Is it possible to think an example of refraction in which Fermat principle involve a maximum without using reflection? In a question What is incorrect about the original statement of Fermat's principle? is showed an example of reflection in which Fermat principle involve a maximum, and in comments is said that it could be interesting an example involving refraction but question is left pending. I can think examples in which Fermat principle with refraction involve minima (Snell's law) or stationary optical path length (lenses converging light from a point source to another point), but I can't find a refraction example that involve a maximum.
 A: *

*First of all, note that a stationary optical path of finite length can never be a local maximum among virtual paths (even if we are allowed to use reflections), because it is always possible to locally device a longer virtual detour.
(The previous sentence does not imply that the set of stationary optical paths of finite length couldn't have a maximal element.)


*From now on let us assume that the refractive index $n({\bf r})\geq 1$ is a smooth function of position ${\bf r}$, and that there are no mirrors.


*It is still possible to artificially mimic reflection (as long as the angle of incidence is not zero) by letting the $n({\bf r})$ become smaller in the "reflection layer" (and bigger in the bulk). Think e.g. of an optical fiber with a smooth gradual fusing of the cladding, cf. Fig. 1.

$\uparrow$ Fig. 1a & 1b. A typical optical fiber. The reader should imagine that refractive index $n({\bf r})$ varies smoothly between the core and the cladding, thereby effectively creating a tubular mirror.


*Similarly, we can approximate/mimic many optical systems that contains mirrors, or say, boundaries between 2 media. E.g. the systems discussed
in OP's linked Phys.SE post and links therein.
