Active Plasma Resonance Spectroscopy (APRS) is a well known diagnostic method, where a radio frequency probe is immersed into a plasma and excites plasma oscillations. The response of the plasma is recorded as frequency dependent spectrum, in which resonance peaks occur. By means of a mathematical model plasma parameters like the electron density or the electron temperature can be determined from the detected resonances. The majority of all APRS probes have in common, that they are immersed into the plasma and perturb the plasma due to the physical presence of the probe. Thus, they are invasive and can at least influence the homogeneity of the plasma. To overcome this problem, the planar Multipole Resonance Probe (pMRP) was invented, which can be integrated into the chamber wall of a plasma reactor. Within this paper, the first analytic model of the pMRP is presented, which is based on a cold plasma description of the electrons. The general admittance of the probe-plasma system is derived by means of functional analytic methods and a complete orthonormal set of basis functions. Explicit spectra for an approximated admittance including a convergence study are shown. The determined resonance frequencies are in good agreement with former simulation results.

Active Plasma Resonance Spectroscopy (APRS) is a well-established plasma diagnostic method: a radio frequency signal is coupled into the plasma via a probe or antenna, excites it to oscillate, and the response is evaluated through a mathematical model. The majority of APRS probes are invasive and perturb the plasma by their physical presence. The planar multipole resonance probe (pMRP) solves this problem: it can be integrated into the chamber wall and minimize the perturbation. Previous work has studied the pMRP in the frame of the Drude model, but it misses important kinetic effects like collision-less damping. In this work, a collision-less kinetic model is developed to further investigate the behavior of the pMRP. This model consists of the Vlasov equation, which is coupled with the Poisson equation under electrostatic approximation. The spectral response of the probe-plasma system is found by calculating the complex admittance. This model covers the kinetic effects and overcomes the limitations of the Drude model.

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Active plasma resonance spectroscopy (APRS) is a process-compatible plasma diagnostic method, which utilizes the natural ability of plasmas to resonate on or near the electron plasma frequency. The multipole resonance probe (MRP) is a particular design of APRS that has a high degree of geometric and electric symmetry. The principle of the MRP can be described on the basis of an idealized geometry that is particularly suited for theoretical investigations. In a pressure regime of a few Pa or lower, kinetic effects become important, which cannot be predicted by the Drude model. Therefore, in this paper, a dynamic model of the interaction of the idealized MRP with a plasma is established. The proposed scheme reveals the kinetic behavior of the plasma that is able to explain the influence of kinetic effects on the resonance structure. Similar to particle-in-cell, the spectral kinetic method iteratively determines the electric field at each particle position, however, without employing any numerical grids. The optimized analytical model ensures the high efficiency of the simulation. Eventually, the presented work is expected to cover the limitation of the Drude model, especially for the determination of the pure collisionless damping caused by kinetic effects. A formula to determine the electron temperature from the half-width [Formula: see text] is proposed.

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