Veselago lens optics

A Veselago lens is a refractive lens with a refractive index of negative one. Though Veselago lenses, like all refractive lenses, obey Snell's Law of Refraction, they are also closely related to mirrors. The chart below summarizes mirror optics.

A short review of mirror optics. Rotating any light ray 180° about the point where it strikes the optical surface yields a diagram of a Veselago lens. In mirror optics the heterogeneous object/image pairs (i.e., real/virtual, virtual/real) are hyperbolic, and the homogeneous pairs are elliptical. In Veselago optics the opposite is the case. Underlying diagram quoted from "Mirrors" by Mike George.

Observe that just two kinds of mirror profile arise in practical cases: elliptical or hyperbolic (that is, conics of eccentricity e<1 or e>1.) There is not a practical need to consider parabolic profiles (e = 1), since there is no practical need to translate an object or image point "all the way" to infinity.

Of the four possible object/image cases, e.g., real/virtual, real/real, etc., only two classes are actually distinct. In two heterogeneous cases, real/virtual and virtual/real, we get the other case by simply reversing the direction of the light rays, and no new case is presents itself when we use mirror's other side. In the two homogeneous cases, real/real and virtual/virtual, we get the other case by simply using the mirror's other side.

A Veselago lens is like a transmissive mirror or "transflector." A mirror optics diagram can be transformed into a Veselago optics diagram by simply rotating one light ray 180° about the point where it intersects the optical surface. Additionally rotating the other light ray 180° would just bring us back to a mirror diagram—one that happens to use the other side of the mirror. The procedure is general: an odd number of 180° ray rotations about the same point carries us into the other domain, an even number of 180° ray rotations carries us back.

Rotating a ray flips the nature of one of the foci, either from real to virtual or from virtual to real, and therefore also flips the object/image type from heterogeneous to homogeneous or vice versa. Thus we these two domains of optics can be identified by the curvatures of their homogeneous realms: mirrors are homogeneous-elliptical, Veselago lenses are homogeneous-hyperbolic. That is, one would use an elliptical mirror to transport light from one point to another; a hyperbolic Veselago lens can accomplish the same end.

Any of the elliptical profiles would serve to transfer light between the two foci by mirror. Any of the hyperbolic profiles would serve to transfer the light by Veselago lens. Note that a planar disk is a degenerate ellipsoid—though not a very useful one; a plane is a degenerate hyperboloid, and a very useful one indeed.

It is sometimes surprising that a flat Veselago lens (i.e., a planar n/-n interface) yields a real image of a real object, but this is just the homogeneous counterpart of a flat mirror yielding a virtual image of a real object. Perhaps we do not take the virtual images that are all around us seriously enough.

A starting point for the design of the beam-down optics for a field of telescopic heliostats would be to place one of the foci on the perimeter of the heliostat field, and the other focus on the farther edge of the oculus, and then find the smallest hyperbola that works. Rotating this hyperbola around the rotational symmetry axis of the oculus give a candidate Veselago lens for the beam-down.

Ronian Siew has shown that amazing things can be accomplished in lens design using negative index metamaterials or NIMs and more than one interface. The two examples reproduced below, though they use refractive indices more negative than minus one, suggest what can be accomplished with multiple-interface Veselago lenses.

A negative refractive index (-1.517) singlet designed by Ronian Siew.

A negative refractive index (-2.0) singlet designed by Ronian Siew.