Equivalent dioptric power asymmetry relations for thick astigmatic systems

For optical systems consisting of separated obliquely crossed toric interfaces, the equivalent dioptric power has principal meridians that are not necessarily orthogonal to each other. In this case it takes four parameters to specify the equivalent power. A set of parameters convenient for ophthalmic optics consists of three traditional spherocylindrical parameters SC x theta together with a dioptric asymmetry parameter g. The parameter g has been described as "so far" being entirely mathematical in nature. The purpose of this paper is to develop further optical knowledge about the equivalent power asymmetry g. The method was a theoretical and numerical study involving optics and the dioptric power matrix theory. Among the results of this study are a number of new equations involving g that clarify the relationships between the nonorthogonal principal meridians and the power and axis meridians of SC x theta, as well as explicitly illustrating the parameters that can increase or decrease g. It is also pointed out that the asymmetry g is formally identical to the circular astigmatism that has previously been presented in discussions of ray vector deflection fields (and is used in the Humphrey Lens Analyzer measurements). In conclusions, the theoretical relations presented here provide optical insight into the equivalent dioptric power asymmetry and the parameter g. The relations and insight can assist further developments.