Topological photonic crystal fibre

Abstract

Photonic crystal fibres (PCFs) are optical fibres that guide light using a modulated dielectric medium. They provide an exceptionally versatile platform for various applications, thanks to the flexibility with which light-guiding can be customised by modifying the fibre geometry. Here, we realise a PCF with guided modes produced by photonic bandstructure topology rather than conventional mode-trapping mechanisms. The design, which is compatible with the stack-and-draw fabrication process, consists of a cross-sectional photonic topological crystalline insulator with a disclination. A bulk-defect correspondence produces degenerate topological modes, lying below the cladding light line. We use various theoretical methods to confirm their topological origins, including a spectral localiser that makes minimal assumptions about the bandstructure. Our experiments on the fabricated topological fibre show it transmitting visible to near-infrared light with low losses of 10--20 dB/km, which do not increase much when the fibre is bent. A comparable solid-core PCF of conventional design exhibits substantially higher bending losses. Optical fibres based on topological modes thus hold promise for improved performance and novel functionalities.

Figures (7)

• Figure 1: Topological photonic crystal fiber (TPCF) implementation. (A) Schematic of cross sectional photonic structure, comprising a topological crystalline insulator (TCI) with a disclination. (B) Unit cell (red lines) for the disclination-free TCI, which has $C_{6v}$ symmetry. Starting from a triangular lattice of equal air holes (dashed circles), the hole radii are alternately increased and decreased. For two different choices of modulation, the Wannier centers (magenta stars) are located at the unit cell's sides (upper plot) or center (lower plot). (C) Photograph of the drawn fiber cane. Inset: stacking arrangment of the preform. (D) Scanning electron microscope image of the TPCF's end face. (E) Calculated intensity profile (power flow in the $z$ direction) for a disclination state of the structure from (A) at $k_z d/2\pi = 2$. The air holes are indicated by white circles. (F) Optical microscope photograph of a $100$$\textrm{m}$ TPCF, with a supercontinuum light source at the opposite end. Inset: infrared camera image of the same. (G) Measured transmittance (upper plot) for a $67$$\textrm{m}$ TPCF and transmission loss (lower plot) using the same source.
• Figure 2: Analysis of guided topological defect modes (GTDMs). (A) Band diagram for the measured TPCF structure (Fig. \ref{['fig:TPCF']}D). The GTDMs are plotted in red (radially polarized) and blue (azimuthally polarized), and bulk band frequencies are drawn as pink areas. In the large-$k_z$ regime on the right, dispersion curves for individual bulk states are plotted in gray. The gray-stroked region contains numerous modes that cannot be resolved numerically. (B) Calculated intensity profiles (normalized power flow in $z$) for exemplary radially and azimuthally polarized GTDMs at $k_z d/2\pi = 20$. Polarization directions are indicated by cyan arrows. (C) In-plane quality ($Q$) factors of the GTDMs versus $k_z$. (D) Mode areas of the GTDMs versus $k_z$. (E) Characterization of bulk TCI bands at fixed $k_z$. Left panel: band spectrum for the periodic TCI structure (corresponding to Fig. \ref{['fig:TPCF']}B, upper plot). Right panel: Berry phases of the Wilson loop operator for different base points, using bulk bands $\#$(1, 2, 3) and $\#$(4, 5, 6). (F) Calculated eigenfrequencies for the preform hole profile (left panel; pink areas denote bulk bands), and the corresponding spectral charges in the five unit cells around the center (right panel). Blue/red dots respectively indicate azimuthally/radially polarized GTDMs. For details, see the Supplementary Materials, section S3. (G) Topological characterization via the spectral localizer (see the Supplementary Materials, section S4). Left panel: eigenfrequencies of a symmetrized structure based on the preform profile. Center panel: the local index $\zeta_\omega^{S}$, where $S$ is mirror symmetry around the $x$ axis. Right panel: the local gap measure $\mu_\omega$. The results for E--G are calculated at $k_z d/2\pi=2$.
