A New Perspective on Matrix Representation of Paraxial Geometric Optics using Two Kinds of Three-Matrix Decompositions of the $2\times 2$ Special-Linear-Group Matrices

TL;DR

This paper introduces two novel three-matrix decomposition methods for ABCD matrices in paraxial optics, enabling flexible optical system design with optimized non-paraxial features.

Contribution

It proposes new decomposition techniques for ABCD matrices and a transformation between them, aiding the design of multi-component optical systems with tailored non-paraxial properties.

Findings

Two types of three-matrix decompositions for ABCD matrices are formulated.

A transformation between the two decompositions allows adjusting the number of refraction surfaces.

The methods facilitate optical system design with fixed paraxial specifications and optimized non-paraxial characteristics.

Abstract

We require decomposition methods for the ABCD-matrix formulation in rotationally symmetric paraxial geometric optics when designing a multi-component optical system from a given single paraxial specification (represented by an ABCD matrix) to optimize non-paraxial specifications (e.g., optical aberrations). In this study, we propose two kinds of three-matrix decomposition of ABCD matrices by focusing on the fact that the ABCD matrices have three real-number degrees of freedom. In addition, we formulate a transformation between the two kinds of decomposition for a single matrix, which can increase or decrease the number of refraction surfaces in the optical configuration while keeping the paraxial specifications fixed. This nature is useful for the optical design of multi-component systems with optimized non-paraxial characteristics.