Uncoupled achromatic tilted S-bend

A particular section of the electron beam transport line, to be used in the e-cooling project [1] of the Relativistic Heavy Ion Collider (RHIC), is constrained to displace the trajectory with both horizontal and vertical offsets so that the outgoing beamline is parallel to the incoming beam-line. We also require that section be achromatic in both planes. This mixed horizontal and vertical achromatic S-bend is accomplished by rotating the two dipoles and the quadrupoles of the line, about the longitudinal axis of the incoming beam. However such a rotation of the magnetic elements may couple the transported beam through the first order beam transfer matrix (linear coupling). In this paper we study a sufficient condition, that the first order transport matrix (R-matrix) can satisfy, so that this section of beam transfer line is both achromatic and linearly uncoupled. We provide a complete solution for the beam optics which satisfies both conditions.