  Very high energy (VHE; energy $E \gtrsim 100\,$GeV) $\gamma$-rays originating
from extragalactic sources undergo pair production with low-energy photons of
background radiation fields. These pairs can inverse-Compton-scatter background
photons, initiating an electromagnetic cascade. The spatial and temporal
structure of this secondary $\gamma$-ray signal is altered as the $e^+e^-$
pairs are deflected in an intergalactic magnetic field (IGMF). We investigate
how VHE observations with the future Cherenkov Telescope Array with its high
angular resolution and broad energy range, can potentially probe the IGMF. We
identify promising sources and simulate $\gamma$-ray spectra over a wide range
of values of the IGMF strength and coherence length using the publicly
available ELMAG Monte Carlo code. Combining simulated observations in a joint
likelihood approach, we find that current limits on the IGMF can be
significantly improved. The projected sensitivity depends strongly on the time
a source has been $\gamma$-ray active and on the emitted maximum $\gamma$-ray
energy.
