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Proceedings of the IEEE

Energy transport velocity of electromagnetic propagation in dispersive media

E. O. Schulz-DuBois

Published Jan 1, 1969• 41 citations
Publisher

Abstract

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.

DOI
10.1109/proc.1969.7389

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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.
The propagation of short pulses in a dispersive medium can be described in simple physical terms by employing the concept of the group velocity provided the medium is lossless. When the medium is absorbing, the group velocity description breaks down. In a previous paper,1 we presented a similar description employing the velocity of energy of monochromatic waves in place of the group velocity which is valid for a particular dispersive medium with absorption known as the Lorentz medium. That model describes the entire dynamics of the pulse as it propagates with the exception of the early stages of the evolution of the second precursor (otherwise known as the Brillouin precursor). Since that region contains the peak values of the pulse, it is important to include it in the model. The present paper describes the arrival of the Brillouin precursor in terms of the group velocity of a propagating field of the form exp [ωttω z] where ω and kω are real and positive.
The Poynting vector, energy density, and energy velocity of light pulses propagating in an anomalous dispersion medium with two closely spaced gain lines are calculated. The results show that a negative energy density in the medium propagates along the opposite direction of incidence with a velocity similar to the negative group velocity vg{v}_{g}vg​ while Poynting vector is in the same direction of incidence. In other words, one might say that a positive energy density in the medium would propagate along the positive direction with a speed ∣vg∣.|{v}_{g}|.∣vg​∣. We further point out that neither energy velocity nor group velocity is a good concept to describe the propagation process of light pulse inside the medium in the Wang, Kuzmich, and Dogaiu [Nature (London) 406, 277 (2000)] experiment owing to the strong accumulation and dissipation effects.
Two theorems on the group velocity are presented in this paper. First a simple proof is given that for any dispersive dielectric, there must be a frequency at which the group velocity of an electromagnetic pulse becomes abnormal, i.e., greater than the vacuum speed of light, infinite, or negative. Second, at the frequency at which the attenuation (or gain) is a maximum, the group velocity must be abnormal (or normal). This second theorem is more widely applicable, e.g., to propagation in waveguides or through multilayer dielectrics. To illustrate these theorems we discuss dispersion in a medium with two resonance lines, one absorption and the other gain. We find that the group velocity is abnormal within the absorption line and in a transparent region outside the gain line.
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 The propagation of light in dispersive media can be described by five different kinds of wave velocities [1-3]: • Phase velocity, which is the speed at which the zero crossings of the carrier wave move • Group velocity, at which the peak of a wave packet moves • Energy velocity, at which the energy is transported by the wave • Signal velocity, at which the half-maximum wave amplitude moves • Front velocity, at which the first appearance of a discontinuity moves Generally, all five velocities can differ from each other, although in a linear passive dispersive media, they coincide and are, usually, less than the light velocity in vacuum. Recently, experimental demonstrations [4-6] in variousmedia show that the group velocity of light canbe reducedby10-100million compared with its vacuum phase velocity. The physics is based on very steep frequency dispersion in the vicinity of narrow resonance of electromagnetically induced transparency (EIT) [7-11].
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 transfer function for transmission through a maser medium is derived. It is valid within the dynamic range of linear maser response and under the assumption of weak dispersion. All signal distortions due to propagation in a maser medium may be canceled by subsequent propagation in a complementary absorbing medium. The signal leaves the cascaded media undistorted with a delay corresponding to propagation at the vacuum velocity of light. This possibility of recovering the speed of light makes it difficult to talk of the propagation of information in a dispersive medium as taking place at a signal velocity below that of light. A potentially useful scheme is shown in which a severely distorting low-noise maser amplifier is used. After further amplification the signal distortions are eliminated by transmission through a complementary absorbing medium. The mathematical form of the transfer functions also leads to a new integral formula involving Bessel functions.