# Real-Time Tomography of Gas-Jets with a Wollaston Interferometer
## 1 Introduction  
The aim of future advanced accelerator concepts is to use accelerating structures that can sustain higher electric field strengths to downsize the structure compared to conventional microwave cavities. Laser Wake Field Acceleration (LWFA) is one such promising technology, which is being investigated by several research groups around the world. The idea of LWFA is to excite a wake field in an underdense plasma by a short energetic laser pulse, the so-called driver. The plasma can be generated by ionizing a gas jet using the driver pulse itself. In this plasma, the driver pulse excites electron density oscillations, which are co-propagating with the driver almost at the speed of light forming a plasma wave, which is moving with a relativistic phase velocity. Due to the local charge separation (the ions remain stationary on the corresponding time scales), longitudinal electric fields with amplitudes in excess of \(100\,\mathrm{GV/m}\) are generated, orders of magnitude higher than in conventional accelerator structures. Electrons can be accelerated in this moving field structure when they are trapped in the wave with the right phase, i.e., at the correct position and with the correct velocity. An inherently synchronized method for trapping electrons is down-ramp injection, which can be induced when the plasma wave propagates across a sudden drop in density, a so-called down-ramp [1; 2]. Here, the plasma wave, which has been generated in a first, high-density region, enters a second region with lower density. The electrons in the high-density region oscillate faster since the plasma frequency \(\omega_{p}=\sqrt{n_{e}e^{2}/(\varepsilon_{0}m_{e})}\) dependson the electron density \(n_{e}\); \(m_{e}\) and \(e\) are the mass and charge of the electron, respectively. Therefore, the wavelength of the plasma wave, \(\lambda_{p}\approx c/\omega_{p}\), is shorter than in the subsequent low-density region. When this density transition occurs over a distance of the order or shorter than the plasma wavelength, some of the electrons forming a density peak in the plasma wave in the high-density region will then no longer be associated with the density peak of the low-density plasma wave, but they will find themselves in the accelerating phase of the wave. It has been shown that a density change by a factor of two to three can be achieved by a shock front in supersonic gas jets [3]. At such a sharp transition, many electrons are loaded into the same phase-space volume of the plasma wave, which will result in a narrow energy spread of the accelerated electrons. The parameters of the generated electron beam (energy spectrum, charge and duration) critically depend on the plasma wave and its evolution, which in turn is very sensitive to the plasma density distribution. With these facts in mind, a precise and fast real-time density measurement is required for controlling the down-ramp process [4] and (when shot-to-shot-fluctuations in the gas jet cannot be sufficiently suppressed) for increasing the shot-to-shot stability of the generated electron beam. The view of high quality beams from LWFA, a fast and accurate characterization of the gas jet is of great importance. A comprehensive overview on density measurements of gas jets can be found in [5]. While cylindrically-symmetric gas jet distributions can easily be characterized from a single interferometric measurement taken in one direction, e.g., using a Michelson-type interferometer or a Nomarski-type interferometer using a Wollaston prism, tomographic methods are necessary when non-symmetric distributions, e.g., caused by a density jump as mentioned above, are required. The latter methods, which include interferometric measurements of the gas density taken along many directions, are more elaborate and sometimes time-consuming to analyze, when high accuracy and spatial resolution are needed [6; 7; 8]. Therefore, analysis methods that can be applied in quasi-real time are highly advantageous.