# One-Body Capillary Plasma Source for Plasma Accelerator Research at e-LABs
## 2 Development of the One-Body Capillary Plasma Source
### Characterization  
The effective utilization of the capillary plasma source requires a comprehensive understanding of the gas density distribution within the capillary. To characterize the capillary density profile, we employed Mach-Zehnder interferometry and computational fluid dynamics (CFD) simulations with a 1 mm diameter capillary of a 7 mm length. The experimental setup for the interferometry is represented in Figure 2a. The capillary system was positioned in a vacuum chamber operated at \(1.2~{}\times~{}10^{-4}\) Torr. The interference images were acquired using a continuous wave He-Ne laser (632 nm wavelength) and a camera (Marlin-F-131B) capture rate of 67.27 frames per second. The laser was separated into two beams (probe and reference) by the first beam splitter. The laser beam was expanded to a size larger than the sapphire block, enabling the acquisition of a reference image of its side. The signal image was obtained via the probe beam passing through the capillary gas cell, while the reference image was obtained by the probe beam bypassing the capillary, effectively reducing the noise induced by the vacuum pump system imprinted on the reference image. The second beam splitter recombined the two beams, and the mirrors positioned after the splitter reflected both the signal and reference images of the probe beam. This setup is designed to yield both a magnified signal and reference images separated spatially in a single image, as is depicted in Figure 2b. Figure 1: (**a**) Designed and (**b**) manufactured one-body capillary gas cell made of a sapphire. For the gas delivery, two one-touch fittings are directly connected to the sapphire block. Electrodes are attached at each end of the capillary and the tapped holes at the bottom connect the holder. The gas pressure inside the capillary cell was determined by analyzing the phase difference between two signal images obtained before and after the gas injection. A reference image, shown in Figure 2b, was used to eliminate the noise data, which can be induced by vibration. The resulting phase shift was achieved using a fast Fourier transform (FFT) method. The relation between the gas pressure (\(P\)) and phase shift is expressed as \(P=(2 RT\Delta\phi)/(3 Ak_{0}L)\), where \(\Delta\phi\) is the total phase shift accumulated along the gas density, \(A\) is the molar refractivity, \(R\) is the gas constant, \(T\) is the temperature in Kelvin, \(k_{0}\) is the wavenumber of the laser, and \(L\) is the gas length [28]. In the measurement, nitrogen gas was utilized with a molar refractivity of \(4.459\) cm\({}^{3}\)/mol [29], and the pressure was varied from 200 to 500 mbar while maintaining a constant gas opening time of 300 ms. In the longitudinal interferometry, a problem arises in specifying the gas length, \(L\). Since the pressure is dependent on the longitudinal position, \(x\), the total phase shift is a function of both pressure and the length. \[L_{eff}=\int_{-\infty}^{\infty}P(x)dx/P_{\text{max}}. \tag{1}\]  
where \(L_{eff}\) is the effective length and \(P_{\text{max}}\) is the maximum pressure. Thus, the position-dependent pressure distribution is required to determine the total phase. To achieve the phase, a three-dimensional (3D) CFD simulation was conducted using ANSYS FLUENT software (version 2019-R1). The simulation conditions involved continuous gas injection at a constant pressure from the inlets and outlets (the exit of capillary) in a vacuum state. Figure 3a shows the CFD simulation results of the obtained gas density distribution. In this example, the capillary length was 7 mm and the effective length was calculated to be 5.69 mm using Equation (1). Using these values, the gas density was calculated as a function of time (Figure 3b). The maximum gas density was reached after about 50 ms and was maintained for the duration of the gas opening time (300 ms). These results suggest that our capillary cell provides a stable gas distribution. Figure 2: (**a**) Experimental setup for the gas density measurement using a Mach–Zehnder interferometer.