One analytic form for four branches of the ABCD matrix

The two-by-two ABCD matrix plays a central role in optical sciences. The four elements of this matrix are all real, and its determinant is one. Thus, it has three independent parameters. Yet, it has a rich mathematical content which could lead to interesting results in physics. It is generally assumed that this matrix can be diagonalized by a rotation, but this is not the case as shown in our previous paper [1]. We demonstrated there that this matrix can be brought to an equi-diagonal matrix by a rotation, and then by a squeeze to one of the following four Wigner matrices.