Self-Similar Nanocavity Design with Ultrasmall Mode Volume for Single-Photon Nonlinearities

We propose a photonic crystal nanocavity design with self-similar electromagnetic boundary conditions, achieving ultrasmall mode volume ($V_{\text{eff}}$). The electric energy density of a cavity mode can be maximized in the air or dielectric region, depending on the choice of boundary conditions. We illustrate the design concept with a silicon-air one-dimensional photon crystal cavity that reaches an ultrasmall mode volume of $V_{\text{eff}}\sim 7.01\times {10}^{-5}{\lambda }^3$ at $\lambda \sim 1550\mathrm{nm}$. We show that the extreme light concentration in our design can enable ultrastrong Kerr nonlinearities, even at the single-photon level. These features open new directions in cavity quantum electrodynamics, spectroscopy, and quantum nonlinear optics.

Figure 1

Cavity field profiles of a slot cavity and slot-bridge (SB) cavities with different width of bridges. [(a) and (b)] Slot cavity achieving enhancement with the type-1 BC ($V_{\text{eff}}=2.5\times {10}^{-2}{\lambda }^3$). Here, $s$ denotes slot width, and $s=40\mathrm{nm}$ is used. [(c)–(e)] SB cavities achieving enhancement with type-2 BCs. Top: Index profile of the structure. Here, $b$ denotes bridge width. Middle: Three-dimensional finite-difference time-domain (FDTD) simulation result. Bottom: Two-dimensional electrostatic simulation result. (c) $b=5\mathrm{nm}$ narrow bridge ($V_{\text{eff}}=2.5\times {10}^{-3}{\lambda }^3$). (d) $b=10\mathrm{nm}$ intermediate bridge ($V_{\text{eff}}=4.3\times {10}^{-3}{\lambda }^3$, $V_{\text{guess}}=3.4\times {10}^{-3}{\lambda }^3$). (e) $b=40\mathrm{nm}$ wide bridge ($V_{\text{eff}}=7.7\times {10}^{-3}{\lambda }^3$, $V_{\text{guess}}=1.3\times {10}^{-2}{\lambda }^3$). For the $V_{\text{eff}}$ calculation, electric energy density at the middle of the bridge is used as a maximum. This is because corners of the bridges produce a singularity of the field (suppressed by mesh size), but do not affect typical light-matter interaction. In other words, this is justified by overlap factors, for example, in the Purcell factor.

Figure 2

Electric field distribution. (a) Slot-bridge-slot (SBS) cavity. (b) Tip cavity as a limiting case of concatenation. Here, we show an air-mode cavity, and the tip is smoothed out. The two-dimensional field distribution in log scale (left). Field distribution along the line cut in linear scale (middle), and log scale (right). For the same reason as in Fig. 1, the field at the middle of the structure is used as a maximum for $V_{\text{eff}}$ calculation. Note that the tip cavity has a higher field intensity at the tip than the middle in the inset.

Figure 3

(a) Quality factors and mode volumes in each case after $Q$ optimization: $S$ cavity ($s_1=40\mathrm{nm}$); SB cavity ($b_1=10\mathrm{nm}$); SBS cavity ($s_2=3\mathrm{nm}$). Tip: Tip cavity (air-mode cavity, 1 nm gap, $r=1\mathrm{nm}$) [16]. (b) Two figures of merit: $Q/V_{\text{eff}}$ as a general criteria and $Q{V}_M/V_{\text{eff}}^2$ for single-photon nonlinearities.