  We study the radiation from charged particles crossing a cholesteric plate in
the shortwave approximation when the wavelength of photons is much smaller than
the pitch of the cholesteric helix whereas the escaping angle of the photon and
the anisotropy of the permittivity tensor can be arbitrary. The radiation of
photons is treated in the framework of quantum electrodynamics with classical
currents. The radiation of the plane-wave photons and the photons with definite
projection of the angular momentum (the twisted photons) produced by charged
particles crossing the cholesteric plate and moving rectilinearly and uniformly
is considered. The explicit expressions for the average number of radiated
photons and their spectra with respect to the energy and the projection of the
angular momentum are obtained in this case. It turns out that in the paraxial
approximation the projection of the orbital angular momentum, $l$, of radiated
twisted photons is related to the harmonic number $n\in \mathbb{Z}$ as
$l=2n+1$, i.e., the given system is a pure source of twisted photons as
expected. It is shown that in the paraxial shortwave regime the main part of
radiated photons is linearly polarized with $l=\pm1$ at the harmonics
$n=\{-1,0\}$. The applicability conditions of the approach developed are
discussed. As the examples, we consider the production of $6.3$ eV twisted
photons from uranium nuclei and the production of X-ray twisted photons from
$120$ MeV electrons.
