Ray-tracing versus Wave-Tracing: Generalized Method for Computing the Magnification Matrix of Spectacle Lenses

For computing the 2 2× magnification matrix of a spectacle lens in front of an eye, there exist well-established methods in the most cases except for the general case of tilted surfaces. In this general case there are in principle two possibilities: either time-consuming numerical calculations have to be performed, or the linear optics approach involving 4 4× transference system matrices has to be extended to the oblique case and applied to the combined optical system of the eye plus the lens in front of the eye. Whereas in the first approach the result is represented as numerical values, the second method yields an explicit analytic expression which is much more descriptive. However, the effort for computing all the matrix element of the respective 4 4× transference matrices increases considerably if a large number of positions on a lens surface have to be examined with respect to the magnification matrix. Even if the symplectic relations among the components of each relevant 4 4× transference matrix can be exploited in order to decrease this effort, complexity can be reduced much more significantly by applying our new method which directly acts on the 2 2× angular subspace. Based on 2 2× wave-tracing methods, we provide a method which directly achieves to establish an analytic expression for the generally nonsymmetric magnification matrix, and to express its matrix elements in terms of magnification, distortion, torsion, and direction of the principal magnification.