Shift and broadening in attenuated total reflection spectra of the hyperfine-structure-resolved ${\mathrm{D}}_2$ line of dense rubidium vapor

We have observed attenuated total reflection at a glass/rubidium vapor interface in high precision. The atom density was from $3.8\times {10}^{20}\text{to}3.7\times {10}^{21}{\mathrm{m}}^{-3}$ and the angle of incidence was from 48.4 to 64.8°. Redshift and broadening of the reflection spectra were observed. The redshift depended on atom density, angle of incidence, light polarization and hyperfine structure. The observed broadening was attributed to the self-broadening of the resonance line. Calculation based on the Fresnel formula with a local index for the susceptibility of the vapor, which included the Lorentz local field correction and the modification of the wave vector of the evanescent light by the optically dense atomic vapor, was found to be in reasonable agreement with the experimental results.

Figure 1

The experimental setup. DAC: digital-to-analog converter, FPI: Fabry-Pérot interferometer as a frequency marker, GTP: Glan-Thompson prism, PD: photo detector.

Figure 2

ATR spectra at ${\theta }_1=48.4°$ for (a) $s$ and (b) $p$ polarizations at the coldest temperature 453, 473, 493 and $513\mathrm{K}$ (the solid curves from top to bottom). The dotted curve is the absorption spectrum of the reference cell at room temperature. The vertical dashed lines show the peak positions of the absorption spectrum. (c) Energy level diagram of the Rb ${\mathrm{D}}_2$ line with hyperfine structure splitting.

Figure 3

ATR spectra at ${\theta }_1=48.4°$ at the coldest temperatures (a) 453 and (b) $513\mathrm{K}$. The dashed curve is the spectrum calculated with the wave vector of the evanescent light at the glass/vacuum interface and the dotted curve is the spectrum calculated with the wave vector modified by the atomic vapor. In both cases the Lorentz local field correction is included.

Figure 4

(a) The $z$ component of the wave vector of the evanescent light and (b) the refractive index of the atomic vapor for the evanescent light at ${\theta }_1=48.4°$ at the coldest point temperature $513\mathrm{K}$. The solid curves are the calculated results with $k_{2z}^{\left(j\right)}$ and the dotted curves are those with $k_{2z}^{\left(i\right)}$. In both cases the Lorentz local field correction is included.

Figure 5

Differences in the spectra between the experiment and the calculations for (a1) $s$ and (a2) $p$ polarizations at $N=3.8\times {10}^{20}{\mathrm{m}}^{-3}$. Those at $N=3.7\times {10}^{21}{\mathrm{m}}^{-3}$ for $s$ and $p$ polarizations are shown in (b1) and (b2), respectively. In each figure, $k_{2z}^{\left(i\right)}$ is used in the curves (1) and (2) while $k_{2z}^{\left(j\right)}$ is used in (3) and (4). The Lorentz local field correction is not included in the curves (1) and (3) while it is included in (2) and (4).

Figure 6

(a) Shift of the absorption peaks in the ATR spectra of ${}^{85}\mathrm{Rb}(F=3\to F^{\prime })$ (circles), ${}^{85}\mathrm{Rb}(F=2\to F^{\prime })$ (squares) and ${}^{87}\mathrm{Rb}(F=1\to F^{\prime })$ (triangles) transitions at ${\theta }_1=48.4°$ as a function of $N$ for (a1) $s$ and (a2) $p$ polarizations. (b) Shift of the absorption peak of the ${}^{85}\mathrm{Rb}(F=2\to F^{\prime })$ transition at $N=3.7\times {10}^{21}{\mathrm{m}}^{-3}$ as a function of ${\theta }_1$ for (b1) $s$ and (b2) $p$ polarizations. The experimental result is shown by closed symbols with a solid line. The calculated results with $k_{2z}^{\left(j\right)}$ and $k_{2z}^{\left(i\right)}$, both of which include the Lorentz local field correction, are shown by open symbols with solid and dashed lines, respectively. Open symbols with dotted and dotted-dashed lines show those with $k_{2z}^{\left(j\right)}$ and $k_{2z}^{\left(i\right)}$, respectively, neither of which include the Lorentz local field correction. The accuracy in determination of the peak positions is several MHz for both the experiment and calculations.

Figure 7

${\theta }_1$ dependence of the integrated absorbance obtained from the observed ATR spectra for $s$ polarization (closed squares) and $p$ polarization (closed circles). The thin curves show the calculated results with Eq. (10). ${\theta }_1$ dependence of the Goos-Hänchen shift is also shown by the thick curves. The solid and dashed curves show the results of $s$ and $p$ polarizations, respectively.

Figure 8

(a) Broadening as a function of ${\theta }_1$ at $N=9.1\times {10}^{20}$ (triangles), $2.0\times {10}^{21}$ (squares) and $3.7\times {10}^{21}$ (circles) ${\mathrm{m}}^{-3}$. The open and closed symbols are for $s$ and $p$ polarizations, respectively. (b) Broadening as a function of $N$. The solid line shows the result of the linear fit with the null intercept.