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# MATHEON Projekt D26

Duration | April 2011 - May 2017 |
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Head | Dr. Kersten Schmidt |

Member | Dr. Dirk Klindworth (bis Januar 2016) Dr. Adrien Semin (ab Februar 2016) |

Cooperation | Juliette Chabassier (INRIA Bordeaux Sud Ouest / Université de Pau) Sonia Fliss (ENSTA ParisTech) Carlos Jerez Hanckes (Pontificia Universidad Católica de Chile) |

Link | MATHEON project webpage |

Description | Photonic crystal wave-guides are devices that allow for exceptional tailoring of the properties of light propagation. Currently, the prediction of the properties relies mainly on models for infinite, perfect photonic crystal wave-guides. For photonic crystal circuits scattering matrix approaches have been proposed. In this project we study imperfect photonic crystal wave-guides and circuits of finite lengths with techniques of asymptotic expansion. |

## Project Background

Photonic crystals (PhC) are materials with a periodic refractive index in the order of the wavelength of the light. They allow for exceptional tailoring of light propagation. Light is guided efficiently in wave-guides which are formed by omitting one or a few rows of holes inside the PhC. Due to the existence of particular slow light modes PhC wave-guides are of particular interest in integrated optics, e.g., in the telecommunication sector, where they might be used as one key element in photonic integrated circuits.

PhC wave-guides are made out of semiconductors in which a periodic array of holes of diameter and distance at the size of the wavelength (several 100 nm) are etched. On the connection between different wave-guides, *e.g.*, a fiber and a PhC wave-guide, the light is scattered mainly into the propagating mode(s), but besides a back-scattered fields light “leaks” (or tunnels) the finite PhC domain towards the sides. PhC wave-guides would be perfect closed wave-guides if the PhC domain had infinite extent.

The closed PhC wave-guide is therefore a “near by” model, neglecting the leakage. Eigenmodes in localised perturbations or line “defects” inside infinite PhCs can be modelled by the super-cell approach leading to the same eigenvalue problems and numerical methods as for the infinite crystal, just on a larger computational domain.

### Dirichlet-to-Neumann approach

- Profile of a transverse electric mode in a photonic crystal wave-guide with hexagonal lattice.
- © Dirk Klindworth

Transparent boundary conditions based on a Dirichlet-to-Neumann approach were applied to the periodic, infinite half-strips of a photonic crystal wave-guide yielding parameter-dependent non-linear eigenvalue problems. This Dirichlet-to-Neumann approach can hence be regarded as an exact approach for the infinite photonic crystal wave-guide compared to the widely used, but inexact computation based on a supercell approach. In particular for guided modes that are not well-confined, *i.e.*, modes whose profile does not decrease to a neglectable value within a few periods, the supercell approach lacks accuracy and the Dirichlet-to-Neumann approach is preferable. The Dirichlet-to-Neumann operators are computed with the help of Dirichlet problems in one periodicity cell of the photonic crystal. That means only one periodicity cell of the infinite, periodic half-strips has to be discretised for the computation of the Dirichlet-to-Neumann operators. The non-linearity of the eigenvalue problem results from a non-linear dependence of the Dirichlet-to-Neumann operators on the eigenvalue. The non-linear eigenvalue problem is posed in the wave-guide cell instead of in the whole photonic crystal wave-guide strip. This reduces the size of the problem dramatically compared to the standard approach of using a supercell. For the solution of the non-linear eigenvalue problem we proposed an iterative approach based on Newton’s method and a direct approach based on a Chebyshev interpolation of the non-linear operator.

### Robin-to-Robin approach

We continued with a study of Robin-to-Robin transparent boundary conditions for the computation of guided modes in photonic crystal wave-guides. This study extends the previous work on Dirichlet-to-Neumann operators since Robin-to-Robin operators can be regarded as preferable compared to Dirichlet-to-Neumann operators since the former are well-defined for all frequencies in the band gaps, while Dirichlet-to-Neumann operators are not well-defined for eigenfrequencies of the homogeneous Dirichlet problem in the infinite half-strip and their computation is ill-posed for eigenfrequencies of the Dirichlet problems in the unit cell of the photonic crystal.

### Computation of the group velocity and adaptive sampling of the band structure

A closed formula for the group velocity of guided modes and modes in photonic crystals was developed that could also be extended to higher derivatives of the dispersion curves. Due to the analyticity of dispersion curves these closed formulas for the derivatives of the dispersion curves allow for an efficient calculation of photonic crystal and photonic crystal wave-guide band structures using Taylor expansions. We developed an adaptive algorithm for the band structure computation that is able to resolve very special phenomena of the dispersion curves such as mini-stop bands. At mini-stop bands two dispersion curves approach and seem to cross but in a very small vicinity of this “crossing” they change their behaviour dramatically and in fact do not cross and form a so-called mini-stop band. The algorithm adaptively selects the location of the nodes around which a Taylor expansion is computed. The larger the magnitude of the derivatives, *e.g.*, near a mini-stop band, the closer the algorithm places the nodes. This implies that for rather simple curves only a very small number of nodes is necessary which explains the efficiency of the method. On the other hand, if there is a mini-stop band and the number of required nodes is large, the algorithm is still much faster than the standard approach of evaluating the band structure for an equidistant grid of the quasi-momentum *k*, if the standard algorithm is supposed to resolve the mini-stop band correctly.