1. INTRODUCTION

Studies of high-density atomic vapor are important for understanding the processes underlying collisional broadening of spectral lines and thermal formation of molecules, phenomena exploited for numerous applications (e.g., laser radiation frequency converters [1], alkali vapor lasers [2,3], photoassociation of hot atoms [4], etc.). Significant information on the subject can be retrieved from the absorption spectra. Meanwhile, increase of density creates an optically thick transmission regime already at moderate temperatures. Alternatively, a selective reflection technique (see [5–8] and references therein) responsible for dispersive rather than absorptive properties of the vapor has been shown to be a powerful tool for characterization of a high-density atomic medium. But even this technique was not relevant to monitor the failure of the binary dipole–dipole interaction and onset of a multiparticle collision regime [9], and so more sophisticated methods, in particular, studies of the magneto-optical response of hot vapor [10], had to be implemented for revealing the signature of a “permanent collision regime.” Precise determination of dispersion, notably, on the far wings of atomic resonance, remains a challenge [11].

On the other hand, embedding the resonant atoms in a high-density buffer gas results in significant changes of the interaction regime, caused by trapping of the atoms inside the laser beam, thus increasing the interaction time [12]. Besides, frequent velocity-changing elastic collisions with buffer gas particles build up an averaged uniform velocity distribution, and, thus, homogeneous contribution from all the atoms in the interaction region, reducing the Doppler broadening and leading to the collisional narrowing of spectral lines (Dicke narrowing) [13]. These accompanying processes may have a significant impact on the relaxation rates and spectral linewidths of the recorded optical resonances.

Enhancement of group refractive index in resonant gaseous media resulting in light slowing was reported in different processes, such as quantum coherence effects [14]; electromagnetically induced transparency in a buffered cell [15]; Faraday effect in a high-density vapor [16]; enhanced nonlinear optical effects in coherently driven hot atomic gases [17]; and for specific conditions of interaction, such as in a gaseous atomic nanolayer [18], a dense cloud of cold atoms with size comparable to the wavelength of light [19], an optically thick atom-filled cavity [20].

In this paper, we propose a technique to retrieve the resonance dispersion curve (dependence of group refractive index on frequency) based on reflection of a frequency-scanned laser radiation from a vapor cell filled with a high-density atomic vapor, which serves as a low-finesse Fabry–Perot cavity. The group refractive index is derived from interference oscillations observable at the wings of resonance lines. A high precision of measurements allows studying interrelated atom–light and atom–atom interaction processes taking place in the high-density vapor. We also address the issue of enhanced Fabry–Perot cavity finesse for non-parallel windows, which is the manifestation of channeling (waveguiding) of radiation resulting from buffer gas-assisted confinement of excited atoms inside the laser beam.

2. EXPERIMENT

In this section, we present the experimental setup, measurement procedure, and experimental results obtained.

The experiment in the present work is done without special addition of a foreign gas to atomic vapor. Instead, we have used a vapor cell containing natural cesium, which contains some rubidium impurity. When heated, the metal vapor produces predominantly saturated cesium vapor, with some admixture of rubidium vapor with partial pressure corresponding to its abundance in the metal melt. We study reflection in the spectral region of Rb atomic $D_2$ line, while the Cs vapor serves as a high-density buffer gas. With this approach, the pressure (and density) of both resonant vapor and buffer gas are strongly dependent on temperature regime.

A schematic drawing of the experimental setup (a modified version of the setup used in [21]) is presented in Fig. 1. A linearly polarized $\oslash \approx 1.5\mathrm{mm}$ radiation beam of a single-frequency CW extended-cavity diode laser, Atrix Management S.A. ECDL-7850R (wavelength 780 nm, radiation power 20 mW, linewidth 1 MHz), was directed at nearly normal incidence onto a 15-mm-long all-sapphire sealed-off cell with a side-arm containing cesium. The power of the incident laser beam was attenuated to 0.45 mW to prevent the development of nonlinear effects. The cell was placed in a two-section oven, allowing individual temperature control of the side-arm ($T_{\mathrm{sa}}$, measured at the metallic cesium–vapor interface) and the temperature of the vapor column in the interaction region ($T_w$, determined as the temperature of the cell’s windows). Metallic pipes mounted at the optical outlets of the oven helped to prevent thermal convections causing beam walk-off and distortions at high operation temperatures (up to 450°C).

 

Fig. 1. Schematic drawing of the experimental setup (optical part).

