# Modal Properties of Photonic Crystal Cavities and Applications to Lasers
## 4 Line-Defect Cavities
### Bloch Modes  
The plane wave expansion method provides the Bloch modes in a fully vectorial form (that is, \(x\), \(y\) and \(z\) components) and with three-dimensional spatial dependence, namely,  
\[\mathbf{E}_{\pm}(\mathbf{r})=\mathbf{e}_{\pm}(\mathbf{r})e^{\pm i\hat{k}_{z}z} \tag{7}\]  
Here, \(\mathbf{E}_{+}\) (\(\mathbf{E}_{-}\)) is the electric field of the forward-propagating (backward-propagating) guided mode. The envelopes \(\mathbf{e}_{\pm}\) are \(z\) periodic, with the period given by the lattice constant, \(a\). \(k_{z}\) is the wavenumber along the \(z\) axis. The Bloch modes are normalized, at each frequency, such that \(\int_{V_{\mathrm{cell}}}\varepsilon_{0}n_{b}^{2}(\mathbf{r})|\mathbf{e}_{+}( \mathbf{r})|^{2}dV_{\mathrm{cell}}=1\). Here, \(V_{\mathrm{cell}}\) is the volume of the supercell and \(n_{b}\) the background refractive index, with \(n_{b}=n_{\mathrm{slab}}\) (\(n_{b}=n_{\mathrm{clad}}\)) in the slab (cladding). For a qualitative understanding of the radiation loss in LN cavities, it suffices to consider the lateral field component along the longitudinal direction [19; 47]. Therefore, in the following we drop the vectorial notation and implicitly refer to the lateral field component along the _centerline_ of the waveguide. This line is denoted by \(x=y=0\) in Figure 6b, with \(y=0\) corresponding to the middle of the slab along the vertical direction. Figure 7 shows (a) the magnitude and (b) phase of the forward-propagating Bloch mode in a unit cell as a function of \(z\). Here, \(z=0\) is the center of the supercell along the longitudinal direction, as indicated in Figure 6b. Each color corresponds to a different value of the wavenumber, reported in units of \(2\pi/a\). These figures highlight that, as the wavenumber approaches the band edge at \(\pi/a\), (1) the peak-to-peak amplitude increases and (2) the phase linearly varies with \(z\) over a larger and larger portion of the unit cell. These features are common to Bloch modes of conventional Bragg gratings [76]. Moreover, whichever the frequency, the phase is zero at the input and output of the unit cell. Being periodic, the Bloch modes can be expanded in a Fourier series, namely \(e_{+}(z)=\sum_{q}b_{q}e^{+iq\frac{q \begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Lattice constant \(a\)\([\mathrm{nm}]\) & 438 \\ Slab refractive index \(n_{\mathrm{slab}}\) & 3.17 \\ Hole radius \(r\) & 0.25a \\ Slab thickness \(h_{\mathrm{slab}}\)\([\mathrm{nm}]\) & 250 \\ Cladding refractive index \(n_{\mathrm{clad}}\) & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters of the line-defect waveguide assumed throughout Section 4. Figure 6: (**a**) Three-dimensional and (**b**) top view of the supercell to compute dispersion relation and Bloch modes of the line-defect waveguide. The spatial harmonics, \(b_{q}\), are reported in Figure 7c. The harmonics with \(q\) other than \(-1\) and \(0\) are negligible. Therefore, we may express \(e_{+}(z)\) as  
\[e_{+}(z)=b_{0}+b_{-1}e^{-i\frac{2\pi}{\pi}z} \tag{8}\]  
Furthermore, as a result of the variation in the peak-to-peak amplitude (see Figure 7a) and the linear \(z-\)dependence of the phase (see Figure 7b), the magnitude of \(b_{-1}\) approaches the spatial average, \(b_{0}\), as the wavenumber moves towards the band edge. The physical interpretation of this behavior is straightforward. Upon insertion of Equation (8) into Equation (7), the forward-propagating Bloch mode reads  
\[E_{+}(z)=\underbrace{b_{0}e^{+ikz}}_{\text{FW}}+\underbrace{b_{-1}e^{-i\left( \frac{2\pi}{\pi}-k_{z}\right)z}}_{\text{BW}} \tag{9}\]  
In other terms, a Bloch mode results from the interference of a backward (BW) and forward (FW) component [77], whose _relative strength_ corresponds to the ratio \(b_{-1}/b_{0}\). If the mode as a whole is forward-propagating, then the forward component is stronger, i.e., \(|b_{-1}/b_{0}|<1\). In the limit \(|b_{-1}/b_{0}|=1\), the backward and forward component balance out.