Light does not need to reflect first to form a caustic.
If you have a light or sound wave front and light
propagates perpendicular to a surface. For each point on the
surface, there will be an instance, where the light ray will focus
with neighboring rays.This place is called the caustic. In the animation
to the left, you see this. One can compute the caustic by drawing at
every point of the curve a line perpendicular to the curve and chose
the point on that line with distance 1/kappa(t) from the original front.
You see the cusp also in the coffecup caustic. It is stable. If you
change the shape of the initial wave front a bit, then the cusp will
still be there.