# 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.

• In non-imaging optics light trapping structures (using refraction) are used to maximise the entropy of incoming light rays so they ergodically occupy the phase space (essentially by randomising their directions). This increases the path length of rays by $4n^2$. This is more of a statistical mechanics observation of the whole beam, rather that single particle principle like least action. Dec 25, 2020 at 12:25

2. 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.
3. 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.