On mathematical aspects of surface correction via ion beam etching

TL;DR

This paper develops a mathematical framework for solving inverse problems related to ion beam etching, using functional analysis, pseudoinverses, and kernel methods to improve surface correction techniques.

Contribution

It introduces a novel approach combining operator theory and kernel methods for inverse problems in ion beam figuring, enhancing solution stability and practical applicability.

Findings

Singular vectors resemble exponential functions, similar to quantum well eigenfunctions.

The operator is close to finite-dimensional, enabling pseudoinverse solutions.

Kernel Hilbert spaces facilitate practical data reconstruction.

Abstract

Motivated by the ion beam dwell time calculation problem in Ion Beam Figuring we suggest a mathematical framework for solving a specific type of inverse problems, which appear in various areas of applied mathematics and physics. From the point of view of functional analysis we deal with a linear operator equation, which, taking advantage of the observation that the associated operator, acting from the space of dwell times into the space of measurement data, is close to a finite-dimensional one, can be solved employing the pseudoinverse operator. For the main case of interest we describe the behavior of the singular vectors inside the domain, on which measurement data are given, which turn out to be close to functions $e^{ikx}$. Heuristically, these singular vectors are similar to eigenfunctions of the infinitely deep quantum well. An alternative problem formulation, closer to practical calculations, utilizes reconstructing kernel Hilbert spaces (RKHS) and radial basis functions, which are used for pointwise approximation of measurement data and ensuing dwell time determination. Depending on the particularities of the concrete problem either framework or a suitable combination of each can be used.