Topological Insulators by Topology Optimization

An acoustic topological insulator (TI) is synthesized using topology optimization, a free material inverse design method. The TI appears spontaneously from the optimization process without imposing explicit requirements on the existence of $\mathrm{pseudospin}\text{−}1/2$ states at the TI interface edge, or the Chern number of the topological phases. The resulting TI is passive, consisting of acoustically hard members placed in an air background and has an operational bandwidth of $\approx 12.5\%$ showing high transmission. Further analysis demonstrates confinement of more than 99% of the total field intensity in the TI within at most six lattice constants from the TI interface. The proposed design hereby outperforms a reference from recent literature regarding energy transmission, field confinement, and operational bandwidth.

Figure 1

(a) Illustration of backscattering protected, pseudospin-dependent directional field propagation through a TI with (green) defects. (b) Left: Design cell showing the (gray) designable region and (dashed lines) mirror symmetry lines. Middle and right: Design example illustrating the symmetry. (c) Model domain ${\mathbf{\Omega }}_A$, PML layer ${\mathbf{\Omega }}_{\mathrm{PML}}$, and domains ${\mathbf{\Omega }}_{\mathrm{dR},1}$ and ${\mathbf{\Omega }}_{\mathrm{dR},2}$ containing the topological phases. Monopolar source ${\mathbf{P}}_s$ in the focal point of a reflector with surface ${\mathrm{\Gamma }}_R$. (d) Domains for computing ${\mathrm{\Phi }}_{\text{Total}}$ and ${\mathrm{\Phi }}_{\mathrm{BG}}$ and port numbering, $P1$–$P4$.

Figure 2

(a) Initial and (b) optimized material configuration in supercells consisting of the two topological phases (black) air, (white) solid, (shades of blue) air and solid mixture. (c),(d) Sound pressure at $f_0=20\mathrm{kHz}$ for the (c) initial and (d) optimized material distribution in ${\mathrm{\Omega }}_{\mathrm{dR},1}\bigcup {\mathrm{\Omega }}_{\mathrm{dR},2}$. Color map: Pressure magnitude, (white) solid material. (d) The targeted backscattering protected edge-state energy transport is observed. (e) Transmission to ports $P2$, $P3$, and $P4$, as a function of frequency, normalized to the energy flowing through port $P1$. [Port numbering in (d).]

Figure 3

Supercell band structure for the (a) initial material configuration and (b) optimized TI design. The edge-state bands are colored indicating (red) positive, and (blue) negative $\mathrm{pseudospin}\text{−}1/2$, and (white) bulk band gap. (c) Supercell used to compute (b). (d),(e) Normalized eigenmodes at $f_0=20\mathrm{kHz}$. (f) Band structures for the two crystal phases.

Figure 4

Investigation of robust energy transport in TI edge state under geometric defects. (a)–(d) Color map: Max-normalized sound pressure at 20 kHz, (white) solid material. (a) Unperturbed TI. (b) TI with bend. (c) TI with cavity. (d) TI with disorder. (e) Max-normalized transmitted power, recorded after the material slab, for the configurations shown in (a)–(d).

Figure 5

Spatial field confinement of the TI edge state. (a) Fraction of the total power contained within a distance $d$ from the TI interface edge $P_f(d)$, at $f_0=20\mathrm{kHz}$, for [Optim] the proposed TI design and [Ref] the TI in Ref. [18]. Map of the distance $d$ versus frequency, within which more than 99% of the total power is confined, $P_f(d)>0.99$ for [Optim] the optimized TI design and [Ref] the reference. (c) TI supercell, (white) solid material, (black) air.