<title>Achieving desired thickness gradients on flat and curved substrates</title>

A mathematical model has been developed for the calculation of the film thickness distribution on flat, spinning, and curved substrates deposited by the magnetron sputtering technique. With the use of the model it is possible to design shield or mask shapes to intercept material between source and substrate to achieve a particular gradient in film thickness. Such considerations have significant relevance in the deposition of multi-layer thin films for x- ray and neutron optics in which the allowable deviation in the measured thickness gradient from the desired is a few tenths of a percent. Examples of the procedure used to obtain uniform coatings on flat and curved substrates has been given. Further, the consequences of target wear on the film thickness distribution has been considered. Finally, the consequence of spinning the substrate through the deposition region to improve uniformity has also been considered. Good agreement between initial experimental result sand the theoretical calculations has been shown.