Rayleigh Multipole Methods for Photonic Crystal Calculations

Abstract—Multipolemethodshaveevolvedtobeanimportantclassof theoretical and computational techniques in the study of photoniccrystalsandrelatedproblems. Inthischapter,wepresentasystematicand unified development of the theory, and apply it to a range ofscatteringproblemsincludingfinitesetsofcylinders,two-dimensionalstacks of grating and the calculation of band diagrams from thescattering matrices of grating layers. We also demonstrate its utilityin studies of finite systems that involve the computation of the localdensityofstates.1 Introduction2 TheoreticalFormulation2.1 BackgroundandContext2.2 GeneralFramework2.3 InfiniteStructures—Arrays2.4 InfiniteStructures—Gratings2.5 InfiniteStructureswithComplexUnitCell2.6 InfiniteStructures—CrossedGratingsand“Woodpiles”3 FromScatteringMatricestoBandDiagrams4 DisorderedPhotonicCrystals5 Green’s Tensor and Local Density of States for 2DPhotonicCrystals5.1 Background5.2 LDOSforTMPolarisation5.3 LDOSforTEPolarisation6 DiscussionandOutlookReferences1. INTRODUCTIONWhile there exist a variety of theories [1–3] for solving generalscattering and propagation problems, methods that are stronglyadapted to particular scattering geometries or profiles can be quiteadvantageous. Such techniques yield highly accurate results withrelatively short computation times, permitting the study of largeror more complex structures, and facilitating asymptotic analyses incertainlimitingcases.