We carried out in our project mathematical analysis to several problems related to Geometric Optics and Free material design.
Geometric Optics studies the trajectories of light having small wavelength compared to the dimensions of all media traversed. In this case light trajectory can be represented by geometric lines. Reflection by mirrors and refractions by lenses are the phenomena studied in this area. The optical components in applications are generally convex due to design limitations. Recent innovations in industrial optics have allowed the design of free form lenses. Optical tasks that were once impossible using convex materials is now achievable using free form lenses and mirrors.
We studied in this project the existence of lenses composed of two surfaces whose task is to illuminate a far field target. The goal of the design is to refract light rays emitted from a planar source and with different directions in such a way to optimize the illumination at the target depending on the requested task. We show that such device can be constructed using free form optical surfaces. In another model, a lens composed with one surface with similar illumination task has been studied. In this case we were interested in the smoothness of such surface. We find conditions on the target for which our surface is smooth making it hence easier to implement.
The problems in geometric optics becomes more challenging when the light source is polychromatic that is composed of different colors. In this cases, as perceived in prisms, different colors are dispersed in the material creating focusing error at the target this is called chromatic aberration. We investigate this limitation in our project and prove that using classical lenses one can eliminate chromatic aberration created by a point source field composed of two colors. The result is negative for more than two colors. We pushed this model further by requiring each color to go into different direction. Such device is called demultiplexer and allows to recover each color emitted from the source separately.
Recently, a group in Harvard lead by F. Capasso was able to design ‘metalens’ that can reduce chromatic aberration. Metalens as a the word suggest is not a classical lens. Nanomaterial are added to the surface that separates the media of the source and the target generalizing hence the law of refraction. We analyzed these devices mathematically and studied how a lens that perfectly focus one color behaves when the source has different color and measure the error obtained.
The other part of the project was devoted to mathematical problems related to free material design, where the goal is to find the optimal shape of an object to withstand a prescribed load. The material might have voids and loads might be only applied to a part of the object. Theses problems appear in bridge construction, aviation as well as biomedicine and tomography. We considered in our project first the case when the load is applied partially on a strictly convex object like a disc. Our next step was to study these problems on more general objects that are not necessarily strictly convex, like squares and convex polygons. We kept the convexity condition however due to the mathematical difficulty of the problem in general.
These projects although addressed mathematically have various implementation in applied mathematics and industry, especially that we are dealing with problems coming directly from optical and civil engineering. Having deeper understanding of the mathematical problems helps improving the existent numerical algorithms used generally in industry to deal with these questions. But we also respond to the question of possibility of design, since we showed in many places necessary conditions on the parameters of the problems that should be satisfied so that a solution to the problem exists.
Ahmad Sabra 