Designing photonic crystals

摘要

The propagation of waves in periodic systems has been investigated since at least the 19th century, when Floquet studied periodic perturbations to planetary motion. The importance and ubiquity of periodic media in many different sub-areas of physics has resulted in the development of independent theories each with its own special flavour. Obvious examples are X-ray diffraction in crystals, electronic band structure, phonon propagation in crystals, acousto-optic diffraction, volume holography, distributed feedback lasers, fibre Bragg gratings, photonic crystal structures on two-dimensional thin-film waveguides and ― most topically― photonic band gap materials. The theories used in each of these fields developed from different approximations, influenced by experimental parameters. For example, in X-ray diffraction the refractive index modulation is very small and the scattering weak, so that single-scattering theories are usually adequate, leading via the first Born approximation to the well-known Bragg conditions. In thick crystals, however, it is essential to take account of multiple scattering, as explained in the early work of Darwin and Ewald. The result became the celebrated dynamical theory of X-ray diffraction, which led to a very beautiful series of experiments oh thick crystals where the roles of group and phase velocity ― and Pendel-loesung fringes (related to photonic Bloch wave interference discussed in subsect. 3.2)? Were highlighted . Bloch's band theory of electrons in crystals was developed to treat a very different case ― when the scattering strength of each unit cell is very high. This led via a very different set of approximations to a theory couched in terms of electronic states rather than the propagation of waves. Jumping ahead, work at the beginning of the 1980s used the tools of photonic band structure to interpret effects seen in thin-film photonic crystal waveguides and to design working devices based on the dispersion of guided photonic Bloch waves.