  The all-optical synchronization systems used in various X-ray free-electron
lasers (XFEL) such as the European XFEL observe the transient fields of passing
electron bunches coupled into one or more pickups in the Bunch Arrival Time
Monitors (BAM). The extracted signal is then amplitude modulated on reference
laser pulses in a Mach-Zehnder type electro-optical modulator. With the
emerging demand for future experiments with ultra-short FEL shots, fs precision
is required for the synchronization systems even with 1 pC bunches. Since the
sensitivity of the BAM depends in particular on the slope of the bipolar signal
at the zero-crossing and thus, also on the bunch charge, a redesign with the
aim of a significant increase by optimized geometry and bandwidth is
inevitable. In this contribution the theoretical foundations of the pickup
signal are aggregated and treated with a focus on ultra-short bunches as well
as a general formulation. A possible new pickup concept is simulated and its
performance is compared to the previous concept. A significant improvement of
slope and voltage is found. The improvement is mainly achieved by the reduced
distance to the beam and a higher bandwidth.
