  Spatially structured light fields applied to semiconductor quantum dots yield
fundamentally different absorption spectra than homogeneous beams. In this
paper, we theoretically discuss the resulting spectra for different light beams
using a cylindrical multipole expansion. For the description of the quantum
dots we employ a model based on the effective mass approximation including
Coulomb and valence band mixing. The combination of a single spatially
structured light beam and state mixing allows all exciton states in the quantum
dot to become optically addressable. Furthermore, we demonstrate that the beams
can be tailored such that single states are selectively excited, without the
need of spectral separation. Using this selectivity, we propose a method to
measure the exciton wave function of the quantum dot eigenstate. The
measurement goes beyond electron density measurements by revealing the spatial
phase information of the exciton wave function. Thereby polarization sensitive
measurements are generalized by including the infinitely large spatial degree
of freedom.
