Group velocity in a medium with optical dispersion

In spite of very different mechanisms, accomplishing the slowdown of light, two systems—(i) dielectric with a resonant absorption line, and (ii) atomic system with electromagnetically induced transparency—have a common feature. This is the strong dispersion of optical parameters near a resonant frequency w_{0 }(Fig. 13.1). Specifically, the variation of the refractive index n = n(u>) results in the slowdown of group velocity v_{g} (w).

The fundamental study of light propagation and its group velocity in a dispersive medium was initiated more than a century ago by A. Sommerfeld and L. Brillouin. In their works a number of interesting effects, such as the emergence of precursors, was predicted, and they drew attention to the variation of

Fig. 13.1. Schematic representation of real (solid lines) and imaginary (dotted lines) parts of a complex dielectric function in (a) a dielectric with a resonant absorption line and (b) an atomic system with the window of electromagnetically induced transparency at frequency w_{0}. The insets present the simplified schemes of levels.

the group velocity in a resonant system (Brillouin, 1960). While the phase velocity v_{p}h(^) = c/n(u>) is the speed at which the phase front of a monochromatic wave propagates through a medium with п(ш), the group velocity characterizes the motion of the envelope of the wave packet of a photon pulse. The commonly used expression for v_{g}(ш) as a function of frequency ш is written as

Here, к(ш) = ш/(cy/Цй)) is a wavevector of light; е(ш) is the dielectric function of a medium, c is the light speed in vacuum. Strictly speaking, eq. (13.1) is valid for transparent (non-absorbing) media only, otherwise the group velocity may become meaningless; see below. The denominator in eq. (13.1) represents the group refractive index, whose real part defines v_{g} (ш). If the spectrum of an optical pulse falls into the range of strong п(ш) dispersion, the factor дп/дш starts to dominate the denominator. As a result, v_{g}(ш) decreases greatly. At the same time, the phase velocity is not influenced by this factor; its variation is much smaller. Both the group index and wavevector are complex in an absorbing dielectric; thus, the wave passing through such a medium is attenuated.

It was recognized long ago that in the medium with an absorption line the group velocity, given by eq. (13.1), may be superluminal in certain frequency ranges, may be negative in others, and may even equal zero. This intricate situation has been discussed in several textbooks (e.g., Born and Wolf, 1965). It was noted that the term of group velocity loses its strict physical meaning for an electromagnetic wave in the region of abnormal dispersion (Landau and Lifshitz, 1984). Recent analysis of the group velocity concept for absorbing media can be found in Sonnenschein et al. (1998). For a Gaussian light-pulse the possibility of the v_{g} (ш) variation from negative to superluminal values has been shown theoretically, along with the unchanged width of the pulse for many exponential absorption depths (Garrett and McCumber, 1970). It is worth mentioning that this abnormal v_{g} (ш) does not imply violation of the causality principle, but is instead a consequence of pulse-shape distortion. The superluminal group velocity of a pulse does not imply energy transfer (Loudon, 1970), which cannot be faster than the phase velocity of light in vacuum (Crisp, 1971). The same is true for the information, whose transfer rate is limited by c (Molotkov, 2010). This limitation is important for quantum cryptography, because it provides security in communication systems.

The passage of light-pulses through the resonant medium was intensively studied during the epoch of the development of communication systems (Vainshtein, 1976). The theory of energy transfer has been developed for spatially dispersive media with exciton-polaritons, both infinite and bounded (Bishop and Maradudin, 1976; Agranovich and Ginzburg, 1984). The remarkable decrease of energy transport velocity down to a velocity of 4 x 10^{3} m-s^{-1} has been predicted for the exciton-polariton at a resonance in a semiconductor, e.g., GaAs (Puri and Birman, 1981). It has been demonstrated that a width of exciton-polaritons resonance Г strongly affects this velocity. Its value may decrease by orders of magnitude with decreasing Гvalue (Loudon et al., 1997).