The all-optical synchronization systems used in various x-ray free-electron lasers (XFEL) such as the European XFEL usually include bunch arrival-time monitors (BAM), which observe the transient fields of passing electron bunches through their pickups. The extracted signal is used to modulate the amplitude of a reference laser pulse in a Mach-Zehnder type electro-optical modulator. The laser pulse is typically much shorter than the voltage signal and interacts only with an instantaneous value of the applied signal. With the emerging demand for future experiments with 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 short bunches as well as a general formulation. The analytical treatment reveals design limitations and suggests several approaches for further development. Based on these considerations a new pickup concept with possible advantages in manufacturing and temporal resolution is simulated and its performance is compared to the previous concept. The design offers good potential and a significant improvement of slope and voltage is found. Nevertheless the target set for 1 pC operation was not fully reached yet and further optimization is necessary. The observed improvement is mainly achieved by the reduced distance to the beam and a higher bandwidth.