Topological transitions in continuously deformed photonic crystals

We demonstrate that multiple topological transitions can occur, with high sensitivity, by continuous change of the geometry of a simple two-dimensional dielectric-frame photonic crystal consisting of circular air holes. By changing the radii of the holes and/or the distance between them, multiple transitions between normal and topological photonic band gaps (PBGs) can appear. The time-reversal symmetric topological PBGs resemble the quantum spin Hall insulator of electrons and have two counterpropagating edge states. We search for optimal topological transitions, i.e., sharp transitions sensitive to the geometry, and optimal topological PBGs, i.e., large PBGs with a clean spectrum of edge states. Such optimizations reveal that dielectric-frame photonic crystals are promising for optical sensors and unidirectional waveguides.

Figure 1

Geometric transitions in the dielectric-frame 2D PCs. Each hexagonal unit cell has six air holes (dark-blue regions) in the dielectric background (gray-blue regions). Each air hole has radius $r$. The distance between the center of an air hole and the center of the unit cell is $l$. The structure has $C_6$ symmetry and the lattice constant is $a$. By tuning the radius $r$ and distance $l$, the PC can experience various geometric transitions. (a) For air holes of small radius, the main transition is between two types of triangle lattices: One has $l<a/3$ (“triangle I”), while the other has $l>a/3$ (“triangle II”). The transition takes place at $l=a/3$ where the PC becomes a honeycomb lattice of air holes. A special limit is $l=a/2$, where the PC is manifested as a kagome lattice. (b) For air holes of large radius, the transition is different: When $l+r<a/2$, the PC is a triangle lattice of air holes. For larger distance $l+r\ge a/2$, the PC becomes a triangle-honeycomb lattice of dielectrics. In the special cases of $l=a/3$ and $l=a/2$, the PCs are still the honeycomb and kagome lattices of air holes, respectively.

Figure 2

Band diagrams and eigenmodes in dielectric-frame 2D PCs with normal and inversed band order. The dielectric background is made of GaP. (a) Band diagram for the 2D PC with $r=0.2a$ and $l=0.275a$. The dipole modes have lower frequency than the quadrupole modes at the $\mathrm{\Gamma }$ point. (b) The magnetic field $H_z$ profiles of dipole and quadrupole modes at the $\mathrm{\Gamma }$ point. The six circles in each unit cell represent the air holes. (c) Band diagram for the 2D PC with $r=0.2a$ and $l=0.475a$. The dipole modes have higher frequency than the quadrupole modes at the $\mathrm{\Gamma }$ point. (d) Magnetic field $H_z$ profiles of dipole and quadrupole modes at the $\mathrm{\Gamma }$ point. In (b),(d), the upper panels represent modes of higher frequency, while the lower panels are for modes of lower frequency.

Figure 3

(a) Phase diagram of parity-inversion induced topological transitions for a small radius $r=0.13a$ with varying distance $l$. The phase transition is characterized by the quantity ${\omega }_{pd}$ (represented by the green curve). The cyan regions stand for the reversed band order, while the yellow region denotes normal band order. The orange curve represents the size of the complete PBG between the $p$ bands and the $d$ bands. (b) Magnetic field $H_z$ profiles of dipole and quadrupole modes at the BZ center for $r=0.13a$ and various $l$. (c) Phase diagram of parity-inversion induced topological transitions for a large radius $r=0.2a$ with varying distance $l$. (d) Magnetic field $H_z$ profiles of dipole and quadrupole modes for $r=0.2a$ and various $l$. In (b),(d) the upper (lower) field profiles are for higher- (lower-) frequency modes. The field profiles of the degenerate $d\left(p\right)$ bands are adopted from one mode out of the $d\left(p\right)$ doublets.

Figure 4

Double Dirac points emerging at the topological phase transition points for $r=0.2a$ and varying distance $l$. (a)–(c) Band diagrams. (d) Zoomed-in dispersion of the double Dirac point in (c).

Figure 5

Phase diagram of the dielectric-frame PCs for (a) $l=0.4a$ and (b) $l=a/2$ for various radii $r$. (b) represents the case of kagome lattices of air holes. The cyan (yellow) regions stand for the reversed (normal) band order, i.e., ${\omega }_{pd}<0\left({\omega }_{pd}>0\right)$. The orange curve represents the size of the complete PBG between the $p$ bands and the $d$ bands. Significant complete PBGs can be achieved in both (a) and (b).

Figure 6

Topological edge states of photons. (a) Band diagram for a supercell calculation of the edge states. The gray regions represent the bulk photonic bands; the orange and blue curves stand for the edge states. The upper panel shows the geometry of the supercell. (b) Magnetic field $H_z$ profiles of the two edge states at the same frequency $\omega =0.688\left(\frac{2\pi c}{a}\right)$ for opposite wave vectors. (c) Zoomed-in graph of the magnetic field profiles for the same edge states as in (b). Poynting vector profiles are also included in (c), as represented by the black arrows.

Figure 7

Photonic edge states exhibit unidirectional propagation along $Z$-shaped boundaries between PCs with complete PBGs of trivial and nontrivial topology. (a) Illustration of the computational cell. The horizontal section of the $Z$-shaped boundary consists of zigzag boundaries between the two PCs. (b) Unidirectional propagation of photons excited by a point source of orbital angular momentum −1, corresponding to the spin-down edge states. (c) Unidirectional propagation of photons excited by a pointlike source of orbital angular momentum +1, corresponding to the spin-up edge states. Electromagnetic waves excited by the source are eventually absorbed by the PMLs surrounding the whole structure.

Figure 8

(a) Frequencies of the $p$ and $d$ bands at the $\mathrm{\Gamma }$ point, ${\omega }_p$ and ${\omega }_d$, as functions of the thickness of the attached biomarker layer, $\delta r$. The yellow and cyan regions represent normal and topological PBGs, respectively. The structure of the unit cell is illustrated in the inset. The dark-blue regions are the air holes. The white region represents the biomarker attached to the dielectric background (the gray-blue regions). The original radius of the air hole is $r=0.21a$. The distance between the center of an air hole and the center of the unit cell is $l=0.4a$. (b) Illustration of photon transmission through a PC region with 20 unit cells. The gray-blue regions represent the dielectric material. The white regions denote the biomarkers, while the dark-blue regions stand for air holes. (c) Photon transmission spectrum for varying thickness of the biomarkers. The blue, red, and black curves correspond to $\delta r/a=0,0.05,0.1$, respectively.