Method of excess fractions with application to absolute distance metrology: analytical solution

Multiwavelength interferometry provides a solution to a number of applications in metrology for the measurement of optical path differences longer than the source wavelength. To this day, the method of excess fractions (EF) has proved to provide very long, unambiguous measurement ranges with the highest reliability for a given set of wavelengths and level of phase noise. This is achieved because EF combines the individual phase values in an equivalent least-square problem and evaluates the correspondence for all possible solutions. However, this procedure can be slow for a number of applications. In this paper, an analytical solution for EF is presented that allows the direct calculation of the unknown integer fringe order. It is shown that this solution is consistent with the other phase unwrapping approaches as beat wavelength or Chinese remainder theorem-based solutions, but moreover, it can be understood as a unified representation and solution of the fringe order problem.