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A theory of dynamic polarizability for trapping relevant states of Sr$^+$ ispresented here when the ions interact with a focused optical vortex. Thecoupling between the orbital and spin angular momentum of the optical vortexvaries with focusing angle of the beam and is studied in the calculation of themagic 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 ourrecent work on Sr$^+$ [Phys. Rev. A \textbf{97}, 022511 (2018)]. We findvariation in magic wavelengths and the corresponding polarizabilities withdifferent combinations of orbital and spin angular momentum of the vortex beam.The variation is very significant when the wavelengths of the beam are in theinfrared region of electromagnetic spectrum. The calculated magic wavelengthswill help the experimentalists to trap the ion for performing the highprecision spectroscopic measurements.

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