Photonic band gaps in periodic dielectric structures: The scalar-wave approximation

Abstract

Using a plane-wave expansion method we have computed the band structure for a scalar wave propagating in periodic lattices of dielectric spheres (dielectric constant e, ) in a uniform dielectric background (e&). All of the lattices studied (simple cubic, bcc, fcc, and diamond) do possess a full band gap. The optimal values of the filling ratio f of spheres and of the relative dielectric contrast for the existence of a gap are obtained. The minimum value of the relative dielectric contrast for creating a gap is also obtained. These results are applicable to the problem of classical-wave propagation in composite media and relevant to the problem of classical-wave localization. I. INTRODUCTION Recently, there has been growing interest' in the studies of the propagation of electromagnetic (EM) waves in three-dimensional (3D) periodic andlor disordered dielectric structures (photonic band structures). The reasons for this interest are both fundamental and practical. The possibility of the observation of Anderson localization of EM waves in disordered dielectric structures, where the strong el-el interaction effects entering the electron-localization problem are absent, is of fundamental interest. ' Also in analogy to the case of electron waves propagating in a crystal, classical EM waves traveling in periodic dielectric structures will be described in terms of photonic bands with the possibility of the existence of frequency gaps where the propagation of EM waves is forbidden. The potential applications of such photonic band gaps are very interesting. It has been suggested that the inhibition of spontaneous emission in such gaps can be utilized to substantially enhance the performance of semiconductor lasers and other quantum electronic devices. Photonic band-gap materials can also find applications in frequency-selective mirrors, bandpass filters and resonators. Moreover, electromagnetic interaction governs many properties of atoms, molecules and solids. The absence of EM modes inside the photonic gap can lead to unusual physical phenomena. For example, atoms or molecules embedded in such a material can be locked in excited states if the photons to be emitted to release the excess energy have frequency within the forbidden gap. In addition, John has proposed that Anderson localization of light near a photonic band gap might be achieved by weak disordering of a periodic arrangement of spheres. It is therefore, very important to obtain structures with a frequency gap where the propagation of EM waves is forbidden for all wave vectors. Yablonovitch and Gmitter have demonstrated the soundness of the basic idea of photonic bands in 3D periodic structures in an experiment using microwave frequencies, where the periodic structures can be fabricated by conventional machine tools. In addition, a photonic gap in a face-centeredcubic (fcc) dielectric structure was reported. During the same period, theoretical studies of the propagation of EM waves in 3D periodic structures began. ' At first, the photonic band structures have been examined theoretically in the scalar-wave approximation ' in which the vector nature of the EM field is ignored. It soon became apparent' that not so many aspects of the experimental photon bands in periodic dielectric structures can be understood in terms of scattering of scalar waves. However, the scalar-wave approach is directly applicable to the scattering of acoustic waves, an area of equally active interest and to the localization of acoustic waves. ' ' Recently, by expanding the EM fields with a plane-wave basis set, Maxwell's equations were solved exactly, ' taking the vector nature of the EM field fully into account. Comparison of the calculated' ' results of the fcc structure with experiment indicated that, while the experimental data and theory agree very well over most of the Brillouin zone, there are two symmetry points (8' and U) where the experiment indicates a gap, while calculations show that propagating modes exist. It is now believed that the fcc structure does not possess a full photonic band gap in the lowest bands, instead there is a region of low density of states rather than a forbidden frequency gap.