Spectral analysis for the generalized least squares phase-shifting algorithms with harmonic robustness.

Area of Science:

• Optical Metrology
• Signal Processing

Background:

• Phase-shifting algorithms (PSAs) are crucial for optical metrology.
• Standard PSAs often assume uniformly spaced phase shifts.
• Non-uniform phase shifts present challenges in data analysis and performance evaluation.

Purpose of the Study:

• To introduce a novel theoretical framework, the frequency transfer function (FTF) formalism.
• To adapt FTF for generalized least squares phase-shifting algorithms (GLS-PSAs) with non-uniform phase shifts.
• To enable spectral analysis and derive key performance metrics for these algorithms.

Main Methods:

• Developed the frequency transfer function (FTF) formalism.
• Applied FTF to generalized least squares phase-shifting algorithms (GLS-PSAs) with non-uniform phase shifts.
• Utilized the Moore-Penrose pseudoinverse to determine the GLS-PSA's impulsive response.
• Conducted simulations to validate the theoretical analysis.

Main Results:

• The FTF formalism provides a method for spectral analysis of GLS-PSAs.
• Key performance figures of merit, including signal-to-noise ratio (SNR) and harmonic rejection, can be readily determined.
• Simulations demonstrated a trade-off: improved harmonic rejection robustness led to a slight decrease in SNR.

Conclusions:

• The FTF formalism offers a powerful tool for understanding and optimizing GLS-PSAs, especially those with non-uniform phase shifts.
• The derived metrics allow for quantitative assessment of algorithm performance.
• The identified trade-off between SNR and harmonic rejection is critical for practical algorithm selection and design.