A New Perspective on Matrix Representation of Paraxial Geometric Optics using Two Kinds of Three-Matrix Decompositions of the 2× 2 Special-Linear-Group Matrices
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 focus on two kinds of three-matrix decomposition of ABCD matrices to formulate a transformation between the two kinds of decomposition, which can increase or decrease the number of refractive surfaces in the optical configuration while keeping the paraxial specifications fixed. This transformation is not a parameterization of ABCD matrices nor a mapping between existing parameterizations, but a canonical structural relation derived from the canonical three-matrix decompositions. This nature is useful for the optical design of multi-component systems with optimized non-paraxial characteristics.
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BibTeX
@misc{itoh2026perspective,
title = {{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}},
author = {Satoshi Itoh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.01249},
eprint = {2605.01249},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.01249},
note = {CC BY-NC-SA 4.0},
}
RIS
TY - UNPB
TI - A New Perspective on Matrix Representation of Paraxial Geometric Optics using Two Kinds of Three-Matrix Decompositions of the 2× 2 Special-Linear-Group Matrices
AU - Satoshi Itoh
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2605.01249
UR - https://arxiv.org/abs/2605.01249
AB - 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 focus on two kinds of three-matrix decomposition of ABCD matrices to formulate a transformation between the two kinds of decomposition, which can increase or decrease the number of refractive surfaces in the optical configuration while keeping the paraxial specifications fixed. This transformation is not a parameterization of ABCD matrices nor a mapping between existing parameterizations, but a canonical structural relation derived from the canonical three-matrix decompositions. This nature is useful for the optical design of multi-component systems with optimized non-paraxial characteristics.
ER -