The group-theoretical treatment of aberrating systems. I. Aligned lens systems in third aberration order

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Abstract
The constituents of a lens system, i.e., slabs of homogeneous transparent material and the refracting surfaces between them, considered to third aberration order, are associated to elements of a nine-parameter aberration group. Three parameters correspond to Gaussian systems and six to group-classified aberrations. The group multiplication operation (through matrix-cum-vector algebra) corresponds to their concatenation, and the linear group action on an eight-dimensional homogeneous space corresponds to the nonlinear action of the system on the optical phase space. This leads to economical computation algorithms that may be extended to aberrating systems of higher order.

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