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The formulation of the interaction of matter with singular light fields needsspecial care. In a recent article [Phys.~Rev.~A {\bf 91}, 033808 (2015)] wehave shown that the Hamiltonian describing the interaction of a twisted lightbeam having parallel orbital and spin angular momenta with a small objectlocated close to the phase singularity can be expressed only in terms of theelectric field of the beam. Here, we complement our studies by providing aninteraction Hamiltonian for beams having antiparallel orbital and spin angularmomenta. Such beams may exhibit unusually strong magnetic effects. We furtherextend our formulation to radially and azimuthally polarized beams. Theadvantages of our formulation are that for all beams the Hamiltonian is writtensolely in terms of the electric and magnetic fields of the beam and as such itis manifestly gauge-invariant. Furthermore it is intuitive by resembling thewell-known expressions in the dipole-electric and dipole-magnetic momentapproximations.

Formulation of the twisted-light--matter interaction at the phase singularity: beams with strong magnetic fields