Is beam-shaping necessary?

A Fresnel Veselago lens, without any beam-shaping, can produce an annular focal zone of modest concentration from a solar highbeams field.

Beam-shaping (reshaping divergence) at the Fresnel Veselago lens can potentially produce very high concentration at the oculus—and thus reduce thermal losses from the hot space—but this would come at a cost of complexity and reflection loss. Is this a marginal improvement we could postpone for later?

Since reflection losses in beam-shaping with lenticular lens arrays will probably subtract at least of 5% of the total power, and improving concentration at the oculus by a factor of four would reduce the radiant heat percentage, q, by the same factor, the improvement offered by beam-shaping is something like

0.75q - 0.05.

Assuming we would put off for now a 10% improvement in system power in the interest of simplification, that is, if:

0.75q - 0.05 < 0.1,

or q < 20% .

Radiant flux in the hot space at the 3/4-full temperature of 1936 °K is 800 suns (calculations included in the figure above too-generously assume radiant transfer at the empty temperature of 1250 °C = 1523 °K) this becomes a 20% loss when the incident flux is 4,000 suns. From the diagram above it looks like that is probably about four times more concentration than can be achieved at an annular focal zone without divergence reshaping. Adding a 2D CPC concentrator to the annulus or 2D radiation traps for thermal radiation escaping at wide angles might effectively double the concentration at the annulus to 2,000 suns, but that would still leave us a factor of two away, and we would really rather be at 8,000 suns so that heat loss is reduced to 10%.

A better approach is to bring light directly to a circular focal zone. Calculations in the diagram suggest we can get above 3500 suns without beam-shaping, and additionally we can  block much of the radiant heat loss by suspending an elliptical mirror over the oculus. (Sunlight is shown directly striking the boiler tubes, but this could be avoided by slanting the tubes, arranging them on the inside of a cone rather than a cylinder.)

A Fresnel Veselago lens, without any beam-shaping, can produce a circular focal zone with high concentration.

Is beam-shaping necessary? Likely not.

Center-coordinated packings of circles and ellipses

Certain "magic" aspect ratios allow a close packing of ellipses to be center-coordinated with a hexagonal close-packing of circles. The aspect ratios that work are central polygonal numbers; in this case 3.

Center-coordinated tessellations or packings are relevant in two ways to telescopic heliostats: they describe a way to simultaneously close-pack the primary mirrors (as seen from the sun's perspective) and the secondary mirrors (as seen from the perspective of the beam-down optics,) and they also describe ways that the lenslet arrays in the beam-down optics can be paired in registration with each other. A previous post showed that a tessellation of rectangles can be centroid-coordinated with a tessellation of squares if the aspect ratio of the rectangle is drawn from the set of integers. This post considers packings of ellipses that are center-coordinated with hexagonal close-packings of circles.

In the figure above, an affine transformation that transforms the bold equilateral triangle into the dashed green triangle will also transform the packed circles into the packed ellipses, and use the same set of points for centers. If the bold triangle, together with its decoration of three circular arcs, is considered a truchet tile that generates the packing of circles, shearing and stretching this truchet triangle (along with its decorations) yields a new truchet triangle that generates the packing of ellipses.

The example illustrated in the figure is just the first in a series of possibilities: for instance, we could have translated the apex of the triangle rightward two steps instead of just one. Taking the sides of the equilateral triangle to be unit length, the transformed left side of the triangle has length √3, and this is identically the length of the major axis of the ellipses. Since the area of the ellipses must be equal to that of the original circles (in order to give a center-coordinated packing) the minor axis of the ellipses is therefore 1/√3, and the aspect ratio of the ellipses is 3—or simply the square of the length of the longest side of the new triangle. From the Pythagorean Theorem, the square of that length is, where n is the number of steps the apex has translated:

(1/2 + n)² + (√3/2)²

= 1/4 + n + n2 + 3/4

= n + n2 + 1

= {1, 3, 7, 13, 21, 31, 43, 57, 73, 91...}.

These numbers are known as the central polygonal numbers. 

Therefore, a close-packing of ellipses can be center-coordinated with a hexagonal close-packing of circles provided the aspect ratio of the ellipses is drawn from the set of central polygonal numbers.

The angle of the major axis is rotated by an angle θ from vertical where

θ = arctan((1 + 2n)/√3)).

