# Modal Properties of Photonic Crystal Cavities and Applications to Lasers
## 5 Fano Laser
### Tuning Characteristics  
To understand the properties of the Fano laser, it is of interest to examine the tuning characteristics of the laser, i.e., the oscillation frequency in dependence of the cavity length or nanocavity resonance frequency. This is useful both for general design purposes and more specific applications. For instance, the Fano laser can be modulated via the mirror, by dynamically changing the nanocavity resonance frequency [94; 97]. The response of the threshold gain and oscillation frequency to variations of the nanocavity resonance frequency gives important indications on the on- and off-state of the laser (and transition from one to the other) in switching applications [100]. The current Fano laser models do not account for waveguide dispersion and, consequently, slow-light effects. In particular, a major assumption is that the Bloch modes traveling in the laser cavity are only coupled at the mirrors [105], thereby neglecting the gain-induced distributed feedback associated with slow-light [88]. The assumption is valid if the laser operates far from the band edge of the waveguide dispersion relation. This can be ensured by designing the nanocavity such that its resonance frequency lies far enough from the slow-light region. However, there is a number of reasons to investigate how slow-light would affect the Fano laser characteristics. As for conventional PhC lasers based on line-defect waveguides, the enhancement of the modal gain per unit length would improve the device compactness and energy efficiency. Furthermore, owing to the extreme sensitivity of the oscillation condition to variations of the cavity length and/or nanocavity resonance frequency [40; 105], the additional strong coupling of the amplitude and phase conditions introduced by slow-light is likely to influence both the static and dynamic characteristics of the laser. In the following, we solve the laser oscillation condition upon tuning of the cavity length, \(L\), and nanocavity resonance frequency, \(\omega_{0}\), with the right mirror reflection coefficient given by Equation (24). We follow the approach of [105] and assume that the laser oscillates at a reference frequency, \(\omega_{r}\), when (1) the nanocavity resonance frequency coincides with \(\omega_{r}\) and (2) the cavity length is set to a reference value, \(L_{r}\). Therefore, for given \(L_{r}\) and \(\omega_{r}\), the left mirror phase is chosen such that a longitudinal mode lies at \(\omega_{r}\) for the _cold_ cavity. Then, we let either \(\omega_{0}\) or \(L\) vary and solve for each value the oscillation condition. As a first step towards the inclusion of slow-light effects, we consider the waveguide complex propagation constant in the absence of gain-induced distributed feedback from Equation (21). Here, we Taylor-expand \(k_{z}\) to first order around \(\omega_{r}\) and evaluate \(n_{g}\) at \(\omega_{r}\), leading to  
\[\tilde{\beta}_{r}(\omega,N_{\text{car}})=k_{z}(\omega_{r})+\tfrac{\omega- \omega_{r}}{c}n_{g}(\omega_{r})-\tfrac{i}{2}\tfrac{n_{g}(\omega_{r})}{n_{ \text{slab}}}\big{[}\Gamma_{\text{FF}}(\omega_{r})\Gamma_{y}g(N_{\text{car}})( 1-i\alpha_{H})-\alpha_{1}\big{]} \tag{25}\]  
We make use of Equation (25) to model the field propagation across the waveguide. Thus, the laser oscillation condition reads \(r_{L}(\omega)r_{R}(\omega)e^{+2i\tilde{\beta}_{r}(\omega,N_{\text{car}})L}=1\). It should be noted that this approach obviously neglects the gain-induced distributed feedback, and employs fixed values for the group index and optical confinement factor. However, in contrast to previous studies [97; 105], these values are directly computed from the dispersion relation and Bloch modes of the line-defect waveguide around \(\omega_{r}\), rather than set a priori. This should provide a more quantitative basis for the estimate of the laser threshold gain. Full inclusion of slow-light effects is possible by applying a coupled-Bloch-mode approach [89], but this is beyond the scope of this paper.