Fitting freeform shapes with orthogonal bases.

Area of Science:

• Optics and optical engineering
• Metrology
• Computational geometry

Background:

• Accurate characterization of freeform optical surfaces is crucial for performance.
• Existing methods for analyzing mid-spatial frequency errors can be computationally intensive.
• Fitting arbitrary shapes requires robust mathematical frameworks.

Purpose of the Study:

• To develop an efficient analytical method for fitting freeform shapes using orthogonal polynomials.
• To enable precise quantification and filtering of mid-spatial frequency errors in optical components.
• To leverage orthogonality for improved metrology of as-built parts.

Main Methods:

• Exploitation of orthogonality for fitting analytically-specified freeform shapes.
• Representation of shapes using orthogonal polynomials.
• Coupling of Fast Fourier Transforms (FFTs) with Gaussian quadrature for analysis.
• Application to measurements of as-built parts with circular domains.

Main Results:

• An explicit and efficient formulation for fitting freeform shapes was derived.
• The method allows for the use of an arbitrary number of polynomial terms.
• Successful quantification and filtering of mid-spatial frequency structures were demonstrated.
• The approach is applicable to real-world measurements of manufactured optical components.

Conclusions:

• The developed method provides an efficient and accurate approach for freeform surface metrology.
• Orthogonal polynomial fitting combined with FFTs offers a powerful tool for error analysis.
• This technique facilitates improved quality control and understanding of mid-spatial frequencies in optical manufacturing.