  Theoretical approaches to strong field phenomena driven by plasmonic fields
are based on the length gauge formulation of the laser-matter coupling. From
the theoretical viewpoint it is known there exists no preferable gauge and
consequently the predictions and outcomes should be independent of this choice.
The use of the length gauge is mainly due to the fact that the quantity
obtained from finite elements simulations of plasmonic fields is the plasmonic
enhanced laser electric field rather than the laser vector potential. In this
paper we develop, from first principles, the velocity gauge formulation of the
problem and we apply it to the high-order harmonic generation (HHG) in atoms. A
comparison to the results obtained with the length gauge is made. It is
analytically and numerically demonstrated that both gauges give equivalent
descriptions of the emitted HHG spectra resulting from the interaction of a
spatially inhomogeneous field and the single active electron (SAE) model of the
helium atom. We discuss, however, advantages and disadvantages of using
different gauges in terms of numerical efficiency.
