Polynomial fitting of interferograms with Gaussian errors on the fringe coordinates. II: Analytical study.

Area of Science:

• Optical Metrology
• Interferometry
• Statistical Optics

Background:

• Twyman-Green interferometry is a key technique for optical surface testing.
• Fringe coordinate errors, often modeled as Gaussian noise, can affect measurement accuracy.
• Polynomial fitting is commonly used to analyze interferogram data.

Purpose of the Study:

• To derive analytical formulas for expected aberration coefficients in the presence of Gaussian fringe errors.
• To investigate the influence of noise level and fringe count on these coefficients.
• To understand the impact of polynomial fitting order on data adjustment accuracy.

Main Methods:

• Statistical analysis of an ideal Twyman-Green interferogram with Gaussian fringe coordinate errors.
• Polynomial fitting to adjust interferogram data points.
• Derivation of analytical formulas for expected aberration coefficients and sum of squares of residuals.

Main Results:

• Expected aberration coefficients are zero, except for tilt about the x-axis and the comatic term.
• Deviation of coefficients increases with noise level and decreases with the number of fringes.
• Incorrect polynomial order selection is linked to erroneous data point adjustment.

Conclusions:

• Gaussian errors in fringe coordinates systematically introduce tilt and comatic aberrations.
• The fidelity of interferogram analysis depends on noise levels and data point fitting accuracy.
• Careful selection of polynomial fitting order is crucial for reliable optical metrology.