Veselago lens optics

A Veselago lens is a refractive lens with a refractive index of negative one. Though Veselago lenses, like all refractive lenses, obey Snell's Law of Refraction, they are also closely related to mirrors. The chart below summarizes mirror optics.

A short review of mirror optics. Rotating any light ray 180° about the point where it strikes the optical surface yields a diagram of a Veselago lens. In mirror optics the heterogeneous object/image pairs (i.e., real/virtual, virtual/real) are hyperbolic, and the homogeneous pairs are elliptical. In Veselago optics the opposite is the case. Underlying diagram quoted from "Mirrors" by Mike George.

Observe that just two kinds of mirror profile arise in practical cases: elliptical or hyperbolic (that is, conics of eccentricity e<1 or e>1.) There is not a practical need to consider parabolic profiles (e = 1), since there is no practical need to translate an object or image point "all the way" to infinity.

Of the four possible object/image cases, e.g., real/virtual, real/real, etc., only two classes are actually distinct. In two heterogeneous cases, real/virtual and virtual/real, we get the other case by simply reversing the direction of the light rays, and no new case is presents itself when we use mirror's other side. In the two homogeneous cases, real/real and virtual/virtual, we get the other case by simply using the mirror's other side.

A Veselago lens is like a transmissive mirror or "transflector." A mirror optics diagram can be transformed into a Veselago optics diagram by simply rotating one light ray 180° about the point where it intersects the optical surface. Additionally rotating the other light ray 180° would just bring us back to a mirror diagram—one that happens to use the other side of the mirror. The procedure is general: an odd number of 180° ray rotations about the same point carries us into the other domain, an even number of 180° ray rotations carries us back.

Rotating a ray flips the nature of one of the foci, either from real to virtual or from virtual to real, and therefore also flips the object/image type from heterogeneous to homogeneous or vice versa. Thus we these two domains of optics can be identified by the curvatures of their homogeneous realms: mirrors are homogeneous-elliptical, Veselago lenses are homogeneous-hyperbolic. That is, one would use an elliptical mirror to transport light from one point to another; a hyperbolic Veselago lens can accomplish the same end.

Any of the elliptical profiles would serve to transfer light between the two foci by mirror. Any of the hyperbolic profiles would serve to transfer the light by Veselago lens. Note that a planar disk is a degenerate ellipsoid—though not a very useful one; a plane is a degenerate hyperboloid, and a very useful one indeed.

It is sometimes surprising that a flat Veselago lens (i.e., a planar n/-n interface) yields a real image of a real object, but this is just the homogeneous counterpart of a flat mirror yielding a virtual image of a real object. Perhaps we do not take the virtual images that are all around us seriously enough.

A starting point for the design of the beam-down optics for a field of telescopic heliostats would be to place one of the foci on the perimeter of the heliostat field, and the other focus on the farther edge of the oculus, and then find the smallest hyperbola that works. Rotating this hyperbola around the rotational symmetry axis of the oculus give a candidate Veselago lens for the beam-down.

Ronian Siew has shown that amazing things can be accomplished in lens design using negative index metamaterials or NIMs and more than one interface. The two examples reproduced below, though they use refractive indices more negative than minus one, suggest what can be accomplished with multiple-interface Veselago lenses.

A negative refractive index (-1.517) singlet designed by Ronian Siew.

A negative refractive index (-2.0) singlet designed by Ronian Siew.

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.

Fresnel-mirror Veselago lenses in solar beam-down optics

The vine above is of a POV-Ray ray-tracing of a spherical Veselago lens, that is, of a sphere with internal refractive index -1. Veselago lenses are well known in metamaterial optics, but a negative refractive index may sound impractical at large scale. As pointed out in an earlier post, Fresnel mirrors with 90-degree facets, under certain symmetry conditions possess exactly the same optics as Veselago lenses. Those particular Fresnel mirrors also have other attractive properties such as zero chromatic aberration, no thermodynamically imposed loss of radiance, identical facet angles, analytical lens profiles (the same as for any refractive lens) and easy ray tracing (in POV-Ray you use "interior { ior -1.0 }".) POV-Ray scene description file here.

