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.

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.