  A method for the evaluation of the angular width of an electron beam
generated by a nanoconstriction is proposed and demonstrated. The approach is
based on analysis of a narrow-width electron flow, that quantizes into modes
inside a confining constriction which is described in the adiabatic
approximation, evolving into a freely propagating electronic state after
exiting the constriction. The method that we developed allows us to find the
parameters and the shape of the constriction that are optimal for generation of
extremely narrow electron beams. In the case of a constriction characterized by
a linear widening shape an asymptotically exact solution for the injection
problem is found. That solution verifies semi-quantitative results related to
the angular characteristics of the beam, and it opens the way for determination
of the distribution function of the electrons in the beam. We have found the
relationship between the angular distribution of the electron density in the
beam and the quantum states of the electrons inside the constriction. Such
narrow electron beams may be employed in investigations of electronic systems
and in data manipulations in electronic and spintronic devices.
