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.
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.
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.
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.
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.
Recently,there has been interest in modifying the group velocity of optical pulses,resulting in stopped,slow,and fast(superluminal)light.According to the physical theory,fast appears in the anomalous medium,and then it so called light medium.There are many experimeuts to produce fast light,this situation opens up new perspectives for the so called dispersion engineering.For example,signal propagating with a group velocity that exceeds the speed of light in vacuum(c) is intriguing for scientists.And then,the traveling of short wave pulses,microwave pulses and light pulses with group velocity vg,whose values exceed infinity,so called negative group velocity(NGV) propagation.Such circumstance occurs when the peak of a pulse traveling through DUT,exits before the peak of the input pulse has reached the beginning of DUT.This effect is counterintuitive,but it is a discover under the experimental basis.When the group velocity of light in a material is negative,it is the NGV medium. Although NGV are possible particularly in gain systems,there have been some experimental demonstrations of that phenomenon in passive systems,such as the coaxial cables,waveguides,micro-strip lines,and optical fibers,etc.If the experimental configuration used with a higher impedance mismatch,then it permits larger group delays,and a corresponding NGV.If n is the effective refraction index of the medium,we wantfn |f/ndn/df|1 to obtain the NGV. Many experiments showed the group velocity can surpass the light speed c,but scientists still lack the further understanding of the results.In 2001,Scientists obtained the formula v-e=|T|2vg,where v-e is average energy velocity,T is transmission coeficient of the system.If we want v-ec,the condition is vgc|T|-2.Hence it can be concluded that only vg/c is very large,the energy velocity can surpass c.This demonstration opens the door to interesting possibilities. The definition of signal velocity by Brillouin,mathemalical meaning exceeds physical meaning.We argue that undistorted wave group(wave packet) propagation are the first consideration.According to this viewpoint,the theoretical analysis show that superluminal propagation(i.e.superluminal communication) was observed by authors of many NGV researchers.……The superluminal group velocity wave propagation,the negative group velocity wave propagation——each of the two cases has its own use.Now,scientists do research works in these fields: such as optical communication,squeezed light and quantum entanglement.The condusion tell us that,a hight efficiency and low-loss fast light scheme may explore potential applications.
For a pulse with carrier frequency detuned less than an atomic plasma frequency but outside a narrow Lorentzian gain line, the group velocity will be negative. Unlike propagation at the center of an absorption line, the energy velocity is approximately equal to the group velocity, and is also negative. We show that a classical Gaussian pulse with such a carrier frequency will propagate at the negative group velocity for many atomic plasma wavelengths, before dispersion deforms the pulse shape. The peak of the transmitted pulse leaves the gain medium before the peak of the incident pulse enters, i.e., the pulse is transmitted superluminally. For a sufficiently long medium the exit pulse is well resolved from a comparison pulse traveling through an equal distance of vacuum. There is no conflict with causality, as numerical simulations with a switched-on Gaussian demonstrate. We propose an experiment to observe this kind of propagation by sending a pulsed probe beam through a Xe gas cell pumped to achieve inversion.
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.