# Modal Properties of Photonic Crystal Cavities and Applications to Lasers
## 6 EDC Cavities  
It is commonly held that in dielectric (non-metallic) cavities the mode volume \(V_{\text{eff}}(\mathbf{r}_{\text{max}})\) cannot be reduced below the so-called diffraction limit [108; 109], \(\left[\lambda/\left(2n\right)\right]^{3}\), corresponding to half a guided wavelength in all three dimensions. Intuitively, this can be understood by thinking of a resonator as a rectangular box with perfect mirrors. Metallic structures supporting plasmons have been demonstrated to feature modes with much smaller volumes [110; 111], but ohmic losses severely limit the Q-factor. Recently, however, new designs [8; 9; 10] for dielectric cavities have been proposed, which can break the diffraction limit while keeping a high Q-factor. These are so-called extreme dielectric confinement (EDC) cavities. As compared to typical PhC cavities, EDC cavities greatly enhance the electric energy density around a given point, thereby maximizing the denominator in Equation (1). The numerator essentially remains unchanged. EDC cavities are targeted at the same realm of applications where PhC cavities have flourished, such as nanolasers, nonlinear photonics and quantum technologies. In particular, a recent work [70] has pointed out that EDC cavities with few emitters embedded may be also utilized as nanolasers and nanoLEDs featuring squeezed intensity noise. Figure 15 summarizes the main design strategies of EDC cavities proposed so far. One approach, based on earlier works on air-slot waveguides and microcavities [112], is embedding a bow-tie in a PhC nanobeam cavity [9], as illustrated by Figure 15a. FDTD simulations of this EDC cavity indicate an ultrasmall mode volume \(V_{\text{eff}}(\mathbf{r}_{\text{con}})\approx 7.01\times 10^{-5}\,\lambda^{3}\) at \(\lambda\approx 1550\,\text{nm}\)[9], with \(\mathbf{r}_{\text{con}}\) denoting the center of the bow-tie. This structure features an air gap in the center, where a nonlinear material could be deposited. Alternatively, one may fabricate a bow-tie with connected tips and simply exploit the nonlinearities of the nanobeam material. As a result of the enhanced light-matter interaction, Kerr nonlinearities may enable a shift of one linewidth in the cavity resonant wavelength with just few photons or even a single photon (depending on the nonlinear material which is employed) [9]. As another strategy (Figure 15b), one might modify the unit cell of a PhC nanobeam by directly embedding a bow-tie into each air hole. Subsequently, a cavity is carved out of this modified nanobeam by varying, for instance, the bow-tie rotation angle [8] or size [113]. Here, a key point is that the cavity resonant mode is derived from the _air_ mode of the original PhC nanobeam, i.e., the higher-frequency guided mode having most of its energy within the air holes. By including bow-ties within the holes, the mode energy density around the bow-tie tips is greatly enhanced. The variation of the bow-tie rotation angle or size misaligns the band edges of the mode dispersion relation in the cavity and mirror regions. Therefore, a cavity is formed in a very similar fashion to photonic heterostructure cavities reviewed in Section 2. The gradual variation of the bow-tie rotation angle or size ensures a gentle evolution of the resonant mode spatial envelope, thereby optimizing the Q-factor. By this approach, an EDC cavity has been experimentally demonstrated [113], with a loaded Q-factor around \(10^{5}\). It should be noted that these EDC cavity designs require minimum feature sizes (e.g., the gap between the bow-tie tips) to be on the order of few nanometres [9], which is at the limit of current fabrication techniques. In this respect, alternative designs based on topology optimization [114] may be advantageous. Topology optimization enables systematic design of EDC cavities [10] while taking into account practical constraints on the minimum manufacturable feature size [61], mainly determined by electron beam lithography or semiconductor dry etching.