Approximate solution for optical measurements of the diameter and refractive index of a small and transparent fiber

When a plane electromagnetic wave is scattered by an optically transparent object, whose size is much larger than the wavelength, a series of bright and dark fringes forms the primary rainbow, which is one of the most splendid phenomena in nature. In this work, an optical technique is discussed for simultaneous measurement of the diameter and refractive index of an axisymmetric and dielectric fiber by studying some rainbow features. This noncontact optical technique uses a beam of light exhibiting low temporal coherence, which enabled us to reduce the detrimental sensitivity of the rainbow features to variations of the fiber properties, thus allowing for high-precision estimates. Approximate mathematical formulas for the diameter and refractive index measurements were derived from the lowest-order complex angular momentum correction to Airy theory of rainbow. Furthermore, sensitivity of the measurement data to small deformation of the fiber's cross section into an ellipse was discussed. Preliminary empirical results provide a qualitative verification.