Group velocity, energy velocity, and superluminal propagation in finite photonic band-gap structures

We have analyzed the notions of group velocity V_g and energy velocity V_E for light pulses propagating inside one-dimensional photonic band gap structures of finite length. We find that the two velocities are related through the transmission coefficient t as V_E=\|t\|²V_g. It follows that V_E=V_g only when the transmittance is unity (\|t\|²=1). This is due to the effective dispersive properties of finite layered structures, and it allows us to better understand a wide range of phenomena, such as superluminal pulse propagation. In fact, placing the requirement that the energy velocity should remain subluminal leads directly to the condition V_g<=c/\|t\|². This condition places a large upper limit on the allowed group velocity of the tunneling pulse at frequencies of vanishingly small transmission.