Ray transfer matrix analysis propagates paraxial ray height and slope with a two-by-two matrix. Free propagation through uses , a thin optical lens uses , and an unfolded concave-mirror reflection uses . Multiply matrices in propagation order, with the first operation at the right.
For forward paraxial slopes and a lossless same-index optical system with and , Fermat's principle gives endpoint derivatives and . Substituting and and integrating proves the displayed optical path. Free space has and increases path length by , fixing the sign. Its quadratic phase generates the associated linear canonical transform; is a separate imaging limit.
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Ray transfer matrix analysis, often referred to simply as matrix analysis in optics, is a mathematical technique used to analyze the propagation of rays through optical systems, such as lenses, mirrors, and other optical components. The fundamental idea is to use matrices to describe the transformations that rays undergo as they pass through different optical elements. This approach is particularly useful in understanding and designing complex optical systems.