Analytical Perspective for Bound States in the Continuum in Photonic Crystal Slabs

We investigate the formation of photonic bound states in the continuum (BICs) in photonic crystal slabs from an analytical perspective. Unlike the stationary at-$\mathrm{\Gamma }$ BICs which originate from the geometric symmetry, the tunable off-$\mathrm{\Gamma }$ BICs are due to the weighted destructive via the continuum interference in the vicinity of accidental symmetry when the majority of the radiation is precanceled. The symmetric compatible nature of the off-$\mathrm{\Gamma }$ BICs leads to a trapping of light that can be tuned through continuously varying the wave vector. With the analytical approach, we explain a reported experiment and predict the existence of a new BIC at an unrevealed symmetry.

Figure 1

(a) Structure: liquid (${ϵ}_l=2.11$), ${\mathrm{Si}}_3{\mathrm{N}}_4$ PC slab (${ϵ}_{sn}=4.08$), silica cladding (${ϵ}_{so}=2.13$), and Si substrate. Setting ${ϵ}_l\simeq {ϵ}_{\mathrm{so}}$ ensures mirror-flip symmetry [20]. The silica and liquid layers are sufficiently thick to be assumed as infinite. (b) Basic lattice: thickness $2d=180\mathrm{nm}$, periodicity $a=336\mathrm{nm}$, hole diameter $2r=160\mathrm{nm}$. (c),(d),(e) Phase matching conditions: (c) $\mathrm{\Gamma }$ point; (d) $k=0.25$ in the $\mathrm{\Gamma }-X$ direction; (e) $k=\sqrt{2}/6$ in the $\mathrm{\Gamma }-M$ direction. Red arrows indicate extra degenerate wave vectors. For square lattice, the ${\mathrm{\Gamma }}^1$, ${\mathrm{\Gamma }}^2$, and ${\mathrm{\Gamma }}^3$ points are defined as wave vectors with lengths of 0, $2\pi /a$, and $2\sqrt{2}\pi /a$, respectively [25].

Figure 2

(a) Identified ${\mathrm{TM}}_{1–4}$ modes near the ${\mathrm{\Gamma }}^2$ point by increasing frequency [25], using both CWT (lines) and finite-difference time domain (FDTD) (dots). (b) CWT-calculated quality factor $Q_r$ of ${\mathrm{TM}}_1$ mode near ${\mathrm{\Gamma }}^2$. (c) Precancellation within a closed channel at ${\mathrm{\Gamma }}^2$. The coupling strength [22] is defined as $⟨G|{\kappa }_{m,n}L|H_{i,m,n}⟩$. For the in-plane coupling, $L=({\partial }^2/\partial z^2)$; for the surface coupling, $L=-\delta (|d|)(\partial /\partial z)$.

Figure 3

CWT-calculated $Q_r$ (a) ${\mathrm{TM}}_1$ mode near $\mathrm{\Gamma }-X$ BIC (e) ${\mathrm{TM}}_2$ mode near $\mathrm{\Gamma }-M$ BIC. CWT-calculated radiation ability at the symmetry (defined by the modulus of the diffraction into the open channel of individual wave vectors with normalized profiles, i.e., $\parallel ⟨G_{i,-m,-n}{H}_{i,m,n}⟩\parallel$, $i=x,y$) for the (b) $\mathrm{\Gamma }-X$ BIC (f) $\mathrm{\Gamma }-M$ BIC. Radiation profile decomposition within the slab of (c),(d) ${\mathrm{TM}}_1$ $\mathrm{\Gamma }-X$ BIC in the $\mathbf{x}$ and $\mathbf{y}$ directions (g),(h) ${\mathrm{TM}}_2$ $\mathrm{\Gamma }-M$ BIC in the $\mathbf{x}$’ ($y=x$) and $\mathbf{y}$’ ($y=-x$) directions. The radiation profiles are calculated by $P_{m,n}\sum _{m^{\prime },n^{\prime }}⟨{\mathbf{V}}_{{\mathrm{TM}}_{1(2)}}|G_{i,-m^{\prime },-n^{\prime }}|H_{i,m^{\prime },n^{\prime }}⟩,i=x,y$, where ${\mathbf{V}}_{\mathrm{T}{\mathrm{M}}_{1(2)}}$ is the eigenvector of the $\mathrm{\Gamma }-X(M)$ BIC and $P_{mn}$ denotes corresponding projection operators, i.e., $|mn⟩⟨mn|$.

Figure 4

Location shift of tunable BICs in the (a) $\mathrm{\Gamma }-X$ and (b) $\mathrm{\Gamma }-M$ direction using CWT (solid lines) and FDTD (dots). ${\epsilon }_c$ is the cladding permittivity. Dashed lines indicate the exact triangular symmetry [Figs. 1 and 1].

Figure 5

(a) TM-like and (b) TE-like band structures in the continuum using FDTD. $\varphi$ and $\chi$ indicate locations of tunable BICs. $\mathrm{\Phi }$ and $\mathrm{\Xi }$ indicate band gaps formed by triangular symmetry.