Energy-flux characterization of conical and space-time coupled wave packets

We introduce the concept of energy density flux as a characterization tool for the propagation of ultrashort laser pulses with spatiotemporal coupling. In contrast with calculations for the Poynting vector, those for energy density flux are derived in the local frame moving at the velocity of the envelope of the wave packet under examination and do not need knowledge of the magnetic field. We show that the energy flux defined from a paraxial propagation equation follows specific geometrical connections with the phase front of the optical wave packet, which demonstrates that the knowledge of the phase fronts amounts to the measurement of the energy flux. We perform a detailed numerical study of the energy density flux in the particular case of conical waves, with special attention paid to stationary-envelope conical waves (X or O waves). A full characterization of linear conical waves is given in terms of their energy flux. We extend the definition of this concept to the case of nonlinear propagation in Kerr media with nonlinear losses.