k_{s}^{2}-k_{0}^{2}e_{m,d}}\) and \(\varepsilon_{m,d}\) are the dielectric permittivities of the metal and dielectric, respectively, \(k_{s}\) is the wave vector of plasmon, and \(a_{w}\) is the radius of the cylinder.  
Our prime interest is the efficiency of the excitation of the plasmon that rotates while propagating along a conductor with a frequency determined by the orbital angular momentum of the exciting beam; therefore, as a model of a plasmon, we will choose an electromagnetic wave that has the following field distribution over the surface  
\[F(z,r,\psi)=\dfrac{K_{1}(p_{m}r)}{K_{1}(p_{m}a_{w})}e^{i\ell\psi}e^{-\gamma(z- f)},\;r>a_{w} \tag{24}\]  
and carries an orbital moment corresponding to the moment of the exciting wave. Here, \(\gamma\) is the decay constant of the amplitude of the surface wave during the propagation of the plasmon along the cylinder. In the terahertz range, for metals such as copper and gold, the plasmon propagation length varies from centimeters to tens of centimeters [34], that is, by an order of magnitude; the attenuation coefficient of the surface wave along the z-axis is \(10^{-2}\) mm\({}^{-1}\). With a conductor diameter of several millimeters, the electric field decays almost exponentially with respect to the radius:  
\[F(z,r,\psi)=e^{-\beta(r-a_{w})}e^{i\ell\psi}e^{-\gamma(z-f)}. \tag{25}\]  
In this case, the decay constant \(\beta\) must be close to the respective value for plasmons on flat surfaces. For a gold surface with a zinc sulfide coating that is 0 to 1.5 um thick, the measured \(\beta\) values are in the range of 0.3 \(\div\) 5 mm\({}^{-1}\)[36]. According to experiments [37], the optimal thickness of the dielectric coating, which ensures good coupling of the plasmon with the surface and reduces parasitic radiation losses, is approximately 1 um. In this case, the characteristic decay length for the evanescent wave in air is approximately 200-300 um, that is, practically all energy of the plasmon field is concentrated within a cylindrical layer that is less than 0.6 mm thick.