  We present an analysis of a \v{C}erenkov free-electron laser (FEL) driven by
a flat electron beam. In this system, an electron beam travelling close to a
dielectric slab placed at the top of an ideal conductor interacts with the
co-propagating electromagnetic surface mode. The surface mode arises due to
singularity in the reflectivity of the dielectric slab for the incident
evanescent wave. Under suitable conditions, the surface mode grows as a result
of interaction with the electron beam. We show that the interaction of the
surface mode with the co-propagating electron beam can be rigorously understood
by analyzing the singularity in the reflectivity. Using this approach, we set
up coupled Maxwell-Lorentz equations for the system, in analogy with
conventional undulator based FELs. We solve these equations analytically in the
small signal regime to obtain formulae for the small signal gain, and the
spatial growth rate. Saturation behaviour of the system is analyzed by solving
these equations numerically in the nonlinear regime. Results of numerical
simulations are in good agreement with the analytical calculations in the
linear regime. We find that \v{C}erenkov FEL under appropriate conditions can
produce copious coherent terahertz (THz) radiation.
