Fresnel-mirror Veselago lenses in solar beam-down optics

The vine above is of a POV-Ray ray-tracing of a spherical Veselago lens, that is, of a sphere with internal refractive index -1. Veselago lenses are well known in metamaterial optics, but a negative refractive index may sound impractical at large scale. As pointed out in an earlier post, Fresnel mirrors with 90-degree facets, under certain symmetry conditions possess exactly the same optics as Veselago lenses. Those particular Fresnel mirrors also have other attractive properties such as zero chromatic aberration, no thermodynamically imposed loss of radiance, identical facet angles, analytical lens profiles (the same as for any refractive lens) and easy ray tracing (in POV-Ray you use "interior { ior -1.0 }".) POV-Ray scene description file here.

One way to describe the optical action at a point on a surface is to center a small sphere on that point and speak in terms of where the incident and exiting rays cross the sphere. If the sphere is infinitesimally small, the surface intersects it in a great circle that we may call its equator. The action to  of an n/-n refractive interface, and thus the action of a Veselago lens, is now easy to describe: an incident ray that enters the sphere at a point A, will exit the sphere at a point A' that is the reflection of A about the equator. A Fresnel mirror facet mounted on the same surface at the same given point can now be characterized by the point M where its outward surface normal exits the sphere. The action of the Fresnel mirror facet is also easily described: an incident ray that enters the sphere at a point A, will exit the sphere at a point A' that is the reflection of A about the point M (i.e., the 180-degree rotation of point A around point M.) Since reflection about a line (mirror reflection) is different from reflection about about a point (180-degree rotation about a point,) no arrangement of Fresnel mirrors is equivalent to a Veselago lens. However, symmetry can create a degenerate case where these two different operations cannot be distinguished. That case is when both the lens and the object being imaged share an axis of rotational symmetry.

In the degenerate case, where both the Veselago lens and the object being imaged share the same axis of rotational symmetry, reflection about a point and a line become indistinguishable, and a Fresnel mirror with 90-degree facets can precisely emulate a Veselago lens.

For example, the yellow circle in the animation above shares an axis of rotational symmetry with the Veselago lens above it, therefore its image is behaving exactly as it would under a spherical Fresnel mirror having 90-degree facets.