PhD Project Hyperbolic Monge-ampere Equation for Freeform Optics

Eindhoven University of Technology, Netherlands is inviting excellent students to apply for a PhD position in computer science. Applications should be sumbmitted as soon as possible.

The Department of Mathematics and Computer Science of Eindhoven University of Technology has a vacancy for a PhD-student in its Centre for Analysis, Scientific computingand Applications (CASA). CASA comprises the chairs Scientific Computing (SC) andApplied Analysis (TA). Its major research objective is to develop new and to improveexisting mathematical (both analytical and numerical) methods for a wide range of applicationsin science and engineering.

Illumination optics plays an important role in modern society. Products like mobilephones, lamps, car headlights, road lighting and even satellites all utilize illuminationoptics. A good optical design determines, for example, the energy efficiency of illumination devices, the minimization of light pollution or the sensitivity of sensors in satellites.The design of novel, sophisticated optical systems requires advances in the mathematicaldescription and numerical simulation methods for these systems.The optics applied in illumination is nonimaging, in contrast to for example a cameralens which is imaging. In nonimaging optics we study the transfer of light from a sourceto a target. The key problem is to design optical systems that convert a given sourceintensity into a desired target intensity.

A modern trend in illumination optics is to use scattering elements in addition to commonlyused refractive (lenses) or reflective (mirrors) optical components. For example,in LED lighting systems scattering surfaces are used to hide too bright light sources andto redistribute the light. The physical description of scattering surfaces, on the one hand,

and refractive/reflective surfaces, on the other hand, is quite different. To bridge the gap between the corresponding subdisciplines scattering and geometricaloptics, an NWO/TTW perspectief project was proposed by UT, TU Delft and TU/e,called Free-Form Scattering Optics. This proposal is supported by leading parties in the illuminationindustry: TNO, ASML, Signify, Lumileds, Demcon and Schott. In this project12 PhD students work on different topics related to a) the fundamentals of scattering, b)the fundamentals of free-form optics, c) homogenization and diffusion and d) control thedirection of light by interference.

This PhD project relates to the work package b) fundamentals of free-form optics.

Freeform optics, a branch of geometrical optics, is concerned with the design of opticalsurfaces, either reflectors or lenses, that convert a given source light distribution into adesired target distribution. An example is a single reflector that transforms the emittanceof an LED source into an intensity distribution in the far field, as used for street lighting.

The governing laws are the principles of geometrical optics (law of reflection/refraction)and conservation of energy. Geometrical optics gives the optical map from source totarget, and combined with energy conservation, this gives rise to the so-called Monge-Ampere equation, which is a second order, nonlinear partial differential equation. The Monge-Ampere equation can be classified as either elliptic or hyperbolic. It is the purposeof this project to develop new numerical solution methods for the hyperbolic equation. The standard case is the elliptic Monge-Ampere equation, for which there exist severalleast-squares solution methods. The hyperbolic equation is fundamentally more difficultsince the solution is propagating along intersecting lines, creating very complicated solutions.The least-squares method developed for the elliptic equation breaks down, soan alternative solution method has to be developed. It is anticipated that solution methodsfor hyperbolic conservation laws could be used. Moreover, issues like existence anduniqueness of the solution is an open question, and needs to be investigated. Finally, thenewly developed method should be applied to some test cases to compute optical surfaces.

As a PhD student your tasks are the following:

We are looking for talented, enthusiastic PhD candidates who meet the following requirements:

The application should consist of the following parts:


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