One way to describe the optical action at a point on a surface is to center a small sphere on that point and speak in terms of where the incident and exiting rays cross the sphere. If the sphere is infinitesimally small, the surface intersects it in a great circle that we may call its equator. The action to  of an n/-n refractive interface, and thus the action of a Veselago lens, is now easy to describe: an incident ray that enters the sphere at a point A, will exit the sphere at a point A' that is the reflection of A about the equator. A Fresnel mirror facet mounted on the same surface at the same given point can now be characterized by the point M where its outward surface normal exits the sphere. The action of the Fresnel mirror facet is also easily described: an incident ray that enters the sphere at a point A, will exit the sphere at a point A' that is the reflection of A about the point M (i.e., the 180-degree rotation of point A around point M.) Since reflection about a line (mirror reflection) is different from reflection about about a point (180-degree rotation about a point,) no arrangement of Fresnel mirrors is equivalent to a Veselago lens. However, symmetry can create a degenerate case where these two different operations cannot be distinguished. That case is when both the lens and the object being imaged share an axis of rotational symmetry.

In the degenerate case, where both the Veselago lens and the object being imaged share the same axis of rotational symmetry, reflection about a point and a line become indistinguishable, and a Fresnel mirror with 90-degree facets can precisely emulate a Veselago lens.

For example, the yellow circle in the animation above shares an axis of rotational symmetry with the Veselago lens above it, therefore its image is behaving exactly as it would under a spherical Fresnel mirror having 90-degree facets.

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.

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.)

Divergence reshapers

A pair of lens arrays can constitute a divergence reshaper.

Arrangements of lens arrays (a.k.a., micro lenses, mini-lenses, lenslets, fly's eye lenses, lenticular arrays) like the one in the image above are known in beam homogenization and integral field spectroscopy. Though the relation between the two lens arrays is actually reciprocal, the first-encountered array is usually termed the field lens array, and the second-encountered array is usually termed the pupil lens array. Each lens array lies in the focal plane of the other, and the reciprocal system works equally well in both directions.

In some applications both arrays have lenses of the same planform, usually either rectangular or hexagonal. However, it is not necessary for both arrays to have lenses of the same planform: the centroid-coordinated tessellations of the plane indicate an infinity of other possibilities. When the two arrays differ in planform, the arrangement can be used to reshape the divergence of a beam of light.

A micro array of lenses having hexagonal planform. Image quoted from Anteryon.com.

In solar energy applications, in order to reduce energy losses, it is advantageous to make both lens arrays plano-convex and cement the two plano surfaces together, or to simply mold the whole apparatus out of one piece of glass or plastic. Since the focal lengths of both arrays are the same, the radius of curvature of the lenses in both arrays must be the same. In effect, all the lenslets are portions of spheres of the same radius.

The lenslets in the two arrays need not have the same planform for their centers to align. Here, lenslets of circular planform align with lenslets of elliptical planform.

For perfect imaging, the curvature of the field should also have the same radius. A well-known solution to this problem exists, one that is used in the design of spherical retroreflectors. When spheres have refractive index 2, they focus light from a distant source onto their own rear surface. This arrangement has spherical aberration, but the f-numbers needed for divergence rotation in solar energy applications are small. The lenslets on each face have dimensions that are small compared to the focal length, so the effect of spherical aberration is slight.

An integrated array of cat eye lenses (refractive index = 2) can serve as a divergence reshaper if the planforms of the lenses on each face differ. 

The small f-numbers also mean that refractive indices less than 2 can also be used with tolerable results, and that defocussing due to chromatic dispersion over the wide solar spectrum is not too severe.

