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Journal of Physics A Mathematical and General

Energy expressions and energy velocity for wave packets in an absorptive and dispersive medium

M. Lisak

Published Jul 1, 1976• 10 citations
Publisher

Abstract

Expressions for the stored energy, energy power, power dissipated and the velocity of energy propagation are derived for wave pulses with slowly varying amplitudes. The results are expressed in terms of the dispersion function D( omega ,k, lambda ) which is assumed to be an explicit function of the dissipational parameter lambda . It is shown that in the case of resonant pulse propagation through a non-inverted atomic medium the energy velocity is always less than the free space velocity of light, c. It is also shown, using the expression for the dissipated power in an atomic medium, that the asymmetric absorption of energy of the pulse results in a motion of the pulse maximum at a velocity greater than c.

DOI
10.1088/0305-4470/9/7/017

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Expressions for the stored energy, energy power, power dissipated and the velocity of energy propagation are derived for wave pulses with slowly varying amplitudes. The results are expressed in terms of the dispersion function D( omega ,k, lambda ) which is assumed to be an explicit function of the dissipational parameter lambda . It is shown that in the case of resonant pulse propagation through a non-inverted atomic medium the energy velocity is always less than the free space velocity of light, c. It is also shown, using the expression for the dissipated power in an atomic medium, that the asymmetric absorption of energy of the pulse results in a motion of the pulse maximum at a velocity greater than c.
The evolution of slowly varying wave pulses in strongly dispersive and absorptive media is studied by a recursive method. It is shown that the resulting envelope function may be obtained by including correction terms of arbitrary dispersive and absorptive orders. Different definitions of pulse velocity are discussed in some detail and finally, qualitative examples of a Gaussian pulse in a cold plasma and an atomic medium are given, illustrating the importance of higher-order terms in the velocity expression. Particularly, the packet velocity is shown to be dependent on pulse width besides dispersion.
It is shown that the prediction of Garrett and McCumber that a light pulse's velocity may exceed the speed of light in a resonantly absorbing medium is due to asymmetric absorption of energy from the light pulse. More energy is absorbed from the trailing half of the pulse than from the front half, causing the center of gravity of the pulse to move at a velocity greater than the phase velocity of light. It is also shown that in certain cases a pulse's maximum will propagate at the group velocity even when the pulse as a whole is distorted by dispersion.
The evolution of slowly varying wave pulses in strongly dispersive and absorptive media is studied by recursive and ''Schrodinger'' methods. It is shown that the resulting envelope function may be obtained by including correction terms of arbitrary dispersive and absorptive order. Different concepts of pulse velocity are discussed in some detail. The initial pulse form and the dispersion relation describing the medium determine the character of the pulse velocity. Finally, qualitative examples of a Gaussian pulse in a cold plasma and an atomic medium are given, which illustrate the importance of higher- order terms. (GRA)
We investigate the propagation of laser pulses in media with non-stationary optical and magnetic response. A new amplitude integro-differential equation, governing the evolution of wave packets with one and few cycles under the envelope, is derived. The application of slowly varying amplitude approximation to this equation reduces it to a nonlinear Slowly Varying Equation of Amplitudes (SVEA) in a second approximation of the linear dispersion. It is important to point out that the SVEA obtained in media with non-stationary optical and magnetic response is similar to the SVEA equation with stationary magnetic response. The only difference is in the expressions for the group velocity and for the dispersion parameters. In particular, in paramagnetic media near to the magnetic resonances and also in experiments with a strong DC longitudinal magnetic field, it is possible to attain negative group velocity and group velocity, higher than the velocity of light in vacuum.
For electromagnetic propagation in linear dispersive media, two definitions of energy transport velocity may be introduced. The first is the transport velocity of total stored energy density, and it was first discussed by Brillouin. This concept does not distinguish different forms of energy, but considers only the total. It is equal to the group velocity for the case of a lossless dispersive medium. Its significance is not fully understood for a maser medium where it is either greater than the velocity of light or negative, or for a dissipative dispersive medium where it does not seem to offer a simple description of the propagation process. The second is the transport velocity of electromagnetic energy. It is based on the distinction of propagating energy which is just electric and magnetic field energy, from localized energy which is stored in the degrees of freedom of the dispersive medium. Thus propagation involves electromagnetic propagation per se, plus a transfer of energy between the propagating and localized forms. This transport velocity is always less than the velocity of light, for absorbing and maser media alike. It may be determined through a gain or attenuation measurement since it enters the gain equation of electromagnetic transmission structures.
Propagation of a Gaussian wave packet in an absorbing medium is examined in order to shed light on the physical basis of group velocity which exhibits a singular behavior. It is found that the velocity of the wave packet, defined as the traveling distance of the peak amplitude divided by its flight time, decreases in the absorption range of frequency, although the group velocity becomes infinite in the same range. Fast pulse propagation, which was observed by Chu and Wong and is characterized by a packet velocity faster than the light velocity, turns out to be a characteristic in the early stage of the flight and is understood in terms of packet distortion due to damping of Fourier-component waves in an anomalous dispersion region. It also turns out that slow pulse propagation characterized by a packet velocity less than the light velocity appears for long traveling distance. The results provided a unified picture of wave-packet propagation in an absorbing medium.