Search

After a brief summary of the main factors causing pulse distortion in optical fibres, expressions are derived for phase velocity, group velocity and dispersion, that is velocity variation with frequency, of each mode. The analysis is carried out by making extensive use of the functiony(x) (y andx being the arguments of the first and second kind Bessel functions that appear in the solutions of Maxwell equations), that we call characteristic function. By using this function we obtain particularly simple and compact formulas that make easier the analysis of the behaviour of velocity and dispersion near and far from cut-off. It will also be seen that the ratio between the power propagating in the core,W 1, and the power in the cladding,W 2, for each mode takes a very simple form if expressed by means of the characteristic function: this fact permits a direct relation to be found between group velocity and power flow in an optical fibre. The study of the ratioW 1/W 2 will show that, contrary to expectation, for some modes, a substantial fraction of energy travels inside the core, even near cut-off. Expressions are given forW 1/W 2 at cut-off, and far from cut-off. Plots of velocity and dispersion are shown. On the grounds of the obtained results pulse distortion in optical fibres is discussed.

The group velocity of 93 GHz phonons in liquid 4 He has been measured directly at T ≲ 0.1 K between 0 and 24 bar. Above ∼ 12 bar the group velocity falls increasingly below the ultrasonic phase velocity, and at 24 bar the dispersion is consistent with the quadratic deviation below linearity as suggested by neutron scattering.

Determination of the group velocity for wave propagation in a dispersive medium is a problem addressed by almost every physics student at some time during a course in optics or electricity and magnetism. A simple derivation is given here for the group velocity of frequency-modulated waves in which the group velocity is equal to dω/dk, where ω is the angular frequency and k is the wave number. The mathematical description used to determine the group velocity is also shown to be applicable to kinematic waves in glacier flow as well as in traffic flow on crowded highways. These analogies thus provide physics students with examples of ways in which familiar ideas can be applied to new problems outside the conventional sphere of physics.

We propose and demonstrate a method to directly measure with high accuracy the group velocity of light. We make use of the natural fluctuations of light and determine in a correlation experiment the coincidence of fluctuations of light passing through a sample path and a reference path with known delay. It is shown that this coincidence can be detected by using amplitude interferometers such as a Michelson interferometer, or intensity interferometers. Accuracies of the order of ${10}^{\ensuremath{-}6}$ are practically possible. That is orders of magnitude better than previous methods allow. Using a Nd:glass laser as light source and an intensity interferometer for detecting coincidence, we demonstrate our method by measuring the group velocity of light in water within 2 \ifmmode\times\else\texttimes\fi{} ${10}^{\ensuremath{-}4}$ using a sample length of only 15 cm. The limitations of this new method caused by second-order dispersion are discussed quantitatively; higher-order effects are discussed qualitatively.

Very-low-frequency radio-wave diurnal phase changes for frequencies between 10.2 and 26.1 kHz, inclusive, are tabulated and plotted as a function of frequency and path distance. Ninety measurements are listed, of which 25 are new data. By using a daytime theoretical phase velocity for a homogeneous, sharply bounded ionosphere at 70 km which agrees well with experimental data, it is possible to obtain a nighttime first-order-mode phase-velocity curve which does not change with frequency in a manner which is easy to predict. First-order-mode phase velocity curves are used to obtain estimates of first-order-mode group velocity. These calculations suggest that for frequencies less than about 16kHz the nighttime group velocity is greater than the daytime group velocity. Above 16kHz it appears that the nighttime group velocity is less than the daytime group velocity. Calculated average group velocity is in agreement with a recent experimental measurement at 17.8 kHz within the experimental error.

The physical meanings of concepts such as group velocity and wave path, are first carefuly examined and mathematically founded, and it is then investigated whether the significant concepts of group velocity and wave path do exist or not, when the Maxwell field energy is lost in a medium. It is concluded that, in a rigorous sense, they generally do not exist in absorbing non-uniform medium, and that, when the energy loss in a medium is sufficiently small that only the first order changes in group velocity and wave path due to the energy loss are required to be taken into account, they approximately exist and are concordant with the ones obtained when the field energy is not lost in that medium, so that it is anyhow meaningless to discuss their changes due to the energy loss in a medium.

Two aspects of power flow associated with electromagnetic waves in plane-stratified, dispersive, anisotropic media that are also lossless and linear are considered. One aspect is the relation between group velocity and the velocity of energy transport of surface waves in such media. It is shown that the group velocity of surface waves is equal to the ratio of the real part of the complex Poynting vector, integrated over the coordinate of stratification, to the corresponding integral of the stored energy density. The second aspect is the relation between the dyadic surface impedance representing either a slab of plane-stratified medium above a perfectly conducting plane or a semi-infinite region, the latter for the case of evanescent fields, and the power flow in the respective structures. The significance of the surface impedance and power relations for surface waves is discussed.

The velocities of relativistic hydromagnetic waves in a compressible, perfect fluid of infinite conductivity are calculated in the framework of general relativity. In the absence of viscuous and Joule heat losses, the flow is isentropic, and, therefore, the wave surfaces are propagated without change of shape. The velocities are first obtained in terms of the four‐vector magnetic field and then in terms of the three‐dimensional field. Several limiting cases are considered, and, in particular, it is shown under what conditions the expressions reduce to the nonrelativistic forms. Finally, the group velocities are calculated. The existence of a group velocity for such waves is based on the fact that the velocities exhibit a directional dependence. The group velocity in this case is of significance because it is the velocity of energy propagation, just as in the case when dispersion exists.

The full adiabatic theory is used to describe wave propagation in a warm magnetoplasma for frequencies around the plasma frequency and the upper hybrid resonance frequency. By comparison with the microscopic theory for directions perpendicular and parallel to the static magnetic field, we can well define the range of validity of the full adiabatic theory, which can then be used for computation of the group velocity. The variation of this group velocity as a function of frequency for several directions of propagation is shown for models corresponding to the ionosphere and the far magnetosphere. Three factors deserve attention: (1) the upper hybrid resonance and its neighborhood can be rigorously studied for far magnetospheric conditions by using the full adiabatic theory; (2) the computed modes present a mixed electrostatic-electromagnetic character; (3) The minima in group velocities for pseudoresonance frequencies progressively fill up as the temperature increases.