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.
Consider a lossless optical system in the paraxial approximation with the same refractive index at its input and output. In ray-height/slope coordinates, writeFor , its optical path length generating function isThe factor is required for physical optical path length; omitting it defines the index-normalized length .
Fermat's principle gives endpoint derivatives and . Substituting and and integrating proves the expression. For uniform free space , the extra optical length is , fixing both sign and normalization. Its quadratic wave phase generates a linear canonical transform. The case is a separate imaging limit.
Articles by others on the same topic
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.