Optical diagram of a divergence reshaper having refractive index 1.5. Incident light strikes the new side. If the incident beam has a maximum divergence of 15°, lenslets on the old side will be shaped in planform like the divergence of the original beam, and will have an f-number  of 5.8 based on their maximum dimension. 

In a divergence reshaper, the light strikes the new side—the side whose lenses are shaped in planform like the new divergence pattern. Light exits from the old side—the side whose lenses are shaped in planform like the old divergence pattern.

Cooling the thermal mirrors

The thermal mirrors (thermal cap, yellow; thermal wall, orange) are exposed to intense thermal radiation from the furnace opening or oculus (black line.) The black fringes represent 0 suns, 1000 suns, 2000 suns, etc., of back radiation when the the furnace is at the temperature of the sun.

The thermal mirrors—the cap and the wall—are exposed to intense thermal radiation. If the oculus were at the temperature of the sun, the cap would see about 9,000 suns of flux, and the wall about 1,000 suns. Of course, for the sake of efficiency, the furnace will actually operate at a much lower temperature, reducing these fluxes by about a factor of ten. The mirrors will reflect most of this heat, but perhaps about 5% will be absorbed. This absorbed heat needs to be dissipated from these surfaces to keep the mirrors cool—a thermal flux amounting to about 45 suns on the cap and 5 suns on the wall. The thermal wall may use passive cooling, but active convection is needed for the cap. For comparison, a 2-kw stove element heating a 20-cm diameter pot produces a thermal flux of about 64 suns.

Water is too dangerous to use directly above the furnace opening, so this significant cooling must be obtained by convected air. If the cap is segmented into smaller mirrors, each shingled over the other, a chimney extending to the top of the lamp may draft enough air between the mirrors to keep the mirrors cool.

Beam-down optics designed with coextensive penumbras

Design for the optics at the center of a field of telescopic heliostats. Beam from farthest heliostat (blue) and nearest heliostat (yellow.) Lamp walls with elliptical profile (red,) thermal cap (yellow,) thermal wall (orange.) Penumbras of the field and of the thermal cap and thermal wall are adjusted to be coextensive.

Cross-section of the entire heliostat field (green) with lamp (red) and target point (yellow.)

Following the advice of the previous post, this design has coextensive penumbras. That is, the penumbra of the thermal cap (the upper portion of the radiant field lying between black and the sharp bends in the fringes) is coextensive along the surface of the lamp (red) with the upper penumbra of the field (the region between the blue rays coming from the farthest heliostat;) and the penumbra of the thermal wall (the lower portion of the radiant field between black and the sharp bends in the fringes) is coextensive along the surface of the lamp with the lower penumbra of the field (the region between the yellow rays coming from the nearest heliostat.)

This design has:

field radius = 1
inner field radius = 0.2
telescopic heliostat power = 6x
oculus radius = 0.0058
thermal cap radius = 2.1 * oculus radius
zenith distance to edge of cap = pi/3.6
thermal wall radius =2.7 * oculus radius
zenith distance to top of wall = pi/2.33
lamp height = 0.060

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.

Approximate dimensions of the beam-down optics

Dimensions of the beam-down optics for a field of 6x telescopic heliostats. The radius of the heliostat field is taken as 1.

Based on the heliostat targeting described in an earlier post, here is a rough approximation of the dimensions of the beam-down optics, the optics in the center of the heliostat field that redirect concentrated sunlight toward the furnace opening or oculus. Taking the radius of the heliostat field to be 1, then, as found in the earlier post, the height of the optics is 0.071. Assuming the beam-down optics, when viewed from the furnace opening (oculus,) fill all but a 30°-zenith distance of the sky, and approximating the beam-down optics as a cylinder, the cylinder diameter is 0.082. If sunlight filled the entire sky at the oculus, geometric concentration would be 40,000x. The 30°-zenith distance hole in the sky reduces this to 30,000x, giving an oculus radius of 1/√30,000 = 0.0058, or a diameter of 0.012. The heliostat field radius is 173x the oculus radius.