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The radiation reflected from the cell was directed to a photodetector PD1 located 110 cm from the cell. It was possible to clearly distinguish all the four reflection beams from the four faces of the vapor cell at room temperature. A pinhole placed in front of the detector was used to pass solely the beam reflected from the inner surface of the front window, yielding the reflection signal contribution. Moreover, the pinhole helped to block the fluorescence signal from the cell (detection solid angle is 3 μsrad). The radiation transmitted through the cesium cell was monitored with a photodetector PD2.

Part of the laser beam was branched to pass through the second, room-temperature cell containing rubidium vapor. The latter allowed having transmission frequency references within the tuning range of the laser covering four spectrally resolved hyperfine transition groups of Rb atomic $D_2$ line, namely, ${}^{87}\mathrm{Rb}$ $F_g=2\to F_e=1,2,3$, ${}^{85}\mathrm{Rb}$ $F_g=3\to F_e=2,3,4$, ${}^{85}\mathrm{Rb}$ $F_g=2\to F_e=1,2,3$, and ${}^{87}\mathrm{Rb}$ $F_g=1\to F_e=0,1,2$ (in order of increasing frequency), recorded with photodetector PD3.

The number density of atomic ($N_{\mathrm{Cs}}$) and molecular ($N_{{\mathrm{Cs}}_2}$) saturated vapor was determined from the values of $T_{\mathrm{sa}}$ and $T_w$ measured by calibrated thermocouples using expressions for vapor pressure derived from the empirical formula presented in [22].

In our experiment, the reflected signal from a Fabry–Perot-type vapor cell in the region of $D_2$ line resonances of atomic rubidium was measured using a high-temperature cesium vapor cell. In spite of the high purity of the metallic Cs loaded into the cell’s side-arm, it contains trace amounts of other alkali metals, in particular, Rb.

The vapor pressure of residual Rb vapor can be derived from Raoult’s law [23] stating that

The number density of rubidium atoms $N_{\mathrm{Rb}}$ (in ${\mathrm{cm}}^{-3}$) can then be determined by

The abundance of residual Rb in the metal filled into the cell’s side-arm was derived from a comparison of the transmission spectra in the spectral region of Rb $D_2$ line with the transmission spectra of a pure Rb cell, and was estimated to be $x_{\mathrm{Rb}}\approx (2.7\pm 0.1)\times {10}^{-3}$.

In the present study, the maximum temperatures were $T_{\mathrm{sa}}=400°\mathrm{C}$ and $T_w=450°\mathrm{C}$, which correspond to $N_{\mathrm{Cs}}=3.03\times {10}^{17}{\mathrm{cm}}^{-3}$, $N_{{\mathrm{Cs}}_2}=4.59\times {10}^{15}{\mathrm{cm}}^{-3}$, and $N_{\mathrm{Rb}}=5.62\times {10}^{14}{\mathrm{cm}}^{-3}$. $T_w$ was always kept somewhat higher than $T_{\mathrm{sa}}$ to prevent condensation of vapor on the windows.

To study reflection in the spectral region of atomic Rb $D_2$ line, the laser wavelength was scanned around $\lambda =780\mathrm{nm}$. It should be noted that for this wavelength, one could also expect a contribution from the $X^1{\mathrm{\Sigma }}_g^{+}\to B^1{\mathrm{\Pi }}_u$ molecular absorption band of ${\mathrm{Cs}}_2$ (750–800 nm; see [21]); but in the temperature conditions of our experiment, this contribution was negligible because of a large number and a weak oscillator strength of the individual rovibronic transitions of the ${\mathrm{Cs}}_2$ dimer.

In order to record the reflection, transmission, and reference spectra, the laser radiation frequency was linearly scanned in a 20 GHz spectral region at a temporal rate allowing the establishment of a steady-state interaction regime. The frequency scanning was controlled with a computer using National Instruments USB-6211 DAQ card. The same DAQ card was employed also to simultaneously record the signals from all the detectors (6250 to 125,000 measurement points per scan, depending on the temporal rate).

A series of reflection spectrum measurements was performed by varying $T_{\mathrm{sa}}$ from 200°C to 380°C while keeping other experimental parameters constant. Typical reflection spectrum covering the $\approx 20\mathrm{GHz}$ mode-hop-free tuning range of laser radiation frequency around atomic Rb $D_2$ line is shown in Fig. 2 along with simultaneously recorded transmission and reference spectra.

