Length optimization of the detector region dipoles in LHeC and FCC-eh

In the LHeC and FCC-eh interaction region designs strong quadrupole septa are used to focus the colliding proton beam at the interaction point. Since the electron beam has a significantly lower beam rigidity it must pass through these magnets’ field free region. This leads to a large angle between the two incoming colliding beams. Dipoles around the Interaction Point (IP) bend the electron beam into head-on collisions with the proton beam and separate the two beams after the IP. Although these dipoles are weak, the high electron beam energy and current lead to a synchrotron radiation power in the range of tens of kW in the immediate vicinity of the detector. This note shows that there is an optimum dipole length that minimizes the synchrotron radiation power for a given distance L* between IP and the first quadrupole.