Negative superluminal velocity and violation of Kramers-Kronig relations in causal optical systems

We investigate nonanalyticities (e.g., zeros and poles) of refractive index n(omega) and group index n(g)(omega) in different optical media. We first demonstrate that while a Lorentzian dielectric has no nonanalyticity in the upper half of the complex frequency plane (CFP), its group index-which governs the pulse-center propagation-possesses zeros in the same region. Thus we classify the nonanalyticities as in the (a) first order (refractive index or reflection) and (b) second order (group index or group delay). The latter contains the derivative of the former. Then, we study a possible connection between the negative superluminal velocities and the presence of nonanalyticities in the upper half of the CFP. We show that the presence of nonanalyticities in the upper half of the CFP for (a) the first-order and (b) the second-order responses are accompanied by the appearance of negative (a) phase and (b) group velocities, respectively. We also distinguish between two kinds of superluminosity, v > c and v < 0, where we show that the second one (v < 0) appears with the violation of KKRs.