 

Fig. 2. Reflection (upper trace) and transmission (middle trace) spectra in the region of $D_2$ line of residual Rb vapor recorded at $T_{\mathrm{sa}}=269°\mathrm{C}$ and $T_w=440°\mathrm{C}$ ($N_{\mathrm{Rb}}=2.02\times {10}^{13}{\mathrm{cm}}^{-3}$, $N_{\mathrm{Cs}}=1.30\times {10}^{16}{\mathrm{cm}}^{-3}$, $N_{{\mathrm{Cs}}_2}=9.24\times {10}^{12}{\mathrm{cm}}^{-3}$), and room temperature Rb reference spectrum (lower trace). $D_2$ line hyperfine transition groups ${}^{87}\mathrm{Rb}$ $F_g=2\to F_e=1,2,3$, ${}^{85}\mathrm{Rb}$ $F_g=3\to F_e=2,3,4$, ${}^{85}\mathrm{Rb}$ $F_g=2\to F_e=1,2,3$, and ${}^{87}\mathrm{Rb}$ $F_g=1\to F_e=0,1,2$ (left to right) are denoted by vertical lines.

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As can be seen from this figure, the reflection spectrum exhibits oscillating behavior with a nearly constant off-resonance amplitude. As the frequency approaches the hyperfine transition lines, the oscillation period sharply decreases, accompanied by an amplitude reduction caused by resonant absorption. The increase of vapor density results in the broadening of the absorption area and movement of the oscillations further away from the resonant lines.

The oscillating nature of the reflection signal has to be attributed to the low-finesse Fabry–Perot cavity behavior of the vapor cell. Tuning the laser radiation frequency across the resonance lines significantly changes the optical length of the cavity, $n_{g}L$ ($L$ is the cell length and $n_g$ is the group refractive index), because of strong dispersion, thus leading to the frequency dependence of the cavity’s free spectral range.

In Fig. 3, the dispersion curve (dependence of group refractive index, $n_g$, on laser radiation frequency) is plotted using the Fabry–Perot formula

 

Fig. 3. Dependence of group refractive index $n_g$ on frequency retrieved from experimental reflection spectra (dots) and calculated by Eqs. (4) and (11) (solid lines; see Section 3) in the region of $D_2$ line of residual Rb vapor for four temperature regimes: (a) $T_{\mathrm{sa}}=220°\mathrm{C}$, $T_w=350°\mathrm{C}$; (b) $T_{\mathrm{sa}}=260°\mathrm{C}$, $T_w=370°\mathrm{C}$; (c) $T_{\mathrm{sa}}=300°\mathrm{C}$, $T_w=400°\mathrm{C}$; (d) $T_{\mathrm{sa}}=350°\mathrm{C}$, $T_w=410°\mathrm{C}$. The zero detuning frequency is set to the $F_g=2\to F_e=1$ hyperfine transition of ${}^{87}\mathrm{Rb}$.

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As one can see from these graphs, the group refractive index, departing from the $n_g=1$ off-resonance value, increases sharply up to $n_g\approx 100$ moving along the line wings toward resonance frequencies. Absence of data points in the immediate vicinity of resonances is caused by the development of an optically thick layer with complete absorption, as can be seen from the middle trace in Fig. 2. Increase of vapor density results in a broadening of the total absorption area, thus moving the dispersion wings away from the resonances.

3. THEORETICAL MODEL

We restrict our theoretical calculations to a model of an idealized Fabry–Perot resonator. Propagation of the laser field is described by a self-consistent system of Maxwell equations and density matrix equations:

Correct boundary conditions for the medium polarization read as follows:

These conditions correspond to the assumption that atoms lose their polarization when colliding with the cell windows.

Taking into account the last relation, we can represent the Maxwell equation, in linear approximation with respect to the probe field, as follows:

In dilute media where the coupling parameter $q$ is small, the dispersion equation can be solved by a simple iteration procedure:

The functions $\tilde{\chi }(s_{1,2})$ in these expressions, and, consequently, the index $n(s_{1,2})$, are complex functions, but they do not bear physical meaning. Indeed, the medium susceptibility must be averaged over all atomic velocities (not only positive velocities). The symmetry of equations for the density matrix leads to the following expression: ${⟨\chi (x)⟩}_u={⟨\chi (x)⟩}_{u>0}+{⟨\chi (x)⟩}_{u<0}={⟨\chi (x)⟩}_{u>0}+{⟨\chi (-x)⟩}_{u>0}$. It is not difficult to show from expressions (10) that $\tilde{\chi }(s_1^{n-1})=\tilde{\chi }(-s_2^{n-1})$ and $\tilde{\chi }(-s_1^{n-1})=\tilde{\chi }(s_2^{n-1})$. Hence, the full complex refractive index $n$, which includes the absorption of medium, is

In the other limiting case, where the number density of atoms in the resonant medium is so high that the collision width exceeds the Doppler broadening, the function $\tilde{\chi }$ is independent of $s$ (quasi-stationary solution). Then we have again expression (11).

