This paper describes a method for computing the field components in a barrel open resonator from a specified electric field on the axis. It shows how the shape of the resonator can be determined from the fields and how the properties of the resonator can be calculated. The method can be applied to other, similar, problems.
The paper describes a novel method for synthesising the shapes of a class of resonators which are important for lasers, particle accelerators and some microwave tubes. It provides highly accurate solutions to problems which had previously been solved by approximate methods. The paper has led to a number of invitations to write papers on related topics for other journals.