  The multipole expansion of a nano-photonic structure's electromagnetic
response is a versatile tool to interpret optical effects in nano-optics, but
it only gives access to the modes that are excited by a specific illumination.
In particular the study of various illuminations requires multiple, costly
numerical simulations. Here we present a formalism we call "generalized
polarizabilities", in which we combine the recently developed exact multipole
decomposition [Alaee et al., Opt. Comms. 407, 17-21 (2018)] with the concept of
a generalized field propagator. After an initial computation step, our approach
allows to instantaneously obtain the exact multipole decomposition for any
illumination. Most importantly, since all possible illuminations are included
in the generalized polarizabilities, our formalism allows to calculate the
total density of multipole modes, regardless of a specific illumination, which
is not possible with the conventional multipole expansion. Finally, our
approach directly provides the optimum illumination field distributions that
maximally couple to specific multipole modes. The formalism will be very useful
for various applications in nano-optics like illumination-field engineering, or
meta-atom design e.g. for Huygens metasurfaces. We provide a numerical open
source implementation compatible with the pyGDM python package.
