Refractive error correction for in situ curvature measurement using laser beam deflection method

Stoney's equation allows one to calculate the internal stress in a thin film deposited on a thick substrate from the curvature of the bilayer beam. The curvature measurement is thus critical in the bending beam method, which is based on this equation. When an optical technique is used in curvature measurement, the refraction at optical boundaries needs to be corrected. An approximate correction to Stoney's equation accounting for refractive error in the bending beam method in electrochemical applications was reported previously. This paper revisits the derivation. A set of diagrams for optical paths in in situ curvature measurement using the laser beam deflection method are presented. These diagrams allow us to establish the triangular relationships among the incident angles of the light beam and the deflection angle of the cantilever beam. Correction solutions are derived for three possible optical arrangements with the consideration of the travel distance of the light beam in liquid and in the container wall. In its approximated form, the correction derived in this paper differs from the previous formula by a factor of 2. It was discovered that an incorrect relationship for curvature was used in the previous derivation.