Global optimization and relectivity data fitting for X-ray
multilayer mirrors by means of genetic algorithms

M. Sanchez del Rio(a), and G. Pareschi(b) 

a)European Synchrotron Radiation Facility
  BP 220, 38043 Grenoble-Cedex, France

b)Osservatorio Astronomico di Brera
  Via E. Bianchi 46, I-23807 Merate (Lc), Italy

ABSTRACT 

The x-ray reflectivity of a multilayer is a non-linear function of many 
parameters (materials, layer thicknesses, densities, roughnesses). Non-linear 
fitting of experimantal data with simulations requires to use initial values 
sufficiently close to the optimum value. This is a difficult task when the 
space topology of the variables is highly structured, as in our case. The 
application of global optimization methods to fit multilayer reflectivity data 
is presented. Genetic algorithms are stochastic methods based on the model of 
natural evolution: the improvement of a population along successive generations.
A complete set of initial parameters constitutes an individual. The population 
is a collection of individuals. Each generation is built from the parent 
generation by applying some operators (e.g. selection, crossover, mutation) on 
the members of the parent generation. The pressure of selection drives the 
population to include "good" individuals. For large number of generations, 
the best individuals will approximate the optimum parameters. Some results on 
fitting experimental hard x-ray reflectivity data for Ni/C multilaters 
recorded at the ESRF BM5 are presented. This method could be also applied to 
the help in the design of multilayers optimized for a target application, like 
for an astronomical grazing-incidence hard X-ray telescopes.

Keywords: genetic algorithms, evolutionary systems, x-ray reflectivity, 
          multilayers, data modelling, global optimization