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American Journal of Physics

Propagation of a step in the amplitude or envelope of a pulse or wave packet

Henry M. Bradford

Published Aug 1, 1979• 2 citations
Publisher

Abstract

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.

DOI
10.1119/1.11743

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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.
The usual textbook discussions of the velocity and spreading of a wave packet are based on special cases and approximations which have a limited range of validity. In this paper more general relations are obtained by considering the time dependence of the moments of a wave packet. Both a packet of real sinusoidal plane waves (a real pulse, e.g., sound) and a packet of complex exponential plane waves (e.g., the quantum-mechanical wave function of a free particle) are considered. The results are similar if the real wave packet contains no standing wave components. It is shown that the centroid of a wave packet moves at a constant velocity which is a weighted average of the group velocity. The spreading of the wave packet depends on the product of the variance of the group velocity and the square of the time interval before or after the time of minimum packet width. The packet moves and spreads like a collection of particles in free flight, in which the velocities of the particles are the group velocities of the wave components of the packet.
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 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.
It is shown that, when a narrow-band signal with a non-Gaussian envelope propagates in a medium with a strong dispersion, the signal envelope being distorted due to the dispersion of the gain (or) absorption in the medium. Nevertheless, the amplitude-modulated signal having a symmetric (even) intensity envelope remains the same. The propagation velocity of the `middle point' (i.e., the symmetry centre) of the signal coincides with the real group velocity of the wave with a given carrier frequency in the medium and can be subluminal, superluminal or negative. Therefore, the concept of a real group velocity (including the superluminal or negative velocity) is also applicable in media with a strong dispersion of the gain (or absorption).
Analysis of an infinitely long train of sine-squared pulses—as a toy model—propagating in a medium exhibiting anomalous dispersion shows that although the shape of the light pulses seems to be almost preserved on a small scale, the pulses are severely distorted on a large scale. The frequency of the sine-squared pulses is doubled before they propagate a tenth of the width (FWHM) of a pulse in the medium. The analysis and a numerical calculation show that the front of a wave with a step-function envelope propagates at the speed of light, c , through a medium with two gain lines. Finally, the propagation of a Gaussian pulse—a more realistic pulse—in a medium exhibiting anomalous dispersion is studied. The distortion of the pulse in a Wang–Kuzmich–Dogariu (WKD) experiment (Wang L J, Kuzmich A and Dogariu A 2000 Nature 406 277) should be observable at a precision of ±1 ns. If a Gaussian pulse travels in the medium for 1/50 of a width (FWHM), it will collapse. 'Snapshots' of a Gaussian pulse show that the shape of the pulse in space will have more than one peak at certain times. A possible picture of light propagation in WKD-like experiments is proposed on the basis on these findings.