Note that the numbering of roots is chosen such that root $s_1$ corresponds to the wave traveling in the positive direction, whereas root $s_2$ corresponds to the wave arising in the resonant medium and propagating in the opposite direction.

Finally, we obtain for the reflection and transmission coefficients that, respectively,

Figure 4 shows the reflection and transmission spectra for $D_2$ line of residual Rb atomic vapor strongly buffered by Cs vapor, calculated with Eq. (12) for the parameters corresponding to the experimental conditions in Fig. 2, assuming natural abundance for rubidium (72.2% ${}^{85}\mathrm{Rb}$ and 27.8% ${}^{87}\mathrm{Rb}$). The numerical calculation was performed using four orders of iteration procedure. The homogeneous width caused by collisions of Rb atoms with buffer gas (Cs) particles was taken into account phenomenologically. The mean value of ${\mathrm{\Gamma }}_k$ homogeneous broadenings of the 12 individual $F_g\to F_e$ hyperfine transitions of ${}^{85}\mathrm{Rb}$ and ${}^{87}\mathrm{Rb}$, denoted by $\overline{\mathrm{\Gamma }}$, was set as a fitting parameter. The best match to the experimental reflection spectrum was obtained for $\overline{\mathrm{\Gamma }}\approx 2\pi \times 23.3\mathrm{MHz}$ and a Doppler broadening of $2\pi \times 690\mathrm{MHz}$ FWHM. As we see, the reflection spectrum obtained well reproduces the experimental one (see upper trace in Fig. 2).

 

Fig. 4. Reflection (upper graph) and transmission (lower graph) spectra in the region of $D_2$ line of Rb atoms calculated by Eq. (12) for the conditions in Fig. 2, with a fitting parameter of $\overline{\mathrm{\Gamma }}=2\pi \times 23.3\mathrm{MHz}\approx 146.7\mathrm{MHz}$. The zero detuning frequency is set to the $F_g=2\to F_e=1$ hyperfine transition of ${}^{87}\mathrm{Rb}{D}_2$ line.

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The frequency dependencies of the real and imaginary parts of refractive index calculated for the same conditions using Eq. (11) are shown in Fig. 5.

 

Fig. 5. Frequency dependencies of the real (upper graph) and imaginary (middle graph) parts of refractive index calculated by Eq. (11) in the region of $D_2$ line of Rb for the conditions in Fig. 2. The lower graph shows the corresponding group refractive index calculated by Eq. (4). The zero detuning frequency is set to the $F_g=2\to F_e=1$ transition of ${}^{87}\mathrm{Rb}$.

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4. DISCUSSION

Several interconnected physical effects occur when alkali metal atoms experiencing resonant interaction with laser radiation are embedded in the environment of a dense buffer gas. We will now discuss the physical processes underlying resonant light reflection from a high-pressure buffered vapor cavity.

Resonant atoms, being surrounded by closely located buffer gas particles, undergo rapid succession of elastic velocity-changing collisions, which keep them trapped in the laser beam, thus increasing the interaction time [12]. This prevents the atoms from spin-exchange collisions with the cell walls [26]. In the conditions of our experiment, the mean free path of the Rb atoms caged by Cs becomes smaller than the wavelength of the resonant radiation, and the lifetime of the excited atoms may exceed the time interval between the elastic collisions. Nevertheless, we do not observe any evidence of coherent Dicke narrowing. This result is not surprising knowing that in the optical domain, this narrowing is masked by prevailing buffer gas broadening, and can be revealed by only employing special configurations such as electromagnetically induced transparency [27] or bufferless nanocells [28].

Considering a vapor cell with a transparent dielectric windows as a low-finesse Fabry–Perot cavity, one may expect interference of the forward and backward beams in the reflection signal, with an oscillation frequency determined by the optical length of the cell. Meanwhile, noteworthily, the cavity effect visible in Fig. 2 is observed for the cell with windows with a significant tilt with respect to each other ($\approx 42\mathrm{mrad}$ in the present experiment), so that the interference Fabry–Perot behavior was completely washed out when the laser radiation frequency was tuned tens of GHz away from the Rb atomic resonances. This is clear evidence of the resonant nature of the observed resonator effect. It was possible to eliminate interference effects in reflection only by increasing the respective wedge angle of the windows to much higher values (160 mrad in [29]).

This unusual behavior has to be caused by a physical mechanism, which makes favorable the development of a strictly backward beam under conditions of non-normal incidence onto the cell window, thus enhancing the $Q$-factor of a Fabry–Perot resonator formed by a vapor cell with sapphire windows (off-resonance reflection per facet is $\approx 7.6\%$) serving as mirrors.

