# Modal Properties of Photonic Crystal Cavities and Applications to Lasers
## 5 Fano Laser
### Fano Mirror  
The Fano mirror can be modeled by temporal coupled-mode theory [103]. The most general implementation of the Fano mirror on the PhC platform may include a blocking air hole in the waveguide below the nanocavity (a so-called PTE, partially transmitting element) [104]. This PTE renders the mirror spectrum asymmetric with respect to the nanocavity resonance frequency, which can be useful in various applications [100; 102]. By displacing the PTE from the mirror plane, the parity of the resonance can also be controlled, thereby blue or red-shifting the reflection maximum as compared to the minimum.  
In the following, we focus on the simplest implementation, without a PTE. In this case, one finds [94]  
\[r_{R}(\omega)=\frac{-\gamma_{c}}{i(\omega_{0}-\omega)+\gamma_{T}} \tag{24}\]  
Here, \(\omega_{0}\) is the nanocavity resonance frequency, \(\gamma_{c}\) is the coupling rate between nanocavity and waveguide, and \(\gamma_{T}=\gamma_{c}+\gamma_{v}+\gamma_{p}\) is the total decay rate of the field in the nanocavity. This rate accounts for coupling with the waveguide (\(\gamma_{c}\)), vertical out-of-plane scattering loss (\(\gamma_{p}\)) and possible coupling to other ports (\(\gamma_{p}\)), if present. For each coupling rate, the corresponding Q-factor is defined as \(Q_{x}=\omega_{0}/(2\gamma_{x})\), with \(x=T,c,v,p\). From Equation (24), the maximum reflectivity is \(|r_{R_{\max}}|=\gamma_{c}/\gamma_{T}=Q_{T}/Q_{c}\) and achieved at \(\omega_{0}\). Therefore, it is clear that the peak reflectivity approaches unity for \(Q_{c}\ll Q_{T}\). Since the waveguide can be strongly coupled to the nanocavity by reducing the distance between the two, this condition can be easily realized in practice.  
The inclusion of another output port above the nanocavity, a so-called _cross-port_, is not strictly necessary, but useful if one desires to improve the laser differential quantum efficiency [40]. In fact, if the laser operates around the nanocavity resonance frequency and no cross-port is included (i.e., \(\gamma_{p}=0\)), the Fano mirror reflection is high and low power is coupled out to the waveguide below the nanocavity, the so-called _through-port_. In the presence of the cross-port, instead, the total quality factor of the nanocavity is reduced somewhat, but the reduction is negligible if one ensures \(Q_{c}\ll Q_{v},Q_{p}\). The cross-port differential quantum efficiency is approximately proportional to \(\eta_{d}=Q_{v}/(Q_{v}+2Q_{p})\)[40] and can be easily maximized by simply ensuring \(Q_{p}\ll Q_{v}\). Therefore, one obtains the practical design rule \(Q_{c}\ll Q_{p}\ll Q_{v}\). Figure 13b,c shows the power reflectivity and phase of a Fano mirror with \(Q_{v}=10^{5}\), \(Q_{p}=1.5\times 10^{4}\) and different values of \(Q_{c}\)  
Figure 13: (**a**) Fano laser implementation on the PhC platform. The active region is denoted in red. (**b**) Power reflectivity and (**c**) phase for a Fano mirror with \(Q_{v}=10^{5}\), \(Q_{p}=1.5\times 10^{4}\) and different values of \(Q_{c}\) versus detuning from the resonance frequency.  
These are typical Fano mirror parameters [97]. As the coupling between nanocavity and waveguide weakens, the peak reflectivity decreases and the mirror bandwidth shrinks. Indeed, from Equation (24) the FWHM of the power reflectivity is given by \(2\gamma_{T}/(2\pi)\), thereby diminishing with decreasing \(\gamma_{C}\). For the sake of convenience, \(\gamma_{T}\) is denoted in the following as the Fano mirror linewidth [101].