# A Simple and Compact Laser Scattering Setup for Characterization of a Pulsed Low-Current Discharge
## 2 Theoretical Considerations
Concluding, for the mean scattered power, it can be written [26; 40]  
\[\frac{\overline{dP_{s}}/(d\omega\,\Omega)}{P_{i}/A}=\Sigma\ n_{q}V\,s(\vec{k}, \omega). \tag{1}\]  
with \(\sigma\) denoting the differential scattering cross section for the scattering mechanism being considered and \(n_{q}\) the density of the scattering particles. This scattering cross section can be understood as the probability that a light quantum of the incident radiation is scattered by a particle of the medium (i.e., the free and bound electrons) into the observation solid angle. The variables \(\overline{dP_{s}}\), \(\sigma\), \(n_{q}\) and \(s(\vec{k},\omega)\) are specific for each scattering mechanism. In case of Thomson scattering, if incoherent scattering is assumed, the scattered signal is a sum of the uncorrelated contributions of each scattering particle and the spectral distribution function \(s_{ThS}(\vec{k},\omega)\) can be described by a Gaussian function [26]. Hence, it can be written as  
\[s_{ThS,incoth}(\vec{k},\lambda,\lambda_{0})=s_{0,ThS}\ \mathrm{e}^{-\left(\frac{ \lambda-\lambda_{0}}{\lambda\lambda_{ThS}}\right)^{2}} \tag{2}\]  
with  
\[s_{0,ThS}=\frac{2\sqrt{\pi}c_{0}}{kv_{e,th}\lambda_{0}^{2}} \tag{3}\]and  
\[\Delta\lambda_{THS}=\frac{k\
u_{e,th}\lambda_{0}^{2}}{2\pi c_{0}}. \tag{4}\]  
where \(\lambda_{0}\) denotes the incident laser wavelength, \(c_{0}\) the speed of light, \(k\) the absolute value of the difference between the wave vector of the scattered radiation \(\vec{k}_{s}\) and the wave vector of incident radiation \(\vec{k}_{i}\) and \(v_{e,th}\) denotes the thermal electron velocity. The spectral distribution function does not in general correspond exactly to the experimentally measured scattering signal. The registered signal at (for instance) the detector, an image intensified charge coupled device (ICCD) camera, depends not only on the spectral sensitivity of the imaging sensor, but also on the sensitivity of the experimental setup and on the spectrometer's transfer function. Hence, for the measured signal \(M(\lambda^{*})\), it can be written  
\[M(\lambda^{*})=K\int S(\lambda)\ S_{F}(\lambda-\lambda^{*})\,d\lambda, \tag{5}\]  
with \(S(\lambda)\) as the scattered light spectrum, \(K\) as the detection sensitivity of the experimental setup and \(S_{F}\) denoting the transfer function of the setup. The detection sensitivity \(K\) takes into account the spectral response of all setup components, i.e., the spectral sensitivity of the ICCD, the lenses, mirrors and spectrometer used. Since the spectral range of the scattering signal is usually smaller than 20 nm [21; 34; 41], the sensitivity \(K\) is assumed to be independent of the wavelength and furthermore to be constant within this range. Both parameters, the detection sensitivity \(K\) and transfer function \(S_{F}\), can be determined by measuring light scattered by a reference object (source) with known properties. Rayleigh scattering of pure nitrogen at room temperature and atmospheric pressure can be used for this purpose. Analogous to Equations (2)-(4), the main parameters determining the scattering spectrum are, in the case of the Rayleigh scattering mechanism, the temperature of heavy particles \(T_{h}\) and the heavy particle number density \(n_{h}\), which for the above conditions can be obtained from the ideal gas law. Since thermal equilibrium can be assumed for room temperature under atmospheric pressure conditions, the spectral line width of the Rayleigh scattered signal is in the range of picometers [42]. However, the resolution of the used spectrometer usually cannot resolve linewidths in the picometer range (see Section 3); hence, for the spectral distribution function of Rayleigh signal, a Dirac delta function can be assumed  
\[s_{RJS}(\vec{k},\lambda,\lambda_{0})=s_{0,RJS}\ \mathrm{e}^{-\left(\frac{ \lambda-\lambda_{0}}{\lambda_{0}RJS}\right)^{2}}\cong\delta(\lambda-\lambda_{ 0}). \tag{6}\]  
Figure 1: Scattering geometry used in this work.