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

TL;DR

This paper reexamines 2x2 matrix optics by introducing complex ray vectors, right-acting operators, and Fermion algebra, revealing new classifications and invariants in optical systems.

Contribution

It introduces complex height-angle vectors, right-acting matrices, and Fermion algebra into matrix optics, providing a novel classification and analysis of optical systems.

Findings

Complex ray vectors improve optical modeling.

Four classes of optical systems identified with distinct invariants.

Only telescopic and imaging systems possess Lagrange invariants.

Abstract

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 2x2 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.