Isotropic photonic structures: Archimedean-like tilings and quasi-crystals

We present a detailed numerical study of Archimedean-like tilings and quasi-crystals. Archimedean-like tilings are periodic structures like conventional 2D photonic crystals. However, the use of two kinds of polygons for the tiling and thus, the use of crystal unit cells of several atoms, can lead to a 12-fold local symmetry. We show that Archimedean-like tilings offer a much higher degree of isotropy than a conventional 2D Bravais lattice because the gap widths are almost independent of the light propagation direction. Besides, the conditions for gap opening and the mean gap widths remain similar to those met for conventional triangular and honeycomb lattices. The comparison between Archimedean-like tilings and quasi-crystals is carried out in two steps according to the level of the refractive index modulation. In the case of strong modulation, we show that Archimedean-like tilings with a few atoms per unit cell are simple and well-working alternatives to complex quasi-crystals. They can present the same degree of isotropy for the same positions and widths of the photonic gaps. A good fit is also found between quasi-crystals and periodic structures constructed from the simplest approximants, thereby showing the importance of short-range wave-lattice interactions in that case. In a second step, we consider weakly (or moderately) modulated structures, where long-range interactions are possible. A quasi-periodic (fractal) structure recently reported in the literature is compared to an Archimedean tiling of very large unit cell (181 atoms). Surprisingly, the unusual quasi-crystal gap at long wavelength cannot be reproduced for the Archimedean tiling in spite of its very large unit cell size.