  We have used high resolution transmission spectroscopy to study the exciton
level spectrum in Cu$_2$O subject to a longitudinal external electric field,
i.e., in the geometry where the transmitted light is propagating along the
field direction. Different experimental configurations given by the field
orientation relative to the crystal and the light polarization have been
explored. We focus on the range of small principal quantum numbers $n \leq 7$.
The number of exciton states belonging to a particular principal quantum number
increases with $n$, leading to an enhanced complexity of the spectra. Still, in
particular for $n = 3 \ldots 5$ a spectral separation of the different lines is
feasible and identification as well as assignment of the dominant state
character are possible. We find a strong dependence of the spectra on the
chosen light propagation direction and polarization configuration, reflecting
the inadequacy of the hydrogen model for describing the excitons. With
increasing the field excitonic states with different parity become mixed,
leading to optical activation of states that are dark in zero field. As
compared with atoms, due to the reduced Rydberg energy states with different
$n$ can be brought into resonance in the accessible electric field strength
range. When this occurs, we observe mostly crossing of levels within the
experimental accuracy showing that the electron and hole motion remains
regular. The observed features are well described by detailed calculations
accounting for the spin-orbit coupling, the cubic anisotropy effects, and the
symmetry-imposed optical selection rules.
