Caustics and wavefronts produced by multiple refraction-reflections through plane-parallel plates

We study the formation of caustics and wavefronts produced by multiple refraction-reflections through a plane-parallel transparent plate, assuming a point source placed at an arbitrary position along the optical axis. The caustic surfaces are obtained by using the envelope's method. Subsequently, the wavefronts are directly related to the involutes, which are associated with the envelopes for all the rays. Hence by using the Malus-Dupin theorem, we obtain their respective wavefronts produced by multiple refraction-reflections through a plane-parallel transparent plate. On the other hand, we implement Huygens' principle to obtain the wavefronts leaving the plate after undergoing multiple reflections inside the plate, which we have called zero-distance phase wavefronts. Finally, we establish the correspondence between the wavefronts obtained by Huygens' principle and the involutes associated with caustic surfaces; they are brought in coincidence assuming parallel curves from each other.