Wavefronts and their propagation in astigmatic optical systems

The geometry of the astigmatic wavefront is derived from the symplectic nature of linear optics. It is shown to be paraboloidal. Equations are derived that govern the propagation of such wavefronts through astigmatic systems in general and through thin lenses and across refracting interfaces and homogeneous gaps in particular. The equations allow the generalization of the concept of wavefront curvature or vergence to astigmatic systems. In particular they show how the step-along method of calculating wavefront curvature is generalized. Not only are Keating's earlier conclusions on this topic confirmed but also they are shown to hold under more general circumstances. They hold even when the system contains gradient-index elements such as the natural lens of the eye. Some of the premises used in the earlier study are shown not to be necessary: they are a consequence of symplecticity. The analysis also provides a step-along procedure for calculating wave-front direction. A numerical example in the Appendix shows the application of the step-along method to a particular separated astigmatic system: the back-vertex power of the system is determined as is the equation of the emergent wavefront for a distant object point.