Transflective beam-down optics

Under certain constraints there are only four kinds of optical surfaces: windows, mirrors, retroreflectors, and transflectors.

As mentioned in the previous post, thermodynamics does not require that light arrive and depart from an optical surface at the same angle to the surface normal, but everything is simpler and more efficient in practice if this is the case. A further sensible and practical constraint is that all three vectors (the surface normal, the incident ray, and the emergent ray) all share the same plane. Under those two constraints, there are only four kinds of optical surfaces:

• window: ray emerges on the opposite side of the surface, and in the same direction as the incident ray,
• retroreflector: ray emerges on the same side of the surface, and in the same direction as the incident ray,
• mirror: ray emerges on the same side of the surface, but not in the same direction as the incident ray,
• transflector: ray emerges on the opposite side of the surface, but not in the same direction as the incident ray.

If we know which of these four types of optical surface we are dealing with, the geometry is completely settled, because, under the assumptions, we know both the plane of the emergent ray and the angle it makes with the surface normal.

Windows and retroreflectors, of course, do not form images. Of the two remaining possibilities— transflector or mirror—the former is more practical for solar beam-down optics because a transflector can be mounted directly on the ground rather than atop a tower.

Note that the reflected ray and the transflected ray are always collinear and oppositely directed—so what is a real image for one becomes a virtual image for the other. Imaging transflectors take on the same conic-section profiles as imaging mirrors, but the real-or-virtual property of the image is switched. Beam-down optics for an array of telescopic heliostats must receive light that is propagating nearly horizontally and form a real image at the oculus. Thus the shape we need is the same as that of a mirror that takes horizontal light and forms a virtual image at the oculus. That shape is a surface of revolution whose profile is a parabola having a nearly horizontal axis.

Parabolic curves (representing the geometric constraint on the beam-down optics) superimposed on the back radiation from the oculus (representing the thermodynamic constraint.) Perhaps the outermost of these profiles would operate at a practical level of flux and divergence.

We don't have a lot of freedom in varying the surface determined by the geometrical constraint. We can hope that there is a focal length for the parabola that gives a surface that approximately satisfies the thermodynamic constraint as well. Failure to satisfy the thermodynamic constraint will mean that we either waste light already collected by the heliostat field, or we waste thermodynamic efficiency in converting that collected light to our intended purpose—which really amounts to the same thing.

From the diagram above, it can be seen that the flux on the beam-down optics must increase gradually from zero at the bottom, reach a maximum about halfway up, and then decrease—but flux is still going to be rather high wherever the parabolic profile tops out. At that point the flux must abruptly drop to zero since that light would be overshooting the beam-down optics. The heliostat field—and its targeting—must produce this kind of flux distribution. Even though the flux at the bottom should build up slowly from zero, it will not be economical to operate beam-down at low flux levels. There needs to be a truncation of the optics near the bottom of the parabola as well, and likewise an abrupt drop-off of flux there as well.

Outside these two cutoff angles spherical mirrors are needed to reflect back-radiation toward the oculus.