A unified paraxial approach to astigmatic optics

In Gaussian optics properties such as dioptric power, lateral and angular magnification and thickness are simple scalar concepts. In linear optics, the optics of thick astigmatic systems, however, these concepts generalize to three-dimensional concepts in some cases (the dioptric power of thin systems, for example) and to four-dimensional concepts in general. As a result, the quantitative treatment of these properties in astigmatic systems presents challenges to the researcher in optometry, ophthalmology, and vision science. Considerable progress has been made only in the case of dioptric power. This paper presents a generalized approach to astigmatic optics which allows different physical properties to be treated in the same way: the theory is unified and, in a sense, complete. Mathematical and statistical methods developed for treating one concept become directly applicable to others. The paraxial optical properties of any optical system are completely defined by the 4 x 4 ray transfer matrix, called here the (ray) transference. The transference defines four fundamental properties of an optical system, tentatively called here positional magnification, optical thickness, divergence, and directional magnification. They are the four 2 x 2 submatrices A, B, C, and D of the transference. Each fundamental property is a modification of a familiar concept. Divergence is the negative of dioptric power expressed as the dioptric power matrix F. The four fundamental optical properties A, B, C, and D, and the derived property F, despite being different physically, all have the same underlying mathematical structure. This fact is exploited in developing a unified theory. The theory is complete in the sense that the fundamental properties fully characterize the paraxial optics of any system. The paper presents a general treatment that applies to any of the five properties. The implications are far reaching and extend beyond what can be described in the paper. Dioptric power of thin systems is treated as a particular application of the general theory. The result is the resolution of a number of issues of current interest to the researcher. It is shown, for example, that root-mean-squared (curvital) power, root-mean-squared torsional power, and length of the power vector (or dioptric strength) have a Pythagorean relationship, the power vector being the hypotenuse. Mean-squared curvital and torsional powers are in effect the area enclosed by polar profiles of curvital and torsional power, respectively. The full character of dioptric power cannot be represented by a single vector in the usual sense of the term. Two vectors are required: they are the meridional (vector) power and the orthogonal (vector) power, both of which are associated with the reference meridian. The power along a meridian (often thought of as a scalar or as two scalars) is a vector, the meridional power. This meridional power has components along (the meridional component of the meridional power) and perpendicular to (the orthogonal component of the meridional power) the meridian. In the literature, these components are the curvital power and the negative of the torsional power, respectively. The paper also examines the generalization of these results to the dioptric power of thick systems. Dioptric power is not a fundamental optical property but a derived property. Divergence, the negative of dioptric power, is the corresponding fundamental property. The theory described here is ray-based. The concept of the wavefront is unnecessary. The many formulas and concepts that apply in the context of dioptric power apply directly to the fundamental properties as well. The theory has the potential to provide a complete framework for future studies of astigmatic systems and could systematize the approach to and enhance the knowledge of astigmatism.