# Modal Properties of Photonic Crystal Cavities and Applications to Lasers
## 3 Microcavity Lasers for Energy-Efficient Communications  
A very important application of PhC cavities is the realization of a new generation of microcavity lasers, e.g., for on-chip optical interconnects [62; 63]. In this section we summarize key laser properties, which are determined by the characteristics of the cavity. The threshold current of an electrically-pumped semiconductor laser may be approximately expressed as [38]  
\[I_{\mathrm{th}}=\frac{1}{\tau}\frac{q}{\eta_{i}}\left(N_{\mathrm{tr}}V_{ \mathrm{act}}+\frac{\omega_{s}}{Q}\frac{\tau_{r}}{\beta_{\mathrm{sp}}}\right) \tag{5}\]  
Here, we have assumed for simplicity the material gain to be linear with carrier density, with \(N_{\mathrm{tr}}\) being the transparency carrier density, \(q\) the electron charge, \(\eta_{i}\) the injection efficiency and \(\omega_{s}\) the lasing angular frequency. The carrier lifetime, \(\tau\), is generally a function of carrier density and related by \(\tau^{-1}=\tau_{r}^{-1}+\tau_{nr}^{-1}\) to the radiative lifetime, \(\tau_{r}\), and nonradiative lifetime, \(\tau_{\mathrm{nr}}\). The spontaneous emission factor, \(\beta_{\mathrm{sp}}\), is the fraction of spontaneous emission coupled to the lasing mode. The threshold current has two contributions. The first contribution scales with the active region volume and represents the current required to reach transparency, where stimulated emission and absorption exactly balance. The second contribution is the current necessary to balance the optical loss once transparency has been reached. This term is inversely proportional to the product of Q-factor and spontaneous emission factor. In PhC lasers, \(\beta_{\mathrm{sp}}\) is orders of magnitude larger than in conventional semiconductor lasers, with values approaching \(1\) being reported recently [64; 65]. This is because a single dominant mode is found within the bandwidth of spontaneous emission. However, the reduction in threshold current with cavity length is, mostly, simply due to the smaller active volume, which reduces the number of carriers necessary to reach transparency [66]. In fact, the (loaded) Q-factor in PhC lasers is typically high enough (\(Q\sim 10^{3}-10^{4}\)[67]) that the threshold carrier density is essentially determined by the transparency carrier density. It should be noted that the optical con  
Figure 4: Comparison of three photonic crystal cavities (respectively, L3, H1 and H0) and a cavity for extreme dielectric confinement (EDC) designed by topology optimization. All cavities are made of InP and have similar footprint. We notice, though, that the Q-factors for L3, H1 and H0 may be further optimized by perturbing the holes, as explained earlier. The topology-optimized design was based on that reported in [10], which however has confinement in air. As compared to [10], we added in the center a \(10\,\mathrm{nm}\times 16\,\mathrm{nm}\) InP bridge. Calculations have been performed by a three-dimensional finite-difference-time-domain (FDTD) method. The center of the cavity is denoted by \(\tau_{\mathrm{con}}\). The effective mode volume of the H0 cavity is not reported, since the field had a minimum in the center. finement factor is embedded implicitly into Equation (5). Indeed, the ratio \(\beta_{\rm sp}/\tau_{r}\) may be equivalently expressed as \(\Gamma v_{g}g_{N}/V_{\rm act}\)[14], with \(g_{N}\) being the differential gain [23] and \(v_{g}\) the group velocity (reflecting material, but not structural dispersion [14]). Therefore, maximizing the optical confinement factor (thus minimizing the mode volume \(V_{p}\)) contributes to reducing the threshold current. However, in the presence of a high Q-factor, the improvement is limited and the threshold current is mostly determined by the number of carriers at transparency [28], as already stated. As an example, we consider the case of a quantum well laser. Figure 5a,b illustrates how the cavity length, \(L\), and optical confinement factor per active layer, \(\Gamma_{\rm well}\), impact the threshold current at a given value of the Q-factor.