Revisiting 2x2 matrix optics: Complex vectors, Fermion combinatorics, and Lagrange invariants

We propose that the height-angle ray vector in matrix optics should be complex, based on a geometric algebra analysis. We also propose that the ray’s 2×2 matrix operators should be right-acting, so that the matrix product succession would go with light’s left-to-right propagation. We express the propagation and refraction operators as a sum of a unit matrix and an imaginary matrix proportional to the Fermion creation or annihilation matrix. In this way, we reduce the products of matrix operators into sums of creation-annihilation product combinations. We classify ABCD optical systems into four: telescopic, inverse Fourier transforming, Fourier transforming, and imaging. We show that each of these systems have a corresponding Lagrange theorem expressed in partial derivatives, and that only the telescopic and imaging systems have Lagrange invariants.