### Fast Light

In a linear medium with anomalous dispersion, the group velocity is larger than the velocity of light in vacuum. Without being at odds with causality and special relativity, it is then possible to propagate without important distortion smooth enough light pulses in such a way that the maximum of the transmitted pulse appears sooner than if the pulse had propagated in vacuum or even before the maximum of the input pulse. The challenge in such experiments is to achieve advances significant with respect to the pulse width.

We reported in 1985 an experiment that remains one of the most convincing demonstrations of the phenomenon (see figure on right). The ado made (including in the tabloid press !) about a paper published in 2000 [Wang *et al*., Nature **406**, 277 (2000)])], showing much smaller and even disputable effects, led us to theoretically study the limit to the actually observable fractional advances. The latter are only significant if the system transmission *T* is minimal at the frequency of the input pulse and takes much larger values outside of the pulse spectrum. In a real experiment, the transmission dynamics *Tmax/Tmin* should not be too large in order to avoid problems of noise and parasitic signals. With the sole constraint of causality, we have determined, as a function of *Tmax/Tmin*, the fractional advance that could be attained in an optimal system for a given level of distortion. This enables one to compare the fractional advances attained with various systems to its upper limit. We also studied the fast light effects that can be obtained by quasi-destructive interference of light-pulses propagating at different velocities. This system is not optimal but it allows fully analytical calculations and provides a simple interpretation of the phenomena evidenced with birefringent systems (fibre or photonic crystal).

### Slow Light

Slow light does not raise problems comparable to those of fast light. It is generally obtained when the medium transmission is maximal at the pulse frequency and very small outside the pulse spectrum. We showed that, under certain general conditions, the light pulse was “normalised” during the propagation to take a *Gaussian* shape, irrespectively of its initial shape and of the exact form of the medium transfer function. This result may be considered as a consequence in a deterministic system of the central-limit theorem in probability theory. We also clarified the difference between strictly-speaking slow-light and apparent slow light evidenced in saturable absorbers. It seems that the current analysis of the latter in terms of coherent population oscillations is not always relevant.

### Optical Precursors

Fast light and slow light are obtained with pulses, the envelope of which is slowly varying with regard to the optical period and characteristic times of the medium. In order to remove the apparent inconsistency between the existence of superluminal wave velocities and special relativity, Sommerfeld and Brillouin studied about one century ago the propagation in an optically very thick Lorentz-medium of a step-modulated sine-wave with a rise-time infinitely short compared to all the previous times.

We succeeded in obtaining *new analytical solutions* to this canonical problem when the propagation distance is such that the Sommerfeld and Brillouin forerunners are well apart (a condition met for the parameters considered by Brillouin). The Sommerfeld forerunner is then mainly determined by the dispersion effects and only depends on the order of the initial discontinuity of the incident field. On the other hand, the Brillouin forerunner originates in the sole effects of the frequency dependence of the medium absorption and has a *Gaussian shape*. This result may again be considered as a consequence of the central-limit-theorem.

In the double limit of weak susceptibility and narrow resonance, it is possible to *analytically* follow how the Sommerfeld and Brillouin forerunners evolve towards the unique precursor actually observed in optics and in the experiments of nuclear coherent forward scattering. The previous results apply when the field frequency is that of a medium absorption line but we also succeeded in obtaining *analytical expressions* of the transmitted signal (precursor and main field) when the pulse frequency lies in a transparency window, natural (between two absorption lines) or electromagnetically induced.

Precursors are obviously not specific to optics. In the microwave domain, the Debye media, opaque (transparent) at high (low) frequencies, enable one to study Brillouin precursors in favourable conditions. We have shown that they only appear at distances at which the medium impulse response is reduced again to a Gaussian. By simple convolution, we obtain then explicit analytical expressions for the output signals generated by various reference input signals and we show notably that, asymptotically, shape and amplitude of the precursor only depend on the integral properties of the incident signal but not on its precise shape.

Fast light, slow light and precursors can be obtained with the same system. We have studied in particular the propagation of a linearly polarized wave in a resonant Faraday medium submitted to a magnetic field parallel to the direction of light propagation. In the case of *adiabatic* incident pulses, we show that that this system enables one to obtain fast light in the polarization parallel to that of the incident field and slow light in the perpendicular polarization. Similarly, when the incident is *suddenly* switched on, precursor and main signal appear in the parallel and perpendicular polarizations, respectively.

In spite of the numerous studies on fast and slow light, some questions remain open. We think in particular to the analyses invoking the “coherent population oscillations” or the notion of “weak measurements”, to the fact that group delay and delay of the pulse maximum may be of opposite sign, etc. We will examine these questions.

### Publications

Works related to this thematic

**Contributors**

Bruno MACKE (Emeritus Professor)

Bernard SEGARD (Emeritus Professor)

**Collaborations**

Franck WIELONSKY, Laboratoire Painlevé, UMR 8524, Lille1 et CNRS.