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In \v{C}erenkov and Smith-Purcell free-electron lasers (FELs), a resonantinteraction between the electron beam and the co-propagating surface mode canproduce copious amount of coherent terahertz (THz) radiation. We perform athree-dimensional (3D) analysis of the surface mode, taking the effect ofattenuation into account, and set up 3D Maxwell-Lorentz equations for boththese systems. Based on this analysis, we determine the requirements on theelectron beam parameters, i.e., beam emittance, beam size and beam current forthe successful operation of a \v{C}erenkov FEL.

Three-dimensional analysis of the surface mode supported in \v{C}erenkov and Smith-Purcell free-electron lasers