Optical magnification matrix: near objects in the paraxial case

For appropriate patient management understanding, the problems due to the change of image size in the near magnification matrix of spectacles are important. The purpose here is to derive an equation for the paraxial 2 x 2 magnification matrix of spherocylindrical two-surface optical instruments (spectacles) in front of an eye at a finite object distance.

A 2 x 2 stepalong method for the contribution factors to the magnification matrix is developed, starting from a definition within a ray-tracing picture before translating it into quadratic-order wave-tracing concepts. Extensive use is made of the wave-tracing description of propagating wavefronts by 2 x 2 vergence matrices for deriving an expression for the magnification matrix in ophthalmic terms.

A 2 x 2 magnification matrix as a function of the dioptric power matrices of the interfaces as well as the object distance is given explicitly. The expression is an intuitive product representation possessing the familiar appearance of a shape factor times a power factor. Additionally, however, the result contains now a scaling prefactor depending on the finite object distance, which would not be present for objects at infinity.

The result is a generalization of the formerly known result for the paraxial magnification matrix for infinite object distance. The effect of the finite object distance correction is small but not negligible. Moreover, the methods for deriving this result serve as a starting point for an even more general treatment of oblique incidence in a future work.