• Figure 3: Polarization dependence and bending resistance. (A) Experimental setup with a tunable filter, fixed linear polarizer, and rotatable half-wave plate (HWP) between the source and the input face of a 0.5 m TPCF. A second linear polarizer, with variable angle $\theta_2$, is placed between the end face and a beam profiler. Inset: a measured output intensity profile concentrated at one of the high-index regions around the central air hole. (B) Measured intensity at the center of the selected spot, versus HWP angle. Results are shown for two values of $\theta_2$ differing by $90^\circ$, with fixed wavelength 1070 nm. (C) Variation of intensity with position, measured along a radial line passing through the selected spot (cyan dashes in the inset of (A)), with HWP fixed at $24^\circ$ (vertical dashes in (B)). (D) Intensity at the center of the selected spot versus HWP angle, for three different input wavelengths and fixed $\theta_2 = 90^\circ$. (E) Measured transmittance for a straight TPCF (brown) and a TPCF with a two-loop bend of radius 1 cm (blue). (F) Output power difference between a straight fiber and one with a two-loop bend of radius 1 cm, for the TPCF (blue) and a comparable solid-core (SC) PCF (yellow). Inset: Scanning electron microscope image of the SC-PCF. (G) Calculated mean in-plane $Q$ factors for the GTDMs in the TPCF (blue) and the fundamental core modes in the SC-PCF (yellow), for different bending radii at $k_z d/2\pi=30$ (corresponding to $\sim$740 nm). For the SC-PCF in (F) and (G), the ratio between the air hole radii and the pitch is $0.38$.
• Figure S1: Lattice design. (A) Wedge with opening angle $\pi/3$, extracted from a triangular lattice with nearest neighbor center-to-center spacing $d$. The discs (air holes) have alternating radii of $0.57d$ and $0.35d$. (B) Wedge with opening angle $2\pi/5$, generated from (A) by uniformly scaling the azimuthal coordinates of the site centers. This corresponds to the structure shown in Fig. 1A. (C) Wedge generated from (B) by gradually enlarging the disc radii and adjusting their positions, with boundaries fixed, until they are jam-packed. (D) Structure generated from (C) by assigning inner air holes of radius $R_1' = 0.49d$ and $R_2' = 0.33d$ to the discs. This corresponds to the target stacking arrangement shown in Fig. 1C. (E) Comparison between the preform hole profile based on (D) (black lines), and the measured hole profile of the fabricated TPCF (pink lines). The air holes radii are slightly enlarged to $R_1 = 0.55d$ and $R_2 = 0.36d$. (F and G) Calculated eigenfrequencies at $k_z d/2\pi = 2$ for the measured hole profile (F) and the preform hole profile (G). Red (blue) data points correspond to GTDMs with radial (azimuthal) polarization, while gray data points are bulk states. The pink-shaded region indicates the bulk band. (H) Scanning electron microscope image of the fiber cross section. The small black specks correspond to non-filled interstitial air holes, which can be seen to mostly occur in the out-of-core region.
• Figure S2: Wilson loop and spectral charge calculations. (A) Phase profiles of the out-of-plane electric field ($E_z$) for the six Bloch states marked in Fig. 2E at high-symmetry momentum points. The symmetry indicators of bulk bands $\#(1,2,3)$ are labeled on the right. (B) Lowest five bands for the periodic photonic structure with trivial crystalline topology, calculated from the unit cell in Fig. 1B (lower plot). (C) Phase profiles of the out-of-plane electric field ($E_z$) for the Bloch states $\#(1,2,3)$ in the trivial structure of (B) at high-symmetry momentum points. (D) Berry phases of the Wilson loop operator at different initial $k$ for the trivial bulk bands $\#(1,2,3)$. Inset: the schematic of the Wilson loop. The base point lies along the path $\Gamma \rightarrow \Gamma'$, and the Wilson loop follows the dashed line running parallel to $\Gamma$--$\Gamma"'$. (E) Left panel: calculated eigenfrequencies for the trivial PCF structure, based on Fig. 2F but with large and small air holes switched. Inset: schematic of the lattice, composed of 30 intact unit cells and a central defect. Right panel: calculated spectral charges (black numbers) for the schematic structure in the left panel. In (A--E), the eigenmodes are calculated at $k_z d/2\pi = 2$. The spectral charge calculations are performed using symmetric preform hole profiles.