# Real-Time Tomography of Gas-Jets with a Wollaston Interferometer
## 3 Data Analysis for Interferometry and Real-Time Tomography
### Numerical Tools  
The undisturbed fringes are mathematically described by a harmonic oscillation with a fixed frequency multiplied by the Gaussian amplitude of the laser beam profile. The wave period of the oscillation is given by the undisturbed fringe width \(S\) (Equation (1)), i.e., the spacing between the fringes. A gas jet will locally change the fringe width according to the phase introduced to the different light rays in the laser beam by the gas flow. The following phase extraction problem arises [15]: given noisy discrete values of a function of the form:  
\[g(y)=b(y)\cdot e^{i[2\pi fy+\alpha(y)]},\]  
where \(\alpha(y)\) represents the phase shift induced by the gas jet. The term \(b(y)\) takes global intensity changes due to the Gaussian profile of the interferometry laser into account; \(y\) is the coordinate perpendicular to the gas flow and interferometry laser; and \(f\) is the oscillation frequency of the undisturbed fringes. This problem can be solved by fitting the measured data to an ansatz for \(\alpha(y)\). However, this is not appropriate, since one has to make (possibly incorrect) assumptions about the form of \(\alpha(y)\). This is particularly true when a shock front has to be characterized by \(\alpha(y)\). A more elegant way to directly unwrap the phase from the noisy data makes use of a Fourier transformation:  
\[\mathcal{F}(g)(\omega)=\int_{-\infty}^{+\infty}e^{-i2\pi\alpha y}\cdot b(y) \cdot e^{i[2\pi fy+\alpha(y)]}\;\mathrm{d}y.\]  
Figure 7: Tomography setup for shock front studies. A razor blade is positioned with a linear and a rotational stage. Since the acquired signal is real, the full information is contained in the positive frequency domain where the spectrum has two significant peaks: one at the fringe frequency \(f\) and another around zero due to the slowly varying Gaussian intensity profile of the beam. A Gaussian window is applied to the spectrum to cut out the low frequency peak. In order to eliminate the \(2\pi fy\) phase factor, we apply a rotation of the Fourier-transformed signal by:  
\[R_{f}\mathcal{F}(g)(\omega)=\mathcal{F}(g)(\omega+f)=\int_{-\infty}^{+\infty}e ^{-\mathrm{i}2\pi\omega y}\cdot b^{\prime}(y)\cdot e^{\mathrm{i}\alpha(y)}\; \mathrm{d}y,\]  
with the result that the rotation by \(R_{f}\) shifts the peak at frequency \(f\) to zero. In order to obtain the sought phase shift \(\alpha\), we apply the inverse Fourier-transform:  
\[\mathcal{F}^{-1}(R_{f}\mathcal{F}(g))(y)=b^{\prime}(y)\cdot e^{\mathrm{i} \alpha(y)}=:A(y).\]  
As a result, \(A\) contains the information about the phase shift \(\alpha\), as well as remaining effects of \(b\) denoted by \(b^{\prime}\). These are minor effects due to imperfections of the mirrors, lenses and polarizers or inhomogeneities of the two windows through which the laser passes. These effects are eliminated by the point-wise product \(A\cdot A_{ref}^{-1}\), where \(A_{\mathrm{ref}}\) is the signal obtained by the same transformation applied to the reference data, i.e., without gas jet. The phase of the remaining term corresponds to the phase caused by the gas jet alone. The term \(e^{\mathrm{i}2\pi fy}\) would arise in \(A_{\mathrm{ref}}\), as well, and is therefore canceled in the last step, even if the rotation \(R_{f}\) is omitted. As explained in Section 2, the Wollaston interferometer measures the phase difference between rays separated by a distance \(d\) in the object plane. Therefore, phase shifts obtained at two points in the image plane that are separated by a distance corresponding to \(d\) have to be added up. We obtain the additional phase \(\phi\) by \(\phi(y)=\frac{2\pi}{\lambda}\int(\eta(y,z)-1)\;\mathrm{d}z\). This is the phase that a ray has picked up when passing through the gas jet, in comparison to a ray that has passed through a vacuum only. The integration is taken along the ray's path at position \(y\); \(\lambda\) is the wavelength of the laser; and \(\eta(y,z)\) refers to the distribution of the refractive index in the gas jet.