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Chapter 20:
Several Controversial Topics in Contemporary Optics: Dispersive Pulse Dynamics and the Question of Superluminal Pulse Velocities
Early considerations of the wave theory of light represented the optical wavefield as a coherent superposition of monochromatic scalar wave disturbances. Dispersive wave propagation was first considered in this manner by Sir William R. Hamilton in 1839, when the concept of group velocity was first introduced. In that paper,Hamilton compared the phase and group velocities of light, stating that “the velocity with which such vibration spreads into those portions of the vibratory medium which were previously undisturbed, is in general different from the velocity of a passage of a given phase from one particle to another within that portion of the medium which is already fully agitated; since we have velocity of transmission of phase = s/k, but velocity of propagation of vibratory motion = ds/dk,” where s denotes the angular frequency and k the wavenumber of the disturbance in Hamilton’s notation. Subsequent to this definition, Stokes posed the concept of group velocity as a “Smith’s Prize examination” question in 1876. Lord Rayleigh then mistakenly attributed the original definition of the group velocity to Stokes, stating that “when a group of waves advances into still water, the velocity of the group is less than that of the individual waves of which it is composed; the waves appear to advance through the group, dying away as they approach its anterior limit. This phenomenon was, I believe, first explained by Stokes, who regarded the group as formed by the superposition of two infinite trains of waves, of equal amplitudes and of nearly equal wavelengths, advancing in the same direction.” Rayleigh then applied these results to explain the difference between the phase and group velocities of light with respect to their observability, arguing that “Unless we can deal with phases, a simple train of waves presents no mark by which its parts can be identified. The introduction of such a mark necessarily involves a departure from the original simplicity of a single train, and we have to consider how in accordance with Fourier’s theorem the new state of things is to be represented. The only case in which we can expect a simple result is when the mark is of such a character that it leaves a considerable number of consecutive waves still sensibly of the given harmonic type, though the wavelength and amplitude may vary within moderate limits at points whose distance amounts to a very large multiple of λ . . . From this we see that . . . the deviations from the simple harmonic type travel with the velocity dn/dk and not with the velocity n/k,” where n denotes the angular frequency and k the wavenumber in Rayleigh’s notation.
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