  The general formula for probability to record a twisted photon produced by a
charged particle moving in an inhomogeneous isotropic dispersive medium is
derived. The explicit formulas for probability to record a twisted photon are
obtained for the radiation of a charged particle traversing a dielectric plate
or an ideally conducting foil. It is shown that, in the case when the charged
particle moves along the detector axis, all the radiated twisted photons
possess a zero projection of the total angular momentum and the probability of
their radiation is independent of the photon helicity. The radiation produced
by helically microbunched beams of charged particles is also considered. The
fulfillment of the strong addition rule for the projection of the total angular
momentum of radiated twisted photons is demonstrated. Thus the helical beams
allow one to generate coherent transition and Vavilov-Cherenkov radiations with
large projections of the total angular momentum. The radiation produced by
charged particles in a helical medium is studied. Typical examples of such a
medium are metallic spirals and cholesteric liquid crystals. It is shown that
the radiation of a charged particle moving along the helical axis of such a
medium is a pure source of twisted photons.
