Step-along vergence procedures in stigmatic and astigmatic systems.

The traditional step-along vergence procedure applies to stigmatic systems, that is, systems that are not astigmatic. Computation is disrupted when a focus coincides with a thin lens or refracting surface. A small change to the procedure results in a modified procedure which overcomes the computational problems. The modified procedure is easier to execute than the traditional procedure and allows one to write down useful equations directly. Among the formulae are those for back-vertex power. A step-along vergence procedure also exists for astigmatic systems. It makes use of the dioptric power matrix and the reduced vergence matrix. Computational problems arise when a point or line focus coincides with a thin lens or refracting surface; however they are not overcome by an analogous modification to the procedure. Nevertheless the modified procedure has some advantages including the fact that, as for stigmatic systems, it allows one to write down useful formulae directly. Stepwise calculations of vergence are sometimes performed backward through a system; the advantages and disadvantages described for step-along procedures holds for such step-back procedures as well.