Alternative computation of the Seidel aberration coefficients using the Lie algebraic method

Abstract

We give a brief introduction to Hamiltonian optics and Lie algebraic methods. We use these methods to describe the operators governing light propagation, refraction and reflection in phase space. The method offers a systematic way to find aberration coefficients of any order for arbitrary rotationally symmetric optical systems. The coefficients from the Lie method are linked to the Seidel aberration coefficients. Furthermore, the property of summing individual surface contributions is preserved by the Lie algebraic theory. Three examples are given to validate the proposed methodology with good results.

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

“…As such, we use the phase-space coordinates (𝒒, 𝒑) as our ray coordinates, cf. [12]. Note that the coordinates of the OAR are at the origin of phase-space both before and after reflection, i.e., the OAR will have coordinates 𝒒 = 0 = 𝒒 ′ and 𝒑 = 0 = 𝒑 ′ .…”

“…We introduce the Hamiltonian 𝐻 ( 𝒑) governing free propagation of light in a medium of constant refractive index 𝑛 [9,12,13], i.e.,…”

“…where ( q, 𝜁 ( q)) is the intersection point of the incoming ray and the reflector. The intersection point q is related to the screen coordinate before reflection 𝒒 and the one after reflection 𝒒 ′ by [9,12,13]…”

“…These operators enable us to derive closed form expressions for the aberration components of an arbitrary optical system. A more detailed description of the Lie algebraic tools used in this work can be found in [9,12,13]. Here, we briefly introduce the main concepts.…”

“…The maps for ray propagation and reflection plus rotation are symplectic and map the origin onto itself [9,12,13]. It is therefore possible to represent them as an infinite, or approximate them by a truncated, concatenation of Lie transformations according to the result in Eq.…”