Classification of lossless first-order optical systems and the linear canonical transformation

Based on the eigenvalues of the ray transformation matrix, a classification of ABCD systems is proposed and some nuclei (i.e., elementary members) in each class are described. In the one-dimensional case, possible nuclei are the magnifier, the lens, and the fractional Fourier transformer. In the two-dimensional case we have-in addition to the obvious concatenations of one-dimensional nuclei-the four combinations of a magnifier or a lens with a rotator or a shearing operator, where the rotator and the shearer are obviously inherently two-dimensional. Any ABCD system belongs to one of the classes described in this paper and is similar (in the sense of matrix similarity of the ray transformation matrices) to the corresponding nucleus. Knowledge of a nucleus may be helpful in finding eigenfunctions of the corresponding class of first-order optical systems: one only has to find eigenfunctions of the nucleus and to determine how these functions propagate through a first-order optical system.