Dynamics of Topological Polarization Singularity in Momentum Space

The polarization singularity in momentum space has recently been discovered as a new class of topological signatures of Bloch modes in photonic crystal slabs concerning the far-field radiations, beyond its near-field description with widely explored topological band theory. Bound states in the continuum (BICs) in photonic crystal slabs are demonstrated as vortex eigenpolarization singularities in momentum space and the circular polarization points ($C$ points) are also obtained based on BICs, opening up more possibilities for exotic light scattering and various topological phenomena of singular optics. Here, focusing on the nondegenerate bands, we report the generation to annihilation of two pairs of $C$ points in momentum space in the photonic crystal slabs with inversion symmetry but broken up-down mirror symmetry. Interestingly, as the $C$ points evolve with the structure parameter, we find two merging processes of $C$ points, where an accidental at-$\mathrm{\Gamma }$ BIC and unidirectional radiative resonances with leaky channels of drastically different radiative lifetime emerge. The whole evolution is governed by the global charge conservation and the sum of topological charges equals to zero. Our findings suggest a novel recipe for dynamic generation and manipulation of various polarization singularities in momentum space and might shed new light to control the resonant and topological properties of light-matter interactions.

Figure 1

The evolution of polarization singularities in momentum space supported by one-dimensional photonic crystal slab. (a) Schematics of a one-dimensional photonic crystal slab composed of two superimposed identical gratings with a misalignment $\delta$. (b) Calculated TE band structure along the $k_x$ axis at $\delta =0$. The blue line is the third TE band (${\mathrm{TE}}_3$) that we focus on. (c) Mode profile in the unit cell for the ${\mathrm{TE}}_3$ band at $\mathrm{\Gamma }$ point. (d) Trajectories traced by two pairs of $C$ points with different topological charges ($q=\pm 1/2$) in momentum space as $\delta$ increases for downward radiation. The blue and red discs marked by $\pm 1/2$ indicate the right-handed and left-handed circular polarizations with a topological charge of $\pm 1/2$. (e)–(h) show the calculated projected polarization vector fields of downward radiation when $\delta =0.036a$ (e), $0.0674a$ (f), $0.092a$ (g), $0.1a$ (h). The black discs are the $V$ points with charge $q=-1$.

Figure 2

The far-field polarization states, radiation losses, asymmetry ratio, and electric field profiles of eigenmodes. (a) Far-field polarization states in upward and downward radiation in momentum space for $\delta =0.0674a$. The blue (red) ellipse denotes the right (left)-handed elliptic polarizations with different major axes orientation. The black lines on the $k_x$ axis represent the linear $y$ polarization. (b) Radiation losses from eigenmodes along the $k_x$ axis toward the top ${\gamma }_u$ (orange) and bottom ${\gamma }_d$ (dashed green) of the structure with $\delta =0.0674a$. (c) The asymmetry ratio between upward and downward radiative loss, $\eta ={\gamma }_u/{\gamma }_d$, for $\delta =0.0674a$. (d) Electric field profiles ($y$ component) of the eigenmodes at ${\mathit{k}}_{\Vert }a/(2\pi )=(0.0523,0)$ for $\delta =0$ (top panel) and $\delta =0.0674a$ (bottom panel).

Figure 3

Evolution of polarization singularities. (a)–(c) show far-field polarization states of eigenmodes in upward radiation of structures with increasing (a) $\delta =0.0674a$, (b) $0.08a$, (c) $0.092a$, respectively. Far-field polarization states of eigenmodes in downward radiation corresponding to (d)–(f).

Figure 4

The radiation losses, $Q$ factor, electric field profile, and far-field polarization states of eigenmodes at $\delta =0.092a$. (a) Radiation losses of eigenmodes along $k_x$ axis toward the top ${\gamma }_u$ (orange) and bottom ${\gamma }_d$ (dashed green) of structures for $\delta =0.092a$. (b) The distribution of $Q$ factors of eigenmodes near the $\mathrm{\Gamma }$ point. (c) Electric field profiles ($y$ component) of the accidental at-$\mathrm{\Gamma }$ BIC [${\mathit{k}}_{\Vert }a/(2\pi )=(0,0)$]. (d) Far-field polarization states of eigenmodes in downward radiation.