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In the optics of charged particle beams, circular transverse modes can be introduced; they provide an adequate basis for rotation-invariant transformations. A group of these transformations is shown to be identical to a group of the canonical angular momentum preserving mappings. These mappings and the circular modes are parametrized similar to the Courant-Snyder forms for the conventional uncoupled, or planar, case. The planar-to-circular and reverse transformers (beam adapters) are introduced in terms of the circular and planar modes; their implementation on the basis of skew quadrupole blocks is described. Various kinds of matching for beams, adapters and solenoids are considered. Applications of the planar-to-circular, circular-to-planar and circular-to-circular transformers are discussed. A range of applications includes round beams at the interaction region of circular colliders, flat beams for linear colliders, relativistic electron cooling, and ionization cooling.

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The wake forces produced by a beam bunch can be reduced by making the vacuum chamber cross section axially asymmetric. Furthermore, the asymmetry results in a betatron tune shift for particles in the tail of the bunch. As a result, transverse instabilities of the bunch should be significantly suppressed for an asymmetric vacuum chamber. [S0031-9007(99)08709-8] PACS numbers: 29.27.Bd, 29.20.Dh An ultrarelativistic charged particle generates electromagnetic fields behind it in the vacuum chamber. The net effect of these fields on a following charge is determined by integrating the force over a structure period of the vacuum chamber L. The integrated transverse force F caused by a slight offset r 0 of the leading particle from the axis of a round chamber is conventionally expressed in terms of the wake function [1]: Z L