# Modal Properties of Photonic Crystal Cavities and Applications to Lasers
## 2 Fundamental Principles of Light Confinement in Photonic Crystal Structures  
It is well known that defects in the crystalline structure of semiconductor materials (such as doping atoms) introduce additional states within the _electronic_ band gap [41]. Similarly, defects in the periodic refractive index pattern of a PhC structure lead to additional states within the photonic band gap [42]. For instance, one may start from a semiconductor slab with an hexagonal lattice of air holes. By removing an entire row of holes, a line-defect is introduced. Thus, one creates a waveguide (see the inset of Figure 2a), where light is efficiently guided. Figure 2a shows the projected [1] band structure of a line-defect waveguide, with the cladding made of air. The band structure has been computed by the plane wave expansion method [43]. Further details on the simulation parameters and procedure are provided in Section 4.1. The \(y\)-axis displays the normalized angular frequency. The \(x\)-axis shows the normalized wavevector along the propagation direction, spanning over the right and left-hand sides of the first and second Brillouin zone, respectively. The line-defect introduces two bands within the band gap, corresponding to _guided modes_. These modes are confined in the vertical direction by total internal reflection, and confinement in the lateral direction is due to distributed Bragg reflection. Theoretically, these modes are truly lossless. Out-of-plane radiation losses only occur due to disorder [44], i.e., unavoidable fabrication imperfections. At the upper limit of the first Brillouin zone, the wavevector is equal to \(\pi/a\), which is often denoted as the _band edge_ and corresponds to the cutoff frequency of the guided modes. Close to the band edge, _slow-light_[45; 46] may be achieved. In the slow-light propagation regime, the group velocity is much smaller than the light speed in vacuum and ideally tends to zero at the band edge. Other regions of the band structure include slab modes and the light cone. The _slab modes_ are a continuum of modes which are confined along the vertical direction by total internal reflection, but are delocalized in the lateral direction. Conventionally, the lower-frequency (higher-frequency) band is denoted as dielectric (air) band, because lower-frequency modes in PhCs tend to concentrate their energy in the high refractive index regions [1]. These bands are analogous to the valence and conduction band of the electronic band structure of semiconductor materials. Finally, the _light cone_ corresponds to modes which do not fulfill the condition of total internal reflection. These modes leak out of the semiconductor-air interface. By removing a finite number, \(N\), of air holes from a given row in a PhC semiconductor slab, a so-called line-defect cavity or LN cavity [47] is created, as shown in Figure 1a. The mode confinement in the longitudinal direction occurs by distributed Bragg reflection. Essentially, the mode guided along the line-defect lies within the photonic band gap of the perfectly periodic crystal stretching to the left and right of the cavity. Therefore, these periodic regions act as mirrors, between which the guided mode bounces back and forth. In contrast to line-defect waveguides, line-defect cavities suffer from out-of-plane radiation loss even in the absence of disorder. The spatial localization of the cavity mode thus gives rise to spatial frequencies, i.e., \(k\)-values, within the light cone. At a given frequency, the time-averaged power radiated out of the cavity is proportional to the integral within the light cone of the _spatial_ Fourier transform of the electric field intensity [48]. Therefore, suppressing the wavevector components of the field within the light cone is an effective strategy to maximize the Q-factor. The concept of light cone is better illustrated by Figure 2b. The Fourier transform of the electric field spatial profile consists of a set of plane waves, each with a given wavevector \(\mathbf{k}\).