# Modal Properties of Photonic Crystal Cavities and Applications to Lasers
## 1 Introduction
### Key Mode Characteristics: The Q-Factor and the Mode Volume  
Broadly speaking, the strength of the light-matter interaction in a cavity increases when light is stored longer in the cavity and when the light is concentrated spatially, so that the intensity in the material increases. The quality factor (Q-factor) reflects the temporal confinement of light. The Q-factor is defined as \(Q=\omega_{0}U/P\)[11], where \(\omega_{0}\) is the mode resonance angular frequency, \(U\) the electromagnetic energy stored inside the cavity and \(P\) the energy lost per unit time. By expressing \(P\) as \(dU/dt\), one finds that the energy stored within the cavity decays with a time constant \(\tau_{p}=Q/\omega_{0}\), which is the _photon lifetime_. It is customary to express the Q-factor as \(Q^{-1}=Q_{c}^{-1}+Q_{i}^{-1}\). Here, \(Q_{i}\) is the _intrinsic_ or _unloaded_ Q-factor, which accounts for radiation losses, and absorption losses due to impurities [12]. \(Q_{c}\) reflects instead losses due to coupling with input and output waveguides. In the presence of a finite \(Q_{c}\), the total Q-factor, \(Q\), is often denoted the _loaded_ Q-factor. The spatial confinement of light is often quantified by a mode volume. The appropriate definition of such a mode volume depends on the type of light-matter interaction considered. For the Purcell-enhanced emission rate of a dipole emitter, it is customary to express the relevant mode volume as [13]  
\[V_{\text{eff}}(\mathbf{r}_{e})=\frac{\int_{V}n^{2}(\mathbf{r})|\mathbf{E}( \mathbf{r})|^{2}d^{3}\mathbf{r}}{n^{2}(\mathbf{r}_{e})|\mathbf{E}(\mathbf{r}_ {e})|^{2}} \tag{1}\]  
Here, \(n\) is the cavity refractive index, \(\mathbf{E}\) the electric field and \(V\) the integration volume. Moreover, \(\mathbf{r}\) is the position vector, with \(\mathbf{r}_{e}\) denoting the point where the emitter (e.g., a quantum dot) is placed. Therefore, for a given cavity, \(V_{\text{eff}}\) generally varies depending on the position of the emitter. If the emitter transition frequency coincides with the mode  
Figure 1: (**a**) Photonic crystal line-defect cavity, (**b**) photonic crystal nanobeam cavity and (**c**) a cavity for extreme dielectric confinement (EDC). In Figure 1c, the gray area is air and the black area is the semiconductor. Figure 1c is reprinted from [10], with the permission of AIP publishing, 2021.  
resonance frequency and the transition dipole moment of the emitter is aligned with the field polarization, the emitter decay rate within the cavity reads [14]  
\[\gamma_{r}=\frac{2d^{2}}{\hbar\epsilon_{0}n^{2}V_{\text{eff}}(\mathbf{r}_{e})} \frac{\omega_{0}}{1/\tau_{p}+2/T_{2}} \tag{2}\]  
where \(d\) is the transition dipole moment and \(T_{2}\) the emitter dephasing time. The same emitter in a homogeneous medium with refractive index \(n\) would decay with a rate \(\gamma_{r,\text{hom}}=d^{2}\omega_{0}^{3}n/(3\pi\epsilon_{0}hc^{3})\)[14]. Thus, the emitter decay rate inside the cavity may be enhanced by a factor \(F_{p}=\gamma_{r}/\gamma_{r,\text{hom}}\)--the Purcell factor [15]. This factor scales with \(Q_{\text{eff}}/V_{\text{eff}}(\mathbf{r}_{e})\), where \(Q_{\text{eff}}=\omega_{0}\big{(}1/\tau_{p}+2/T_{2}\big{)}^{-1}\) is an effective Q-factor including the effect of the emitter and cavity broadening. Equation (1) is also widely employed as a loose figure of merit of the light spatial confinement. In this case, \(\mathbf{r}_{e}\) is often taken at an antinode, \(\mathbf{r}_{\text{max}}\), of the cavity field. Notice that the optical modes of a cavity with finite Q-factor are intrinsically leaky. Consequently, they diverge in space at sufficiently large distances from the cavity [16]. As a result, the integral in the numerator of Equation (1) diverges as the integration volume increases [17]. For this reason, a rigorous and unambiguous definition of the mode volume entering the Purcell-enhanced emission rate is that based on the theory of quasinormal modes [17; 18]. However, the large Q-factor of PhC cavities makes the cavity modes slowly divergent [17].