Group-classified polynomials of phase space in higher-order aberration expansions

Abstract

Polynomial functions of the classical phase variables p 2 , p×q , and q 2 , are used in higher-order perturbation expansions in various fields of physics, including geometric aberration optics of axially-symmetric lens and mirror systems. These polynomials participate in operations such as linear combination, multiplication, Poisson brackets, and a Baker-Campbell-Hausdorff compounding that corresponds to concatenation of optical elements. We are interested in handling the polynomials through structures and in bases where the above operations are as short as possible. The monomial basis is one obvious choice that performs efficiently under multiplication. For Poisson brackets and aberration-group products however, the symplectic basis that uses solid spherical, harmonics in three variables is shown to be a better choice. In the last operation, for seventh aberration order, we halve the computation complexity in the symplectic basis.