Zernike-like systems in polygons and polygonal facets

Area of Science:

• Optics and Photonics
• Computational Optics
• Optical Engineering

Background:

• Zernike polynomials are standard for circular apertures due to their complete and orthonormal properties.
• Existing methods for non-circular apertures often lack a unified mathematical framework.
• Previous work extended Zernike bases to elliptic and annular apertures.

Purpose of the Study:

• To generalize Zernike polynomials for arbitrary optical apertures, focusing on polygons.
• To develop a robust Zernike-like basis for polygonal and segmented optical systems.
• To ensure mathematical and physical property invariance in the new basis.

Main Methods:

• Utilizing a piecewise diffeomorphism to map the unit disk onto polygonal apertures.
• Defining a Zernike-like orthonormal system over polygons using this mapping.
• Applying the method to ensembles of polygonal facets for segmented mirror telescopes.

Main Results:

• A generalized Zernike basis applicable to polygons and segmented mirrors.
• The method provides a unique solution, unlike ad hoc approaches.
• The new basis maintains key mathematical and physical properties of the original Zernike polynomials.

Conclusions:

• The proposed generalization offers a unified approach for Zernike expansions on diverse optical apertures.
• This method is particularly beneficial for designing complex optical systems like segmented mirror telescopes.
• The invariance of properties ensures reliable wavefront analysis and optical system performance.