A unified paraxial approach to astigmatic optics

In Gaussian optics properties such as dioptric power, lateral and angular m agnification and thickness are simple scalar concepts. In linear optics, th e optics of thick astigmatic systems, however, these concepts generalize to three-dimensional concepts in some cases (the dioptric power of thin syste ms, for example) and to four-dimensional concepts in general. As a result, the quantitative treatment of these properties in astigmatic systems presen ts challenges to the researcher in optometry, ophthalmology, and vision sci ence. Considerable progress has been made only in the case of dioptric powe r. This paper presents a generalized approach to astigmatic optics which al lows different physical properties to be treated in the same way: the theor y is unified and, in a sense, complete. Mathematical and statistical method s developed for treating one concept become directly applicable to others. The paraxial optical properties of any optical system are completely define d by the 4 x 4 ray transfer matrix, called here the tray) transference. The transference defines four fundamental properties of an optical system, ten tatively called here positional magnification, optical thickness, divergenc e, 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 express ed as the dioptric power matrix F. The four fundamental optical properties A, B, C, acid D, and the derived property F, despite being different physic ally, all have the same underlying mathematical structure. This fact is exp loited in developing a unified theory. The theory is complete in the sense that the fundamental properties fully characterize the paraxial optics of a ny system. The paper presents a general treatment that applies to any of th e 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 th e general theory. The result is the resolution of a number of issues of cur rent interest to the researcher. It is shown, for example, that root-mean-s quared (curvital) power, root-mean-squared torsional power, and length of t he power vector (or dioptric strength) have a Pythagorean relationship, the power vector being the hypotenuse. Mean-squared curvital and torsional pow ers are in effect the area enclosed by polar profiles of curvital and torsi onal power, respectively. The full character of dioptric power cannot be re presented by a single vector in the usual sense of the term. Two vectors ar e required: they are the meridional (vector) power and the orthogonal (vect or) power, both of which are associated with the reference meridian. The po wer 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 (t he meridional component of the meridional power) and perpendicular to (the orthogonal component of the meridional power) the meridian. In the literatu re, these components are the curvital power and the negative of the torsion al 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 fundamen tal property. The theory described here is ray-based. The concept of the wa vefront is unnecessary. The many formulas and concepts that apply in the co ntext of dioptric power apply directly to the fundamental properties as wel l. The theory has the potential to provide a complete framework for future studies of astigmatic systems and could systematize the approach to and enh ance the knowledge of astigmatism.