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arXiv:0805.4240

A common model for superluminal propagation in absorbing atomic media and lossy metastructures

Pedro Chamorro‐Posada, Francisco Javier Fraile-Pelaez

Published May 28, 2008• 1 citation
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Abstract

We analyse the superluminal propagation of narrow-band pulses at resonances in dissipative media. The output waveform is an attenuated, undistorted, time-advanced version of the input which can be interpreted as the result of the interference of two scaled replicas of the input having a positive relative delay. The proposed analysis applies both to the propagation in a passive bulk medium at an electronic resonance and in a dielectric waveguide coupled to a lossy micro-ring resonator. This latter case provides a clear insight of the underlying physical phenomena.

DOI
10.48550/arxiv.0805.4240

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We analyse the superluminal propagation of narrow-band pulses at resonances in dissipative media. The output waveform is an attenuated, undistorted, time-advanced version of the input which can be interpreted as the result of the interference of two scaled replicas of the input having a positive relative delay. The proposed analysis applies both to the propagation in a passive bulk medium at an electronic resonance and in a dielectric waveguide coupled to a lossy micro-ring resonator. This latter case provides a clear insight of the underlying physical phenomena.
Pulse propagation through 1-dimensional stratified media is studied. The general features of pulse propagation in relation to the delay or advancement of the pulse are discussed. It is shown that in a lossless system with spatial symmetry the reflected and the transmitted pulses have the same delay or advancement. The phase times and the explicit output pulse profiles for an input Gaussian pulse are calculated. While recalling recent results for a variety of systems (resonant absorbers or heterogeneous medium in a Fabry-Perot cavity, a slab of negative index medium, a left-handed/right-handed periodic structure, etc.), new results involving coupled waveguides and a high finesse cavity containing an electromagnetically induced transparency (EIT) medium are presented. Calculations clearly show that for pulses with a carrier tuned at the resonance of the stratified media, one generally has subluminal propagation. Alternatively, in stop gaps or away from the resonances one ends up with superluminal transit. In the case of the EIT cavity it is shown that the large dispersion offered by the intracavity medium can lead to substantial superluminality (ng≃−106) with peak transmission as high as 20%. This is in sharp contrast to large signal attenuation for usual superluminal transit. Note also that EIT, to the best of the authors knowledge, has so far been used to achieve only slow light.
I derive a simple model for the propagation of a smooth pulse detuned outside a gain line. The pulse moves superluminally and with little distortion provided the medium is not longer than the saturation length. If the medium is longer than this, two effects occur: amplified spontaneous emission interferes with the pulse (although this interference averages to zero over many experiments) and the magnitude of the group delay increases only logarithmically with the length. I argue that both of these effects are inherent to any optical system with gain. Numerical simulations of the superluminal pulse agree with the model presented. \textcopyright{} 1996 The American Physical Society.
We present a simple electronic circuit which produces negative group delays for base-band pulses. When a band-limited pulse is applied as the input, a forwarded pulse appears at the output. The negative group delays in lumped systems share the same mechanism with the superluminal light propagation, which is recently demonstrated in an absorption-free, anomalous dispersive medium [Wang et al., Nature 406, 277 (2000)]. In this circuit, the advance time more than twenty percent of the pulse width can easily be achieved. The time constants, which can be in the order of seconds, is slow enough to be observed with the naked eye by looking at the lamps driven by the pulses.
The propagation of pulses through dispersive media was investigated by solving Maxwell's equations without any approximation. We show that the superluminal propagation of pulses through anomalous dispersive media is a result of the interference of different frequency components composed of the pulse. The coherence of the pulse plays an important role for the superluminal propagation. With the decrease of the coherence of the pulse, the propagation changes from superluminal to subluminal. We have shown that the anomalous dispersion (the real part of the susceptibility) not the amplification (the imaginary part of the susceptibility) plays the essential role in the superluminal propagation. Although the superluminality always exists as long as the spectrum of the coherent pulse is within the anomalously dispersive region, both the energy propagation velocity and the frontal velocity never exceed the light speed in the vacuum. The output pulse through the medium is not the original pulse; instead it carries the information of the original pulse and the information of the prepared medium.
A model-independent theorem demonstrates how a causal linear dielectric medium responds to the instantaneous spectrum, that is, the spectrum of the electric field pulse that is truncated at each new instant (as a given locale in the medium experiences the pulse). This process leads the medium to exchange energy with the front of a pulse differently than with the back as the instantaneous spectrum laps onto or off of nearby resonances. So-called superluminal pulse propagation in either absorbing or amplifying media as well as highly subluminal pulse propagation are understood qualitatively and quantitatively within this context.
The optical density of a resonant medium significantly affects the output signal in time resolved nonlinear optical interactions. The resonant interaction with a medium reshapes any propagating short pulse, and the mutual interaction of several such pulses, as in four-wave mixing, must be treated self-consistently. We present a theoretical framework for the proper handling of the propagation of all pulses, input as well as generated, in the small area limit. For optically thick resonant absorbers, negative time delay signals are observed, and the apparent decay rate of the induced polarization is faster than the rate observed for thin samples. We experimentally measure degenerate four-wave mixing in an atomic medium as an example, and demonstrate the quality of the theoretical model by the excellent fit to measured signals over several orders of magnitude. The improved understanding enables us to provide a simple, but surprisingly accurate, estimate for the apparent decay rate in homogeneously broadened optically thick media: If the absorption is given by \ensuremath{\alpha}, the propagation length is L, and the transverse relaxation time is T2{\mathit{T}}_{2}T2​, the apparent decay rate 2/Ta{\mathit{T}}_{\mathit{a}}Ta​ of a time resolved four-wave mixing signal is given by 2/Ta{\mathit{T}}_{\mathit{a}}Ta​=(2/T2{\mathit{T}}_{2}T2​)(1+\ensuremath{\alpha}L/2).