Iterative unfolding of intensity data, with application to molecular beam scattering

An efficient iterative technique is presented for solving the Fredholm integral equation of the first kind, ${\mathrm{h(x)\; =\; \int }}_a^b\mathrm{dx\prime \; g(x,\; x\prime )f(x\prime )}$ , which occurs in various forms in experimental intensity measurements; h is the measured intensity distribution, f is the ideal distribution, and g is a bandpass function representing apparatus resolution. The scheme is based on the equations f₀(x) = h(x) and f_n+1(x) = f_n(x)[h(x)/∫dx′g(x, x′)f_n(x′)] and is shown to result in a nonnegative distribution free of diffraction effects. The method is easily extended to multidimensional abscissas x and multidimensional integrals, and an application to molecular beam data reduction for doubly differential reaction cross sections is presented.