Generalized theory of phase unwrapping: approaches and optimal wavelength selection strategies for multiwavelength interferometric techniques

Abstract

Multi-wavelength interferometry (MWI) has a long tradition in the field of optical metrology and is used as a solution to a number of applications. MWI phase unwrapping procedures are usually based on beat wavelength approaches, Chinese Remainder Theorem (CRT) techniques, or the method of Excess Fractions (EF). Each of these unwrapping approaches have a distinct advantage for a given application: Beat wavelength and CRT based approaches offer a direct calculation of integer fringe order, and EF offers many alternative sets of wavelengths to achieve a large unambiguous measurement range (UMR) with high reliability. Nevertheless, a drawback of Beat wavelength and CRT based approaches is that they have a limited UMR due to the available measurement wavelengths, and the alternative approach the EF is often impractical in practice, because the calculation of the integer fringe order involves a large number of computational steps. Recently, we have reported a unified theory of beat wavelength, EF and CRT approaches, which enables the derivation of phase unwrapping approaches with low computational effort, which hitherto had only been possible for CRT and beat wavelength approaches, whilst offering flexibility in choosing the measurement wavelengths for a given UMR, which had previously only been the case for EF. In this work, we briefly summarize the previous developed framework that determines the UMR and measurement reliability and derive optimization criteria that are based on parameters, which are dependent on the choice of the measurement wavelengths. The developed optimum wavelength selection strategies maximize the dynamic range of interferometer for a given value of phase noise the dynamic range of interferometer.