The Legendre transformation as a way of constructing a phase portrait of beams of light rays

The Legendre transformation is used to describe limits of beams of light rays in homogeneous space and optical systems. It is shown that by means of a Legendre transformation, a light ray, a beam of rays and an envelope of beams of light rays are converted into a point, a set of points and a border of a set of points, respectively, in a phase space (phase portraits). Application of a Legendre transformation to an envelope of the beam of light rays passing through an optical fiber results in a border of the phase stop. It is found that phase portraits of beams of light rays simplify description of vignetting in high aperture and wide angle objectives.