Zero-Index Bound States in the Continuum

Metamaterials with an effective zero refractive index associated with their electromagnetic response are sought for a number of applications in communications and nonlinear optics. A promising way that this can be achieved in all-dielectric photonic crystals is through the design of a Dirac cone at zero Bloch wave vector in the photonic band structure. In the optical frequency range, the natural way to implement this design is through the use of a photonic crystal slab. In the existing implementation, however, the zero-index photonic modes also radiate strongly into the environment due to intrinsic symmetry properties. This has resulted in large losses in recent experimental realizations of this zero-index paradigm. Here, we propose a photonic crystal slab with zero-index modes which are also symmetry-protected bound states in the continuum. Our approach thus eliminates the associated radiation loss. This could enable, for the first time, large-scale integration of zero-index materials in photonic devices.

Figure 1

(a) Schematic of the photonic crystal slab. The inset shows the daisy hole shape with its inner and outer radii (dashed lines), as well as the in-plane elementary cell (solid line). (b) Photonic bands for the quasi-TM modes computed with GME. (c) Enlargement within the dashed lines of (b), with dots showing the modes computed with COMSOL. (d) Isofrequency contours in 2D reciprocal space for the top band of the dirac cone [blue band in (c)]. (e): Radiative quality factors for the three corresponding bands in (c).

Figure 2

(a)–(c) Electric field $\mathrm{Re}(E_z)$ in the center of the slab ($z=0$) for a plane-wave excitation from the left, with a wave vector along the $y$ direction ($\mathrm{\Gamma }\mathrm{M}$), at reduced frequency $\overline{\omega }=\omega a/(2\pi c)=0.48$ (a), 0.486 (b), and 0.492 (c). (d) Phase of the electric field at the sampling points marked with black dots in panels (a)–(c). (e) Effective index computed from the slope of the phase evolution with position (dots), and from the Bloch band structure (line). (f) Group index and propagation loss of the Dirac-cone bands along the two crystallographic directions as a function of wavelength, assuming $a=760\mathrm{nm}$. The $x$-axis scale and range in panels (e) and (f) are the same. Plots as in (a)–(e) for $\mathrm{\Gamma }K$ are in the Supplemental Material [27].

Figure 3

Electric field $\mathrm{Re}(E_z)$ for a simulated triangular region of zero-index PhC slab surrounded by unpatterned slab. (a) 3D view and (b) cross section in the center of the slab for a dipole source excitation in the center of the PhC region (black dot), at frequency $\overline{\omega }=0.486$. Arrows indicate the direction of the emitted waves. (c) Dipole-source excitation in the center of a homogeneous triangular region with relative permittivity and permeability ${ϵ}_r={\mu }_r={10}^{-6}$. The simulated structure is infinite in the $z$ direction. (d)–(f) Same as (a)–(c), but with a Gaussian wave packet excitation from below.