The cylinder diameter is about 6.8x the oculus diameter, and the cylinder height is about 5.9x the occults diameter. The cylinder has an area of pi * 0.082 * 0.071, while the heliostat field has an area of pi, giving a geometric concentration on the beam-down optics of 172.

In summary, geometric concentration factors are approximately:

telescopic heliostat primary: 1x

telescopic heliostat secondary: 36x

beam-down optics: 172x

oculus: 30,000x

From the earlier post, the aiming target is 0.84 behind the center of the field, and at a height of 0.084.

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. 

Back radiation from the target

Far-field approximation of the radiant flux from a horizontal disk. The region closer than one disk radius from the center of the disk has been blacked out. If the disk is at the temperature of the sun, the remotest black fringe is at 0 suns of flux, the next at 1000 suns, etc.

In a perfect solar concentrator, the target reaches thermal equilibrium (radiative balance) upon reaching the temperature of the sun's surface. Radiant flux is then net zero throughout the system due to the presence of thermal radiation radiated back from the target. In the forward direction, sunlight uniformly illuminates the target; in the reverse direction, back radiation from the target is collimated and beamed toward the sun. When a perfect optical system is in thermodynamic equilibrium, these two fluxes cancel out everywhere.

At any point in empty space, the divergence pattern of the beam of sunlight will be exactly mimicked by the back radiation traveling in the opposite direction. At an optical surface—where light might be redirected and its divergence pattern reshaped—the fluxes on the two sides of the surface must still sum to zero.

Flux on a surface is proportional to cosθ, the cosine of the angle the incident rays make with the surface normal. This is Lambert's Law. The divergence pattern of light at a particular point, P, on a surface can be described as the interior of region drawn on a unit sphere. In geometrical terms, Lambert's Law says that the flux at the surface is not the area of this region on the unit sphere, but rather it is the area of this region after it has been projected orthogonally onto the surface at point P. The orthogonal projection supplies the factor of cosθ.   Our requirement for net zero flux through the optical surface is then that the projected areas for both sides of the optical surface are equal.

There is one part of the optical design we have no control over: the pattern of thermal radiation that will be emitted from the target. In the case of a solar glass-melting furnace, the target is the opening of the furnace, the oculus, which is effectively a horizontal disk. Calculating the flux of back radiation from this disk is merely a problem in visual perspective because it is a perspective view of the disk that determines the shape and area of the region on the unit sphere. According to Lambert's Law, we should follow upon that exercise in visual perspective with an orthogonal projection onto some surface (whose orientation would be somewhat unclear at this point,) but, if we assume the angles of incidence to be the same on both sides of the optical surface (a desirable condition for both efficiency and simplicity,) that step is unnecessary. We will make that assumption.

The problem in visual perspective gets a bit complicated close to the disk, but in a far-field approximation the horizontal disk appears as an ellipse, one angular dimension being proportional to 1/r, and the other to cosθ/r.

More precisely, taking the radius of the oculus = 1, and r = sqrt(x² + y² + z²),

the angular extent of the ellipse in the horizontal (non-foreshortened) dimension = 2 * arctan(1/r), or approximately 2/r,

and the angular extent of the ellipse in the vertical (foreshortened) dimension, is approximately cosθ * 2/r.

Therefore, our perspective view of the disk translates approximately into an elliptical region on the unit sphere having an area, measured in steradians, of cosθ * 4/r² * π/4 = πcosθ/r².

The angular diameter of the sun is .01 radians, so a flux of 1000 suns corresponds to an image of the sun covering 1000 * π * (.005)²  =  .079 steradians.

The graphic above shows the far-field approximation of the thermal flux from the oculus. When the disk is at the sun's blackbody temperature, each fringe represents 1000 suns, that is, the remotest black fringe is at 0 suns, the next black fringe is at 1000 suns, etc. To give a sense of scale, points closer to the center of the disk than one disk radius have been blacked out, but the far-field approximation is not actually good enough to go in that close.

