# Strongly Focused Circularly Polarized Optical Vortices Regulated by a Fractal Conical Lens
## 2 Theory Model  
Fractal conical lens (FCL) is known as a more general diffractive optical device than conical lens (CL), which has a rotational symmetric structure and special radial phase envelope designed following a kind of Cantor set (CS) of a certain stage \(S\)[22; 23]. In order to visualize the structure of a FCL, Figure 1a plots the schematic of a one-dimensional (1-D) fractal structure of a FCL following the triadic CS, and Figure 1b presents the two-dimensional (2-D) fractal structure of a FCL (\(S=3\)) based on the triadic CS. Here, it is notable that a CS other than triadic could also be formed with a similar process. According to its special fractal structure, the Cantor function \(F_{S}(x)\) could be defined by [22; 23]  
\[F_{S}(x)=\left\{ \begin{array}{ll}\frac{l}{2^{S}}&\text{if $p_{S,l}\leq x\leq q _{S,l}$}\\ \frac{2^{S}}{p_{S+1}-q_{S,l}}+\frac{l}{2^{S}}&\text{if $q_{S,l}\leq x\leq p_{S,l+1}$} \end{array} \right., \tag{1}\]  
where \(S\) and \(l\), respectively, represent the Cantor function's stage and the number of disjoint gaps intervals [\(p_{S,l}\), \(q_{S,l}\)], with \(S=0\), \(1\), \(2\),... and \(l=1\),..., \(2^{S-1}\). \(q\) and \(p\), respectively, stand for the start and end points for each subsection of Cantor function. As illuminated by monochromatic light waves, the FCL's transmittance function could be given by  
\[T(\theta)=\exp\left[-\text{i}2^{S+1}\pi F_{S}\left(\frac{\sin\theta}{\sin \alpha}\right)\right], \tag{2}\]  
where \(\alpha=\sin^{-1}(NA/n_{1})\) represents the maximum value of convergence angle \(\theta\) related to NA and \(n_{1}\). Here, \(n_{1}\) denotes the refractive index in the image space. In this paper, tight focusing characteristics of a circularly polarized optical vortex through a high NA optical system in cases with and without a FCL are analytically examined and numerically  
Figure 1: (**a**) Schematic of a 1-D fractal structure of a FCL following the triadic CS, starting from the stage \(S=0\) to \(S=3\); (**b**) schematic of 2-D fractal structure of a FCL (\(S=3\)) following the triadic CS. illustrated. Following the generalized vector Debye integral [22], for circularly polarized optical vortex incidence, the electric field in the focal vicinity has the form [25]  
\[ \begin{array}{ll}E(r,\varphi,z)=&\left[\begin{array}{c}E_{x}\\ E_{y}\\ E_{z}\end{array}\right]=-\frac{ikf}{2\pi}\int\limits_{0}^{\alpha}\int\limits_{0}^ {2\pi}A(\theta)T(\theta)\exp(im\phi)\sin\theta\sqrt{\cos\theta}\\ &\times\exp[ik(z\cos\theta+r\sin\theta\cos(\phi-\varphi))]\\ &\times\left[\begin{array}{c}(\cos^{2}\phi\cos\theta+\sin^{2}\phi)\pm i\cos \phi\sin\phi(\cos\theta-1)\\ \cos\phi\sin\phi(\cos\theta-1)\pm i(\cos^{2}\phi+\sin^{2}\phi\cos\theta)\\ \sin\theta\exp(\pm i\phi)\end{array}\right]\\ &\times d\phi d\theta.\end{array}  \tag{3}\]  
where \(A(\theta)\) denotes the apodization function. \(T(\theta)\) represents the transmittance function of a FCL. After tedious integral calculations, the \(x\)-, \(y\)-, and \(z\)-polarized components of an electric field in the focal vicinity could be obtained, respectively, as  
\[ \begin{array}{ll}E_{\pm,x}(r,\varphi,z)=&-\frac{ikf}{2}\int\limits_{0}^{ \alpha}A(\theta)T(\theta)\sin\theta\sqrt{\cos\theta}\exp(ikz\cos\theta)\\ &\times[(1+\cos\theta)^{jm}J_{m}(kr\sin\theta)\exp(im\varphi)\\ &+(\cos\theta-1)^{jm\pm 2})J_{m\pm 2}(kr\sin\theta)\\ &\times\exp[i(m\pm 2)\varphi]]d\varphi,\\ \\ E_{\pm,y}(r,\varphi,z)=&-\frac{ikf}{2}\int\limits_{0}^{\alpha}A(\theta)T( \theta)\sin\theta\sqrt{\cos\theta}\exp(ikz\cos\theta)\\ &\times[i\times i\times(1+\cos\theta)^{jm}J_{m}(kr\sin\theta)\exp(im\varphi)\\ &\pm i\cdot(\cos\theta-1)^{jm\pm 2})J_{m\pm 2}(kr\sin\theta)\\ &\times\exp[i(m\pm 2)\varphi]]d\varphi,\\ \\ E_{\pm,z}(r,\varphi,z)=&-ikf\int\limits_{0}^{\alpha}A(\theta)T(\theta)\sin^{2 }\theta\sqrt{\cos\theta}\exp(ikz\cos\theta)\\ &\times i^{m\pm 1}J_{m\pm 1}(kr\sin\theta)\exp[i(m\pm 1)\varphi]d\varphi, \\ \end{array}