# Modal Properties of Photonic Crystal Cavities and Applications to Lasers
## 4 Line-Defect Cavities
### Slow-Light Photonic Crystal Lasers
On the contrary, \(\Gamma_{\text{FF}}\) does depend on frequency and diminishes towards the band edge [88; 92], as shown by Figure 12a. This is due to spreading of the Bloch modes in the lateral direction as the frequency approaches the band edge. The laser oscillation condition, i.e., \(r_{L}r_{R}e^{+2i\beta_{\text{eff}}(\omega,N_{\text{car}})L}=1\), leads to  
\[\Gamma_{\text{FF}}\Gamma_{y}g_{\text{th}}=\alpha_{1}+\alpha_{ \text{mir}}/S \tag{22a}\] \[2\big{(}k_{z}-\Gamma_{\text{FF}}\Gamma_{y}g_{\text{th}}\alpha_{ H}/2-\pi/a\big{)}L+\phi_{L}+\phi_{R}=2\pi m \tag{22b}\]  
Here, \(\alpha_{\text{mir}}=\frac{1}{L}\ln\Big{(}\frac{1}{|r_{L}||r_{R}|}\Big{)}\) is the mirror loss in the absence of slow-light, \(\phi_{L}\) (\(\phi_{R}\)) is the phase of the left (right) mirror and the integer \(m\) is the order of the resonant mode. Equations (22a) and (22b) are the amplitude and phase condition, respectively. All quantities are evaluated at the oscillation frequency. As an example, we have assumed \(r_{L}=1\), \(r_{R}=0.99\), \(\phi_{L}=\phi_{R}=0\), \(\alpha_{1}=5\,\text{cm}^{-1}\) and \(\alpha_{H}=1.5\). The other parameters of the line-defect waveguide are summarized in Table 1. For a given \(m\), as the cavity length increases, a smaller detuning from the band edge at \(\pi/a\) is required, in order to continue to fulfill the phase condition. Therefore, the modes move towards the band edge with increasing cavity length, as shown by Figure 12b, where line colors correspond to different modes. Figure 12: (**a**) Optical confinement factor \(\Gamma_{\text{FF}}\) of a line-defect PhC laser with a buried heterostructure active region (inset). \(\Gamma_{\text{FF}}\) was computed as if the active layers entirely filled the slab in the vertical (\(y\)) direction. The \(x\)-axis shows the wavenumber \(k_{z}\) of the associated line-defect waveguide without active material. (**b**) Wavenumber and (**c**) modal gain at the lasing threshold of the various longitudinal modes versus cavity length. The modal gain only included the optical confinement factor \(\Gamma_{y}\), which was approximately independent of frequency. \(\Gamma_{y}\) accounts for the optical confinement along the vertical direction within the active layers. (**d**) Output power (**left**) and slope efficiency (**right**) versus the slow-down factor. The cavity length was \(L=5a\) and the bias current was \(50\,\upmu\)A. Further details on the simulation parameters are provided within the text. This trend is in agreement with experimental results [75]. Correspondingly, the modal threshold gain \(\Gamma_{y}\delta_{\text{th}}\) is reduced, due to the diminishing _effective_ mirror loss, \(\alpha_{\text{mir}}/S\). This is illustrated by Figure 12c. The effective loss decreases because of the larger cavity length (as for conventional semiconductor lasers) _and_ enhanced slow-down factor, due to the shift of the oscillation frequency towards the slow-light region. This benefit outweighs the reduction of the optical confinement factor \(\Gamma_{\text{FF}}\). The slow-light enhancement of the gain per unit length equivalently manifests itself as a reduction of the cavity loss per unit time, i.e., a longer photon lifetime. Specifically, one finds [75; 93]  
\[\frac{1}{\tau_{p}}=\frac{\omega_{s}}{Q}=\underbrace{(c/n_{\text{slab}})\alpha _{\text{mir}}/S}_{\omega_{s}/Q_{c}}+\underbrace{(c/n_{\text{slab}})\alpha_{1} }_{\omega_{s}/Q_{c}} \tag{23}\]  
Therefore, slow-light increases the Q-factor associated with out-coupling losses, \(Q_{c}\). With these definitions, one may utilize conventional rate equation models [23] to compute key laser properties, such as threshold current and output power. In particular, the threshold current reads (cf. Equation (5)) \(I_{\text{th}}=\frac{1}{\tau_{p}! }\left(N_{\text{tr}}V_{\text{act}}+\frac{ \omega_{s}}{\Omega}\frac{V_{\text{act}}}{(c/n_{\text{slab}})\beta_{N}}\right)\). Here, \(\Gamma\) is equal to \(\Gamma_{\text{FF}}\Gamma_{y}\) and \(Q\) is given by Equation (23).