Temperature sensitivity of the cavity scale factor enhancement for a Gaussian absorption resonance

We derive analytic expressions for the on-resonant cavity scale factor enhancement dependence on temperature, $S_0\left(T\right)$, for an intracavity medium with a Gaussian absorption resonance. Results are expressed as functions of the cavity parameters and the two resonance parameters: ${\alpha }_0\left(T\right)$, the peak absorption coefficient, and ${\mathrm{\Gamma }}_{\alpha }^R\left(T\right)$, the resonance width. A semiempirical model is developed for the temperature-dependent absorption coefficient, ${\alpha }_F(\mathrm{\Delta },T)$, in an alkali-metal-atom vapor cell, and is used to compare the predicted behavior of ${\alpha }_0\left(T\right)$ and ${\mathrm{\Gamma }}_{\alpha }^R\left(T\right)$ with the measured values for the $D{}_{2}F=2\to F^{\prime }$ resonance in ${}^{87}\mathrm{Rb}$, over the temperature range 298–325 K. Measurements of $S_0\left(T\right)$ in a low-finesse ring cavity, using the same vapor cell as the intracavity dispersive medium, were performed and found to be in agreement with the temperature-dependent behavior predicted by our theory, with quantitative agreement to 2 K for the critical temperature. The practical range of $S_0$ is found to be limited by the achievable temperature stability of the resonance parameters of the dispersive medium.

Figure 1

Ring cavity geometry for the general treatment of $S_0^{\Im }\left(T\right)$ and $S_0^{\Re }\left(T\right)$. The laser enters the input coupler (BS) from only one direction, restricting the ring cavity to unidirectional operation. The intracavity, temperature-stabilized rubidium vapor cell provides anomalous dispersion at its $D$-line resonances. The liquid crystal variable retarder (VR) permits tuning the cavity mode across the atomic resonance, by changing the optical path length in the cavity. We measure for each temperature the ${\mathrm{\Delta }}_p$ versus ${\delta }_p$ curve for the cavity mode $p$ closest to the atomic resonance frequency. In our experiment, the output coupler for the cavity is a high-reflectance mirror, $r_2=1$. Therefore, only the cavity reflectance $\left(\Re \right)$ modes are observed at the photodetector (PD). The remaining two mirrors (M) of the ring cavity are high-reflectance mirrors.

Figure 2

(a) Single-pass transmittance spectra, $\mathcal{T}(\mathrm{\Delta },T)$, through the rubidium vapor cell for three different temperatures: (i) 300 K, (ii) 311 K, and (iii) 323 K. The corresponding fits to $exp[-{\alpha }_{F=2}(\mathrm{\Delta },T){\ell }_m]$ are superimposed. The dip in transmittance at a detuning of +1100 MHz is due to absorption from ${}^{85}\mathrm{Rb}$, and this region has been excluded from the fit. (b) Absorption coefficient spectra, ${\alpha }_{F=2}(\mathrm{\Delta },T)$, corresponding to the fitted curves in (a). Note the asymmetry in the absorption curves, arising from the multiple hyperfine transitions contributing to the resonance.

Figure 3

(a) Peak absorption coefficient, ${\alpha }_0\left(T\right)$, and (b) resonance FWHM, ${\mathrm{\Gamma }}_{\alpha }^R\left(T\right)$, for the $D_{2}F=2\to F^{\prime }$ resonance. Points are experimental values while the lines are computed from the semiempirical model [Eq. (15)]. In (a), the solid line uses the vapor pressure parameters, $A=3.485$ and $B=-3805$, determined from measurements for our atomic vapor cell, while the dashed line uses the parameters from Ref. [5] $(A=4.312$ and $B=-4040)$ for the vapor pressure of liquid rubidium. In (b), the solid line shows the width computed from theory, with no adjustable parameters. The average error between the measurements and theory is 2.8%.

Figure 4

(a) Measurement of dispersive-cavity mode detuning $\left({\mathrm{\Delta }}_p\right)$ vs empty cavity-mode detuning $\left({\delta }_p\right)$ from the ${}^{87}\mathrm{Rb}{D}_{2}F=2\to F^{\prime }$ resonance at a vapor temperature of 314 K. The on-resonance scale factor enhancement, $S_0^{\Re }$, at this temperature is found by a linear fit to the data near resonance. FSR denotes free spectral range. (b) Comparison of the general analytical model prediction of $S_0^{\Re }$ vs $T$ for a Gaussian resonance (solid), computed using Eq. (28) and Eq. (15), with the measured values of $S_0^{\Re }$ (points), as in (a), at different temperatures. The dot-dash line is a fit to the data using the phenomenological expression, Eq. (32), which gives the measured critical temperature, $T_c=315.23\pm 0.18\mathrm{K}$. The dotted line is the high-finesse model prediction of $S_0\left(T\right)$, from Eq. (25). The general model predicts a critical temperature of $T_c=317.35\mathrm{K}$, and the high-finesse approximation predicts $T_c=325.21\mathrm{K}$. (c) Comparison of the phenomenological model of $S_0\left(T\right)$ (dot-dash line), Eq. (32) using $G=6.823$ and $T_c=317.35$ K, with the exact theoretical expression (solid line), Eq. (29).