# Modal Properties of Photonic Crystal Cavities and Applications to Lasers
## 6 EDC Cavities
Essentially, the design starts from a reference structure serving as initial guess, e.g., an H1 PhC cavity or an unstructured membrane. Maxwell equations in the frequency domain are solved for a given resonant mode, excited by an electric dipole source [10]. The structure is iteratively updated by using a gradient-based optimization method, with the aim of maximizing a frequency-averaged local photonic density of states (LDOS) [115]. In fact, the LDOS scales proportionally to the ratio between Q-factor and mode volume. A constraint on the minimum feature size can be also introduced in the optimization algorithm [10]. The final result is a bow-tie surrounded by anelliptic ring grating [10] (Figure 15c). This grating effectively acts as a distributed Bragg reflector, which increases the Q-factor by reducing the in-plane radiation loss. For given targets (essentially, resonant frequency, type of resonant mode, domain size and minimum feature size), nearly identical final designs are obtained regardless of the initial guess [116], outlining the robustness of this design strategy. The physical mechanisms behind the strong field concentration in a bow-tie is illustrated by Figure 16. A first mechanism is the so-called _slot effect_ (Figure 16a), which is also exploited in air slot PhC cavities [112; 117]. Essentially, [PERSON]'s equations require the normal component of the electric displacement field to be continuous across a dielectric interface. Across a slot, the electrical permittivity, \(\epsilon\), changes abruptly from a high \(\epsilon_{h}\) to low \(\epsilon_{l}\) value, thereby being discontinuous. Consequently, the normal component of the electric field is forced to be discontinuous. As result, by assuming the electric field to be highly polarized in the normal direction, the electric energy density within the slot (\(W_{S}\)) is roughly enhanced by a factor \(e_{h}/\epsilon_{l}\) as compared to that in the high refractive index medium (\(W_{0}\)). By building a bridge across the slot, the energy density can be further enhanced. This is the so-called _antislot effect_[8] (Figure 16b). In this case, one leverages the continuity of the tangential electric field component at the boundaries between slot and bridge (or antislot, from now on). As a result, the electric energy density within the antislot (\(W_{A}\)) is roughly enhanced by an additional factor \(\epsilon_{h}/\epsilon_{l}\) as compared to that inside the slot. Air is conveniently used as the low permittivity medium in order to maximize the refractive index contrast. By interlocking slots (S) and antislots (A), the energy density can be progressively enhanced [9]. This strategy is illustrated by Figure 16c. The largest enhancement is achieved in the limit of a bow-tie structure. In this case, the energy density is maximum within the gap between the two tips (bow-tie with disconnected tips) or inside the tiny dielectric strip connecting them (bow-tie with connected tips), depending whether the last element of the concatenation is either a slot or antislot. As a rule of thumb, the smaller the minimum feature size is and the larger the energy density enhancement becomes [9]. Figure 15: Cavities for extreme dielectric confinement (EDC). EDC cavities realized (**a**) by embedding a bow-tie in a PhC nanobeam cavity [9] and (**b**) through sub-wavelength engineering with a bow-tie shape of the unit cell of a PhC nanobeam [8; 113]. (**c**) EDC cavity designed by topology optimization [10]. In Figure 15c, the gray area is air and the black area is semiconductor. Figure 15c is reprinted from [10], with the permission of AIP publishing, 2018.