# Modal Properties of Photonic Crystal Cavities and Applications to Lasers
## 4 Line-Defect Cavities
### Resonant Modes: Reciprocal Space Distribution and Radiation Loss  
The reciprocal or \(k\)-space distribution is the Fourier transform of the spatial field. From Equation (18), the Fourier transform of \(E(z)\) reads  
\[E(\xi) =\tilde{F}[|b_{0}|W(\xi-k_{2})+|b_{0}|W(\xi+k_{2}) \tag{19}\] \[\quad+\ |b_{-1}|W(\xi-k_{1})+|b_{-1}|W(\xi+k_{1})],\quad \text{for $N$ odd}\] \[E(\zeta) =\tilde{F}[|b_{0}|W(\xi-k_{2})-|b_{0}|W(\xi+k_{2})\] (20) \[\quad+\ |b_{-1}|W(\xi-k_{1})-|b_{-1}|W(\xi+k_{1})],\quad \text{for $N$ even}\]  
where \(\zeta\) is the spatial angular frequency and \(W(\zeta)=L\frac{\sin(\zeta L/2)}{\zeta L/2}=L\text{sin}\Big{(}\frac{\zeta L} {2\pi}\Big{)}\) the spatial Fourier transform of \(w(z,L)\). For a given cavity length, the light cone is given by \(|\zeta|<(\omega_{0}/c)n_{\text{clad}}\), with \(\omega_{0}\) being the angular frequency of the resonant mode. As an example, we select an L9 cavity and plot in Figure 10 the various components of the mode spectrum from Equation (19). The blue (red) solid line is for the sinc function centered at \(+k_{1}\) (\(+k_{2}\)), and the blue (red) dotted line corresponds to that centered at \(-k_{1}\) (\(-k_{2}\)). The total spectrum is the black line. Only the positive frequencies are shown due to the symmetry  
Figure 9: (**a**) L15 cavity and (**b**) electric field profile of the fundamental resonant mode. (**c**) Resonance condition of LN cavities: normalized wavevector \(k_{2}\) (**left**) and magnitude of the ratio between the Bloch modes spatial harmonic \(b_{-1}\) and spatial average \(b_{0}\) (**right**). of the spectrum. We note that the destructive interference of the various components contributes to reducing the field spectral content within the light cone. We define the _light cone power fraction_ as the relative fraction of the electric field intensity within the light cone, namely \(\eta=\int_{-\zeta_{0}}^{+\zeta_{0}}|E(\zeta)|^{2}d\zeta/\int_{-\infty}^{+\infty} |E(\zeta)|^{2}d\zeta\), where \(\zeta_{0}=(\omega_{0}/c)n_{\text{clad}}\) is the upper limit of the light cone for a given cavity length. By computing the light cone power fraction as a function of the cavity length, one obtains an approximate measure of the scaling of the radiation loss with the size of the cavity [75]. The absolute radiation loss rate and, therefore, the Q-factor, may be quantified through full 3D-simulations of the structure (see, e.g., [79] for a comparison of different numerical methods and their accuracies), but this falls outside the scope of this paper. The result is illustrated in Figure 11a. The light cone power fraction decreases as the size of the cavity increases, albeit with some local maxima. Depending on the ratio \(|b_{-1}/b_{0}|\), the various components of the field \(k\)-space distribution (shown in Figure 10 for an L9 cavity) interfere _destructively_ within the light cone, more or less effectively. This mechanism is the origin of the local maxima in Figure 11a, which were also found in [75]. The general reduction of the light cone power fraction with increasing cavity length is due to the \(k\)-space distribution being shifted outside the light cone. This is emphasized by Figure 11b, depicting \(|E(\zeta)|^{2}\) in dB normalized to its maximum value for various LN cavities. The spectrum is computed through Equations (19) and (20). The dashed, vertical line denotes the upper limit of the light cone for each cavity. As the cavity becomes longer, (1) \(k_{1}\) approaches \(\pi/a\) and (2) the width of each of the sinc functions in Equations (19) and (20) is reduced. As a result, the field spectrum departs from the light cone and the light cone power fraction generally decreases. We note that FDTD simulations have demonstrated monotonic increase in Q-factor with cavity length, without local maxima [47; 79]. This is because, in practice, the destructive interference of the various spatial frequency components of the field is less effective than predicted by Equations (19) and (20).