# A Simple and Compact Laser Scattering Setup for Characterization of a Pulsed Low-Current Discharge
## 2 Theoretical Considerations
The angle \(\varphi=90^{\circ}\) is chosen to maximize the scattered intensity, whereas the scattering angle \(\theta=90^{\circ}\) is chosen to acquire an incoherent scattering signal. The incident power \(P_{i}\) of the laser beam is distributed across an area \(A\), which defines the scattering volume \(V\). The detection apparatus records the mean scattered power \(\overline{dP_{s}}\) over a spectral frequency range \(d\omega\) observed through solid angle \(\Omega\). According to the above discussion, the following relation can be written for the measured Rayleigh scattered signal by combining Equation (1) with Equations (5) and (6),  
\[M_{RyS}(\lambda^{*})=\sigma_{R}\;n_{h}V\;K\int\delta(\lambda-\lambda_{0})\;S_{F} (\lambda-\lambda^{*})\,d\lambda \tag{7}\]  
or equivalently,  
\[M_{RyS}(\lambda^{*})=\sigma_{R}\;n_{h}V\;K\;S_{F}(\lambda^{*}). \tag{8}\]  
Assuming that the transfer function \(S_{F}(\lambda^{*})\) is normalized to 1, the above equation can be solved for the detection sensitivity \(K\) in the case of \(\lambda^{*}=\lambda_{0}\) and yields  
\[\frac{\overline{dP_{s}}/(d\omega\,\Omega)}{P_{i}/A}=\Sigma\ n_{q}V\,s(\vec{k}, \omega). \tag{1}\]0  
Accordingly, the transfer function of the whole detection apparatus, which is mainly determined by the transfer function of the spectrometer, is given by  
\[\frac{\overline{dP_{s}}/(d\omega\,\Omega)}{P_{i}/A}=\Sigma\ n_{q}V\,s(\vec{k}, \omega). \tag{1}\]1  
With both the detection sensitivity \(K\) and transfer function \(S_{F}\) defined based on the Rayleigh scattering signal, the spectral distribution of the measured Thomson scattering signal \(M_{ThS}(\lambda^{*})\) can be expressed similarly to Equation (7) as  
\[\frac{\overline{dP_{s}}/(d\omega\,\Omega)}{P_{i}/A}=\Sigma\ n_{q}V\,s(\vec{k}, \omega). \tag{1}\]2  
The substitution of Equation (9) for \(K\) yields  
\[\frac{\overline{dP_{s}}/(d\omega\,\Omega)}{P_{i}/A}=\Sigma\ n_{q}V\,s(\vec{k}, \omega). \tag{1}\]3  
If the spectral width of the transfer function \(S_{F}\) is significantly smaller than the spectral width of the Thomson scattered spectrum \(s_{0,ThS}\), then the convolution integral, which represents the spectral distribution of the measured signal \(M_{ThS}\), is accurately reproduced by a Gaussian distribution function. Hence,  
\[\frac{\overline{dP_{s}}/(d\omega\,\Omega)}{P_{i}/A}=\Sigma\ n_{q}V\,s(\vec{k}, \omega). \tag{1}\]4  
Thereafter, the electron temperature \(T_{e}\) can be determined from the characteristic width \(\Delta\lambda_{m,ThS}\) of the measured signal fitted by a Gaussian curve. The electron density \(n_{e}\), on the other hand, can be determined by rearranging the Equation (12) to  
\[\frac{\overline{dP_{s}}/(d\omega\,\Omega)}{P_{i}/A}=\Sigma\ n_{q}V\,s(\vec{k}, \omega). \tag{1}\]5  
The heavy particle temperature, on the other hand, is determined from the intensity ratio of the Rayleigh peaks of the cold gas and plasma measurements via the ideal gas law after the respective measurements have been corrected for stray light beforehand (discussed in detail in Section 3). This is possible because the recorded Rayleigh scattered intensity is directly proportional to the particle density according to the Equation (8). According to the above considerations, the choice of the spectrograph is a key point for an accurate functioning of the proposed method. Its spectral resolution should be significantly smaller than the spectral width of the Thomson scattered signal, but low enough to satisfy the assumption of a Dirac delta function for the Rayleigh signal. These requirements are easily achievable with most of today's instruments. Under these conditions,the setup can be calibrated with the Rayleigh signal. If then the electron number density of the measurement object is high enough so that a sufficient intensity of the Thomson signal can be detected and the stray light, i.e., the light scattered by the immediate vicinity of the measurement volume, is low, a measurement can be performed with constant acquisition settings.