A theoretical analysis of the laser melt-level monitoring system for the Czochralski crystal growing furnaces

This paper discusses the operating theory of a trigonometrical laser melt-level monitoring system as applied to Czochralski crystal pullers. The surface movement of the silicon melt is modeled with three-dimensional random harmonic waves. The effect of this motion on the spatial distribution of the laser signal is simulated and studied. Means of separating the laser signal from the noise caused by the background radiation from the melt is described. Statistical analysis is used to determine measurement accuracy. It has been found that with a relatively tranquil melt surface, characterized by a small wave amplitude to wavelength ratio and a large wavelength to laser beam size ratio, the laser signal is confined to a limited area, thus can be effectively acquired by a photo detector. The accurate measurement can then be achieved with statistical means. Experimental data generated with water and molten silicon surfaces are in good conformity with the theoretical results.