Effect of x-y coupling on the beam breakup instability

In solenoidal beam transport systems, motions in the x and y directions are coupled by the vXB force. A two-dimensional coupled mode description of the beam breakup (BBU) instability is presented; its dispersion relation is derived and compared with the one-dimensional BBU dispersion relation. In the two-dimensional description, instability growth is doubled and two additional wave modes are found in the regime of strong focusing. In the weak focusing regime, the two-dimensional description gives the same dispersion relation as the one-dimensional model.