# Modal Properties of Photonic Crystal Cavities and Applications to Lasers
## 4 Line-Defect Cavities
### Slow-Light Photonic Crystal Lasers  
In Section 3, we have outlined the main advantages of microcavity lasers, such as PhC lasers. Slow-light is a further, potential advantage of PhC lasers. In fact, the gain coefficient per unit length is enhanced when operating in a regime of slow-light, as originally pointed out in [80]. This effect has been verified experimentally in active PhC waveguides [81]. Intuitively, the gain enhancement results from a longer effective length, due to the multiple back-and-forth scattering across the periodic structure. As a result, PhC lasers [75] and on-chip optical amplifiers [82] may benefit from slow-light in terms of compactness and energy efficiency. Notice that light slow-down in translationally invariant waveguides, e.g., close to a mode cut-off, does not enhance the spatial gain [83]. A critical limitation to the use of slow-light is the presence of disorder [44], i.e., fabrication imperfections which inevitably affect real devices. These imperfections can be, for instance, roughness at the hole boundary or displacement of holes from the position of the ideal periodic lattice, on the order of few nanometers. Disorder may induce two types of loss [84], namely, out-of-plane radiation loss and backscattering loss. In the latter case [85; 86], a guided mode propagating in a given direction is partly reflected by the waveguide imperfections, thereby losing power due to coupling with guided, counter-propagating modes. Furthermore, an intrinsic limitation to the slow-light gain enhancement is posed by the gain itself [87]. This effect has been analysed in [88] for slow-light semiconductor optical amplifiers. As shown therein, the presence of material gain induces a distributed backscattering, which ultimately limits the amplifier gain. In the following, we briefly discuss the impact of slow-light on threshold gain and out-coupling efficiency of line-defect PhC lasers. We neglect the influence of the gain-induced distributed feedback, which has been shown to be small, unless the lasing frequency is very close to the band edge [89]. Furthermore, we also neglect the disorder-induced backscattering loss, the impact of which is currently a matter of debate [75; 90]. Proper inclusion of this effect is possible by combining descriptions of gain-induced [88] and disorder-induced [85] distributed feedback, which is beyond the scope of this paper. We consider line-defect PhC lasers with a buried heterostructure active region, as shown in the inset of Figure 12a. In the absence of gain-induced distributed feedback, the waveguide complex propagation constant reads [88]  
\[\tilde{\beta}_{\text{eff}}(\omega,N_{\text{car}})=k_{z}(\omega)-\frac{i}{2}S( \omega)\big{[}\Gamma_{\text{FF}}(\omega)\Gamma_{y}g(N_{\text{car}})(1-i\alpha _{H})-\alpha_{1}\big{]} \tag{21}\]  
Figure 11: (**a**) Light cone power fraction as a function of the size of the cavity. (**b**) Squared magnitude of the electric field \(k\)-space distribution in dB normalized to its maximum for various LN cavities. The dashed, vertical line indicates the upper limit of the light cone for each cavity. Here, \(S=n_{g}/n_{\text{slab}}\) is the _slow-down factor_, where \(n_{g}\) is the group index of the line-defect waveguide without active material (see Figure 7d, right axis); \(g\) is the material gain, which depends on the carrier density, \(N_{\text{car}}\); \(\alpha_{H}\) is the linewidth enhancement factor; and \(\alpha_{1}\) is the disorder-induced out-of-plane radiation loss [91]. Optical confinement along the vertical direction within the active layers is accounted for by \(\Gamma_{y}\). On the other hand, \(\Gamma_{\text{FF}}\) is the optical confinement factor computed _as if_ the active layers entirely filled the slab along the vertical direction. Owing to the high refractive index contrast between slab and cladding, \(\Gamma_{y}\) is essentially independent of frequency. For instance, for a single active layer of 8 nm placed in the middle of the slab, we estimate \(\Gamma_{y}\approx 4\%\).