# Modal Properties of Photonic Crystal Cavities and Applications to Lasers
## 2 Fundamental Principles of Light Confinement in Photonic Crystal Structures
The light cone is determined by the component of \(\mathbf{k}\) which is _tangential_ to the interface between slab and cladding [49]. We denote the magnitude of this component by \(k_{\parallel}\), with \(k_{\parallel}^{2}=k_{x}^{2}+k_{z}^{2}\). If \(k_{\parallel}\) cannot be conserved upon transmission through the interface, total internal reflection occurs. Otherwise, the plane wave with that wavevector can leak out of the slab. This is only possible if the angle of incidence, \(\theta_{s}\) (see Figure 2b), is _smaller_ than the critical angle \(\theta_{s,cr}=\sin^{-1}(n_{\text{clad}}/n_{\text{slab}})\). Here, \(n_{\text{clad}}\) and \(n_{\text{slab}}\) are the refractive indices of the cladding and semiconductor slab, respectively. As a consequence, the light cone corresponds to the region in the reciprocal space defined by \(k_{\parallel}<(\omega_{0}/c)n_{\text{clad}}\), with \(\omega_{0}\) being the angular frequency of the resonant mode. The rule of thumb to maximize the Q-factor is that the spatial envelope of the field should vary as smoothly as possible at the edges of the cavity [49]. Intuitively, this can be better understood by assuming the cavity to be one-dimensional. In this case, the field  
Figure 2: (**a**) A dispersion diagram of TE-like modes of a line-defect waveguide in a photonic crystal membrane, as depicted in the inset. The \(y\)-axis shows the angular frequency, \(\omega\), normalized to \(2\pi c/a\), with \(c\) being the light speed in vacuum and \(a\) the lattice constant. The \(x\)-axis shows the wavevector along the propagation direction, \(k_{z}\), normalized to \(2\pi/a\). Modes in the gray area (the so-called light cone) are not confined by total internal reflection in the membrane. Details on the simulation parameters are provided in Section 4.1. (**b**) A ray-optics illustration within the light cone of the reflection and transmission at the interface between semiconductor slab and air cladding. (**c**) Optimization of a line-defect cavity by displacement of the holes at the edges of the cavity. The hole shift reduces the out-of-place radiation loss. of the resonant mode in real space may be expressed as the product of an envelope and a sinusoidal wave, with spatial period given by the guided resonant wavelength. Therefore, abrupt changes in the envelope lead to high frequency components in its spatial Fourier transform. These components are then transferred to the light cone upon convolution of the envelope Fourier transform with the Fourier transform of the sinusoidal wave, thereby degrading the Q-factor. As a consequence, these abrupt changes should be avoided [12]. When the spatial extent of the cavity is decreased, the mode extends further in reciprocal space, with a considerable fraction of wavevector components lying within the light cone. Consequently, if the cavity is not properly designed, the out-of-plane radiation loss can severely limit the Q-factor. By displacing the holes at the edges of the cavity, the fraction of wavevector components within the light cone can be significantly reduced and the Q-factor thereby increased. This optimization procedure is illustrated by Figure 2c. Indeed, by displacing the holes, the periodicity of the crystal on either side of the cavity is perturbed and its reflection somehow weakened. Therefore, the field penetrates deeper within the crystal and is more gently confined within the cavity. By this technique, the theoretical, unloaded Q-factor is increased from few thousands up to around \(10^{7}\)[50]. Importantly, the mode volume \(V_{\text{eff}}(\mathbf{r}_{\text{max}})\) is almost unchanged by this small displacement of specific holes and remains on the order of the cubic guided wavelength in the case of an L3 cavity [19; 50]. The experimental values of the Q-factor are usually lower (owing to fabrication imperfections and impurities [12; 50]), but well confirm the optimization trend.