My research interests are Below are some examples of my recent work.

Shape Reconstruction:
An inclusion inside an opaque solid can be detected from the temperature and heat flux on the surface. The figure shows the inclusion (a cat) and the reconstructed shape. The algorithm finds the shape that is a best fit of the given data by iteration. Each step involves solving the heat equation in the solid. Thanks to fast boundary integral methods the optimal shape can be obtained in about half an hour.

Wave propagation in dielectric waveguides:
Dielectric slab waveguides are characterized by their dispersion relations, which are the propagation constants of the modes as a function of the frequency. Below cutoff, the propagation constants are complex and the characteristic equation is a four-valued function. To better understand the coupling of modes near cutoff, it is instructive to consider the change of variables

Here, z is the new variable, k is the frequency n0, nJ are the refractive indices in the semiinfinite layers and beta the propagation constant. This is a movie that shows how the modes z(k)/k evolve for k from 0 to 15. This waveguide has five interior layers with refractive indices between 1.53 and 1.66. The indices in the semiinfinite layers are 1.5 and 1.0. The guided modes appear to the right of the circle. The other modes are either leaky or non-physical. Every frame in this movie corresponds to a different frequency.

Wave propagation in biperiodic media:
We look at a dielectric waveguide that has a biperiodic perturbation. Due to coherent scattering, guided modes can better propagate in certain directions than in others. This is illustrated in the following figure, that shows the Bloch wave-vector of a guided mode (black) for different frequencies. The corresponding curves for a planar structure (light blue) are circles. The gray regions are occupied by the continuous spectrum.
This is a similar picture for a two mode structure. One can see nicely how the two modes couple.