  The term "Active Plasma Resonance Spectroscopy" refers to a class of
diagnostic methods which employ the ability of plasmas to resonate on or near
the plasma frequency. The basic idea dates back to the early days of discharge
physics: An signal in the GHz range is coupled to the plasma via an electrical
probe; the spectral response is recorded, and then evaluated with a
mathematical model to obtain information on the electron density and other
plasma parameters. In recent years, the concept has found renewed interest as a
basis of industry compatible plasma diagnostics. This paper analyzes the
diagnostics technique in terms of a general description based on functional
analytic (or Hilbert Space) methods which hold for arbitrary probe geometries.
It is shown that the response function of the plasma-probe system can be
expressed as a matrix element of the resolvent of an appropriately defined
dynamical operator. A specialization of the formalism for a symmetric probe
desing is given, as well as an interpreation in terms of a lumped circuit model
consisting of series resonators. We present ideas for an optimized probe design
based on geometric and electrical symmetry.
