  Optical responses of graphene in the presence of intense circularly and
linearly polarized terahertz fields are investigated based on the Floquet
theory. We examine the energy spectrum and density of states. It is found that
gaps open in the quasi-energy spectrum due to the single-photon/multi-photon
resonances. These quasi-energy gaps are pronounced at small momentum, but
decrease dramatically with the increase of momentum and finally tend to be
closed when the momentum is large enough. Due to the contribution from the
states at large momentum, the gaps in the density of states are effectively
closed, in contrast to the prediction in the previous work by Oka and Aoki
[Phys. Rev. B {\bf 79}, 081406(R) (2009)]. We also investigate the optical
conductivity for different field strengths and Fermi energies, and show the
main features of the dynamical Franz-Keldysh effect in graphene. It is
discovered that the optical conductivity exhibits a multi-step-like structure
due to the sideband-modulated optical transition. It is also shown that dips
appear at frequencies being the integer numbers of the applied terahertz field
frequency in the case of low Fermi energy, originating from the quasi-energy
gaps at small momentums. Moreover, under a circularly polarized terahertz
field, we predict peaks in the middle of the "steps" and peaks induced by the
contribution from the states around zero momentum in the optical conductivity.
