Dipole photonic resonators and analogs of matter

High power RF deflecting cavities have found a wide range of applications which include particle separation, emittance exchange, X-ray pulse compression, temporal beam diagnostics, and crab crossing in colliders. However, conventional deflecting cavities suffer from beam degrading parasitic modes that are also confined within these cavities. Several mode damping/coupling schemes have been developed to solve this problem but they add to the cost, size, and complexity of the resonant cavities. Photonic Crystal (PC) accelerators and klystrons have benefited from the high mode selectivity of PCs where a specic EM state are confined to a defect in the PC while all other states are forced to propagate away. This work presents a systematic approach to designing a PC that confines only the TM110-like dipole Bloch state and forces the propagation of all other EM states. This dipole PC resonator was tuned and optimized for crabbing application at 11.9942 GHz (operational frequency of the crab cavity at CLIC). Also in this thesis, a carefully designed experimental measurement of the transmission spectrum of microwaves in a photonic crystal was used to demonstrate a well pronounced exhibition of the Dirac point in photonic analogues of graphene. This work adds to previous literature by studying the sensitivity of the Dirac point to the incident and receiving angles of waves propagating away from the Gamma - K direction at the input and output boundaries of the PC respectively. In addition, the challenges associated with the experimental retrieval of the dispersion plot of a photonic crystal were pointed out