Topological phase transition in a stretchable photonic crystal

We design a setup to realize tunable topological phases in elastic photonic crystals. Using the Su-Schrieffer-Heeger (SSH) model as a canonical example, we show how a system can be continuously tuned across its topological phase transition by stretching. We examine the setup both analytically and numerically, showing how the phase transition point may be identified from the behavior of bulk modes. Our design principle is generic, as it can be applied to a variety of systems, and enables multiple new theoretical predictions to be experimentally tested by continuously strain-tuning system properties, such as the shape of the band structure and the topological invariant. In addition, it allows for cost-effective device fabrication, since a wide range of parameter space can be accessed on a single photonic crystal chip.

Figure 1

(a) The band structure of the SSH chain Eq. (2) using $t=1$ is plotted for $\delta =0$ (solid lines) and for $\left|\delta \right|=0.5$ (dashed lines). The band structure is symmetric with respect to $\delta \to -\delta$. (b) The contour of the $\vec{d}$ vector is plotted in the $d_x,d_y$ plane for $\delta =0.5$, 0, and $-0.5$, using again $t=1$. Darker shades indicate larger values of $\delta$, and the arrows indicate the winding direction as $k$ is increased. When the contour encircles the origin, the winding number Eq. (4) is nonzero, signaling the presence of zero-energy end states.

Figure 2

(a) Front and perspective view of the stretchable photonic crystal. Pairs of waveguides arranged to form a Y-shaped structure are placed on an elastic substrate, allowing one to tune the intercell hopping ($w$) while keeping the intracell hopping ($v$) fixed. (b) Field confinement in the plane perpendicular to the propagation direction at two different positions along the waveguides as calculated numerically by FDTD simulations (see text). Light is confined within one of the two Y arms, without leaking to the supporting post. The dimensions of the Y structure are indicated with arrows. Here as well as in Figs. 3 and 4, the intensity is divided by its maximum such that the color scale is dimensionless and runs from 0 to 1.

Figure 3

(Top) Strength of the nearest-neighbor hopping $J$ as a function of waveguide separation. The strengths of intracell ($v$, squares) and intercell ($w$, circles) hopping are different, reflecting the different media between the waveguides. Dashed lines show the value of $v$ used in subsequent simulations. (Bottom) Oscillation of the light field in an isolated pair of waveguides using a separation of 75 nm. The light is confined to each of the waveguides forming the Y structure and jumps to the neighboring waveguide periodically. The frequency of jumps is used to extract the hopping strength.

Figure 4

Light intensity profile for the edge (top) and bulk (bottom) modes in the topologically nontrivial phase, corresponding to a separation of 25 nm between waveguides of neighboring Y structures. The waveguides are arranged horizontally, such that in the simulated SSH model the vertical direction represents space while the horizontal direction represents time. Most of the light remains confined to the edge, signaling the presence of a topologically protected boundary state, while it disperses in the bulk owing to the finite velocity of bulk states.

Figure 5

(a) Contours of the $\vec{d}$ vector as momentum is advanced from $-\pi$ to $\pi$, computed using the numerically extracted values of intracell and intercell hopping, $v$ and $w$. Each contour winds in an anticlockwise fashion as momentum is increased. Lighter shades and smaller radii correspond to larger separation between the waveguides of adjacent Y structures (from 25 to 400 nm in steps of 25 nm). (b) Mean chiral displacement computed as in Eq. (8), plotted as a function of the separation between unit cells. The dashed black line shows the result expected in the infinite time limit. As the photonic crystal is stretched, a topological phase transition to a trivial phase takes place. The latter occurs when the contours of panel (a) no longer encircle the origin and is marked by a sharp drop in the mean chiral displacement of panel (b), from values close to 0.5 to values close to 0.

Figure 6

Mean-squared displacement $d$ computed as in Eq. (9) and plotted as a function of the distance between unit cells. The value of $d$ has a maximum at the topological phase transition, since at that point the bulk mode velocity is maximal.

Figure 7

Light intensity profiles after a propagation distance of $40\mu \mathrm{m}$, starting from a bulk state. The light intensity (color scale) is plotted as a function of waveguide position relative to the initially excited waveguide (horizontal axis) and distance between unit cells (vertical axis). The overall spread of light increases as one approaches the phase transition (dashed green line) and decreases afterwards. The light intensity is plotted on a logarithmic scale to better show the behavior close to the critical point.

Figure 8

Spectrum of a photonic crystal consisting of 40 waveguides (meaning $N=20$ unit cells), obtained by solving Eq. (A8) with $v=0.6\mu {\mathrm{m}}^{-1}$ and $w=1\mu {\mathrm{m}}^{-1}$. The spectrum is gapped and contains two topological midgap modes at ${\beta }_z=0$.