Sag- and slope-orthogonal bases to characterise nominally rectangular freeform surfaces

Area of Science:

• Optical engineering
• Computational optics
• Surface metrology

Background:

• Sag- and slope-orthogonal bases are effective for circular apertures.
• Characterizing rectangular apertures with these bases presents challenges.
• Existing methods struggle with aspect ratio dependency and computational efficiency.

Purpose of the Study:

• To investigate challenges in characterizing rectangular apertures using orthogonal bases.
• To introduce and evaluate the Radial Distribution Function (RDF) basis for this purpose.
• To demonstrate the advantages of the RDF basis over traditional methods.

Main Methods:

• Gram-Schmidt process for constructing orthogonal bases.
• Analysis of separation of variables for rectangular domains.
• Application and evaluation of the RDF basis for surface characterization.

Main Results:

• Separation of variables fails for rectangular apertures, leading to aspect ratio-dependent bases.
• The Gram-Schmidt method lacks efficient recurrence relations for computation.
• The RDF basis successfully avoids these issues, providing sag- and slope-orthogonality.

Conclusions:

• The RDF basis is a superior method for characterizing rectangular optical surfaces.
• It offers computational efficiency and independence from aspect ratio.
• This method is ideal for real-time optical design constraints.