# Real-Time Tomography of Gas-Jets with a Wollaston Interferometer
## 2 The Single Beam Wollaston Interferometer Setup
### Estimation of Phase Shift Due to a Gas Jet Density Distribution  
A local density gradient imposes a varying refractive index \(\eta\), resulting in a phase shift \(\Delta\phi\) of rays passing through that particular region in comparison to rays that would have propagated through a vacuum. Therefore, the undisturbed fringe spacing is locally shifted by \(\Delta S\). The relationship between the atomic density \(\rho\) and refractive index \(\eta\) of a gas is given by the Lorentz-Lorenz equation [12]:  
\[\rho=\frac{\eta^{2}-1}{\eta^{2}+2}\frac{N_{A}}{A}\approx(\eta-1)\frac{2}{3} \frac{N_{A}}{A}, \tag{2}\]  
where \(N_{A}\) is Avogadro's number and \(A\) is the molar refractivity, e.g., \(A_{Ar}\approx 4.20\times 10^{-6}\) m\({}^{3}\)/mol. The approximation is valid for \(\eta\approx 1\), which is fulfilled for a gas with densities \(n<10^{19}\)/ cm\({}^{3}\). Inverting Equation (2) yields the dependency of the refractive index on the density. The phase shift between interfering rays is given by \(\Delta\phi=\phi(y)-\phi(y-d)\), where \(\phi\) is the phase imposed by the neutral gas atoms and \(y\) is the coordinate along the direction perpendicular to the interference fringes. Assuming the linear approximation (Equation (2)), a homogeneous density within a gas jet of diameter \(l\) yields:  
\[\phi\approx l(\eta-1)\frac{2\pi}{\lambda}=l\frac{3\pi}{\lambda}\frac{A}{N_{A}}\rho. \tag{3}\]  
This expression yields an upper limit for the phase shift, since the gas jet has a circular throat and, hence, produces a cylindrically-symmetric gas flow, i.e., the path length of a ray propagating through the gas is at most \(l\). As described above, the fringe spacing is affected by the phase shift in the FFW setup; the fractional fringe distance shift is given by:  
\[\frac{\Delta S}{S}=\frac{\Delta\phi}{2\pi}. \tag{4}\]  
For comparison purposes, and because higher pressures than 15... 20 bar are possible, the gas density using the solenoid valve (Parker Miniature High-Speed Valve) will be estimated. The pressure of the gas before leaving the nozzle, the backing pressure \(P_{b}\), as well as the nozzle design play crucial roles with respect to the gas density distribution in the jet. The gas jet used in the present experiment allows for backing pressures up to 80 bar and has a conical nozzle with a half opening angle \(\theta=45^{\circ}\) (see Figure 2). As described by [PERSON] [13], the on-axis particle density at a height \(x\) of a conical gas jet is approximately given by:  
\[\frac{\rho}{\rho_{0}}\approx 0.15\left(\frac{0.74\,d}{x\,\tan\theta}\right)^{2},\]  
where \(x\) is the distance to the throat of the nozzle (see Figure 2), \(\rho_{0}\) is the atomic number density of the gas before leaving the nozzle and \(d\) is the throat diameter of the valve. The gas jet used in this study has a throat diameter \(d=500\)\(\upmu\)m and \(L=250\)\(\upmu\)m. Using the ideal gas law, which is a good approximation for the monoatomic gas argon at room temperature, \(\rho_{0}\) is determined by the backing pressure \(P_{b}\), which refers to the pressure in the chamber before the nozzle, as well as the temperature of the nozzle \(T_{0}\) via: \(\rho_{0}=P_{b}/(k_{B}T_{0})\), where \(k_{B}\) is the Boltzmann constant. Figure 3 shows the on-axis density \(\rho\) with respect to \(h\) (distance from the end of the nozzle) for different backing pressures from 10-40 bar for the dimensions of the Parker solenoid valve. A practical example for gas jets to produce a plasma for LWFA is as follows: in order to excite the plasma wake resonantly with a laser pulse of a 15-fs duration, an electron density in the order of \(1.75\times 10^{18}\) cm\({}^{-3}\) is necessary [14]. The required density (assuming argon being ionized eight-fold) at \(h=2.5\) mm (distance from the nozzle) is achieved with backing pressures of 30 bar. According to Equation (3), the phase shift of a gas jet with an average density ranging from \(1\times 10^{18}\)-\(1\times 10^{19}\) cm\({}^{-3}\) liesbetween 0.1 rad and 1 rad.