# Modal Properties of Photonic Crystal Cavities and Applications to Lasers
## 2 Fundamental Principles of Light Confinement in Photonic Crystal Structures
By proper design, PhC cavities with a relatively high Q-factor can be obtained by removing even a single hole (H1 cavity) or no hole at all (H0 cavity) [51]. In particular, an H0 cavity is carved out by slightly shifting two holes in a given row into opposite directions. In addition, the Q-factor of H1 and H0 cavities is optimized by tuning position and radius of the holes surrounding the point-defect. Among PhC cavities, H0 cavities offer the smallest mode volume (\(V_{\text{eff}}(\mathbf{r}_{\text{max}})\approx 0.29(\lambda/n)^{3}\)), while keeping a relatively high, unloaded Q-factor (\(Q_{i}\approx 10^{5}\)) [51]. In terms of Q-factor, better performances can be achieved by photonic heterostructure cavities. A so-called _photonic_ heterostructure is a connection in series of two or more PhC waveguides with different _photonic_ band gaps. The photonic band gap can be engineered by structural changes, such as tuning the lattice constant [52; 53], waveguide width [37] or slab refractive index [54]. The working principle is summarized by Figure 3. In this case, the cavity is formed by connecting line-defect waveguides with different lattice constants. Essentially, the larger lattice constant in region II as compared to regions denoted by I (see Figure 3c) induces a relative shift of the band edges corresponding to the various regions (see Figure 3b,d). This ensures an effective confinement of photons along the waveguide direction. In fact, photons which are allowed to propagate in region II find themselves within the band gap of regions I. Various modifications and improvements of this cavity have been demonstrated. For instance, the lattice constant can be varied more smoothly, by increasing the number of regions which make up the heterostructure. This approach has led to a theoretical, unloaded Q-factor with a record value of \(10^{9}\), while keeping a mode volume \(V_{\text{eff}}(\mathbf{r}_{\text{max}})\) around one cubic guided wavelength [55]. More recently, an experimental, unloaded Q-factor with a record value of \(1.1\times 10^{7}\) has been even demonstrated [56]. Heterostructure cavities may be also induced via _dynamic_ refractive index modulation. This can be done, for instance, by shining an optical pulse at a given location along a line-defect waveguide. Through optical nonlinearities, the refractive index is locally changed and a high-Q heterostructure cavity is formed, which effectively traps photons injected from the waveguide input. With this scheme, a theoretical, unloaded Q-factor even larger than \(10^{9}\) has been envisioned [57]. Other types of PhC cavities are nanobeam cavities, as shown in Figure 1b. By perforating a ridge waveguide with a single row of holes, a PhC nanobeam is created [1]. This is a periodic dielectric waveguide, whose guided modes are confined in the lateral and vertical direction by total internal reflection. A cavity is carved out of a PhC nanobeam by introducing a defect. This may consist in one or more missing holes [58], modulation of the hole radius [59; 7] or tuning of the lattice constant [60] along the longitudinal direction. As compared to PhC cavities based on line-defect waveguides, PhC nanobeam cavities offer a reduced footprint. EDC cavities may also be obtained out of PhC nanobeam cavities, as shown in Section 6. Figure 4 compares three PhC cavities (respectively, L3, H1 and H0) against an EDC cavity designed by topology optimization. This structure is very similar to that reported in [10], the difference only being the dielectric bridge in the center, as detailed in the figure caption. We have extracted Q-factor and mode volumes by three-dimensional finite-difference-time-domain (FDTD) calculations, with all cavities made of InP. For a fair comparison, the PhC cavities have been tailored to a similar footprint as the EDC cavity, which resulted in the Q-factor of the PhC cavities being strongly limited by in-plane radiation loss.