Pulsed field transmission by atomic frequency combs and random spike media: The prominent role of dispersion

A theory of pulse transmission is presented in which the medium through which the pulse propagates is characterized by an inhomogeneous distribution that is either an atomic frequency comb (AFC) or a medium of randomly spaced frequency spikes (RSM). For an AFC, we obtain analytic expressions for the transmitted field amplitude, which is composed of the (partially) transmitted incident pulse, plus a train of equally spaced echoes. For RSM, we derive expressions for the average transmitted field amplitude and field intensity. In the limit that the spike width is much less than the spike separation, normally encountered with AFC, the overall atom-field dynamics is dominated by dispersion—absorption plays a negligible role. The importance of the average group time delay of the pulse is discussed.

Figure 1

Inhomogeneous frequency distribution $g\left(\delta \right)$. The spikes or teeth are separated by ${\delta }_0$ and the spike width is ${\gamma }_s$. The dashed red curve is the frequency spectrum of the input pulse, whose temporal width is $T_p$. Not shown is the very broad inhomogeneous envelope of the frequency distribution, having width ${\gamma }_w\gg 1/T_p$.

Figure 2

Square root of the averaged transmitted intensity $\sqrt{〈T_r\left(\tau \right)〉}/F_0$ (red, solid curve) for a RSM as a function of $\tau$ for ${\tilde{\alpha }}_c=2, {\mathrm{\Gamma }}_s=0,$ and an average spike spacing ${\mathrm{\Delta }}_0=0.06$. The blue, dashed curve is the corresponding AFC result.

Figure 3

Transmitted AFC energy (blue, dashed curve) and average RSM transmitted energy (red, solid curve) for ${\tilde{\alpha }}_c=2, {\mathrm{\Gamma }}_s=0,$ and ${\mathrm{\Delta }}_0=0.06.$ The transmitted energy is normalized to the input field energy.

Figure 4

Exact transmitted field amplitude $F_G\left(1,\tau \right)/F_0$ (red, solid curves) and approximate (including only dispersion) transmitted field amplitude $F_G^{ap}\left(1,\tau \right)/F_0$ (blue, dashed curves) for the third echo are plotted for ${\tilde{\alpha }}_c=1, {\mathrm{\Delta }}_0=0.06$, and ${\mathrm{\Gamma }}_s=0.001$.

Figure 5

Same as Fig. 4, except that ${\mathrm{\Gamma }}_s=0.0005$.