Shear representations of beam transfer matrices - art. no. 056606

The beam transfer matrix, often called the ABCD matrix, is one of the essen tial mathematical instruments in optics. It is a unimodular matrix whose de terminant is 1. If all the elements are real with three independent paramet ers, this matrix is a 2 X 2 representation of the group Sp(2). It is shown that a real ABCD matrix Can be generated by two shear transformations. It i s then noted that, in para-axial lens optics, the lens and translation matr ices constitute two shear transformations. It is shown that a system with a n arbitrary number of lenses can be reduced to a system consisting of three lenses.