Determining the refractive index of a {lambda}/4 thin film on a thick substrate from a transmittance measurement

The refractive index is a key optical constant required by optical thin film designers. The coater-designer team must constantly verify the refractive indices of the non-absorbing optical thin films since the refractive index of a deposited material can vary by switching coating systems or operators as well as expected changes during the course of a coating run. A transmittance measurement on a spectrophotometer is an easy and accurate (usually to within {+-}0.5% spread) method of determining the refractive index. In one technique, the refractive index is obtained from visually curve-fitting a calculated (using a thin film design program and selecting the refractive index) transmittance spectrum to the measured transmittance spectrum. There are two other techniques which are discussed in this report: A quick, approximate method vs. the exact derivation. What the authors have not been able to find in the literature is the exact transmittance dependence as a function of the refractive indices (layers) through which the light passes and which accounts for the substrate back reflections. There are undocumented approximation methods as well as one in the literature by Cheremukhin and Rozhnox. Otherwise, most texts either derive the transmittances through optical multilayers or just the effect of back reflections on the transmittance for thick substrates. Without correcting for substrate back reflections, the derived refractive indices from the measured transmittances are in error by as much as 10%. In this work, the authors have utilized both an exact and an approximate method of determining the refractive index of the film. It is found that both the exact and approximate methods of determining the refractive index of thin optical coatings are within the measurement errors of commercially available spectrophotometers.