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.