Susceptibility of systematic error-compensating algorithms to random noise in phase-shifting interferometry

Abstract

In phase-shifting interferometry, many algorithms have been reported that suppress systematic errors caused by, e.g., nonlinear motion of the phase shifter and nonsinusoidal signal waveform. However, when a phase-shifting algorithm is designed to compensate for the systematic phase-shift errors, it becomes more susceptible to random noise and gives larger random errors in the measured phase. The susceptibility of phase-shifting algorithms to random noise is analyzed with respect to their immunity to phase-shift errors and harmonic components of the signal. It is shown that for the most common group of error-compensating algorithms for nonlinear phase shift, both random errors and the effect of high-order harmonic components of the signal cannot be minimized simultaneously. It is also shown that if an algorithm is designed to have extended immunity to nonlinear phase shift, simultaneous minimization becomes possible.

Phase-shifting interferometry corrupted by white and non-white additive noise

“…But now we extend previous analysis [3][4][5][6][7][8] to phase-shifting interferometry corrupted by no-white (pink) additive noise. Moreover using Parseval's theorem we unify in a single theory our spectral approach [3,4] with previous non-spectral methods [5][6][7][8] which give the estimated phase variance as a formula depending upon the coefficients of the phase-shifting algorithm.…”

“…Here we use the PSA's FTF to find the estimated phase variance due to a sequence of interferograms corrupted by non-white additive noise. Moreover, for the special case of white additive noise, the equivalence between the noise analysis based on the PSA's spectrum [3,4], and on its coefficients [5][6][7][8] is also shown, and finally two interesting examples of its application is given.…”

“…Note that the amplitude of our input signal (b/2)exp(iφ) is now multiplied by the PSA filter's response H(ω 0 ) at + ω 0 . The PSA's response H(ω 0 ) at ω 0 is a very important piece of information that has not been taken explicitly into account before [5][6][7][8]. Also note that n H (t m ) is the output noise after filtering, which is now complex and non-white with variance [9],…”

“…Equation (11) shows (for the first time in optical interferometry literature) the two sides of the coin, namely; the PSA's phase variance based on its algorithm's spectrum H(ω) [3,4], and on its coefficients c k [5][6][7][8] for the white-noise case. The noise-to-signal input's ratio 4σ n 2 /b 2 depends only on the interferogram' data; the rest depends, on the spectrum, or on the coefficients of the PSA.…”

“…Non-spectral methods for noise analysis in PSI have been reported before [5][6][7][8] using: Taylor expansion of the fringe irradiance [5]; joint statistical distribution of the noise [6]; PSA's x-characteristic polynomial [7]; the derivative of the PSA's arctangent ratio [8]. All these works assume white-additive corrupting noise, and ends-up with formulas for the variance of the PSA's phase-noise using the algorithm's coefficients.…”