# Real-Time Tomography of Gas-Jets with a Wollaston Interferometer
## 2 The Single Beam Wollaston Interferometer Setup  
### Theory  
A Wollaston prism (Figure 1B) is a combination of two prisms made of a birefringent crystal (e.g., quartz), which has a single optical axis. The two prisms are cut and combined such that the optical axes of the prisms are perpendicular to each other. The main feature of such a birefringent crystal is that the refractive index for light polarized parallel to the optical axis (\(\eta_{e}\)) is different from that for light polarized perpendicular to the optical axis (\(\eta_{o}\)). Hence, a light ray polarized at \(45^{\circ}\) with respect to the optical axis is split up into two equally-intense beams with perpendicular polarization, which experience different refractive indices and hence propagate with different phase velocities. In a Wollaston prism, one of the two beams experiences a higher and the second a lower refractive index in the first birefringent prism, while the situation is reversed in the second prism. Hence, the two rays, which still propagate collinearly in the first prism, will be refracted differently at the interface between the two prisms and then propagate under an angle in the second prism. After exiting the second prism at its back surface, the two rays will diverge with an angle \(\epsilon\) (dispersion angle), which depends on the geometric and optical properties of the crystals forming the Wollaston prism. As shown by Small [9],  
\[\epsilon=2\beta(\eta_{e}-\eta_{o}),\]where \(\beta\) is the prism angle of both prisms (see Figure 1A). The Wollaston prism used in this experiment was manufactured by Societe d'Optique de Precision Fichou (Figure 1B) and has a dispersion angle of \(20^{\prime\prime}=5.8\) mrad. If two light rays with initial relative angle \(\epsilon\) pass through the prism, one polarization component of the first ray will coincide with the polarization component of the second ray (compare Figure 1A). Before passing through the lens, these two rays are separated by a distance \(d=\epsilon f_{\circ}\). For the Wollaston prism and a lens with focal length \(f_{\circ}=30\) cm, \(d=1.74\) mm. If a second polarizer at \(-45^{\circ}\) is placed after the prism, these two rays can interfere. This concept is illustrated in Figure 1A. Depending on the position of the prism relative to the focal point of the lens, the interfering rays will pick up a phase difference. If the prism is centered at the focal point, every optical path runs equal distances through the two halves of the prism, i.e., no relative phase difference is generated. This scenario is called normal mode or the Infinite Fringe Width (IFW) setup. Displacing the prism by a distance \(b\) from the focal point results in different path lengths of interfering rays, as they travel different distances in each half of the prism. These phase shifts lead to regular interference patterns on the screen. This setup (\(b\
eq 0\)) is called differential mode or the Finite Fringe Width (FFW) setup [10]. The FFW setup is used in this work and is further described in Section 2.3.1. The spacing \(S\) between the undisturbed fringes is given by:  
\[S=\frac{\lambda}{\epsilon}\frac{p}{b}, \tag{1}\]  
where \(p\) is the distance from the Wollaston prism to the screen [11]. The spacing can be decreased by increasing \(b\), i.e., by changing the position of the Wollaston prism with respect to the focal plane. If the position of the screen remains fixed, which is the case when the lens images a certain plane on the screen, \(p\) changes accordingly. Placing a medium with refractive index \(\eta\
eq 1\) that covers only parts of  
Figure 1: Working principle of the Wollaston prism (**A**) and used prism (**B**). (**A**) Two parallel rays separated by distance \(d\) before the lens are brought to interference after the prism; (**B**) Wollaston prism \(30\times 30\times 5\) mm, \(\epsilon=5.8\) mrad, Societe d’Optique de Precision Fichou.