Active plasma resonance spectroscopy: A functional analytic description

TL;DR

This paper provides a mathematical framework using functional analysis to describe active plasma resonance spectroscopy, enabling better understanding and design of plasma diagnostic probes across various geometries.

Contribution

It introduces a general Hilbert space formalism for plasma-probe systems, extending the analysis to arbitrary probe geometries and offering insights for optimized probe design.

Findings

Response function expressed as a matrix element of the resolvent

Specialized formalism for symmetric probe design

Circuit model interpretation with series resonators

Abstract

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.