Introduction

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:

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

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

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