Concentric spherical DNG double refractive index lenses

Views from the heliostat field of three lamps that are DNG spindle hyperboloids. The oculus is colored yellow. Made in POV-Ray.

As the image above indicates, it is not sufficient to consider only the meridional rays at the beam-down optics. It is possible (e.g., the middle lamp above) to image the oculus over the entire height of the lamp, but little of its width.

(I have found that it is more common in the metamaterial literature to speak of double negative or DNG optics when the lens is not a classic Veselago flat lens. So I am switching to those terms.)

Since the lamp must be rotationally symmetric about a vertical axis, the only possible profiles in the sagittal plane are concentric circles. When light rays in the sagittal plane strike two consecutive DNG interfaces, the action is the same as a double mirror. In 2-D optics, a double mirror rotates all rays by twice the angle between the mirror lines. In this case the angle between the mirror lines is the subtense of the interior ray between where it enters the DNG material and where it exits. The direction of the bending is the same as for a circular air cavity in a "glass" medium (such is of course a diverging lens.) The narrower the spacing between the two DNG interfaces, the closer to 1.0 is the refractive index of the surrounding "glass."

Two concentric negative refractive index or DNG interfaces (composing a single-layer spherical DNG lens,) form a diverging lens.

Adding a third concentric circular surface to make a double-layer DNG spherical lens, would get us back to a converging lens at the cost of some optical loss.