Analytical approximations for the higher energy structure in strong field ionization with inhomogeneous electric fields

Recently, the emergence of a higher energy structure (HES) due to a spatial inhomogeneity in the laser electric field, as is typically found close to a nano tip, was reported in Phys.~Rev. Letter {\bf 119}, 053204 (2017). For practical applications, such as the characterization of near-fields or the creation of localized sources of monoenergetic electron beams with tunable energies, further insight into the nature of this higher energy structure is needed. Here, we give a closed form analytical approximation to describe the movement of the electron in the inhomogeneous electric field. In particular, we derive a simple scaling law for the location of the HES peak and give a scheme to analytically tune the width of the peak, both of which will prove useful in optimizing the nanostructure size or geometry for creating the HES in experimental settings.