Photonic Crystals: Computational Studies


Two-dimensional photonic crystals

R. Hillebrand and W.Hergert


At the MPI of Microstructure Physics Halle there exists a well-established technique to fabricate macroporous silicon by electrochemical pore etching.

Fig. 1 shows part of a two-dimensional (2D) photonic crystal (PC) prepared in this way. The inset shows some of the pores more in detail. These highly periodic structures reveal extraordinary optical properties. The direction of light propagation studied for 2D PCs is perpendicular to the pore axes (yellow arrow).


The theoretical results presented in Figs. 2-4 were calculated applying the program PHOTON (plane-wave approach, authors: W. Hergert, R. Hillebrand).

Fig. 2 presents calculations of the band structure of Si-based photonic crystals (cf. Fig. 1). In the hexagonal pore lattice the pore radius r/a [0.4, 0.5] is varied. It is obvious that the size of  the complete band gap (red bar) can be controlled via the pore radius. No bands in the band gap means no light transmittance for the related frequencies and all directions.


Fig. 3 shows two plots that display the calculated densities of states (DOS) for a 2D hexagonal photonic crystal. The pores have a radius of r/a=0.458. In 2D photonic crystals the polarizations decouple into components being parallel or perpendicular to the pore axis. They are displayed separately here (TM-, TE modes). In the band gaps there are NO states, i.e., DOS=0.


Two-dimensional PCs of dielectric media are routinely assumed to be formed by circular rods or pores, respectively.

In Fig. 4 the effect of elliptic deformations of the pores is studied. Therefore we compare the gap maps of photonic crystals with elliptic and circular pores having the same air filling ratio. The area of intersection of the red and blue gap maps strongly decreases from right to left! Circular pores provide the largest complete band gaps (cf. [1], [2]).


Fig. 5 presents a direct comparison between experimental measurements and theoretical results.


The measured reflectivity of a 2D-macroporous silicon photonic crystal (a = 500 nm, r/a = 0.425) is given in the right panel (left) of the graph. The reflectivity calculations (left) were carried out with the transfer matrix method. The spectra (cf. large figure) were convoluted with Gaussians. The  Γ-M direction of the related band structures (right) fits very well with the experiments.


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