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

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