  Using the Kubo linear response formalism, we study the effects of intrinsic
graphene optical and surface polar phonons (SPPs) on the optical conductivity
of doped graphene. We find that inelastic electron-phonon scattering
contributes significantly to the phonon-assisted absorption in the optical gap.
At room temperature, this midgap absorption can be as large as 20-25% of the
universal ac conductivity for graphene on polar substrates due to strong
electron-SPP coupling. The midgap absorption, moreover, strongly depends on the
substrates and doping levels used. With increasing temperature, the midgap
absorption increases, while the Drude peak, on the other hand, becomes broader
as inelastic electron-phonon scattering becomes more probable. Consequently,
the Drude weight decreases with increasing temperature.
