The cap

Inside the lamp, or beam-down optics, of a telescopic heliostat field, the thermal cap is a mirror in the shape of a portion of a sphere centered on the oculus. Its purpose is to prevent thermal back radiation from the occulus from escaping skyward. 

In the far field, the thermal cap produces an umbral region (where view of the oculus is fully blocked) and also penumbral regions where view of the oculus is partially blocked.

A field of telescopic heliostats has its own sort of penumbral region due to the abrupt truncation of the heliostat field at its maximum radius. In order to balance forward and back radiation at the surface of the lamp, these two different kinds of penumbra must be coextensive.  

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.