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.

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.

The Veselago lens, transflection, and negative-one refractive index (nori) optics

Optical diagram of a single-interface Veselago lens. Image quoted from Yee Sin Ang et al., "Retro reflection of electrons at the interface of bilayer graphene and superconductor."

I just learned of the Veselago lens, an imaging application of negative-one refractive index. The diagram above shows the simplest case, a single refractive index interface producing a real image: in fact, every object in n_i will have a real image in -n_i, and vice-versa. Uses of the Veselago lens (typically with two interfaces, as shown below) are being actively investigated by nano-materials scientists interested in genuinely propagating waves in a negative index medium. As we have seen, such is not necessary in solar power optics, if the divergence pattern has the right symmetry, a Fresnel mirror with its facets tilted 90° has optics identical to an n_i/-n_i interface.

Optical diagram of a two-interface Veselago lens. Image quoted from Lukas Novotny, "Principles of Nano-Optics."

It is helpful to look at the Veselago lens from all four perspectives: retroflection, transmission, reflection, and transflection. If a room has a retro-reflective plane instead of a window, light will be returned directly back to each object in the room, forming a real image that happens to coincide in every case with the object itself, and thus has the same handedness. If there is indeed a transmissive plane where the window should be, an observer on the outside perceives a virtual image that happens to coincide in every case with the object itself, and thus has the same handedness. If there is a mirror (reflective plane) where the window should be, there is virtual image of the room (turned inside-out) in the space outside, and thus handedness is reversed. If there is a transflective plane (or equivalently, the n_i/-n_i interface of a Veselago lens) there is a real image of the room (turned inside-out) in the space outside, and thus handedness is reversed.

Transflection as a negative-one refractive index lens

Negative refractive index materials have not been found in nature so far, but the equivalent effect has been produced in artificially structured materials, or metamaterials. Substituting a negative refractive index into Snell's Law of Refraction shows that the angle of refraction also becomes negative: the refracted light will be bent back to the same side of the surface normal as the incident light. For solar power optics the case of a refractive index equal to negative one is of unique interest: such a lens can have an air-filled interior and exterior (the only structured material needed is at the interface) and the throughput is potentially 100% (at least the Second Law of Thermodynamics will not offended if it is.) Lens optics with negative-one refractive index (nori for short) is both a more precise and more concise description of what I have earlier described as transflection.

In talking about the divergence pattern of optical rays passing through a point, a useful homely analogy is to place the point in question at the center of a transparent earth: then incident rays can be specified by their geographic source, i.e., the geographic point they pass through on their way toward the earth's center; and the emergent rays can be described by their geographic sink, the geographic point they pass through on their way out. I further adopt an orientation of the globe such that the surface normal at the point of interest is a ray from the earth's center toward its north pole.

Using the analogy, we can describe the action of the four basic kinds of optical surfaces—window, mirror, retroreflector, and transflector—visually in terms of where they send rays sourced by a familiar geographic area, let's say the lower 48 American states.

Rays incident on a retroreflector completely reverse course and exit through the same geographic point they entered through:

Retroreflector

Rays incident on a window pass through undeflected, so each ray exits through the antipodes of its source (which, it turns out, is not Australia after all!)

Window

Rays reflecting from a mirror (having its surface normal pointing toward the north pole) will exit at the same latitude as their source, but 180° opposite in longitude.

Mirror

Mirror (with surface normal in center of view)

Rays exiting a transflector have the same longitude as their source, but their latitude is reflected to the opposite hemisphere.

Transflector

Note the similarity of the exit patterns for a transflector and a mirror when seen in the polar view (i.e., with the mirror normal in the center of the image.) For the mirror, the relationship between the incident and emergent ray distributions is a point reflection about the mirror's surface normal. This is equivalent to a 180° rotation about the point—left and right are not reversed. For the transflector, the emergent ray distribution is a reflection over a line (the equator)—and left and right are reversed. (Of course, in one sense, a mirror really does reverse left and right, but our way of describing ray directions is producing its own reversal as the plot for a window shows.)

Reflection about a point. (Image quoted from wikimedia.)

Reflection over a line. (Image quoted from http://geometry.freehomeworkmathhelp.com)

Reflection over a line can be equivalent to reflection about a point if the object has an axis of symmetry perpendicular to the line of reflection.

As diagrammed in the image above, if a divergence pattern has an axis of symmetry that perpendicularly intersects the mirror line, then we have a special case where reflection about the point of intersection is indistinguishable from reflection over the line. Therefore, given such a symmetry in the beam divergence, a Fresnel mirror with 90° facets can act as a nori lens. The radial symmetry of the heliostat field and the central optics about a common axis guarantees that the needed symmetry will exist.

Beam divergence at the lamp (diagrammed in pink) will approximate a segment of a circle of latitude. In this view, the Fresnel mirror's surface normal will lie somewhere on the 0° line of longitude, fulfilling the constraint that it lie on an axis of symmetry for the divergence pattern. Therefore, for this divergence pattern, a Fresnel mirror with 90° facets can precisely simulate a nori lens.

 Gunnar Dolling and Martin Wegener, and R.Varalakshmi have demonstrated that POV-Ray correctly ray-traces in media with negative index of refraction.