The density $ \rho$ of a plasma cannot be measured directly, but its absorbance $ A$ of a laser beam can be obtained experimentally. The absorbance along a straight line is proportional to the corresponding line integral over the density $ \rho(\boldsymbol{r})$. Assuming a radial symmetric plasma column (see Fig. 1), absorbance and density are related via the Abel transformation:

$\displaystyle A(y) = \int\limits_y^R \frac{2r}{\sqrt{r^2-y^2}}\,\rho(r)\,dr,$ (1)

which can be rewritten as linear integral equation of the form:

$\displaystyle A(y) = \int\limits_0^R K_y(r)\rho(r) \, dr$   with$\displaystyle \qquad K_y(r) = \Theta(r-y) \frac{2r}{\sqrt{r^2-y^2}}.$ (2)

Due to the singularity of the integral kernel $ K_y(r)$ at $ r=y$, the inversion of (2) turns out to be an ill-conditioned problem which we will tackle by applying the Maximum Entropy-Method.
Figure 1: Experimental geometry for Abel inversion

Danilo Neuber 2003-10-03