PhD on mathematical modeling in imaging optics

Are you eager to use your mathematical skills to model and design future optical systems for sustainable high-tech devices for billions of people? Do you like to apply advanced math in the field of optics?

Information
The Computational Illumination Optics group is one of the few mathematics groups worldwide working on mathematical models of optical systems. They develop and analyze numerical methods to solve the resulting differential equations. The team has a healthy portfolio of PhD positions and close collaborations with industrial partners. It consists of four full FTEs at Eindhoven University of Technology and one part-time professor.

The group has three research tracks: freeform design, imaging optics and improved direct methods; for more details see https://martijna.win.tue.nl/Optics/. The following mathematical disciplines are important in our work: geometrical optics, ray tracing, (numerical) PDEs, transport theory, nonlinear optimization, Lie operators and Hamiltonian systems.

PhD vacancy
For an imaging optical system, the deviations of a perfect system are the so-called optical aberrations. An optical system can be described by an optical map that connects positions and directions at the object plane with positions and directions at the target or image plane. For an ideal system, this map is linear. Any deviation from this linear map is an aberration.

To derive aberration theory for (complex) optical systems, the mathematically most sound approach is to use Lie operators. This mathematical framework was developed in the nineties of the previous century and successfully applied to normal lens and mirror systems. The relation between the Lie algebraic approach and the well-known Seidel aberrations are known for normal lenses.

In gradient-index (GRIN) optics the refractive index depends on position. GRIN lenses and gradient-index fibers with a rotationally symmetric profile are known for decades and used commercially. Upcoming manufacturing techniques allow freeform-GRIN (F-GRIN) media that can replace traditional optical components. For F-GRIN there is hardly any literature on aberration. The challenge in this project is to develop the aberration theory and use this for the optimization for imaging applications.

Research line on freeform design: The goal in freeform design is to compute the shapes of optical surfaces (reflector/lens) that convert a given source distribution, typically LED, into a desired target distribution. The surfaces are referred to as freeform since they do not have any symmetries. The governing equation for these problems is a fully nonlinear PDE of Monge-Ampère type.

Key publication: Anthonissen, M. J. H., Romijn, L. B., ten Thije Boonkkamp, J. H. M., & IJzerman, W. L. (2021).

Unified mathematical framework for a class of fundamental freeform optical systems. Optics Express, 29(20), 31650-31664. https://doi.org/10.1364/OE.438920.

Research line on imaging optics: The second research track is imaging, where the goal is to form a very precise image of an object, minimizing aberrations. Light propagation is described in terms of Lie transformations.

Key publication: Barion, A., Anthonissen, M. J. H., ten Thije Boonkkamp, J. H. M., & IJzerman, W. L. (2022). Alternative computation of the Seidel aberration coefficients using the Lie algebraic method. Journal of the Optical Society of America A, Optics, Image Science and Vision, 39(9), 1603-1615. https://doi.org/10.1364/JOSAA.465900.

Research line on improved direct methods: Direct methods, such as ray tracing, compute the target distribution given the source distribution and the layout of the optical system. These methods must be embedded in an iterative procedure to compute the final design and are based on Monte-Carlo simulation. They are known to have slow convergence. Using the Hamiltonian structure of the system and advanced numerical schemes for PDEs, we are working on more efficient and accurate methods.

Key publication: van Gestel, R. A. M., Anthonissen, M. J. H., ten Thije Boonkkamp, J. H. M., & IJzerman, W. L. (2021). An energy conservative hp-method for Liouville’s equation of geometrical optics. Journal of Scientific Computing, 89, [27]. https://doi.org/10.1007/s10915-021-01612-x

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

• A master’s degree in (applied) mathematics or (applied) physics with a background in mathematical modeling and scientific computing
• Experience with solving ordinary and partial differential equations numerically
• Experience with programming (C, C++, Python, Matlab or alike)
• Creative pro-active team player with good analytical skills
• A research-oriented attitude
• Ability to work in an interdisciplinary team and interested in collaborating with industrial partners
• Motivated to develop your teaching skills and coach students
• Fluent in spoken and written English (C1 level)

A meaningful job in a dynamic and ambitious university, in an interdisciplinary setting and within an international network. You will work on a beautiful, green campus within walking distance of the central train station. In addition, we offer you:

• Full-time employment for four years, with an intermediate assessment after nine months. You will spend a minimum of 10% of your four-year employment on teaching tasks, with a maximum of 15% per year of your employment.
• Salary and benefits (such as a pension scheme, paid pregnancy and maternity leave, partially paid parental leave) in accordance with the Collective Labour Agreement for Dutch Universities, scale P (min. € 3,059 max. € 3,881).
• A year-end bonus of 8.3% and annual vacation pay of 8%.
• High-quality training programs and other support to grow into a self-aware, autonomous scientific researcher. At TU/e we challenge you to take charge of your own learning process.
• An excellent technical infrastructure, on-campus children's day care and sports facilities.
• An allowance for commuting, working from home and internet costs.
• A Staff Immigration Team and a tax compensation scheme (the 30% facility) for international candidates.