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Journal of the Optical Society of America A

Role of the instantaneous spectrum on pulse propagation in causal linear dielectrics

Justin B. Peatross, Michael J. Ware, Scott Alan Glasgow

Published Jul 1, 2001• 21 citations
Publisher

Abstract

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.

DOI
10.1364/josaa.18.001719

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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.
We examine the energy exchanged between an electromagnetic pulse and a linear dielectric medium in which it propagates. While group velocity indicates the presence of field energy (the locus of which can move with arbitrary speed), the velocity of energy transport maintains strict luminality. This indicates that the medium treats the leading and trailing portions of the pulse differently. The principle of causality requires the medium to respond to the instantaneous spectrum, the spectrum of the pulse truncated at each new instant as a given locale in the medium experiences the pulse.
When a dielectric medium has a finite temperature, the atoms that constitute the medium are in a state of incessant motion. Propagation of a light pulse through such a medium is discussed theoretically in this paper in the classical domain. When the pulse is in resonance with the medium, it is found that the medium becomes dispersiveless when its temperature reaches a critical value, and that as the medium's temperature increases from below to above the critical value, the pulse's propagation changes from superluminal (subluminal) to subluminal (superluminal).
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.
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.
A new context for the group delay function (valid for pulses of arbitrary bandwidth) is presented for electromagnetic pulses propagating in a uniform linear dielectric medium. The traditional formulation of group velocity is recovered by taking a narrowband limit of this generalized context. The arrival time of a light pulse at a point in space is defined using a time expectation integral over the Poynting vector. The delay between pulse arrival times at two distinct points consists of two parts: a spectral superposition of group delays and a delay due to spectral reshaping via absorption or amplification. The use of the new context is illustrated for pulses propagating both superluminally and subluminally. The inevitable transition to subluminal behavior for any initially superluminal pulse is also demonstrated.
The propagation of pulses through dispersive media was investigated by solving Maxwell's equations without any approximation. It was demonstrated that the superluminal or subluminal pulse propagation originates essentially from the interference of the different frequency components of the pulse. The coherence of the different frequency components plays an important role in superluminal propagation. When the pulse becomes partially coherent, the propagation changes from superluminal to subluminal. Since superluminality always exists as long as the spectrum of the coherent pulse is within the anomalously dispersive region whatever the shape of the pulse is, such a coherent pulse must be transmitted superluminally.