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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 contributionof anisotropic tensor polarizabilities. The magic wavelength, at which thedifferential Stark shift of an atomic transition nullifies, depends on thisangle. We identified the magic wavelengths for the$4s_{\frac12}\to3d_{\frac32,\frac52}$ transitions of Ca$^{+}$ ions at differentangles between $\hat{e}_z$ and $\hat{\mathbf{\epsilon}}$ in the case oflinearly polarized light. We found that the magic wavelengths near 395.79 nm,which lie between the $4s_{\frac12}\to4p_{\frac12}$ and $4s_{\frac12}\to4p_{\frac32}$ transition wavelengths, remain unsensitive to the angle, whilethe magic wavelengths, which are longer than the $3d_{\frac52}\to 4p_{\frac32}$resonant transition wavelength (854.21 nm), are very sensitive to the angle.

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