Fourier spectra for nonuniform phase-shifting algorithms based on principal component analysis

Area of Science:

• Optical metrology
• Signal processing
• Interferometry

Background:

• Principal component analysis (PCA) algorithms often introduce phase-demodulation errors with nonuniform phase-shifted interferograms.
• Existing methods for nonuniform phase-stepping algorithms (nPSA) rely on visual demonstrations, which can be biased.

Purpose of the Study:

• To develop an error-free nonuniform phase-stepping algorithm (nPSA) based on principal component analysis (PCA).
• To provide a straightforward method for correcting PCA-based phase-demodulation errors.
• To establish objective figures-of-merit for evaluating PCA-nPSA performance.

Main Methods:

• Development of mathematical formulas for comprehensive analysis of PCA-based nPSA (PCA-nPSA).
• Analysis of PCA-nPSA as a linear quadrature filter to derive its frequency transfer function (FTF).
• Application of linear systems and stochastic process theories to derive figures-of-merit.

Main Results:

• Formulas derived for PCA-nPSA FTF, corrected Lissajous figure, corrected PCA-nPSA formula, harmonic robustness (R_H), and signal-to-noise-ratio (SNR).
• Demonstration of why standard PCA fails with nonuniform phase-shifted fringes using the FTF.
• Avoidance of biased fringe pattern selection through objective, formula-based performance metrics.

Conclusions:

• The proposed PCA-nPSA offers an error-free solution for phase demodulation with nonuniform phase shifts.
• The developed analytical framework provides a robust and unbiased method for evaluating nPSA performance.
• This work establishes a rigorous foundation for PCA-based phase demodulation in optical metrology.