Theoretical analysis and optimisation of the method of excess fractions for long-range metrology

The method of excess fractions has a long history in metrology. More recently excess fractions has been exploited to resolve the fringe order ambiguity in interferometric metrology with varying degrees of success. There are a variety of reports detailing the performance of excess fractions, for example, using 4 wavelengths an unambiguous measurement range of 2.4 mm was achieved with a phase noise of 1/900th of a fringe. In an independent report a 4 wavelength interferometer gave an unambiguous measurement range of 17 mm with a phase noise of 1/200th of a fringe. It has been found that the unambiguous range of an excess fractions multi-wavelength interferometer depends on the wavelengths used within the system. A theoretical model is reported in this paper that can be used in a predictive way to determine the unambiguous measurement range based on three wavelength dependent parameters. The excess fractions model is consistent with beat wavelength techniques but offers many alternative sets of wavelengths to achieve, for a given phase noise, a particular unambiguous measurement range with a given reliability.

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