Immersion graded index optics: theory, design, and prototypes

Immersion optics enable creation of systems with improved optical concentration and coupling by taking advantage of the fact that the luminance of light is proportional to the square of the refractive index in a lossless optical system. Immersion graded index optical concentrators, that do not need to track the source, are described in terms of theory, simulations, and experiments. We introduce a generalized design guide equation which follows the Pareto function and can be used to create various immersion graded index optics depending on the application requirements of concentration, refractive index, height, and efficiency. We present glass and polymer fabrication techniques for creating broadband transparent graded index materials with large refractive index ranges, (refractive index ratio)2 of ~2, going many fold beyond what is seen in nature or the optics industry. The prototypes demonstrate 3x optical concentration with over 90% efficiency. We report via functional prototypes that graded-index-lens concentrators perform close to the theoretical maximum limit and we introduce simple, inexpensive, design-flexible, and scalable fabrication techniques for their implementation.


Basic Relationship:
The reflection invariance follows directly from the shape of the AGILE. The indices and widths of the input and output apertures are related as follows In other words, the gradient index is varied such that the index*width product is constant The slope of the sidewalls are given by the height and the difference in the widths Equation 3 proves the first of the requirements. The slope of the sidewalls, and therefore the reflection angles, are unchanged under a linear scaling in all three dimensions. It is convenient to rewrite the index in the following form

Ray equation in gradient index:
We require that Snell's law applies everywhere: where θ is the angle that the ray makes with the optical axis. The starting point of the ray is at xs,ys,zs, and the starting angle of the ray is designated θs. Our treatment is valid for all points of origin of the ray.
Snell's law tells us that the path of the ray is given by From the equation for the gradient index (Eq. 4) Now we arrive at the ray equation for the AGILE Using Snell's law to express the ray angle in terms of the y coordinate and the initial conditions, we find the desired ray equation: shows that the second requirement for scale invariance is fulfilled. As the AGILE is linearly scaled in all dimensions, all rays with the same initial conditions (xs, ys, zs, P, sinϴs) follow the same path in the similarly scaled x,y,z coordinates. Furthermore, we see that the scaled ray path is only dependent on the scale factor given by the height (h), the initial position, and the angle of the ray.  Table B: Glasses with different refractive indices selected such that they have broadband transparency in the solar spectrum i.e., high optical transmission across the whole solar spectrum say from ~300nm to beyond ~1200nm, and similar thermal expansion and glass transition temperature so that they are compatible once bonded with each other in a stack (data and glasses supplied by Ohara Inc.).

Appendix C: Overlapping Conical Array Concentration Ratio
We calculate the exact area ratio, i.e., geometric concentration of the cluster array fabricated (input area / output area). There is difference between the nominal design as seen in Fig. C (a) and what was fabricated as seen in Fig. C (b). This difference was the result of over-cutting by the reamer as it pulled extra metal with it when machining the conical shape from the fragile metal island in between the overlapping cones at the top input surface. This was anticipated in the design and fabrication trials; and hence what was fabricated was a tile-able/tessellated structure with almost no input aperture area wasted. The CAD design of seven cones going from a radius of 3.5mm to a radius of 2mm with a spacing of 6.35mm between the centers, became seven cones of 3.6mm input radius with the same output radii and spacing. Appendix D Creating a graded index stack using polymers: 1.
Material search was done across various types of optically transparent polymers: silicones, acrylate polymers, polyimides, and polyurethanes (heat and/or UV curable resins). Ellipsometer and spectrophotometer measurements were done for various optical polymer film samples made of a fixed thickness to characterize the properties by measuring the film transmission and RIs across the solar spectrum. Polymers with broadband transparency (i.e., high optical transmission across the whole solar spectrum say from ~300nm to beyond ~ 1200nm) and having refractive indices evenly distributed in as large an index range as possible were chosen. 2.
UV curable optical polymers from Norland Products with RIs 1.46, 1.51, 1.52, 1.54, 1.56, and 1.625 were chosen for fabricating the AGILEs. The single AGILE and 10 layer cluster were made using polymers with index 1.46, 1.51, 1.52, 1.54, and 1.56. The RI=1.625 layers, which require curing in an inert atmosphere (glovebox) was used in the 12-layer cluster fabricated.

3.
Molds were made by reaming cone shaped cavities in aluminium metal plates. These cavities were polished using decreasing grit size sandpapers and polishing agents to make them optically reflective. After cleaning steps, a flat PDMS (Polydimethylsiloxane) film was attached using water soluble glue at the edges at the bottom of the AGILE reflective mold to seal the output in order to start filling in the graded index polymer layers (the UV curable optical polymers do not stick to PDMS and at the end of fabrication the device can be peeled off from the PDMS substrate). If curing is done first from the top there is the issue of uncured gel trapped below a cured top crust. To ensure a complete cure from the base, curing was first done with UV light incident through the PDMS layer from below the device, which was placed on a stand. . Later the cure was finished off from the top. 4.
The UV cure parameters were tuned, such as, power and wavelength of the source, distance from the source, and the duration of cure. Some layers were fabricated in a glovebox due to need for inert atmosphere during the cure to avoid yellowing/clouding in air, e.g., oxygen inhibition in some of the polymers . Air gaps and bubbles were removed pre-cure by intermediate vacuum treatment steps. Some polymers shrunk during the cure and some needed age hardening after cure to achieve the required transparency and RI. These material differences were taken into account to make uniform and broadband transparent layers. Compatibility of the polymers with their neighbouring layers in the stack sequence was also tested before final fabrication. 5.
Layers were filled volumetrically using pipettes to have a fixed layer height in order to complete the conical geometry. Each layer was filled and cured in several steps to ensure thin layers and hence a complete cure. UV curable polymer layers were formed one at a time starting from low-index to high-index polymers in order to fill half of the back-to-back shape; this process was followed by filling the upper part of the AGILE with high-index to low-index polymers. The single AGILE was filled from high-index to low-index. This completed the multi-stage polymer deposition and curing steps to create a graded index material with a large index variation.