| Summary: | Here we describe some relationships between cavity shape and fields on and near its surface that can be used for better understanding of the surface field properties. The problem of accelerating cavity optimization lies in the search of the shape with minimal peak magnetic or electric field for a given acceleration rate. This problem became especially important due to widespread use of superconducting cavities where the maximal magnetic field appeared to have a hard limit. The peak magnetic field can be lowered if one can increase the peak electric field but the electric field is also limited because of field emission. The problem of minimal losses in a cavity is very close to the problem of minimal peak magnetic field, though it is not the same, it relates to the lowest average field for a given acceleration rate. The field configuration on the cavity surface is also responsible for the phenomenon of multipactor. Cavities with these properties—minimal peak fields, minimal losses, and absence of multipactor—are found within the set of elliptic cavities. Further improvement of these properties is possible if we step out of the limits of elliptic shapes.
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