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 computing and Applications (CASA). CASA comprises the chairs Scientific Computing (SC) and Applied Analysis (TA). Its major research objective is to develop new and to improve existing mathematical (both analytical and numerical) methods for a wide range of applications in science and engineering.

Background

Illumination optics plays an important role in modern society. Products like mobile phones, lamps, car headlights, road lighting and even satellites all utilize illumination optics. 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 mathematical description and numerical simulation methods for these systems. The optics applied in illumination is nonimaging, in contrast to for example a camera lens which is imaging. In nonimaging optics we study the transfer of light from a source to a target. The key problem is to design optical systems that convert a given source intensity into a desired target intensity. 

A modern trend in illumination optics is to use scattering elements in addition to commonly used refractive (lenses) or reflective (mirrors) optical components. For example, in LED lighting systems scattering surfaces are used to hide too bright light sources and to 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 geometrical optics, 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 illumination industry: TNO, ASML, Signify, Lumileds, Demcon and Schott. In this project 12 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 the direction of light by interference.

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

Project Description

Freeform optics, a branch of geometrical optics, is concerned with the design of optical surfaces, either reflectors or lenses, that convert a given source light distribution into a desired target distribution. An example is a single reflector that transforms the emittance of 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 to target, 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 purpose of 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 several least-squares solution methods. The hyperbolic equation is fundamentally more difficult since the solution is propagating along intersecting lines, creating very complicated solutions. The least-squares method developed for the elliptic equation breaks down, so an alternative solution method has to be developed. It is anticipated that solution methods for hyperbolic conservation laws could be used. Moreover, issues like existence and uniqueness of the solution is an open question, and needs to be investigated. Finally, the newly developed method should be applied to some test cases to compute optical surfaces. 

As a PhD student your tasks are the following:

Perform scientific research in the described domain;

Present results at international conferences;

Publish results in scientific journals;

Participate in activities of the group and the department;

Assist SC-staff in teaching undergraduate and graduate courses (at most 20 % of the time).

Requirements

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

A MSc in (applied) mathematics, physics or a related discipline with a strong background in computational physics;

Experience with Matlab and preferably C or C++;

Creative, pro-active team player with good analytical skills;

Good communicative skills in English, both written and oral.

Appointment and salary

We offer:

A full-time appointment for a period of four years, with an intermediate evaluation after nine months;

A gross salary of Euros 2,222 per month in the first year increasing up to Euros 2,840 per month in the fourth year;

Support for your personal development and career planning including courses, summer schools, conference visits, etc.;

A research position in an enthusiastic and internationally renowned research group;

A broad package of fringe benefits (e.g. excellent technical infrastructure, saving schemes, excellent sport facilities, and child daycare).