Computation and visualization of photonic quasicrystal spectra via Bloch’s theorem

Previous methods for determining photonic quasicrystal (PQC) spectra have relied on the use of large supercells to compute the eigenfrequencies and/or local density of states. In this paper, we present a method by which the energy spectrum and the eigenstates of a PQC can be obtained by solving Maxwell’s equations in higher dimensions for any PQC defined by the standard cut-and-project construction, to which a generalization of Bloch’s theorem applies. In addition, we demonstrate how one can compute band structures with defect states in the same higher-dimensional superspace. As a proof of concept, these general ideas are demonstrated for the simple case of one-dimensional quasicrystals, which can also be solved by simple transfer-matrix techniques.

Figure 1

Unit cell of the Fibonacci superspace dielectric. The physical dielectric is obtained by taking a slice at an angle $\tan \varphi =\tau$. Black and white are the dielectric constants of the structure factor material and air, chosen to be $\epsilon =4.84$ and $\epsilon =2.56$, respectively.

Figure 2

Left: Frequency spectrum $\omega$ of the Fibonacci quasicrystal vs “wave vector” $k_x$. The blue lines indicate spurious states which arise due to finite-resolution effects (see text). Right: Corresponding density of states $\rho \left(\omega \right)$ computed using a transfer-matrix technique with a supercell of ${10}^4$ layers.

Figure 3

(Color online) Integrated density of states (DOS) vs band index (normalized by resolution) for various resolutions. The dashed red, diamond blue, and solid black lines denote resolutions of 20, 50, and 200, respectively.

Figure 4

(Color online) Plot of the magnetic field amplitude $\mid H_z\mid$ for a band-edge state taken along a slice of the two-dimensional superspace (in the $\varphi$ direction). Inset: Two-dimensional superspace field profile (red/white/blue indicates positive/zero/negative amplitude).

Figure 5

(Color online) Electric field energy distribution of the band-edge states of gaps 1 and 2 in Fig. 2. Although they have a complex small-scale structure, the large-scale variation is easily understood in terms of the structure of the superspace.

Figure 6

(Color online) Enlarged view of the Fibonacci spectrum (Fig. 2) showing a gap with a spurious band crossing it. Insets show the magnetic field $\mid H_z\mid$ for the spurious band at various $k_x$—the localization of this mode around the $X$-parallel edges of the dielectric indicates that this is a discretization artifact.

Figure 7

Dielectric for the Fibonacci chain with $\epsilon =2.56$ (bottom left), and a defect—an additional ${\epsilon }_d=8.0$ layer; shown in gray.

Figure 8

(Color online) Varying the defect epsilon for resolutions 50 (blue) and 100 (red). The thickness of the defect is fixed to 0.02 lattice constants. The number of spurious modes increases with the resolution, the true defect state being the lowest of these modes.

Figure 9

(Color online) Semilogarithmic plots of the magnetic field magnitude $H_z$ for the lowest (top) and highest (bottom) defect state for the configuration shown in Fig. 7. Insets: Two-dimensional superspace visualizations of the defect states. Note the additional node in the lower figure (corresponding to an unphysical oscillation).

Figure 10

(Color online) Projected band structure vs cut angle $\varphi$, showing different one-dimensional quasicrystal realizations. The vertical red line indicates the spectrum when the slope is the golden ratio $\tau$ (the spectra of $\varphi$ and $\pi -\varphi$ are equivalent).

Figure 11

(Color online) Schematic showing (left) the superspace slice $X$ and (right) the projected slice modulo 1 into the unit cell $\overline{X}$, along with the intersection $\overline{T}$ of $\overline{X}$ with the $s=0$ hyperplane.