  Photoexcitation of trapped ions by Hermite-Gaussian (HG) modes from guided
beam structures is proposed and investigated theoretically. In particular,
simple analytical expressions for the Rabi frequencies of induced atomic
transitions are derived that depend both on the parameters of HG beams and on
the geometry of an experiment. By using these general expressions, we
investigate the $^{2}S_{1/2} \to \; ^{2}F_{7/2}$ electric octupole (E3)
transition in an Yb$^{+}$ ion, localized in the low--intensity center of the
HG$_{10}$ and HG$_{01}$ beams. We show how the corresponding Rabi frequency can
be enhanced by properly choosing the polarization of incident light and the
orientation of an external magnetic field, which defines the quantization axis
of a target ion. The calculations, performed for experimentally feasible beam
parameters, indicate that the achieved Rabi frequencies can be comparable or
even higher than those observed for the conventional Laguerre-Gaussian (LG)
modes. Since HG-like modes can be relatively straightforwardly generated with
high purity and stability from integrated photonics, our results suggest that
they may form a novel tool for investigating highly-forbidden atomic
transitions.
