On-resonance material fast light

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.

Figure 1

Comparison of the envelopes $e_1(0,t)$ (solid red line), $e_G(0,t)$ (dashed blue line), and $e_2(0,t)$ (dash-dotted green line) of the incident pulses for the same full width at half maximum ${\tau }_p$ of the corresponding intensity profiles. The thin lines are the periodic continuations of $e_1(0,t)$ (solid red line) and $e_2(0,t)$ (dash-dotted green line). (Inset) The corresponding intensity profiles ${\left|e_1(0,t)\right|}^2$ (solid red line) and ${\left|e_G(0,t)\right|}^2$ (dashed blue line). The intensity profile ${\left|e_2(0,t)\right|}^2$ (not shown) is quasiconfused with the Gaussian profile ${\left|e_G(0,t)\right|}^2$.

Figure 2

Normalized intensity profiles of the transmitted pulse obtained for $e(0,t)=e_1(0,t),L=5.756$ (transmission contrast $C=50$ dB), and ${\tau }_p=0.06/\gamma$ (${\mathrm{\Omega }}_1=38\gamma$). The solid red line (the dashed blue line) is the exact solution obtained by FFT (the analytical solution obtained in the short pulse limit). As in all the following figures, the intensity profile of the incident pulse (dotted black line) is given for reference. Its full width at half maximum ${\tau }_p$ is taken as time unit. (Inset) Comparison of the envelopes of the transmitted pulse obtained for $e(0,t)=e_1(0,t)$ (solid red line) and $e(0,t)=e_G(0,t)$ (dashed blue line). The pulse tail displays a first zero for $t/{\tau }_p\approx j_{1,1}^2/\left(4L\gamma {\tau }_p\right)\approx 10.6$ (see text).

Figure 3

Same as Fig. 2 when $L=1$ ($C\approx 8.7$ dB, dip depth $D\approx 86\%$), and ${\tau }_p=0.2/\gamma$ (${\mathrm{\Omega }}_1=11.4\gamma$). The tail of the envelopes (inset) has now an exponential decrease (see text).

Figure 4

Normalized intensity profiles of the transmitted pulse obtained for $e(0,t)=e_1(0,t),L=1$ ($C=8.7$ dB, dip depth $D\approx 86\%$), and ${\mathrm{\Omega }}_1=\gamma /4$ (${\tau }_p\approx 9.15/\gamma$). The solid red line (the dashed blue line) is the exact solution obtained by FFT (the analytical solution obtained in the long pulse or adiabatic limit). (Inset) Comparison of the envelopes of the transmitted pulse obtained for $e(0,t)=e_1(0,t)$ (solid red line) and $e(0,t)=e_G(0,t)$ (dashed blue line).

Figure 5

Same as Fig. 4 when $L=5.756$ ($C=50$ dB) instead of $L=1$. Note the presence of a precursor and a postcursor which had negligible amplitudes for $L=1$ and are obviously absent in the case of a Gaussian incident pulse (see inset).

Figure 6

Comparison for $L=1$ ($C=8.7$ dB, dip depth $D\approx 86\%$) of the exact intensity profile of the transmitted pulse (solid red line) with the analytical profile obtained by periodically continuing the incident pulse $e(0,t)=e_1(0,t)$ (thin blue line). The pulse duration is that maximizing the fractional advance (${\mathrm{\Omega }}_1=\gamma ,{\tau }_p\approx 2.29/\gamma$). It leads to $a=a_g/2$ and to $F=L/2\gamma {\tau }_p\approx 22\%$. (Inset) Comparison of the envelopes of the transmitted pulse obtained for $e(0,t)=e_1(0,t)$ (solid red line) and $e(0,t)=e_G(0,t)$ (dashed blue line).

Figure 7

Comparison for $L=3$ ($C=26$ dB) of the exact intensity profiles ${\left|e_2(l,t)\right|}^2$ (solid red line) and ${\left|e_1(l,t)\right|}^2$ (dashed blue line) with the analytical periodic profile ${\left|{\widehat{e}}_2(l,t)\right|}^2$ (thin blue line). The pulse duration (${\tau }_p\approx 2.55/\gamma$) is that maximizing the fractional advance of ${\left|{\widehat{e}}_2(l,t)\right|}^2$. (Inset) Comparison of the envelopes of the transmitted pulse obtained for $e(0,t)=e_2(0,t)$ (solid red line) and for $e(0,t)=e_G(0,t)$ (dashed blue line).

Figure 8

Comparison for $L=5.756$ ($C=50$ dB) of the exact intensity profiles s ${\left|e_2(l,t)\right|}^2$ (solid red line) with the analytical periodic profile ${\left|{\widehat{e}}_2(l,t)\right|}^2$ (thin blue line). The pulse duration is that maximizing the fractional advance, that is, ${\tau }_p\approx 3.18/\gamma$ leading to $F\approx 94\%$ and $a=0.52a_g$. (Inset) Comparison of the envelopes of the transmitted pulse obtained for $e(0,t)=e_2(0,t)$ (solid red line) and for $e(0,t)=e_G(0,t)$ (dashed blue line). Note that the slow light regime is attained for the Gaussian pulse (see text).

Figure 9

Same as Fig. 8 for ${\tau }_p\approx 7.5/\gamma$ instead of $3.18/\gamma$. The relative intensity of the $s$econd maximum is so reduced to $R=2\%$ whereas the fractional advance falls to $F\approx 67\%$. The inset shows that $e_2(0,t)$ and $e_G(0,t)$ both lead to fast light with very close pulse advances.

Figure 10

Effect on the transmitted intensity of a frequency detuning of the incident pulse for $e(0,t)=e_2(0,t)$. Parameters $L=3$, ($C=26$ dB), ${\tau }_p\approx 2.55/\gamma$ (as in Fig. 7), and, from top to bottom, $\mathrm{\Delta }/\gamma =\pm 0.6,\pm 0.5,\pm 0.3$ and 0. The solid red lines (the thin blue lines) are the exact solutions (the analytical periodic solutions).

Figure 11

Phase of transparent resonant media as a function of the detuning. The solid red line (the dashed blue line) is obtained for $\beta =1/2$ ($\beta =0$). The phase (the detuning) is expressed in units of $lnC$ ($\gamma$).