We believe that even if the cell windows are not perfectly parallel to each other, the reflected beam may seed a channeled backward beam propagating along the laser beam, waveguided by a high-refractive-index “fiber” structure (see Fig. 6). The waveguiding effect can be expected only if the excited resonant atoms, and, hence, the enhanced refractive index, remain sharply confined within the laser beam, entrapped by a buffer gas. Effectively, this process behaves as an increase of the $Q$-factor of the vapor-filled Fabry–Perot cavity.

 

Fig. 6. Buildup of channeled backward beam under non-normal incidence $\varphi \ne 0$, waveguided along a high-refractive-index laser beam. $I$, $T$, and $R$ represent incident, transmitted, and geometrically reflected beams, respectively; intense color—overlapping area of forward and backward beams.

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The issue of coherent backscattering enhancement in refracting media was addressed in [30]. The channeling effect can be affected by a concomitant process of radiation trapping [31,32], which becomes effective at a relatively high density of resonant atoms. In the radiation trapping regime, photons emitted by one atom get absorbed by a neighboring atom, thus keeping radiation imprisoned in the atomic medium for a long time until it escapes outside. However, the role of this effect is not yet clarified.

The frequent succession of absorption and re-emission by atoms along the laser beam may increase the effective radiative decay rate, thus leading to $Q$-factor enhancement because of buffer gas-assisted imprisonment of atoms resonantly interacting with the laser radiation inside the laser beam. Consequently, also the resonantly enhanced refractive index will remain confined within the laser beam, forming a fiber waveguide. For this reason, at relatively small incidence angles, the waveguiding dominates over the geometrical reflection law. Buildup of such a propagation mechanism may also have an impact on slowing of resonant light, resulting in a significant enhancement of the group refractive index, as can be seen from Fig. 3.

Comparing Figs. 2 and 4, one can see that the interference effect in the transmission spectra clearly visible on the resonance wings in the theoretical model is not observed in the experiment. A possible reason for this discrepancy is contrast damping in the experiment due to a large detection solid angle for the transmitted beam (900 μsrad versus 3 μsrad for the reflection), taking into account non-negligible divergence of the laser radiation beam ($\approx 1.2\mathrm{mrad}$).

Finally, the $\overline{\mathrm{\Gamma }}=2\pi \times 23.3\mathrm{MHz}$ collisional broadening of the Rb line in a dense Cs buffer environment, deduced from the fitting of the experimental spectrum of Fig. 2 by the theoretical model (see Fig. 4), is consistent with the known broadening rates. Although we could not find data on Rb–Cs inelastic (quenching) collisions in the literature, the known rates of non-resonant collisions for the Rb atomic $D_2$ line ($\sim 10–20\mathrm{MHz}/\mathrm{Torr}$ for noble buffer gases and $\sim 20–30\mathrm{MHz}/\mathrm{Torr}$ for molecular buffer gases [33]) are in agreement with the $22.3\mathrm{MHz}/\mathrm{Torr}$ value obtained in our study (the Cs vapor pressure under the conditions in Fig. 2 is 0.96 Torr).

5. CONCLUSIONS

In summary, the technique of reflection from a high-temperature spectroscopic cell serving as a low-finesse Fabry–Perot cavity filled with a dense atomic vapor, yielding frequency-dependent interference oscillations, was implemented to retrieve the dispersion curve in the spectral region of the hyperfine structure of Rb atomic $D_2$ line buffered by a high-density Cs vapor. Strong enhancement of group refractive index (up to $n_g\approx 100$) was obtained under certain temperature conditions. The experimental results are consistent with the results of numerical modeling.

It is deduced that trapping of excited atoms inside the laser beam by buffer gas atoms results in waveguided channeling of the reflection from the cell’s inner windows along the laser beam. Dominating over the geometrical reflection law at small incidence angles (up to $\sim 100\mathrm{mrad}$), this effect causes the enhancement of the effective quality factor of the Fabry–Perot cavity, thus increasing the robustness of the dispersion retrieval technique. Contributions from other interrelated processes taking place in the conditions of the present experiment (radiation trapping, Dicke narrowing, collisional cooling, etc.) are also discussed.

The technique presented can be used for dispersion measurements in dense buffered gases, quantitative studies of transition from a dipole–dipole binary to a multiparticle collisional regime with increase of vapor density, as well as for realization of optical effects in coherently driven hot atomic gases. The results obtained can find applications in the control of laser beam propagation in a dense resonant medium and the development of high-density alkali vapor lasers.