Any two close-packings of ellipses can be center-coordinated provided that the two varieties of ellipses have equal areas and their respective aspect ratios are drawn from the central polygonal numbers. As mentioned in the previous post, given a centroid-coordinated pattern, it can be transformed by any affine transformation into another centroid-coordinated pattern. In particular, ellipse/ellipse patterns can be transformed to a center-coordinated pattern where one variety of the ellipses are circles. Thus the circle/ellipse patterns are the fundamental ones, the rest can be generated from these by affine transformation.

Center-coordinated ellipse and circle packings: aspect ratio = 3.

Center-coordinated ellipse and circle packings: aspect ratio = 7.

Center-coordinated ellipse and circle packings: aspect ratio = 13. 

Center-coordinated ellipse and circle packings: aspect ratio = 21. 

Centroid-coordinated tessellations

Inequivalent affine transformations can produce identical results when they act on an array of unlabeled points. In this case, shear plus one-dimensional magnification (not show) acts like the identity transformation.

Coordinated tessellations occur in two practical problems that arise in relation to telescopic heliostats: first, the problem of simultaneously packing the primary mirrors (as seen from the sun) and the secondary mirrors (as seen from the beam-down optics;) and second, the problem of simultaneously packing the primary and secondary lenses in the divergence reshapers that are a necessary part of the beam-down optics.

Definition: Two tessellations of the plane are centroid-coordinated when they are overlain such that each tile has its centroid coincident with that of a tile in the other tessellation.

Non-trivial centroid coordination (when the two tessellations are not identical) is possible because, when acting on an array of unlabeled points (like the tile centroids of a tessellation,) inequivalent affine transformations may act identically. In other words, if we apply two inequivalent affine transformations to a tessellation we will generate two different tessellations as a result, but applying the same two transformations to an array of unlabeled points (e.g., the tile centroids) may give identical results.

For example, consider a set of tile centroids that are arrayed in a brick-layer's "stretcher bond" pattern as in the top arrangement in the figure above. Shearing this array of points takes them out of proper stretcher bond order (middle arrangement)—but shearing them some more can bring them back into a new stretcher bond order (bottom arrangement.) The height of the bricks is now too short, but that is easily corrected by composition with another affine transformation that is a magnification in one dimension only. This newly contrived composite transformation and the identity transformation now act identically on the array of tile centroids, but differently on the tessellation. The two tessellations generated by the transforms are centroid-coordinated.

The shearing-plus-magnification transform we need is just a shearing in which only x-coordinates are altered. The magic angles that work for this shearing, are ones for which dots translate in the x-dimension by an integer multiple, n, of the inter-dot distance, d.

For the stretcher bond pattern with shearing angle θ:

(d/ √3) * tanθ = n * d,

tanθ = n √3.


For the Cartesian grid pattern

d * tanθ = n * d,

tanθ = n.

Rectangle tesellations

A centroid coordination of a tessellation of squares with a tessellation of 4:1 rectangles. In this kind of arrangement, the centroids are arranged along oblique lines that rise one rectangle's height in traveling the other rectangle's width. Both kinds of rectangles have the same area.

When two rectangle tessellations, a and b, are centroid coordinated, their heights, h_a and h_b, and widths, w_a and w_b, are related by two integers, m and n. Taking the rectangles of type a to be the taller, and those of type b the wider:

h_a = m h_b

w_b = n w_a.

But the rectangle areas are equal:

h_a * w_a = h_b * w_b

so,

m h_b * w_a = h_b * n w_a

m = n.

This is just the commonsense notion that, if two rectangles have the same area, and one is a factor f times taller, it must also be a factor of f times narrower. For rectangles in centroid-coordinated tessellations the constraint is added that f must be an integer.

Since the affine transformation of a polygon’s centroid is the centroid of the affine transformation of the polygon, once we have found a pair of centroid-coordinated tessellations, applying an affine transformation to both will generate a new pair of centroid-coordinated tessellations. For centroid-coordinated rectangle tessellations, their integer characteristic m is not altered by affine transformation, but we can vary the size and aspect ratio one species of rectangle just as we please by choosing different values of linear magnification in each dimension.

For example, we can choose an affine transformation that turns one species of rectangles into squares. Therefore, every pair of centroid-coordinated rectangle tessellations is related by an affine transformation to another pair of centroid-coordinated rectangle tessellations having the same value of m, but where one species of rectangles are squares and the aspect ratio of the other species of rectangles is m squared.

For example, in designing a divergence reshaper having optical inputs and outputs that are delimited by solid angles of rectangular shape, if the output solid angle must be square in shape, the input solid angle must be a rectangle of aspect ratio m², where m is an integer.