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Physical Review A

Wave packets in an absorptive and strongly dispersive medium

Dan Anderson, Jan I. H. Askne, M. Lisak

Published Oct 1, 1975• 38 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.12.1546

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International Journal of Modern Physics B•Apr 26, 2018•0 citations
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.
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)
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.
We derive the propagation equation for the slowly varying envelope (SVE) of the electric field in a homogeneous dispersive nonisotropic medium. Terms in the propagation equation of the field envelope that originate because of finite beam size are obtained and explained, including generalized Fresnel diffraction terms and a term that rotates the SVE wave packet in a uniaxial crystal about an axis perpendicular to the optical plane. Numerical estimates of these terms in the propagation of short pulses in the uniaxial crystal β-barium borate are presented. We obtain an analytic expression for the SVE for an initial Gaussian wave packet by keeping terms up to second order in the wave equation for the SVE.
It is shown that the velocity of propagation of a frequency-modulated wave packet through a strongly dispersive absorbing medium can be significantly different from (either higher or lower than) that of a non-frequency-modulated wave packet. This difference is attributed to the absorption dispersion of the medium. The easiest way to take the absorption dispersion into account is to use the formalism of the complex group velocity of a wave packet. This paper considers the propagation of a linear frequency-modulated wave packet, whose carrier frequency is close to the frequency of a spectral absorption line of the medium.
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.
The semiclassical Maxwell–Bloch equations beyond the slowly varying envelope approximation (SVEA) are derived in order to study the propagation of the ultrashort pulse in a two-level medium. The evolutions of the pulse shape, the group velocity and the pulse peak of 2[Formula: see text] ultrashort pulse are investigated as well as the pulse splitting of 4[Formula: see text] ultrashort pulse. By comparing the numerical results with those obtained using the SVEA, the error introduced by the approximation is quantitatively evaluated with respect to the pulse width and the applicable condition of the SVEA is discussed consequently. The envelope solutions presented by us allow clear observation on the evolution of the envelope.