## 3 Analytical Expressions for Vortex Beams Formed with Binary Axicons
### Transformation of Bessel Beams Formed by Binary Axicons into Perfect Vortex Beams Using a Lens  
Below is the expression for a beam that has passed through a thin lens with a focal distance \(f\)[16; 32]:  
\[E(z,r,\psi)=-\frac{ike^{ikz}}{2\pi z}e^{\frac{ikz}{2}}\int\limits_{0}^{a_{L}} \int\limits_{0}^{2\pi}\rho d\rho d\theta E(z_{L},\rho,\theta)\,e^{\frac{ik\rho ^{2}}{2}(\frac{1}{z}-\frac{1}{f})}e^{-\frac{ik\rho}{z}\cos(\theta-\psi)}, \tag{16}\]  
where \(E(z_{L},\rho,\theta)\) is the amplitude of the wavefield incident on the lens and \(a_{L}\) is the lens radius. Placing a thin lens at the distance \(z^{\prime}=z_{L}\) from the axicon and taking into account the fact that \(k-\kappa^{2}/2k\approx k_{z}\), after some transformations, we get the following expression at the distance \(z\) behind the lens:  
\[ \begin{array}{c}E(z,r,\psi)=\frac{(-i)^{2l+1}4k^{2}}{mzz_{L}}e^{ik(z+z_{L}+ \frac{r^{2}}{2z})}\times\\ \times\int_{0}^{a_{L}}\rho d\rho\,J_{\ell}\left(\frac{k\rho}{z}\right)e^{\frac {ik\rho}{2}(\frac{1}{z_{L}}+\frac{1}{z}-\frac{1}{f})}\left\{\int_{0}^{a}J_{ \ell}\left(\frac{k}{z_{L}}r^{\prime}\rho\right)\sin(\kappa r^{\prime}-\ell \psi)\,e^{ikr^{\prime 2}/2z_{L}}r^{\prime}dr^{\prime}\right\}\end{array}  \tag{17}\]  
Here, \(r^{\prime}\) and \(\psi\) are the polar coordinates in the plane of the lens and \(a_{L}\) is the radius of the lens. Since  
\[F_{1}(r^{\prime},\rho,r)=\int\limits_{0}^{2\pi}\sin[(\kappa r^{\prime}-\ell \theta)]\,e^{-\frac{ik\rho}{z}\cos(\theta-\psi)}d\theta=2\pi(-i)^{\ell}I_{ \ell}\left(\frac{k\rho r}{z}\right)\sin\kappa r^{\prime}-\ell\psi), \tag{18}\]  
we have  
\[ \begin{array}{c}E(r,\psi,z)=\\ =-\frac{(-i)^{2l-1}4k^{2}e^{ik(z+z_{L}+\frac{r^{2}/2z}{2z})}}{\pi z_{L}}\int_ {0}^{a_{L}}\left\{\int\limits_{0}^{a_{L}}J_{\ell}\left(\frac{k\rho r}{z} \right)I_{\ell}\left(\frac{k\rho^{2}}{z}I_{\ell}^{\prime}\rho\right)e^{i\frac {ik\rho^{2}}{2}(\frac{1}{z_{L}}+\frac{1}{z}-\frac{1}{f})}\rho d\rho\right\} \sin(\kappa r^{\prime}-\ell\psi)e^{ikr^{\prime 2}/2Z_{L}}r^{\prime}dr^{\prime}\end{array}  \tag{19}\]  
Making a trivial substitution \(\rho^{2}=x\) in the integral with respect to \(\rho\), introducing the variable \(Z_{Lf}=\left(1/z_{L}+1/z-1/f\right)^{-1}\), and expanding the upper limit of integration \(a_{L}\) to infinity, we get the following integral, the value of which is given in [33] (see Appendix A):  
\[ \begin{array}{c}F_{2}(r,r^{\prime})=\int_{0}^{\infty}\left[J_{\ell}\left( \frac{k}{z_{L}}r^{\prime}\rho\right)I_{\ell}\left(\frac{k\rho r}{z}\right)e^{ i\frac{k\rho^{2}}{2Z_{Lf}}}\right]\rho d\rho=\frac{1}{2}\int_{0}^{\infty} \left[e^{ix(k/2Z_{Lf})}J_{\ell}\left(\sqrt{x}\frac{k\rho}{z}\right)J_{\ell} \left(\sqrt{x}\frac{k\rho^{\prime}}{z_{L}}\right)\right]dx=\\ \qquad\qquad\qquad\qquad\qquad\qquad=i\frac{Z_{Lf}}{k}I_{\ell}\left(iZ_{Lf} \frac{k\rho^{\prime}}{zz_{L}}\right)\exp\left\{-i\frac{Z_{Lf}k}{2}\left[\left( \frac{r^{\prime}}{z_{L}}\right)^{2}+\left(\frac{r}{z}\right)^{2}\right] \right\}=\\ \qquad\qquad\qquad\qquad\qquad\qquad\qquad=i^{\ell+1}\frac{Z_{Lf}}{k}I_{\ell }\left(Z_{Lf}\frac{k\rho^{\prime}}{zz_{L}}\right)\exp\left\{-i\frac{Z_{Lf}k}{2 }\left[\left(\frac{r^{\prime}}{z_{L}}\right)^{2}+\left(\frac{r}{z}\right)^{2} \right]\right\}.\end{array}  \tag{20}\]Substituting the resulting expression (20) into (19), we obtain the final expression for the field behind the lens:  
\[E(z,r,\psi)=\frac{(-i)^{\ell}4 kZ_{Lf}}{\pi zz_{L}}e^{ik(z+z_{L}+\frac{r^{2}}{2z}- \frac{Z_{Lf}r^{2}}{2z^{2}})}\times\int_{0}^{a}J_{\ell}\left(Z_{Lf}\frac{krr^{ \prime}}{zZ_{L}}\right)e^{-ik\frac{Z_{Lf}r^{\prime\prime 2}}{2z_{L}^{2}}}\sin( \kappa r^{\prime}-\ell\psi)\,e^{ikr^{\prime 2}/2z_{L}}r^{\prime}dr^{\prime} \tag{21}\]  
For the distance \(z=f\), which is the main interest to us, we get the field amplitude  
\[E(f,r,\psi)=\frac{(-i)^{\ell}4k}{\pi f}e^{\frac{ikr^{2}}{2f}(1-\frac{z_{L}}{f} )}\times\int_{0}^{a}J_{\ell}\left(\frac{krr^{\prime}}{f}\right)\,\sin(\kappa r ^{\prime}-\ell\psi)\,r^{\prime}dr^{\prime}, \tag{22}\]  
where we omitted the insignificant constant phase factors in the exponent.