Effect of wave-number error on the computation of path-length delay in white-light interferometry

We analyze the error in computed optical path-length delay when using a phase-shifting interferometry (PSI) algorithm with an error in the operating wavelength. The delay error decomposes into two terms. The first is the error in the conversion from a phase measurement to the delay because of the incorrect wavelength, and the second is the error made in the phase measurement itself that is due to the wavelength error. The most important aspect of this investigation is to ascertain this latter error. A general characterization is obtained, and a particularly simple formula is developed for the special case of least-squares estimation involving only the ratio of the wave-number error to the wave number and a multiplicative factor that is an a priori computable nonlinear function of the ratio of the modulator stroke length to the operating wavelength. Because the ultimate path-length error is a function of the two terms, a new set of PSI algorithms that compensate the computed phase error to cancel the conversion error is developed. Numerical simulations are presented to validate the analysis and establish the insensitivity of the new algorithms to wave-number error.