Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1005.0441 · Physical Review A

From fast to slow light in a resonantly driven absorbing medium

B. Macke, Bernard Ségard

Published Aug 23, 2010• 4 citations
arXivHTMLPDF

Abstract

We theoretically study the propagation through a resonant absorbing medium of a time-dependent perturbation modulating the amplitude of a continuous wave (cw). Modeling the medium as a two-level system and linearizing the Maxwell-Bloch equations for the perturbation, we establish an exact analytical expression of the transfer function relating the Fourier transforms of the incident and transmitted perturbations. It directly gives the gain and the phase shift undergone in the medium by a harmonic modulation. For the case of a pulse modulation, it enables us to determine the transmission time of the pulse center of mass (group delay), evidencing the relative contributions of the coherent and incoherent (population) relaxations. We show that the group delay has a negative value (fast light) fixed by the coherent effects when the cw intensity is small compared to the saturation intensity and becomes positive (slow light) when this intensity increases, before attaining a maximum that cannot exceed the population relaxation time. The analytical results are completed by numerical determinations of the shape of the transmitted pulses in the different regimes.

DOI
10.1103/physreva.82.023816

Related papers

Slow light in saturable absorbers : towards the end of a controversy?

B. Macke, Igor M. Razdobreev, Bernard Ségard

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.

•Apr 10, 2017•0 citations

Slow light in saturable absorbers: Progress in the resolution of a controversy

B. Macke, Igor M. Razdobreev, Bernard Ségard

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.

arXiv:1704.02820•Physical Review A•Jun 20, 2017•4 citations

On two models of light pulse delay in saturable absorber

Valerii S Zapasskii, G. G. Kozlov

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.

arXiv:1103.5868•Optics and Spectroscopy•Sep 1, 2010•7 citations

Slow light in saturable absorbers

B. Macke, Bernard Ségard

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.

arXiv:0802.0636•Physical Review A•Jul 10, 2008•29 citations

On-resonance material fast light

B. Macke, Bernard Ségard

We theoretically revisit the problem of the propagation of coherent light pulses through a linear medium when the carrier frequency of the pulses coincides with the minimum of a narrow dip in the medium transmission. Considering realistic contrasts between the maximum and minimum transmission of the medium and incident pulses of strictly finite duration, we combine temporal and spectral approaches to obtain analytical expressions of the transmitted pulse that reproduce the main features of the exact numerical solutions derived by fast Fourier transform. A special attention is paid to the advance of the pulse maximum over that of a pulse covering the same distance in vacuum and the ratio of this advance to the pulse duration (fractional advance) is optimized.

arXiv:1804.06227•Physical Review A•Jun 14, 2018•5 citations

Resonant self-focusing of periodically modulated laser beams

Vladimir Leonardovich Derbov, Inna L. Plastun

On-resonance self-action of a beam with periodically modulated amplitude in a two-level saturable absorber is studied numerically. When the period of modulation is comparable with relaxation times, one cannot describe the medium by means of susceptibility, and the response of the medium to the amplitude variation is not immediate. We solve the Bloch equations in each point of the medium to find this response correctly. Transient behavior followed by stable oscillations of population difference and polarization vector is shown to depend strongly on the frequency and amplitude of modulation. Beam propagation is considered using a simplified model of a thin absorber followed by free space, the propagation distance being small compared with the modulation wavelength. Manifestations of non-stationary response of the medium in the output intensity behavior are analyzed.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Jun 4, 2002•5 citations
We theoretically study the propagation through a resonant absorbing medium of a time-dependent perturbation modulating the amplitude of a continuous wave (cw). Modeling the medium as a two-level system and linearizing the Maxwell-Bloch equations for the perturbation, we establish an exact analytical expression of the transfer function relating the Fourier transforms of the incident and transmitted perturbations. It directly gives the gain and the phase shift undergone in the medium by a harmonic modulation. For the case of a pulse modulation, it enables us to determine the transmission time of the pulse center of mass (group delay), evidencing the relative contributions of the coherent and incoherent (population) relaxations. We show that the group delay has a negative value (fast light) fixed by the coherent effects when the cw intensity is small compared to the saturation intensity and becomes positive (slow light) when this intensity increases, before attaining a maximum that cannot exceed the population relaxation time. The analytical results are completed by numerical determinations of the shape of the transmitted pulses in the different regimes.
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.
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.
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.
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.
We theoretically revisit the problem of the propagation of coherent light pulses through a linear medium when the carrier frequency of the pulses coincides with the minimum of a narrow dip in the medium transmission. Considering realistic contrasts between the maximum and minimum transmission of the medium and incident pulses of strictly finite duration, we combine temporal and spectral approaches to obtain analytical expressions of the transmitted pulse that reproduce the main features of the exact numerical solutions derived by fast Fourier transform. A special attention is paid to the advance of the pulse maximum over that of a pulse covering the same distance in vacuum and the ratio of this advance to the pulse duration (fractional advance) is optimized.
On-resonance self-action of a beam with periodically modulated amplitude in a two-level saturable absorber is studied numerically. When the period of modulation is comparable with relaxation times, one cannot describe the medium by means of susceptibility, and the response of the medium to the amplitude variation is not immediate. We solve the Bloch equations in each point of the medium to find this response correctly. Transient behavior followed by stable oscillations of population difference and polarization vector is shown to depend strongly on the frequency and amplitude of modulation. Beam propagation is considered using a simplified model of a thin absorber followed by free space, the propagation distance being small compared with the modulation wavelength. Manifestations of non-stationary response of the medium in the output intensity behavior are analyzed.