Breakdown of Spin-to-Helicity Locking at the Nanoscale in Topological Photonic Crystal Edge States

We measure the local near-field spin in topological edge state waveguides that emulate the quantum spin Hall effect. We reveal a highly structured spin density distribution that is not linked to a unique pseudospin value. From experimental near-field real-space maps and numerical calculations, we confirm that this local structure is essential in understanding the properties of optical edge states and light-matter interactions. The global spin is reduced by a factor of 30 in the near field and, for certain frequencies, flipped compared to the pseudospin measured in the far field. We experimentally reveal the influence of higher-order Bloch harmonics in spin inhomogeneity, leading to a breakdown in the coupling between local helicity and global spin.

Figure 1

(a) Scanning electron micrograph (SEM) of the topological interface in the fabricated sample with the color-coded regions depicting the shrunken $L_S$ (red) and expanded $L_E$ (blue) lattices. The lattice periodicity is $a=880\mathrm{nm}$. (b) Schematic representation of the TPC lattice with overlaid near- and far-field amplitudes of the electromagnetic field. Inset: Schematic representation of the reciprocal space of an edge state (frequency $\omega ={\omega }_0$) with its typical hexagonal pattern of intensity peaks together with the 3D light cone of air ($\omega =ck$). The dashed circle at the intersection of the light cone and the reciprocal space of the edge states represents the largest spatial frequencies of the edge state that can couple to the far field. Close-up of the normalized in-plane electric (c) near-field and (d) radiative field amplitude in the TPC featuring an armchair interface at $\lambda =1520\mathrm{nm}$ with the light launched into the structure from the left (indicated by the red arrow). The dashed hexagonal pattern outlines the underlying crystal lattice.

Figure 2

(a) Experimentally measured NF spin density ${\sigma }_z$ of the AC-edge mode over two unit cells for an excitation wavelength of $\lambda =1520\mathrm{nm}$. (b) Experimentally measured FF spin density over the same extent as in (a), realized by filtering out the nonradiative wave vectors of the field shown in (a). (c) Numerically calculated spin density for $k_x=0.16(2\pi /\mathrm{a})$ with (d) displaying the spin density for numerical simulations in the FF. The solid gray line indicates the armchair interface while the dashed hexagonal pattern outlines the underlying crystal lattice.

Figure 3

Numerically simulated dispersion relation where the edge state eigenfrequencies are color coded with the estimated (a) near-field and (b) far-field optical spin $S$.

Figure 4

Two-dimensional Fourier representation of the experimentally measured electric field amplitude of the TPC edge mode at a wavelength of $\lambda =1560\mathrm{nm}$, with the amplitude shown in logarithmic scale. The reciprocal lattice vectors are periodically separated in the propagation direction ($k_x$) in units of $2\pi /a$, where $a$ is the lattice constant of the armchair edge. The white dashed circle represents the light cone in air. Right inset: Zoom-in of momentum space restricted to show the fundamental ($k_{\overline{0}}$) and first higher-order BH ($k_{\overline{1}}$).

Figure 5

Buildup of spin-dependent contributions of the BHs from (a) experiment and (b) numerical simulations, evaluated over several higher-order BH ranges. Right inset: Spin-dependent contribution for the largest evaluated BH range which extends from $k_x=[-6.5,6.5]$ with a saturated color scale. The gray dashed line indicates the frequency of the $\mathrm{\Gamma }$-point crossing. The grayed out region extending below 184.4 THz depicts the lower band edge.