Computing aberration coefficients for plane-symmetric reflective systems: A Lie algebraic approach

Abstract

We apply the Lie algebraic method to reflecting optical systems with plane-symmetric freeform mirrors. Using analytical ray-tracing equations we construct an optical map. The expansion of this map gives us the aberration coefficients in terms of initial ray coordinates. The Lie algebraic method is applied to treat aberrations up to the desired order. The presented method provides a systematic and rigorous approach to the derivation, treatment and composition of aberrations in plane-symmetric systems. We give the results for second- and third-order aberrations and apply them to three single-mirror examples.

Lie algebraic methods for aberrations of plane-symmetric mirror systems

“…The main tool, that encodes the aberration information, is the Lie transformation. [3][4][5][6] Let f, g be two functions in phase-space variables, then their Poisson bracket reads…”

“…Utilizing g 2 , g 3 , g 4 , for example, it is possible to determine the third-order aberrations of the system. 3 The same approach has been applied in case of rotationally symmetric systems for the Seidel coefficients. 6 Second, since our approach utilizes analytic expressions of the aberration coefficients, it is possible to analyze the direct dependence of aberration terms on the geometric parameters of the system.…”

“…This allows to develop surface optimization routines to determine optimal mirror systems for a given problem. 3 For validation we used the expansion formulas of the optical map as an approximate ray-tracer for a single biconic reflector with incidence angle θ = π/6 and compare our results to OpticStudio (OS) with excellent agreement; see Figure 1…”

“…1,2 Alternatively, we use Lie algebraic methods to encode aberrations and their compositions. 3 Lie methods enable us to represent the interactions of both intrinsic and extrinsic aberrations. They also provide a more efficient storage of the aberration coefficients compared to the matrix formalism and a more rigorous and systematic mathematical framework.…”