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Wave packets in an absorptive and strongly dispersive medium. Research report

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

Published Sep 1, 1974• 0 citations
OpenAlex

Abstract

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)

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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)
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.
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 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.
The propagation of a steplike change in the amplitude of a dc pulse and a step in the envelope of a plane sinusoidal wave are investigated. It is shown that a step (e.g., the beginning or end of a pulse or wave) propagates like a pulse in the medium; i.e., its velocity is a weighted average of the group velocity, and its rate of spread is determined by the distribution of group velocities within the spectrum of the step. The results apply to a real pulse (e.g., sound) in an infinite, homogeneous, dispersive, lossless medium, and to the quantum-mechanical wave function of a free particle.
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.
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.