Dynamic polarizabilities and magic wavelengths of Sr$^+$ for focused vortex light

A theory of dynamic polarizability for trapping relevant states of Sr$^+$ is presented here when the ions interact with a focused optical vortex. The coupling between the orbital and spin angular momentum of the optical vortex varies with focusing angle of the beam and is studied in the calculation of the magic wavelengths for $5s_{{1}/{2}}\rightarrow 4d_{{3}/{2}, {5}/{2}}$ transitions of Sr$^+$. The initial state of our interest here is $5s_{{1}/{2}}$ with $m_J = -1/2$ of which is different possible trapping state compare to our recent work on Sr$^+$ [Phys. Rev. A \textbf{97}, 022511 (2018)]. We find variation in magic wavelengths and the corresponding polarizabilities with different combinations of orbital and spin angular momentum of the vortex beam. The variation is very significant when the wavelengths of the beam are in the infrared region of electromagnetic spectrum. The calculated magic wavelengths will help the experimentalists to trap the ion for performing the high precision spectroscopic measurements.