# Modal Properties of Photonic Crystal Cavities and Applications to Lasers
## 4 Line-Defect Cavities
### Resonant Modes: Real Space Distribution  
In the following, we derive the spatial dependence of the electric field of a resonant mode. First of all, we assume that an integer number of periods of the Bloch modes \(e_{\pm}(z)\) fits into the cavity length, \(L\), as denoted in Figure 8. Secondly, we assume the mirrors to have (1) frequency-independent reflection coefficients and (2) zero penetration length. In practice, the field is evanescent within the mirrors, with a frequency-dependent decay constant [53]. However, here we are only interested in the field spatial dependence within the cavity length, which is enough to capture the essential physics. Figure 8: LN cavity, with N denoting the number of missing holes. The figure illustrates our choice of the reference planes for the cavity length \(L\). Under these simplifying assumptions and by usage of Equation (10), the electric field within the cavity may be expressed as  
\[E(z)=w(z,L)\left[\bar{F}e^{+ik_{z}z}e_{+}(z)+\bar{B}e^{-ik_{z}z}e_{-}(z)\right] \tag{14}\]  
Here, the window function \(w(z,L)\) accounts for the field confinement, with \(w=1\) (\(w=0\)) for \(|z|\leq L/2\) (\(|z|>L/2\)). We denote the wavenumber \(k_{z}\) of a resonant mode as \(k_{2}\) and insert Equation (8) into Equation (14), leading to  
\[E(z)=w(z,L)\left[\bar{F}\left(b_{0}e^{+ik_{2}z}+b_{-1}e^{-i\frac{2\pi}{2}}e^{+ ik_{2}z}\right)+\bar{B}\left(b_{0}^{*}e^{-ik_{2}z}+b_{-1}^{*}e^{+i\frac{2\pi}{ 4}}e^{-ik_{2}z}\right)\right] \tag{15}\]  
where \(*\) denotes the complex conjugate. By using \(\phi_{+}(\pm L/2)=0\), the boundary condition at the right mirror from Equation (11a) can be recast as \(\bar{B}=\bar{F}e^{+ik_{2}L}r_{R}\). By inserting this expression into Equation (15), the electric field of a resonant mode reads  
\[E(z)=\bar{F}w(z,L)\left[\left(b_{0}e^{+ik_{2}z}+b_{-1}e^{-ik_{1}z}\right)+ \left(b_{0}^{*}e^{-ik_{2}z}+b_{-1}^{*}e^{+ik_{1}z}\right)r_{R}e^{+ik_{2}L}\right] \tag{16}\]  
where we have defined the wavenumber  
\[k_{1}=2\pi/a-k_{2} \tag{17}\]  
In addition, since \(L=Na\), the resonance condition from Equation (13) can be expressed as \(k_{2}L=(m+N)\pi\) with \(N\) being the number of unit cells. This condition, together with Equations (16) and (17), provide the electric field distribution of the \(m\)-th longitudinal mode in an LN cavity. For the sake of brevity, in the following we focus on the fundamental mode, i.e., \(m=1\), but the same analysis can be easily applied to higher-order modes. We assume the magnitude of the right mirror reflection coefficient \(r_{R}\) to be equal to 1, which is a reasonable approximation for photonic band gap mirrors [73]. Concerning the phase, we take it to be equal to zero. We discuss later the implications of a phase equal to \(\pi\). Since \(m=1\), one finds \(e^{+ik_{2}L}\) to be equal to 1 (\(-1\)) if \(N\) is odd (even). Moreover, if \(N\) is odd (even), the center of the cavity at \(z=0\) is aligned with a maximum (minimum) in the magnitude of the Bloch mode \(e_{+}(z)\). Thus, one finds the spatial harmonic \(b_{-1}\) to be real and positive (real and negative) if \(N\) is odd (even). The spatial average \(b_{0}\) is real and positive irrespective of \(N\) being either odd or even. In light of these considerations, from Equation (16) the fundamental resonant mode reads  
\[E(z)= \begin{cases}2\bar{F}w(z,L)[|b_{0}|\cos(k_{2}z)+|b_{-1}|\cos(k_{1}z)],& \text{for $N$ odd}\\ 2i\bar{F}w(z,L)[|b_{0}|\sin(k_{2}z)+|b_{-1}|\sin(k_{1}z)],&\text{for $N$ even}\end{cases}  \tag{18}\]  
For a given \(N\), the fundamental mode has the smallest detuning from the band edge and thus the lowest frequency. For \(N\) being odd (even), the mode is even (odd) with respect to the center of the cavity, as also found in [47]. Figure 9a,b shows an L15 cavity and the corresponding electric field profile for the fundamental resonant mode. The field pattern results from the interference of two pairs of plane waves, with wavevectors \(\pm k_{1}\) and \(\pm k_{2}\).