## 2 Techniques for Bessel and Perfect Vortex Beam Formation
### Methods of Forming Quasi-Bessel Beams  
Equations (2)-(4) show how beams with properties close to those of ideal Bessel beams can be formed in real life. Here, we will call them quasi-Bessel beams (QBBs). To this end, it is necessary to use conically converging beams. A zero-order Bessel beam, for example, is formed behind an axicon (conical lens) illuminated by a beam with a plane wavefront. If a spiral phase plate is added to the axicon [13], then a higher-order Bessel beam can be formed (Figure 1a). The phase transmission function of such a hybrid element, which can be called a spiral axicon (SA), is described by the expression  
\[t_{SA}(r,\varphi)=-kar(n-1)+\ell\varphi\equiv\ell\varphi-\kappa r, \tag{5}\]  
where \(\alpha\) is the cone-apex angle, and \(n\) is the refractive index of the material. Such axicons were fabricated using a 3D printer to generate Bessel beams at a radiation frequency of 0.3 THz in [18]. Since we are interested in radiation frequencies that are an order of magnitude higher and greater radiation powers, such volumetric plastic elements cannot be used because of the strong absorption of radiation and thermal destruction of plastic (see Figure 8 in [14]). In additions, the elements have the drawback of the ability to operate only at one given wavelength. This problem can be solved by forming quasi-Bessel beams via the use of phase diffraction gratings with a given phase shift distribution, which are also called diffractive axicons [17]. Figure 1b-d presents three types of diffractive axicons. The phase function of a binary axicon (BA) is described by a stepwise profile  
\[\Phi_{BA}(r,\varphi)=\frac{\pi}{2}\text{sign}(\sin(\ell\varphi-\kappa r)), \tag{6}\]  
where  
\[\text{sign}(x)=\left\{ \begin{array}{cc}+1&x>0\\ 0&x=0\\ -1&x<0\end{array} \right., \tag{7}\]  
and \(p=2\pi/\kappa\) is the axicon period. A kinoform axicon (KA), as seen in Figure 1c, differs from the previous one in the fact that within each period, the phase incursion decreases linearly along the radius by the value \(2\pi\)  
\[\Phi_{KA}(r,\varphi)=\ell\varphi-\kappa\rho-2\pi\text{fix}\left(\frac{\ell \varphi-\kappa\rho}{2\pi}\right), \tag{8}\]  
where the function \(\text{fix}(z)\) converts a real number into an integer, discarding the fractional part. Essentially, this axicon is a diffractive analogue of a refractive spiral axicon (SA) (see Equation (5)). Another axicon, which we, following [23], will call the holographic axicon (HA), has a profile that is step-wise in radius, but continuous in azimuth (Figure 1d). The width of the zones of this axicon, unlike the previous ones, is not constant, but is determined by the position of the zeros of the Bessel function [24]. In our notation, this function can be written as follows:  
\[\Phi_{HA}(r,\varphi)=\ell\Big{(}\varphi+\pi+\text{H}(\int_{|\ell|}(\kappa r) )\cdot\pi/|\ell|\Big{)}-2\pi\text{fix}\left(\frac{\ell(\varphi+\pi+\text{H}( \int_{|\ell|}(\kappa r))\cdot\pi/|\ell|)}{2\pi}\right). \tag{9}\]  
This expression is derived from (5), but the spiral plate is divided into zones along the radius. The zones are between the zeroes of a Bessel function of a given order, in which the point of reference for the azimuth angle \(\varphi\), which is an argument of the function \(\Phi(r,\varphi)\), is shifted by \(\Delta\varphi=\pi/|\ell|\) for odd zones using the Heaviside function  
\[H(x)=\frac{\text{sign}(x)+1}{2}=\left\{ \begin{array}{ll}+1&x>0\\ 0.5&x=0\\ 0&x<0\end{array} \right.. \tag{10}\]  
For the sign of the beam topological charge to change, the transparency must be rotated through 180 degrees about the axis lying in the plane of the diffractive element and passing through its center.