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Physical review. B, Condensed matter

Linear pulse propagation in an absorbing medium: Effect of film thickness

J. A. Gaspar, P. Halevi

Published Apr 15, 1994• 5 citations

Abstract

We compute the peak velocity of a broad Gaussian pulse propagating in an absorbing thin film modeled with a one-oscillator dielectric function. The exact numerical calculation is done for a GaP:N film in the frequency region of the bound A exciton. For absorption lengths greater than the film thickness, the pulse velocity essentially coincides with the group velocity, and with the experimental results of S. Chu and S. Wong [Phys. Rev. Lett. 48, 738 (1982)]. However, outside the region of strong absorption the pulse velocity depends sensitively upon the thickness of the slab. Our results give a better account of the Chu and Wong experiment than the calculation based on the group velocity. For film thickness with smaller than one absorption length the pulse velocity disagrees with the group velocity even in the strong absorption region, as a consequence of Fabry-P\'erot reflections of the pulse inside the film.

DOI: 10.1103/physrevb.49.10742

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We compute the peak velocity of a broad Gaussian pulse propagating in an absorbing thin film modeled with a one-oscillator dielectric function. The exact numerical calculation is done for a GaP:N film in the frequency region of the bound A exciton. For absorption lengths greater than the film thickness, the pulse velocity essentially coincides with the group velocity, and with the experimental results of S. Chu and S. Wong [Phys. Rev. Lett. 48, 738 (1982)]. However, outside the region of strong absorption the pulse velocity depends sensitively upon the thickness of the slab. Our results give a better account of the Chu and Wong experiment than the calculation based on the group velocity. For film thickness with smaller than one absorption length the pulse velocity disagrees with the group velocity even in the strong absorption region, as a consequence of Fabry-P'erot reflections of the pulse inside the film.

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 propagation of a Gaussian light pulse through a medium having a positive or negative absorption line is examined. Analytical approximations are obtained for the case where the spectral width of the pulse is much smaller than that of the line. It is shown that the pulse remains substantially Gaussian and unchanged in width for many exponential absorption depths, and that the locus of instants of maximum amplitude follows the classical expression for the group velocity, even if this is greater than the velocity of light, or negative. Numerical calculations have been used to examine what happens beyond the limit of usefulness of the analytical approximations.

The pulse velocity in the linear regime in samples of GaP: N with a laser tuned to the bound

A

-exciton line is measured with use of a picosecond time-of-flight technique. The pulse is seen to propagate through the material with little pulse-shape distortion, and with an envelope velocity given by the group velocity even when the group velocity exceeds 3\ifmmode\times\else\texttimes\fi{}

{10}^{10}

cm/sec, $\mathrm{equals}\ifmmode\pm\else\textpm\fi{}\ensuremath{\infty}$, or becomes negative. The results verify the predictions of Garrett and McCumber.

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.

In this paper the effect of photonic band gap on the group velocity of reflected pulse from a dielectric slab doped with two-level or three-level atoms has been investigated. It is assumed that the slab is sandwiched between a uniform medium (like vacuum) and a one-dimensional photonic crystal. It is shown that the reflected pulse from the slab doped with two-level (three-level) atoms will be superluminal (subluminal) if the carrier frequency of the incident Gaussian pulse is in the photonic band gap. In contrast, for the incident pulse with the carrier frequency at the edge of photonic band gap, the reflected pulse from the slab doped with two-level (three-level) atoms is subluminal (superluminal).