Revealing the Topology of Quasicrystals with a Diffraction Experiment

Topological properties of crystals and quasicrystals is a subject of recent and growing interest. This Letter reports an experiment where, for certain quasicrystals, these properties can be directly retrieved from diffraction. We directly observe, using an interferometric approach, all of the topological invariants of finite-length Fibonacci chains in their diffraction pattern. We also quantitatively demonstrate the stability of these topological invariants with respect to structural disorder.

Figure 1

Experimental setup. (a) Principle of the experiment. A collimated laser beam is sent to a diffraction grating programmed on a digital micromirror device (DMD). We measure the far-field diffraction pattern on a CCD camera [29]. (b) Characteristic diffraction pattern for a grating following a Fibonacci sequence along the horizontal $x$ direction (and uniform along the vertical $y$ direction).

Figure 2

Influence of the phason degree of freedom on the diffraction pattern of a Fibonacci grating. (a) Structure composed by the juxtaposition of two identical Fibonacci chains along $x$ (repeated along $y$). White lines correspond to reflecting mirrors ($B$ letters in the Fibonacci chain) and black to nonreflecting mirrors ($A$ letters). We show for clarity a chain of length $F_6=13$. (b) Measured diffraction pattern for two side-to-side Fibonacci chains of length $F_{10}=89$. Each line corresponds to the diffraction pattern for a given value of the phason degree of freedom $\mathrm{\Phi }$. (c) Structure composed by the juxtaposition of a Fibonacci chain and of the reversed chain along $x$ (repeated along $y$). (d) Measured diffraction pattern for the structure depicted in Fig. 2 with length $2F_{10}=89$. (e) Sketch of a transmission experiment. (f) Transmission of a 1D Fibonacci chain as calculated in [27]. Gaps appear at the same position as the peaks of the diffraction pattern. (g)–(i) Cuts along vertical lines of Fig. 2 for three values of $k_x\approx k_q\approx (2\pi /a)$ [0.618 (g), 0.146 (h), and 0.472 (i)] corresponding respectively to topological numbers $q=1$, $-3$, and 4.

Figure 3

Two-dimensional map of topological numbers. (a) Structure of the diffraction grating. White (respectively, black) pixels correspond to reflective (respectively, nonreflective) mirrors. The bottom line is the Fibonacci sequence for $\mathrm{\Phi }=0$. The other lines correspond to the phase-shifted Fibonacci chains scanning $\mathrm{\Phi }$ from 0 to $2\pi$. The length of each chain in this illustration is $F_7=21$ for clarity. (b) Two-dimensional diffraction pattern for 89 phason-shifted chains with $F_{10}=89$ letters. We observe peaks at the same positions along $k_x\approx k_q$ as in Fig. 2 but shifted along the vertical axis to $k_{\mathrm{\Phi }}=q$. (c) Zoom on Fig. 3.

Figure 4

Robustness of the topological features against structural noise. (a)–(c) Examples of 2D diffraction patterns for noise levels $\eta =0.1$, 0.5, and 0.8. (d) Position of the diffraction peaks along the $k_{\mathrm{\Phi }}$ axis for $q=1$ (circles), $q=2$ (squares), and $q=-4$ (triangles). (e) Corresponding amplitude of the diffraction peaks normalized to the amplitude of the $q=1$ peak without noise. Statistical error bars in (d)–(e) are about the size of the markers.