The spontaneous emission dynamics of atoms embedded in an omnidirectional waveguide (ODWG), a novel optical waveguide, is studied on the basis of the complete reflection of one-dimensional photonic crystals. With the dispersion curve of the single waveguide mode within the photonic band gap and various extents of background dissipation, we characterize the photon-atom interaction in the ODWG. The photon emitter of the system is a two-level atom embedded in the low-index medium of the multilayer-film ODWG or the atom-ODWG system. Fractional calculus, an innovative mathematical method in optical systems, is applied to solve the equation of motion for this atom-ODWG system. Two kinds of states with different group velocities exhibit totally distinctive dynamical behavior. The high frequency waveguide mode with a fast group velocity shows fast exponential decay in propagation while the band-edge mode with a slow group velocity displays non-Markovian dynamics with non-exponential oscillating time evolution. We therefore suggest different functions of this atom-ODWG system for these two kinds of states. The richness of the physical content of the system is also revealed through investigating the dynamical behavior of the band-edge mode. These results aid in further application and fundamental understanding of the atom-ODWG system.
