Cubic spline interpolation for inverse Abel transform integral equation - application to plasma spectroscopy

Summary form only given. The purpose of this paper is to determine the radial distribution of the emission coefficient from the measured intensity distribution emitted by an extended source of radiation, particularly a plasma source. The source is assumed to be optically thin and axially symmetrical. This problem is solved by inverting Abel's integral equation. A smoothing procedure is made on the experimental curve in order to attenuate the random errors before computing the derivative. The integral is calculated analytically in a small interval on the right of the discontinuity point, the other part is estimated numerically in Scilab program. Abel's integral equation is frequently applied in the study of extended radiation sources with cylindrical symmetry. A measurement of the transverse distribution I(y) of the intensity emitted perpendicularly to the source axis allows the calculation of the emission coefficient radial distribution F(r). If the source is optically thin, the intensity I(y) is connected to the emission coefficient by the formula: I(y) int
_-x

^x
F(r)dx. F(r) can be deduced form I(y) by the inverse formula: F(r)=-1/piint
_y

^R
(dI(y)/dy)(1/radic(r
²
-y
²
))dy, known as Abel's integral equation. The integral is calculated using a polynomial of second degree for the approximation of dl(y)/dy in a small interval on the right of the discontinuity point, the other part is calculated using an approximate numerical method given by the function intsplin of the Scilab program.