On the Solid‐State Theoretical Description of Photonic Crystals

Abstract

Since the invention of the laser, progress in Photonics has been intimately related to the development of optical materials which allow one to control the flow of electromagnetic radiation or to modify light–matter interaction. Photonic Crystals (PhCs) represent a novel class of optical materials which elevates this principle to a new level of sophistication. These artificial structures are characterized by two–dimensional (2D) or three–dimensional (3D) periodic arrangements of dielectric material which lead to the formation of an energy band structure for electromagnetic waves propagating in them. Recent advances in micro–structuring technology provide an enormous flexibility in the choice of material composition, lattice periodicity and symmetry of these arrangements allowing one to fabricate PhCs with embedded defect structures. As a consequence, the dispersion relation and associated mode structure of PhCs may be tailored to almost any need. This results in a potential for controlling the optical properties of PhCs that may eventually rival the flexibility in tailoring the properties of their electronic counterparts, the semiconducting materials. One of the most striking features of PhCs is associated with the fact that suitably engineered PhCs may exhibit frequency ranges over which ordinary linear propagation is forbidden, irrespective of direction. These photonic band gaps (PBGs) [1–3] lend themselves to numerous applications in linear, nonlinear and quantum optics. For instance, in the linear regime novel optical guiding characteristics through the engineering of defects such as microcavities, waveguides and their combination into functional elements, such as wavelength add-drop filters [4, 5] may be realized. Similarly, the incorporation of nonlinear materials into PBG structures is the basis for novel solitary wave propagation for frequencies inside the PBG. In the case of lattice–periodic Kerr–nonlinearities, the threshold intensities and symmetries of these solitary waves depend on the direction of propagation [6–8], whereas in the case of nonlinear waveguiding structures embedded in a 2D PBG material, the propagation characteristics strongly depend on the nature of the waveguides [9]. Finally, the existence of complete PBGs allows one to inhibit spontaneous emission for atomic transition frequencies, deep in the PBG [1] and leads to strongly non–Markovian effects, such as fractional localization of the atomic population for atomic transition frequencies in close proximity to a complete PBG [10, 11].