Angle-dependent magic wavelengths for the $4s_{1/2}\to3d_{5/2,3/2}$ transitions of Ca$^{+}$ ions

The dynamic polarizabilities of the atomic states with angular momentum $j> \frac12$ are sensitive to the angle between the quantization axis $\hat{e}_z$ and the polarization vector $\hat{\mathbf{\epsilon}}$ owing to the contribution of anisotropic tensor polarizabilities. The magic wavelength, at which the differential Stark shift of an atomic transition nullifies, depends on this angle. We identified the magic wavelengths for the $4s_{\frac12}\to3d_{\frac32,\frac52}$ transitions of Ca$^{+}$ ions at different angles between $\hat{e}_z$ and $\hat{\mathbf{\epsilon}}$ in the case of linearly polarized light. We found that the magic wavelengths near 395.79 nm, which lie between the $4s_{\frac12}\to4p_{\frac12}$ and $4s_{\frac12}\to 4p_{\frac32}$ transition wavelengths, remain unsensitive to the angle, while the magic wavelengths, which are longer than the $3d_{\frac52}\to 4p_{\frac32}$ resonant transition wavelength (854.21 nm), are very sensitive to the angle.