  Higher order modes (HOMs) are simulated with finite element and finite
difference computer codes for the ILC superconducting cavities. In particular,
HOMs in KEK's Ichiro type of cavity and Cornell University's Reentrant design
are focused on in this work. The aim, at these universities and laboratories,
is to achieve an accelerating gradient in excess of 50 MV/m in 9-cell
superconducting cavities whilst maintaining a high quality and stable electron
beam. At these gradients, electrical breakdown is an important cause for
concern and the wakefields excited by the energetic electron beams are also
potentially damaging to the beam's emittance. Here we restrict the analysis to
performing detailed simulations, on emittance dilution due to beams initially
injected with realistic offsets from the electrical centre of the cavities. We
take advantage of the latest beam dynamics codes in order to perform these
simulations.
