# Modal Properties of Photonic Crystal Cavities and Applications to Lasers
## 5 Fano Laser
### Tuning Characteristics
We assume \(|r_{L}|=1\), \(\alpha_{1}=5\,\text{cm}^{-1}\) and \(\alpha_{H}=1.5\). The reference wavelength, \(\lambda_{r}\), is \(1571\,\text{nm}\), leading to \(n_{g}(\omega_{r})\approx 20\) and \(\Gamma_{\text{FF}}(\omega_{r})\approx 62\%\). The reference cavity length is \(L_{r}=8.76\,\text{\mu m}\), corresponding to \(20\) unit cells. The Q-factorsdefining the Fano mirror are \(Q_{v}=10^{5}\), \(Q_{p}=1.5\times 10^{4}\) and \(Q_{c}=780\). The parameters of the line-defect waveguide are summarized in Table 1.  
We start by illustrating the frequency tuning characteristics. Figure 14 shows (a) the threshold modal gain, \(\Gamma_{y}g_{\text{th}}\), and (b) effective detuning of \(\omega_{0}\) from the oscillation frequency, \(\omega_{s}\), normalized to the Fano mirror linewidth. Both the lasing mode (in blue) and second-order mode (in red) are reported. A number of features can be noted. The modal gain shows a succession of minima, occurring where \(\omega_{0}\) (and thus the Fano mirror peak) aligns with a longitudinal mode of the laser cavity. The spacing of these modes is determined by the free spectral range of the cold cavity (within a small correction due to the linewidth enhancement factor). Since the group index is fixed (within the approximation of Equation (25)), the minima are equally spaced. Figure 14b confirms that the minima do correspond to zero effective detuning. The gain and effective detuning are periodic, with the same periodicity. As \(\omega_{0}\) departs from a gain minimum, the threshold gain increases, because the oscillation frequency is detuned from the mirror peak. However, the gain increases more strongly in one detuning direction as compared to the other. The asymmetry is due to the non-zero linewidth enhancement factor [97]. Finally, we note that the lasing mode is always the closest to the Fano mirror peak and nearly tracks the nanocavity resonance frequency within few mirror linewidths, as evident from Figure 14b. We now move to the length tuning characteristics. Figure 14c reports the threshold modal gain, which shows a succession of minima similarly to Figure 14a. As the cavity length is varied, the oscillation frequency has to change as well, in order to continue to fulfill the phase condition. The gain minima occur whenever the oscillation frequency aligns with the nanocavity resonance frequency, where the Fano mirror reflectivity is maximum. This is confirmed by Figure 14d, reporting the effective detuning. Again, the asymmetry in the response arises from the non-zero linewidth enhancement factor. We also note that the threshold gain at each minimum decreases with increasing cavity length, which is hard to see on the figure due to the scale. Finally, the extreme sensitivity of the response to the cavity length tuning should be emphasized [97]. A variation of 1% of \(L\) with respect to \(L_{r}\) is enough to more than double the threshold gain as compared to one minimum. This illustrates that the mode of the Fano laser has the characteristics of a bound mode in the continuum [106; 107], which was recently exploited to realize a microlaser with ultranarrow linewidth. Figure 14: **(a,c)** Threshold modal gain and (**b,d**) effective detuning of the Fano laser upon tuning of (**a,b**) the nanocavity resonance frequency and (**c,d**) cavity length. The lasing mode, M\({}_{1}\), is in blue. The second and third-order modes are in red and yellow, respectively.