The three dimensional (3-D) nature of antennas for fusion applications in the ion cyclotron range of frequencies (ICRF) requires accurate modelling to design and analyse new antennas. In this article, analysis and design tools for radiofrequency (RF) antennas are successfully benchmarked with experiment, and the 3-D physics of the launched waves is explored. The systematic analysis combines measured density profiles from a reflectometer system, transmission line circuit modelling, detailed 3-D magnetostatics modelling and a new 3-D electromagnetic antenna model including plasma. This analysis gives very good agreement with measured loading data from the Tokamak Fusion Test Reactor (TFTR) Bay-M antenna, thus demonstrating the validity of the analysis for the design of new RF antennas. The 3-D modelling is contrasted with 2-D models, and significant deficiencies are found in the latter. The 2-D models are in error by as much as a factor of 2 in real and reactive loading, even after they are corrected for the most obvious 3-D effects. Three dimensional effects play the most significant role at low parallel wavenumbers, where the launched power spectrum can be quite different from the predictions of 2-D models. Three dimensional effects should not be ignored for many RF designs, especially those intended for fast wave current drive