Master Thesis proposal

Guided modes in a locally perturbed hexagonal periodic graph-like domain Sonia Fliss, (POEMS, CNRS-ENSTA Paristech-INRIA, Universit´e Paris-Saclay) B´erang`ere Delourme (LAGA, Universit´e Paris XIII) [email protected], [email protected]

Photonic crystals, also known as electromagnetic bandgap metamaterials, are 2D or 3D periodic media designed to control the light propagation (Figure 1a). First, the multiple scattering resulting from the periodicity of the material can give rise to destructive interferences at some range of frequencies. It follows that there might exist intervals of frequencies (called gaps) wherein the monochromatic waves cannot propagate. Then, a local perturbation of the crystal can produce defect mid-gap modes, that is to say solutions to the homogeneous time-harmonic wave equation at a fixed frequency, located inside one gap, that remains strongly localized in the vicinity of the perturbation. This localization phenomenon is of particular interest for a variety of promising applications in optics, in particular the design of highly efficient waveguides.

(a) An exemple of Photonic crystal

(b) Rectangular and hexagonal graph-like perturbed periodic structures (perturbed on the red line)

Figure 1: Schematic representation of photonic crystals In her PhD work ([1]), E. Vasilevskaya exhibits a simple configuration where such a localization phenomenon occurs: this configuration is made of a rectangular periodic fattened graph perturbed on a line (see Figure 1b, left). Diminishing the thickness of one line creates guided modes. The aim of this master thesis is to extend the previous results to the case of a locally perturbed hexagonal periodic fattened graph (see [2]) (see Figure 1b, right). From a practical point of view, periodic media having this type of structure, called a honeycomb lattice structure (such as the famous graphene), are known to have amazing properties. The analysis will rely an asymptotic analysis argument explained in [3]. As the distance between two consecutive hexagonal obstacles tends to 0, the domain shrinks to a periodic hexagonal graph. Proving the existence of guided modes in the limit configuration, where explicit computations can be carried out, is then sufficient to prove existence of guided modes in the overall structure. The work can be more theoretical or more numerical-oriented, depending on the interests of the student. Location: ENSTA-ParisTech, Universit´e Paris-Saclay, Palaiseau, FRANCE. 1

References [1] Vasilevskaya, Elizaveta, Open periodic waveguides. Theory and computation, PhD Thesis, Universit´e Paris XIII, 2016 [2] Kuchment, P., Post, O. (2007). On the spectra of carbon nano-structures. Communications in Mathematical Physics, 275(3), 805-826. ISO 690 [3] Olaf Post. Spectral convergence of quasi-one-dimensional spaces. Ann. Henri Poincare, 7(5):933973, 2006.

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