Friday, March 14, 2014

Oculus radius calculations

Spreadsheet of oculus calculations for theta = 1.5°, R = 2600 m.

A simplistic model for the oculus radius is as follows:

R = fieldRadius, r = focalRadius, x = lampRadius 
focalRadius = fieldRadius * sin(theta) 
phi = arcsin(focalRadius/lampRadius) 
Divergence aspect ratio, AR, at lampRadius 
AR = phi/theta = (arcsin(r/x))/theta 
Oculus aspect ratio, due to foreshortening, is ~2 
gamma = reshaped beam’s angular major axis 
gamma = phi / sqrt(AR/2) 
corresponding oculus radius = xtan(gamma)
Geometrical relations at the focal zone.


As the spreadsheet indicates, the beam-shaping optics needs to operate at the widest possible field angular radius, phi. Phi in the range of 20°-30° permits a land concentration factor (gross field area/oculus area) of 9,000 to 13,000 and a lamp D/H of 1.8-1.3.

Thursday, March 13, 2014

Beamshaping is necessary

A representation of beam divergence that is wide in "longitude"

If the divergence of the sunlight reaching the lamp is too wide in "longitude," then, short of beamshaping, there is nothing a rotationally symmetric lens can do to fix it. It is therefore important to accurately model the divergence of this light.

Relations for the 180° and the general locus near the beam down optics. R is the field radius and r is the radius of the 180-degree locus. Theta is the beam angular radius. 

At a radius r = Rsinθ, the converging sunlight fills a full 180° of longitude. At a larger radius x, the sunlight fills an angle 2φ, where φ = sin-1(r/x).

For θ = 1.5°, r/R = 0.026. For R = 2600m, r = 68 m.

For φ = 10°, x/r = 5.8

For R = 2600m and x = 120m, φ = 35°; x = 190m, φ = 21°.

Since the width of the beam in the latitude dimension is not likely to be more than 4 degrees, the beam aspect ratio is 10 or greater. Thus beamshaping is necessary. 

Wednesday, March 12, 2014

Concentric spherical DNG double refractive index lenses

Views from the heliostat field of three lamps that are DNG spindle hyperboloids. The oculus is colored yellow. Made in POV-Ray.

As the image above indicates, it is not sufficient to consider only the meridional rays at the beam-down optics. It is possible (e.g., the middle lamp above) to image the oculus over the entire height of the lamp, but little of its width.

(I have found that it is more common in the metamaterial literature to speak of double negative or DNG optics when the lens is not a classic Veselago flat lens. So I am switching to those terms.)

Since the lamp must be rotationally symmetric about a vertical axis, the only possible profiles in the sagittal plane are concentric circles. When light rays in the sagittal plane strike two consecutive DNG interfaces, the action is the same as a double mirror. In 2-D optics, a double mirror rotates all rays by twice the angle between the mirror lines. In this case the angle between the mirror lines is the subtense of the interior ray between where it enters the DNG material and where it exits. The direction of the bending is the same as for a circular air cavity in a "glass" medium (such is of course a diverging lens.) The narrower the spacing between the two DNG interfaces, the closer to 1.0 is the refractive index of the surrounding "glass."

Two concentric negative refractive index or DNG interfaces (composing a single-layer spherical DNG lens,) form a diverging lens.

Adding a third concentric circular surface to make a double-layer DNG spherical lens, would get us back to a converging lens at the cost of some optical loss.

Tuesday, March 11, 2014

Quartic equation of a spindle hyperboloid

2c = 2ae, the distance between the foci of the hyperbola, must equal the field radius plus the oculus radius.

When the parameters a, b, c and e are given their usual meanings, the equation of a hyperbola is:

x*x/a*a - y*y/b*b = 1

with

c = a*e

e = c/a

and

a*a + b*b = c*c .

(At a given c, the closer the vertex lies to the center, the more eccentric the hyperbola. An infinitely eccentric hyperbola is a straight line through the center.)

In designing a spindle hyperboloid lens for Veselago beam-down optics, the parameter that must be fixed at the outset is the distance, D, that separates the two foci,

D = 2c = heliostat field radius + oculus radius.

Once we fix c at c = D/2, variation in the eccentricity of the hyperbola varies the radius of the lamp.

The offset, R, of the rotational axis from the center of the hyperbola is

R = c - oculus radius .

Given an offset R, the quartic equation for a spindle hyperboloid can be derived as follows:

Derivation of the quartic equation for a spindle hyperboloid.

In a POV-Ray scene description a spindle hyperboloid Veselago lens becomes something like:

#declare a = 1.0; //distance from hyperbola center to vertex
#declare b = 1.0; //asymptote of hyperbola slope = b/a
#declare a2 = a*a;
#declare b2= b*b;
#declare ab2 = a2/b2;
#declare R = 1.8; // offset of hyperboloid rotational axis
#declare R2 = R*R;
….
 quartic {
//     x^4   x^3y   x^3z   x^3   x^2y^2
     < 1,    0,     0,     0,    -2*ab2,
//     x^2yz  x^2y   x^2z^2   x^2z   x^2
       0,    0,    2,     0,   2*R2-2*a2-4*R2,
//     xy^3  xy^2z  xy^2   xyz^2  xyz
       0,    0,     0,     0,    0,
//     xy    xz^3   xz^2   xz    x
       0,    0,     0,     0,    0,
//     y^4   y^3z   y^3    y^2z^2  y^2z
      ab2*ab2,    0,     0,    -2*ab2,   0,
//     y^2   yz^3   yz^2   yz    y
       2*a2*ab2-2*R2*ab2,    0,    0,    0,   0
//     z^4   z^3    z^2    z     Const
       1,    0,     2*R2-2*a2-4*R2,   0,   R2*R2-2*R2*a2+a2*a2 >
           clipped_by
    {sphere {<0,0,0>1.0 }}
   bounded_by{clipped_by}
    sturm
    pigment {Clear}
  interior { ior -1.0 }
  scale <100.0,100.0, 100.0>
  translate<0,0,0>
  }

Sunday, March 9, 2014

Applying Veselago lenses to solar beam-down optics

A spindle hyperboloid Veselago lens is formed by revolving the right branch of the hyperbola about the dashed line.  
What an ellipse accomplishes in mirror optics (mapping a real object to a real image) is accomplished by a hyperbola in Veselago optics. The most basic way to use a hyperbolic Veselago lens for beam-down optics is to place one focus at the outer edge of the heliostat field and the other focus at the far edge of the occulus. In this way, locations on step outside the heliostat field will (ideally) not be able to see the oculus at all.

The spindle hyperboloid Veselago lens images the outer edge of the heliostat field onto the far edge of the oculus.

Friday, March 7, 2014

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.

Thursday, March 6, 2014

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.