# Probing Collisional Plasmas with MCRS: Opportunities and Challenges
## 2 Theory  
### Resonant Behavior of a Standing Microwave  
A standing wave--also referred to as a stationary wave--is a wave of which the nodes and antinodes do not move in space over time. Examples of this phenomenon can be found in, e.g., the oscillation of a drum's membrane and that of a guitar's string.  
The Microwave Cavity Resonance Spectroscopy method is based on the physical principle that the resonant behavior of an electromagnetic standing wave in a hollow space--the cavity--surrounded by a metal structure depends on the permittivity \(\varepsilon\) and permeability \(\upmu\) of the cavity volume. The computed spatial profile of the electric field amplitude \(|E|\) of the fundamental resonant mode (i.e., TM\({}_{010}\)) in a cylindrical cavity is shown in Figure 1.  
The standing wave in a cavity is typically excited by microwaves via an antenna that protrudes only slightly into this cavity. A typical frequency response of a cavity (red marks) is shown in Figure 2. From a response like this, the resonant behavior, i.e., the spectral position and the width of the resonance peak, is determined by optimized fitting algorithms (blue line).  
The spectral position of a resonance peak does not only depend on the material properties \(\varepsilon\) and \(\upmu\) but also on the dimensions of the cavity and the excited resonant mode. Typical frequencies used for MCRS as a plasma diagnostic are 1-10 GHz [7; 9; 31]. The cavity is only able to absorb microwaves when the applied frequency matches the resonance frequency \(f_{\text{res}}\) (\(f=\omega/_{2\pi}\)) of a probed resonant mode. This frequency can be found by locating the spectral position where the least power is reflected.  
The second property of the resonant behavior is the width of a resonance peak, which is proportional to the losses in the cavity. The losses and width of a specific mode are indicated by the quality factor \(Q\) (or in short '\(Q\)') [32],  
\[Q\equiv\frac{f_{\text{res}}}{\Gamma}, \tag{1}\]  
Figure 1: The amplitude of the electric field \(|E|\) of the fundamental resonant mode (TM\({}_{010}\)) in a cylindrical resonant cavity.  
where \(\Gamma\) is the Full-Width-at-Half-Maximum (FWHM) of the peak (the frequency and power axes need to be in linear scale). When the losses in a cavity are high, the resonant mode is able to respond fast to changes inside the cavity volume. The characteristic time \(\tau\) of a resonant mode is given by:  
\[\tau=\frac{Q}{\pi f_{\text{res}}}. \tag{2}\]  
This 1/e response time is used to indicate the time it takes for the resonant field to adapt 63% to a new situation. For this reason, it takes a few \(\tau\) for the field to 'fully' adapt to, e.g., a sudden presence of plasma. Typical values in experiments for \(\tau\) vary from \(\sim\)0.01 to 1 \(\upmu\)s.  
The temporal resolution of the diagnostic technique is fundamentally limited by the fact that forming a standing wave requires a number of passes of the electromagnetic waves in the cavity. The quality factor of a resonant mode is a good indication for this number of passes. The technological limitations related to the temporal resolution are the rise and fall times of the microwave power detector and the sampling frequency of the microwave power detector's output.