Effects of rotation errors on goniometric measurements

Goniometric measurements are essential for the determination of many optical quantities, and quantifying the effects of errors in the rotation axes on these quantities is a complex task. In this paper, we show how a measurement model for a four-axis goniometric system can be developed to allow the effects of alignment and rotation errors to be included in the uncertainty of the measurement. We use three different computational methods to propagate the uncertainties due to several error sources through the model to the rotation angles and then to the measurement of bidirectional reflectance and integrated diffuse reflectance, a task that would otherwise be intractable. While all three methods give the same result, the GTC Python package is the simplest and intrinsically provides a full uncertainty budget, including all correlations between measurement parameters. We then demonstrate how the development of a measurement model and the use of GTC has improved our understanding of the system. As a consequence, taking advantage of negative correlations between measurements in different geometries allows us to minimise the total uncertainty in integrated diffuse reflectance, lowering the standard uncertainty from 0.0029 to 0.0015.