On the validity of the paraxial eikonal in catastrophe optics

The paraxial eikonal Psi =Z-f(r)+(R-r) 2 /2Z is frequently used in catastrophe optics. The authors show that, except in the far-field limit, it can lead not only to quantitative errors, but to major qualitative errors unless the singularity involved is three-determinate. If at some point on the local optical axis (R=0) the exact eikonal has a cuspoid singularity or an umbilic singularity with nonzero four-jet, then the paraxial eikonal generically has a four-determinate singularity at that point. The consequences for cusp and swallowtail foci are explored in detail. They show that such errors do not result from any obvious failure to satisfy the conditions under which Psi is derived.