Meron Spin Textures in Momentum Space

We show that a momentum-space meron spin texture for electromagnetic fields in free space can be generated by controlling the interaction of light with a photonic crystal slab having a nonzero Berry curvature. These spin textures in momentum space have not been previously noted either in electronic or photonic systems. Breaking the inversion symmetry of a honeycomb photonic crystal gaps out the Dirac cones at the corners of Brillouin zone. The pseudospin textures of photonic bands near the gaps exhibit a meron or antimeron. Unlike the electronic systems, the pseudospin texture of the photonic modes manifests directly in the spin (polarization) texture of the leakage radiation, as the Dirac points can be above the light line. Such a spin texture provides a direct approach to visualize the local Berry curvature. Our work highlights the significant opportunities of using photonic structures for the exploration of topological spin textures, with potential applications towards topologically robust ways to manipulate polarizations and other modal characteristics of light.

Figure 1

(a) A photonic crystal slab with a honeycomb lattice of circular air holes. The dielectric constant of the slab $\epsilon =4$. The thickness of the slab is $d=0.25a$, where $a$ is the lattice constant. The lower right shows the Brillouin zone. The wave vector $\mathit{q}=(q_x,q_y)$ measured from $K$ and $K^{\prime }$ are defined individually so that $q_y$ axis points towards $\mathrm{\Gamma }$. (b) The band structure near $K(K^{\prime })$. The two bands form a Dirac cone when $d_A=d_B=0.22a$ (black dotted lines), while the degeneracy is lifted when $d_A=0.18a$, $d_B=0.26a$ (red). The blue dashed lines plot the fit from the effective Hamiltonian. (c) Pseudospin textures: core-up (down) meron for the lower (upper) band near $K$, and core-down (up) meron for the lower (upper) band near $K^{\prime }$.

Figure 2

(a) Diffraction scheme. Light with a specific frequency and polarization is incident at a specific angle to excite desired photonic modes around one Brillouin zone corner ($K_1$). The excited mode radiates out to both zeroth-order ($K_1$) and first-order ($K_2$ and $K_3$) diffraction channels on both the transmission and reflection sides. Inset: first Brillouin zone with corners $K_1$, $K_2$ and $K_3$ indicated. (b)–(e) Calculated reflection spectra for incident light with fixed parallel wave vector $K_1$. The shaded regions include the spectral range of the photonic band gap at $K$. (b) and (c) zeroth-order reflection for right (b) and left (c) circularly polarized incident light. (d) and (e) first-order reflection for right (d) and left (e) circularly polarized incident light. The dashed lines show the fit with Lorentzian line shapes.

Figure 3

(a) Isofrequency contours of the lower band near $K$ point are studied frequency by frequency. (b)–(e) Stokes parameters as functions of $(q_x,q_y)$ at frequency $\omega =0.855\times 2\pi c/a$ which is in the lower band. (b) $S_0$ is the intensity of first-order reflected light. (c)–(e) Normalized Stokes parameters $S_1/S_0$, $S_2/S_0$, and $S_3/S_0$.

Figure 4

In the main figure of each panel, the arrow tail positions indicate the band dispersion $\omega (q_x,q_y)$. The arrow direction indicates the spin $\mathbf{n}$ at that point. The inset plots the in-plane spin component $(S_1/S_0,S_2/S_0)(q_x,q_y)$. (a) $K$ valley, reflection. The spin texture is a core-down antimeron (Skyrmion number $Q=1/2$, polarity $p=-1$, vorticity $w=-1$). (b) $K^{\prime }$ valley, reflection. The spin texture is a core-up antimeron ($Q=-1/2$, $p=1$, $w=-1$). (c) $K$ valley, transmission. The spin texture is a core-up meron ($Q=1/2$, $p=1$, $w=1$). This spin texture is identically mapped from the pseudospin texture near $K$ [Fig. 1]. (d) $K^{\prime }$ valley, transmission. The spin texture is a core-down meron ($Q=-1/2$, $p=-1$, $w=1$). This spin texture is identically mapped from the pseudospin texture near $K^{\prime }$ [Fig. 1].