Anomalous momentum states, non-specular reflections, and negative refraction of phase-locked, second-harmonic pulses

We simulate and discuss novel spatio-temporal propagation effects that relate specifically to pulsed, phase-mismatched second-harmonic generation in a material characterized by simultaneously negative permittivity and permeability, having finite length. Using a generic Lorentz model for the dielectric permittivity and magnetic permeability, the fundamental and second-harmonic frequencies are tuned so that the corresponding indices of refraction are negative for the pump and positive for the second-harmonic signal. A phase-locking mechanism causes part of the second-harmonic signal generated at the entry surface to become trapped and dragged along by the pump and to refract negatively, even though the index of refraction at the second-harmonic frequency is positive. These circumstances culminate in the creation of an anomalous state consisting of a forward-moving second-harmonic wave packet that has negative wave vector and momentum density, which in turn leads to non-specular reflections at intervening material interfaces. The forward-generated second-harmonic signal trapped under the pump pulse propagates forward, but has all the attributes of a reflected pulse, similar to its twin counterpart generated at the surface and freely propagating backward away from the interface. This describes a new state of negative refraction, associated with nonlinear frequency conversion and parametric processes, whereby a beam generated at the interface can refract negatively even though the corresponding index of refraction at that wavelength is positive.