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 $V_E=|t{|}^2{V}_g.$ It follows that $V_E{=V}_g$ only when the transmittance is unity $(|t{|}^2=1).\mathrm{}$ 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.