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.

Modeling parabolic primary mirrors in Povray

Simulation of a bendable parabolic primary mirror in Povray.

This povray file ray-traces the sunlight reflected from a bendable parabolic primary at any chosen time and latitude. Actual bending of the mirror is not simulated, but the focal length of the parabola adjusts to the sun's zenith distance.

I am rather uncertain of the ray-trace (photons) settings in the .pov file, but it works. 

Two-mirror heliostats

Optical diagram of a two-mirror, or telescopic, heliostat. All profiles are parabolic and confocal. The primary mirror is gimbaled and bendable to accommodate the changing angles of the sun. The secondary mirror is fixed.

Definition: In solar engineering, a two-mirror heliostat, or telescopic heliostat, is a heliostat composed of two off-axis, parabolic mirrors, arranged in the configuration of a Mersenne telescope.

The larger primary mirror of a two-mirror heliostat is gimbaled and thin-shell bendable to accommodate the apparent movement of the sun. The small secondary mirror, which redirects concentrated sunlight toward the target, is rigid and fixed. The advantage of two-mirror heliostats over conventional one-mirror heliostats is that they can be packed closely together without incurring blocking losses—even when aiming at a target of low angular elevation. The optics and tracking motions of two-mirror heliostats are identical over the entire field.

The telescopic heliostat as a Mersenne telescope

Optical diagram of a Mersenne telescope. This is Mersenne's illustration of 1636 reoriented to be observing the zenith. Image quoted from Fred Watson, "Stargazer: the life and times of the telescope."

I learned recently that the afocal, parabola/parabola Cassegrain telescope is more properly termed a Mersenne telescope. Marin Mersenne's 1636 design for a two-mirror, reflecting telescope is the basis of the telescopic heliostat. In a telescopic heliostat, both mirrors of the Mersenne telescope (the objective, or primary; and the eyepiece, or secondary) are off-axis parabolas. The objective of a telescopic heliostat also operates with varying off-axis angles of incidence and in consequence must be able to bend to adjust its profile.

Divergence rotation

Every centroid-coordinated tessellation of circles and ellipses (left) can be affine-transformed into a centroid-coordinated tessellation of ellipses that are identical but rotated 90° (right.)

A special case of divergence reshaping occurs when the new divergence pattern actually has the same shape, but is oriented differently.

Any coordinated tessellation of circles and ellipses can be transformed by affine transformation into a design for a divergence reshaper that rotates an incident elliptical beam 90°. If the ellipses in the coordinated tessellation have aspect ratio A, an affine shrinking of both circles and ellipses in the direction of the major axis of the ellipses by √A, yields a coordinated tessellation of identically shaped ellipses, all of aspect ratio √A, lying at right angles to each other.

From the previous result that hexagonally-packed coordinated tessellations of circles and ellipses exist for aspect ratios drawn from the central polygonal numbers, {1, 3, 7, 13, 21, 31, 43, 57, 73, 91…}, the magic aspect ratios for a tessellation of hexagonally-packed ellipses that can centroid coordinate with a 90° rotation of itself are the square roots of the central polygonal numbers, {√1, √3, √7, √13, √21, √31, √43, √57, √73, √91…}.

Another way to rotate the divergence pattern of a beam by 90° is to reflect it in a planar Rabl mirror, placing the dihedral line of the mirror at 45° to the major axis of the incident and rotated beams.

Cross-section of a planar Rabl mirror (3M prismatic film.) Image quoted from K.G. Kneipp, "Use of prismatic films to control light distribution."

A divergence rotating mirror: the divergence of a beam of light is rotated 90° by normal reflection from a Rabl prismatic mirror when the prisms'  dihedral lines lie at a 45° angle to the principal axes of the incident—and, as well, the reflected—beams.

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.