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A rectangular waveguide feed to the dielectric resonator antenna (DRA) is proposed. A three-layer DRA structure is used to achieve enhanced and broadband coupling to the waveguide. The antenna topology is easy to realise and does not need additional coupling or matching elements to effectively excite the DRA. The configuration also affords ease in frequency tuning without sacrificing bandwidth or radiation performance. The full-wave Green's function approach is presented for the analysis of such multilayer structures with an arbitrary number of layers with greatly reduced computational overhead. The modes of the multilayer DRA can be identified from the analysis, which can be used to explain the broadband nature of the coupling. The double-summation representation of the homogeneous Green's function has been extended to the multilayer case, leading to a substantial reduction in computation time for the input impedance.