Uniform illumination: the inverse problem of solar beam-down optics

The design of solar beam-down optics is closely related to the following lighting problem:

Given a light source that is a horizontal, incandescent disk, design a luminaire to uniformly illuminate an annulus-shaped parking lot.

This problem is the ray-reversed version of collecting light uniformly from a heliostat field and concentrating all of it onto a circular target. Though we might like to add some more constraints to correspond more exactly to the properties of a field of telescopic heliostats, getting to an optical design that simply provides uniform illumination would be a big first step, illuminating in more ways than one.

The distribution of light from an incandescent disk is Lambertian, which means the disk appears just as bright no matter what angle we observe it from, but we do not have to utilize all of this light. Placing over the incandescent disk an oblate ellipsoidal mirror that images the edge of the disk back onto itself will return nearly all of the emitted light back to the disk. Since there is such an oblate ellipsoid profile passing through every point in space, we can truncate our luminaire wherever need be, switching at that point to the profile of an oblate ellipsoidal mirror, and thus preventing any unnecessary loss of light.

Looking at the inverse problem makes it obvious that we do indeed have here the degenerate case where lens and object share an axis of rotational symmetry—the object we are imaging is the oculus. So we are free to design with Veselago lenses and then use Fresnel mirror optics to precisely emulate them.

Visualizing telescopic heliostat beaming angles

Beaming angle and beam diameter visualized in comparison with the angular diameter of the full moon.

The nominal condition for a telescopic heliostat is 6x linear magnification (producing a beam diameter of six solar or lunar diameters—they are nearly the same) beamed at an angular elevation of 3° (also equal to six lunar or solar diameters.)

The lamp

Elliptical locus for the beam-down optics or lamp. The target point, in yellow, is a focal point of the ellipse. The telescopic heliostat field is in green, and the approximating cylinder is in cyan.

Zoom in on the center of figure above. The black fringes correspond to 0 suns, 100 suns, 200 suns, etc., of thermal radiation when the oculus is at the temperature of the sun. If the profile of the lamp follows the red, elliptical contour up to the height of the cylinder, it will be under about 200 suns of thermal back radiation over most of its surface. In an ideal concentrating system, there would be under an equal, counterbalancing, forward flux of solar radiation.

The profile of the transflective beam-down optics, or, more simply the lamp, is determined by both geometric and thermodynamic constraints.

When telescopic heliostats are aimed at a target point, the desirable geometric constraint that the angle of incidence equal the angle of transflection locates the surface of the lamp on an ellipse having one focus at the oculus and the other at the target point. The top figure shows the elliptical locus in red using the geometry of the previous post. The vertical cyan lines indicate the approximating cylinder.

The lamp needs to be supplemented by a thermal cap, a mirror in the shape of a portion of a sphere centered on the oculus, that reflects thermal radiation back to the oculus rather than letting it escape through the approximately 30° angular radius of open sky.