High-$Q$ impurity photon states bounded by a photonic band pseudogap in an optically thick photonic crystal slab

We show that, taking a two-dimensional photonic crystal slab system as an example, surprisingly high quality factors ($Q$) over ${10}^5$ are achievable, even in the absence of a rigorous photonic band gap. We find that the density of in-plane Bloch modes can be controlled by creating additional photon feedback from a finite-size photonic-crystal boundary that serves as a low-Q resonator. This mechanism enables significant reduction in the coupling strength between the bound state and the extended Bloch modes by more than a factor of 40.

Figure 1

(a) A 3D rendering of the photonic crystal cavity to be studied in this paper. The number of photonic crystal hole layers ($N$ $\mathit{layer}$) and the boundary termination ($p$) are to be varied. $a$ is the lattice constant of the 2D triangular-lattice photonic crystal with the background air holes, $R$, of $0.35a$. The refractive index of the slab is assumed to be 3.4. The six nearest neighbor holes are reduced ($Rm=0.25a$) and slightly pushed away (by a scale factor of 1.1) from the defect center. (b) ${\left|\mathbf{E}\right|}^2$ distribution of the hexapole mode in a $T$ $=$ $1.731a$ slab. (c) Cross-sectional views of the first five lowest Bloch modes at $M$ point. These modes all have even symmetries with respect to the plane at $z$ $=$ 0. (d) The photonic band structure for all even Bloch modes. The hexapole mode resonance shown in (b) (${\omega }_n\sim 0.265$) is represented by the horizontal line, in which ${\omega }_n$ is the normalized frequency defined by ${\omega }_n\equiv a/\lambda$.

Figure 2

$Q_{\mathrm{tot}}$, $Q_{\mathrm{vert}}$, and $Q_{\mathrm{horz}}$ of the hexapole mode for two different boundary terminations, (a) $p=\infty$ and (b) $p=1.2a$. When the mirror boundary effect is weak ($p=\infty$), $Q_{\mathrm{horz}}$ is limited by about 60 000 due to the coupling to the in-plane Bloch modes. However, in the case of the abrupt termination by air, $Q_{\mathrm{horz}}$ can be made over 2 600 000.

Figure 3

(a)–(c) Top-down views (${\left|\mathbf{E}\right|}^2$) of the hexapole resonances in a $T=1.731a$ slab as we increase $N$ $\mathit{layer}$ from 4 to 9 (in the clockwise direction) for three different boundary terminations of $p=0.8,1.0$, and $1.2a$, respectively. (d) The hexapole resonances in a thin ($T=0.488a$) slab for $p=\infty$ are presented in the same manner used in (a)–(c). Note that this thin slab possesses the in-plane photonic band gap. (e) $Q_{\mathrm{horz}}$ and (f) $Q_{\mathrm{tot}}$ for the hexapole resonances shown in (a)–(d).

Figure 4

(a) In our coupled-mode model, the hexapole mode is assumed to couple only to the $\Gamma$-$M$ wavevectors. For further simplification, the solution for the entire system is projected into the $x$ direction. The hexapole symmetry ensures the odd mirror symmetry with respect to the $x=0$ plane. (b) $Q_{\mathrm{horz}}$ enhancement factors in Eq. (4). All plots assume $r_M$ $=$ 0.6.

Figure 5

The air boundary condition is now replaced with a metal slab. The metal is assumed as gold, which is modeled as a single-pole Drude medium (Ref. 28). We calculate $Q_{\mathrm{tot}}$ [(a) and (b)] and $|\mathbf{E}({\mathbf{r}}_{\left|\right|},z=0){|}^2$ [(c) and (d)] as we vary $N$ $\mathit{layer}$ (while $p$ is fixed to $1.4a$) and $p$ (while $N$ $\mathit{layer}$ is fixed to 7) for the photonic crystal slab ($T=1.731a$) shown in (e). (f) A schematic diagram shows how the metal reflector on the side can be designed to function as a current injection electrode. The conical area shaded in red (blue) denotes a $p$-doped (an $n$-doped) region by using ion implantation, forming a laterally $p$-$i$-$n$ doped structure (Ref. 29).