Topological homeomorphism extends orthogonal polynomials to arc-shaped apertures for an extremely-low aberration...

Orthogonal polynomials are indispensable in optical engineering for aberration control and surface machining, yet their application in non-circular apertures remains a persistent challenge. While Zernike polynomials effectively decompose wavefront errors into physically meaningful terms within circular domains, their orthogonality fails in arc-shaped apertures-a critical geometry for extreme ultraviolet lithography (EUVL) systems employing curved fields to avoid light obscuration. This breakdown severely limits high-precision surface correction and aberration control in freeform optics. To address the breakdown of orthogonality for traditional orthogonal polynomials in irregular domains, this study employs the principle of topological homeomorphic transformation to establish a bidirectional mapping function between irregular and regular regions, thereby extending the orthogonality domain of conventional orthogonal polynomials. Taking the irregular effective aperture in EUVL objective lens systems as an example, we derived arc-shaped Legendre orthogonal polynomials (ALOP) and verified their advantages over traditional polynomials through two distinct application scenarios. The introduction of ALOP into the optical design of the six-mirror system has led to a further enhancement of the system's imaging quality, and the wavefront aberration root mean square (RMS) has been optimized from the original 0.148 nm to 0.098 nm. In the context of surface figure refinement applications, we simultaneously established a deterministic mapping between the ALOP parameters of individual mirror surfaces and optical aberrations. This mapping enabled the reduction of the full-field wavefront aberration RMS in the projection optical system from 1.77 nm to 1.05 nm via single-mirror refinement, achieving a refinement efficiency exceeding 40%. Furthermore, the methodology established in this study-linking irregular domains with regular domains through topological homeomorphic transformation-addresses the challenges posed by non-conventional geometries and pupils in advanced optical systems, and provides fresh perspectives for the design and fabrication of future high-precision, highly complex optical systems.