Showing papers on "Metamaterial published in 1994"

An analytical approach to photonic band structure

Abstract: Ever since the first practical demonstration by Yablonovitch and Gmitter' of the existence of a complete three-dimensional photonic band gap, strenuous efforts have been made by many researchers to characterise and quantify the propagation properties of proposed photonic crystal structures. With the notable exception" of one-dimensionally-periodic dielectric structures, however, analysis of such materials has been confined to numerical treatments. Although numerical routines provide the most versatile way forward for the study of twoand three-dimensional periodic media, they are subject to the usual problems of algorithm convergence and accuracy associated with finite wordlength arithmetic, the termination of infinite series, or finite discretisation of continuous functions. Clearly, an exact analysis of the electromagnetic fields in a photonic crystal would be of value for several reasons: these include numerical benchmarking, optimisation, and the direct computation of further necessary quantities such as densities of states. In this paper we present an analytical treatment of electromagnetic propagation in a two-dimensional photonic crystal composed of a particular pure dielectric basis. The dielectric function can be distributed in both rectangular and hexagonal periodicities, and in the first instance, propagation is confined to the plane of periodicity. Exact transfer matrices for both sand p-polarised light are derived and presented, and, with the aid of Bloch's theorem, are used to derive analytical expressions for the dispersion relations for each polarisation. A particular feature of the resulting two-dimensional dispersion curves is the role played by locally evanescent waves within the structure. As an example we show below three sets of two-dimensional dispersion curves for propagation within the plane of periodicity in a square lattice structure. Those in Fig. (1) are for p-polarised light and for a range of frequencies starting from zero (represented by the dot at the origin). There is a band gap beginning at w / c = 1.9. All crystal wave propagation for this polarisation arises from locally travelling waves. Figures (2) and (3) show twodimensional dispersion curves for s-polarised light, and for two frequency ranges separated by a full twc-dimensional band gap. The dotted curves signify that crystal wave propagation origi0 (12 e r 1 6 0 8 1