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arXiv:0802.0636 · Physical Review A

Slow light in saturable absorbers

B. Macke, Bernard Ségard

Published Jul 10, 2008• 29 citations
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Abstract

In connection with the experiments recently achieved on doped crystals, biological samples, doped optical fibers, and semiconductor heterostructures, we revisit the theory of the propagation of a pulse-modulated light in a saturable absorber. Explicit analytical expressions of the transmitted pulse are obtained, enabling us to determine the parameters optimizing the time-delay of the transmitted pulse with respect to the incident pulse. We finally compare the maximum fractional delay or figure of merit so attainable to those which have been actually demonstrated in the experiments.

DOI
10.1103/physreva.78.013817

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arXiv:physics/0601046•Physical Review A•Apr 5, 2006•14 citations
In connection with the experiments recently achieved on doped crystals, biological samples, doped optical fibers, and semiconductor heterostructures, we revisit the theory of the propagation of a pulse-modulated light in a saturable absorber. Explicit analytical expressions of the transmitted pulse are obtained, enabling us to determine the parameters optimizing the time-delay of the transmitted pulse with respect to the incident pulse. We finally compare the maximum fractional delay or figure of merit so attainable to those which have been actually demonstrated in the experiments.
There are two opposing models in the analysis of the slow transmission of light pulses through saturable absorbers. The canonical incoherent bleaching model simply explains the slow transmission by combined effects of saturation and of noninstantaneous response of the medium resulting in absorption of the front part of the incident pulse larger than that of its rear. The second model, referred to as the coherent-population-oscillations (CPO) model, considers light beams whose intensity is slightly pulse modulated and attributes the time delay of the transmitted pulse to a reduction of the group velocity. We point out some inconsistencies in the CPO model and show that the two models lie in reality on the same hypotheses, the equations derived in the duly rectified CPO model being local expressions of the integral equations obtained in the incoherent bleaching model. When intense pulses without background are used, the CPO model, based on linearized equations, breaks down. The incoherent bleaching model then predicts that the transmitted light should vanish when the intensity of the incident light is strictly zero. This point is confirmed by the experiments that we have performed on ruby with square-wave incident pulses and we show that the whole shape of the observed pulses agrees with that derived analytically by means of the incoherent bleaching model. We also determine in this model the corresponding evolution of the fluorescence light, which seems to have been evidenced in other experiments.
Two models are currently opposed to analyse the slow transmission of light pulses through sat-urable absorbers. The canonical incoherent bleaching model simply explains the slow transmission by combined effects of saturation and of non-instantaneous response of the medium resulting in absorption of the front part of the incident pulse larger than that of its rear. The second model, referred to as the coherent-population-oscillations (CPO) model, considers light beams whose intensity is slightly pulse modulated and attributes the time delay of the transmitted pulse to a reduction of the group velocity. We point out some inconsistencies in the CPO model and show that the two models lie in reality on the same hypotheses, the equations derived in the duly rectified CPO model being local expressions of the integral equations obtained in the incoherent bleaching model. For large modulation index, the CPO model, based on linearized equations, breaks down. The incoherent bleaching model then predicts that the transmitted light should vanish when the intensity of the incident light is strictly zero. This point is confirmed by the experiments that we have performed on ruby with square-wave incident pulses and we show that the whole shape of the observed pulses agrees with that derived analytically by means of the incoherent bleaching model. We also determine in this model the corresponding evolution of the fluorescence light, which seems to have been evidenced in other experiments.
A comparative analysis of two approaches to the description of the light modulation pulse in a saturable absorber is presented. According to the simplest model, the delay of the optical pulse is a result of the distortion of its shape due to the self-modulation of absorption in the nonlinear medium. The second model of the effect, arisen at the beginning of our century, relates the pulse delay to the so-called “slow light,” attributing this delay to the group velocity reduction under the condition of coherent oscillations of the population. It is shown that the data of all known experiments on the light pulse delay in saturable absorbers can be comprehensively described in terms of the simplest model of saturable absorber and do not require invoking the effect of population coherent oscillations with spectral hole-burning and anomalous modifications of the light group velocity. It is concluded that the effect of the group velocity reduction under the condition of population coherent oscillations has not received so far any experimental confirmation, and the assertions about real observation of the slow light based on this mechanism are unfounded.
Quantitative analysis of slow light experiments utilising coherent population oscillation (CPO) in a range of saturably absorbing media, including ruby and alexandrite, Er 3+ :Y 2 SiO 5 , bacteriorhodopsin, semiconductor quantum devices and erbium-doped optical fibres, shows that the observations may be more simply interpreted as saturable absorption phenomena. A basic two-level model of a saturable absorber displays all the effects normally associated with slow light, namely phase shift and modulation gain of the transmitted signal, hole burning in the modulation frequency spectrum and power broadening of the spectral hole, each arising from the finite response time of the non-linear absorption. Only where hole-burning in the optical spectrum is observed (using independent pump and probe beams), or pulse delays exceeding the limits set by saturable absorption are obtained, can reasonable confidence be placed in the observation of slow light in such experiments. Superluminal (“fast light”) phenomena in media with reverse saturable absorption (RSA) may be similarly explained.
We analytically study the linear propagation of arbitrarily shaped light pulses through an absorbing medium with a narrow transparency window or through a resonant amplifying medium. We point out that, under certain general conditions, the pulse acquires a nearly Gaussian shape, irrespective of its initial shape and of the spectral profile of the line. We explicitly derive in this case the pulse parameters, including its skewness, responsible for a deviation of the delay of the pulse maximum from the group delay. We illustrate our general results by analyzing the slow-light experiments having demonstrated the largest fractional pulse delays.