  As any $e^+e^-$ scattering process can be accompanied by a hard photon
emission from the initial state radiation, the analysis of the energy spectrum
and angular distributions of those photons can be used to search for hard
processes with an invisible final state. Thus high energy $e^+e^-$ colliders
offer a unique possibility for the most general search of dark matter (DM)
based on the mono-photon signature. We consider production of DM particles at
the International Linear Collider (ILC) and Compact Linear Collider (CLIC)
experiments via a light mediator exchange. Detector effects are taken into
account within the DELPHES fast simulation framework. Limits on the light DM
production in a simplified model are set as a function of the mediator mass and
width based on the expected two-dimensional distributions of the reconstructed
mono-photon events. The experimental sensitivity is extracted in terms of the
DM production cross section. Limits on the mediator couplings are then
presented for a wide range of mediator masses and widths. For light mediators,
for masses up to the centre-of-mass energy of the collider, coupling limits
derived from the mono-photon analysis are more stringent than those expected
from direct resonance searches in decay channels to SM particles.