Beam-down optics for telescopic heliostat arrays

The focal zone of an array of telescopic heliostats scaled to 140 m height—yielding about 750 MWe.

Save for the intervention of the beam-down optics, light from a field of telescopic heliostats would converge onto a focal zone shaped something like an ice cream cone (the orange region in the figure above.)

Taking parameters from an earlier post:

Height to top of the beam-down optics: 140 m

Field radius: 1085 x 1.59 = 1725 m

Beam divergence: 3°

Angular elevation of beam center = 1.6° + 1.5° = 3.1°

[For comparison: the height of the Washington Monument is 169 m (152 m to the base of the pyramidal cap); the distance between the monument and the U.S. Capitol is 1800 m. The Capitol's dome is 88 m in height and 29 m in diameter.]

If it received only light from the outermost ring of heliostats, the focal zone would be a sphere centered at 1725 m x tan(3.1°) = 93 m high, with a radius of 1725 m x tan(3°) / 2 = 45 m. That is the "ice cream" at the top of the cone; superimposing many smaller and lower spheres for each ring of heliostats yields the "cone" itself. 

The beam divergence of a 6x telescopic heliostat (the pink spot) drawn as an area on the globe.

The beam divergence found near the center of an array of telescopic heliostats. Divergence has increased in longitude, but not in latitude.

We will describe the directions light propagates in by reference to an earth globe oriented with north pointing up, and with the prime meridian facing the heliostats we are concerned with. So, for example, close to the heliostat, the light lies within a circle 3 degrees in diameter that is centered on 0 degrees longitude and 3.1 degrees S latitude (the top picture above.) Closer to the focal zone, the light will have the same latitudinal extent, but its longitudinal extent will have increased because the light of many heliostats is being concentrated along that dimension (lower picture above.) The exact longitudinal extent can be calculated by taking a look in the opposite direction and calculating the angular subtense of the cone-shaped focal zone.

At times we need to know the divergence of light where the beam-down optics are located. We can get the same answer by reversing the light rays and considering the focal zone—rather than the heliostats—as the source of light.

Some considerations in designing beam-down optics:

1. Placing beam-down optics too close to the focal cone will make the incident light too widely spread out in the longitudinal direction to be handled efficiently by the optics.
2. We want the surface normal of the beam-down structure to make roughly equal angles with the incoming and outgoing light. This will avoid the same thermodynamic "gotcha" that afflicts conventional heliostat fields that direct high sun to a low target. 
3. We want the incident light to cover the same area on the globe (i.e., subtend the same solid angle) as the opening of the furnace, because that is what we can do most efficiently, reshape, not resize, divergence. 

Reshaping divergence with lenslet arrays

Two lenslet arrays, separated by one mutual focal length, can be used to reshape the divergence of a light beam.

Optics near the central focus of a solar plant operates at high energy flux, for example, at the beam-down optics of a glass-making solar furnace the flux will be on the order of 100 suns. This creates both the need to operate at high optical efficiency, and the opportunity to employ more expensive optical solutions than are possible on surfaces of larger area.

Each point on the beam-down optics of a solar furnace will see the circular opening of the furnace (because it is distorted by perspective) as approximately an ellipse. The beam-down optics needs to redirect approximately horizontal radiation coming from the field of telescopic heliostats downward toward the furnace opening. It also needs to reshape the divergence of the radiation to fit inside the approximately elliptical angular extent of the furnace opening.

A simple way to reshape divergence is with two lenslet arrays of the same focal length, separated by one focal length. In effect, each array acts as a field lens for the other. As the figure above illustrates, the lenslets in the two arrays do not have to have the same shape (though they must have the same area) in order to be in perfect correspondence with the lenslets in the other array. For example, in the figure above, circularly divergent light entering through the elliptical lenslets, would exit the round lenslets with elliptical divergence.