Abstract: Differential algebra is a new mathematical method and a promising tool in charged particle optics. As a kind of automatic differentiation techniques, it is based on the nonstandard analysis and normal power series theory. It presents a straightforward method for the study of arbitrary order property of nonlinear systems, with accuracy limited only by the computer. The differential algebraic method is widely applicable, and it is an effective and elegant tool for the tracking, analysis and correction of high-order aberrations, sensitivity analysis and optimization of structure design, etc., of the charged particle optical systems. The principle and virtues of differential algebra are summarized. The application of differential algebra method in charged particle optics and its implementation in various computer languages are analysed. Some new research subjects in this field are discussed.

Key words: differential algebra, automatic differentiation, charged particle optics, nonstandard analysis