# OPTICS OF

# COMPLEX SYSTEMS

## Quasicrystal

Quasicrystals represent a class of solids which lack translational symmetry, but nevertheless exhibit perfect long-range order and reveal well-defined rotational symmetries, which are not necessarily consistent with periodicity. The existence of long-range order, the possibility of high-fold rotational symmetries and the distinct local environment of each quasicrystal lattice point promise richer optical properties than for photonic crystals. Moreover, the analogy with electronic quasicrystals suggests these kind of structures as good candidate to observe anomalous diffusion of light.For centuries crystals were thought of as solids exhibiting flat surfaces (facets), that intersect each others at characteristic angles. During the 17th century, initial ideas regarding the microscopic structure of crystals began to emerge and were formalized into a theory of crystallography by R. J. Haüy in the early 19th century. The basic concept contained in the theory is that crystals are solids which are ordered at a microscopic level. It was assumed that the only way to achieve order is by having periodicity, that is, some basic structural unit which repeats itself infinitely in all directions, filling up all space.

The presence of order may be described in terms of the symmetries respected by the structure. Among the most well known consequences of periodicity, is the fact that the only allowed rotational symmetries are around the 2-, 3-, 4-, and 6-fold axes. Five-fold rotations (and any n-fold rotation for n > 6) are incompatible with periodicity.

The first observation of a diffraction pattern showing a five-fold symmetry was reported by Shechtman in 1982 as a result of an electron diffraction experiment, on an alloy of Aluminum and Manganese and represent the moment in which quasicrystals were discovered.

However, while nature provides us with three-dimensional quasicrystals for electrons, corresponding structures for light need to be fabricated artificially.

## Examples recent papers on this subject

**Multiple scattering of light in three-dimensional photonic quasicrystals**
- Alexandra Ledermann, Diederik S. Wiersma, Martin Wegener, and Georg von Freymann, Optics Express, to be published (2009).

**Three-dimensional silicon inverse
photonic quasicrystals for infrared wavelengths**
- Alexandra Ledermann, Ludovico Cademartiri, Martin
Hermatschweiler,
Costanza Toninelli, Geoffrey A. Ozin, Diederik S. Wiersma, Martin
Wegener, Georg von Freymann, Nature Materials, 5 942 (2006).