Lateral magnification matrix from the dioptric power matrix formalism in the paraxial case

Previous studies have highlighted that power matrices fully characterize the concept of dioptric power of any astigmatic surface. Thus, the basic equations in physiological optics can be generalized using the matrix formalism of the dioptric power. Among others, lateral magnification has also been interpreted as a matrix but mainly concerning magnification modification induced by spectacle correction of refractive error.

To provide a fresh look into a novel paraxial formulation for the assessment of the lateral magnification using power matrices and in presence of astigmatism for thin and thick imaging systems in general.

Linear optics provides the frame to generalize into a matrix the lateral magnification concept. Using the power matrix formalism, a lateral magnification matrix is derived in virtue of the dioptric power matrix and the object's reduced axial object distance for the paraxial case. In addition, two different degrees of approximation (thin lens and distant object approximations) are analyzed to further simplify the calculations.

A general formulation of the lateral magnification matrix is obtained and validated by numerical examples showing its applicability to different examples in geometrical and physiological optics. As particular case of interest, the degree of asymmetry of the lateral magnification matrix has been derived from the degree of asymmetry of the dioptric power matrix when dealing with obliquely crossed astigmatic thick lenses.

The new formulation is applicable under paraxial approximation and is useful for arbitrary thin and thick imaging systems in any media of homogeneous index of refraction (air and others) and including obliquely crossed astigmatic surfaces. The proposed formulation also yields in a novel interpretation of the lateral magnification matrix concept.