  We propose a free-space, inverse design of nanostructure's effective
mode-matching fields via a backward propagation of tightly focused vector beams
to the pupil plane of an aplanatic system of high numerical aperture. First, we
study the nanostructure's eigenmodes without considering any excitation fields
and then extract the modal near fields in the focal plane. Each modal field is
then taken as the desired focal field, the band-limited waves of which are
backward propagated to the pupil plane via a reversal of the Richards--Wolf
vector diffraction formula. The pupil fields can be designed to be genuinely
paraxial by associating the longitudinal electric/magnetic field component with
the radial one on the reference sphere. The inversely designed pupil field in
turn is propagated forwardly into the focal region to generate the designed
focal field, whose distribution over the nanostructure's surface is used to
evaluate the overlap between the designed focal field and the modal fields,
i.e., the modal expansion coefficients. Studies for a silicon nanodisk monomer,
dimer, and tetramer demonstrate the ability of our inverse approach to design
the necessary tightly focused vector field that can effectively and exclusively
match a certain eigenmode of interest. Compared with the forward beam-shaping
method, the inverse design approach tends to yield quantitatively more precise
mode-matching field profiles. This work can have a significant impact on
optical applications that rely on controllable and tunable mode excitation and
light scattering.
