Generalized reverse theorems for multipass applications in matrix optics

The reverse theorem that governs backward propagation through optical systems represented by transfer matrices is examined for various matrix theories. We extend several reverse theorems to allow for optical systems represented by matrices that may or may not be unimodular and that may be 2 × 2 or take on an augmented 3 × 3 form. As an example, we use the 3 × 3 form of the reverse theorem to study a laser with intracavity misaligned optics. It is shown that, by tilting one of the laser’s mirrors, we can align the laser output arbitrarily, and the mirror tilt angle is calculated.