  Modal expansion is an attractive technique for solving electromagnetic
scattering problems. With the one set of resonator modes, calculated once and
for all, any configuration of near-field or far-field sources can be obtained
almost instantaneously. Traditionally applied to closed systems, a simple and
rigorous generalization of modal expansion to open systems using
eigenpermittivity states is also available. These open modes are suitable for
typical nanophotonic systems, for example. However, the numerical generation of
modes is usually the most difficult and time-consuming step of modal expansion
techniques. Here, we demonstrate efficient and reliable mode generation,
expanding the target modes into the modes of a simpler open system that are
known. Such a re-expansion technique is implemented for resonators with
non-uniform permittivity profiles, demonstrating its rapid convergence. Key to
the method's success is the inclusion of a set of longitudinal basis modes.
