Exploring Space Groups for Three Dimensional Photonic Band Gap Structures Via Level Set Equations: The Face Centered Cub
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Exploring Space Groups for Three Dimensional Photonic Band Gap Structures Via Level Set Equations: The Face Centered Cubic Lattice Martin Maldovan, Chaitanya K. Ullal, Craig W. Carter, and Edwin L. Thomas* Massachusetts Institute of Technology, Cambridge, Massachusetts, 02139, USA ABSTRACT A level set approach was used to study photonic band gaps for dielectric composites with symmetries of the eleven face centered cubic lattices. Candidate structures were modeled for each group by a 3D surface given by f(x,y,z)-t=0 obtained by equating f to an appropriate sum of structure factor terms. This approach allows us to easily map different structures and gives us an insight into the effects of symmetry, connectivity and genus on photonic band gaps. It is seen that a basic set of symmetries defines the essential band gap and connectivity. The remaining symmetry elements modify the band gap. The eleven lattices are classified into four fundamental topologies on the basis of the occupancy of high symmetry Wyckoff sites. Of the fundamental topologies studied, three display band gaps--- including two: the (F-RD) and a group 216 structure that have not been reported previously. INTRODUCTION Much has been written on the promise of photonic crystals and their potential applications especially in the visible and near infrared wavelength range [1,2]. Considerable attention has been directed towards the establishment of periodic dielectric structures that in addition to possessing robust complete gaps, can be easily fabricated with current techniques. A number of such structures have been proposed [3-15]. However, a systematic approach to arrive at structures with large band gaps has not been reported. The existence and characteristics of photonic band gaps depend on such factors as the dielectric contrast, volume fractions, as well as the symmetry, connectivity and topology of the periodic dielectric structure. While the nature of the dependencies is not clearly established, any attempt at a structured approach to find three dimensional photonic band gap structures should incorporate these parameters. Here we propose one such structured approach based on the 230 space groups as a map to explore the effects of symmetry and connectivity. As an example of this approach we examine the 11 FCC space groups and establish some fundamental classes. THEORY An important aspect of photonic crystals is its periodicity. The symmetric properties of a periodic structure are characterized by its space group. Our approach is to ensure a specified space group via a level set approach based on structure factors [17].The photonic band gap is calculated as a function of the remaining parameters, such as volume fraction, topology, etc. In particular the level set approach allows us to easily map structures with different connectivities since subsequent terms fill and connect Wyckoff sites of decreasing symmetry. Each particular structure is modeled by a 3D surface f(x,y,z)-t=0 which defines the interface that separates regions of dissimilar diel
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