Ion beam figuring (IBF) is established for the final precision figuring of high-performance optical components, where the figuring accuracy is guaranteed by the stability of the removal function and the solution accuracy of the dwell time. In this deterministic method, the figuring process can be represented by a two-dimensional (2D) convolution operation of a constant removal function and the dwell time. However, we have found that the current 2D convolution operation cannot factually describe the IBF process of curved surfaces, which neglects the influences of the projection distortion and the workpiece geometry on the removal function. Consequently, the current 2D convolution algorithm would influence the solution accuracy for the dwell time and reduce the convergence of the figuring process. In this part, based on the material removal characteristics of IBF, a mathematical model of the removal function is developed theoretically and verified experimentally. Research results show that the removal function during IBF of a curved surface is actually a dynamic function in the 2D convolution algorithm. The mathematical modeling of the dynamic removal function provides theoretical foundations for our proposed new algorithm in the next part, and final verification experiments indicate that this algorithm can effectively improve the accuracy of the dwell time solution for the IBF of curved surfaces.

Ultra-precision and ultra-smooth surfaces are vitally important for some high performance optical systems. Ion beam figuring (IBF) is a well-established, highly deterministic method for the final precision figuring of extremely high quality optical surfaces, whereas ion sputtering induced smoothing, or roughening for nanoscale surface morphology, strongly depends on the processing conditions. Usually, an improper machining method would arouse the production of nanoscale patterns leading to the coarsening of the optical surface. In this paper, the morphology evolution mechanism on a fused silica surface during IBF of high-slope optical components has been investigated by means of atomic force microscopy. Figuring experiments are implemented on two convex spherical surfaces by using different IBF methods. Both of their surface errors are rapidly reduced to 1.2 nm root mean square (RMS) after removing similar deep material, but their surfaces are characterized with obviously different nanoscale morphologies. The experimental results indicate that the ion incidence angle dominates the microscopic morphology during the IBF process. At near-normal incidence, fused silica achieves an ultra-smooth surface with an RMS roughness value $R_q$ down to 0.1 nm, whereas nanoscale ripple patterns are observed at a large incidence angle with an $R_q$ value increasing to more than 0.9 nm. Additionally, the difference of incidence angles on various machined areas would influence the uniformity of surface quality, resulting from the interplay between the smoothing and roughening effects induced by ion sputtering.

Ion beam figuring (IBF) is established for the final precision figuring of optical components. In this deterministic method, the figuring process is represented by a two-dimensional (2D) convolution operation of a constant removal function and the dwell time, where the figuring precision is guaranteed by the stability of the removal function as well as the solution accuracy of the dwell time. However, the current 2D convolution equation cannot factually reflect the IBF process of curved surfaces, which neglects the influence of the projection distortion and the workpiece geometry. Consequently, the current convolution algorithm for the IBF process would influence the solution accuracy for the dwell time and reduce the convergence of the figuring process. In this part, we propose an improved algorithm based on the mathematical modeling of the dynamic removal function in Part A, which provides a more accurate dwell time for IBF of a curved surface. Additionally, simulation analysis and figuring experiments are carried out to verify the feasibility of our proposed algorithm. The final experimental results indicate that the figuring precision and efficiency can be simultaneously improved by this method.

Nanoscale laser damage precursors generated from fabrication have emerged as a new bottleneck that limits the laser damage resistance improvement of fused silica optics. In this paper, ion beam etching (IBE) technology is performed to investigate the evolutions of some nanoscale damage precursors (such as contamination and chemical structural defects) in different ion beam etched depths. Surface material structure analyses and laser damage resistance measurements are conducted. The results reveal that IBE has an evident cleaning effect on surfaces. Impurity contamination beneath the polishing redeposition layer can be mitigated through IBE. Chemical structural defects can be significantly reduced, and surface densification is weakened after IBE without damaging the precision of the fused silica surface. The photothermal absorption on the fused silica surface can be decreased by 41.2%, and the laser-induced damage threshold can be raised by 15.2% after IBE at 250 nm. This work serves as an important reference for characterizing nanoscale damage precursors and using IBE technology to increase the laser damage resistance of fused silica optics.