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Analytical Theory of Arbitrary-Order Achromats


Abstract

An analytical theory of arbitrary-order achromats for optical systems with mid-plane symmetry is presented. It is based on the repeated use of identical cells; but besides mere repetition of cells, mirror symmetry is used to eliminate aberrations. Using mirror imaging of a cell around the x-y and x-z planes, we obtain four kinds of cells: the forward cell (F), the reversed cell (R), the cell in which the direction of bend is switched (S), and the cell where reversion and switching is combined (C). Representing the linear part of the map by a matrix, and the nonlinear part by a single Lie exponent, the symplectic symmetry is accounted for and transfer maps are easily manipulated.

It is shown that independent of the choice and arrangement of such cells, for any given order, there is a certain minimum number of constraint conditions that has to be satisfied. It is shown that the minimum number of cells necessary to reach this optimum level is four, and out of the sixty-four possible four-cell symmetry arrangements, four combinations yield such optimal systems. As a proof of principle, the design of a fifth-order achromat is presented.


W. Wan, M. Berz, Physical Review E 54,3 (1996) 2870-2883


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