First-order optics: operator representation for systems with loss or gain

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Abstract
The canonical operator theory introduced recently for the description of lossless first-order optics is extended here to first-order systems with loss or gain, elucidating the relation between canonical operator and complex ray methods. The spread functions in the space, frequency, and hybrid domains are derived in terms of the ABCD ray-transfer matrix as well as in terms of four other new matrix descriptions of geometrical optics. The fourfold correspondence among these matrices, the Hamilton characteristics, and the spread functions in the space, frequency, and hybrid domains leads to the derivation of four fundamental explicit canonical representations of the transfer operator and to their parameterization in terms of characteristic matrix elements. The relation between adjoint operators and bidirectional propagation is derived within canonical operator theory of first-order optics as well as within a more-general setting for all reciprocal (but not necessarily lossless) optical systems.

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