Nonlinear magneto-optical rotation in optically thick media

Resonant nonlinear magneto-optical rotation (NMOR) [1, 2] has been the subject of extensive theoretical and experimental studies because it provides a very sensitive way to measure magnetic fields (see, e.g., Ref. [3]). While NMOR experiments are usually carried out with vapor samples of moderate optical thickness (no more than ∼ 2 absorption lengths), the intense research of recent years on electromagnetically-induced transparency (EIT; see, e.g., Ref. [4] for a review) has motivated investigation of optically thick media [5, 6]. Recently, NMOR in the vicinity of the D1 and D2 lines by an optically thick vapor of rubidium was studied [7– 11]. The authors of Ref. [8] used a 5 cm-long vapor cell containing Rb and a laser beam tuned to the maximum of the NMOR spectrum near the D1 line (laser power was 2.5 mW, beam diameter ∼ 2 mm). They measured maximum (with respect to laser frequency and magnetic field) polarization rotation φmax as a function of atomic density. It was found that φmax increases essentially linearly up to n ≈ 3.5 × 10 cm−3. At higher densities, dφmax/dn decreases and eventually becomes negative. The maximum observed rotation was ≈ 10 rad with the applied magnetic field of ≈ 0.6 G. (This shows the effect of power broadening on the magnetic field dependence; at low light power, the maximum rotation would occur at a magnetic field about an order of magnitude smaller.) Rotation slope dφ/dB|B=0, an important parameter for magnetometry, was also measured as a function of atomic density in Ref. [8]. The goal of the present contribution is to analyze the scaling of optimal magnetometric sensitivity with respect to optical thickness of the sample. We analyze NMOR for a simple system—an isolated J = 1 → J = 0 transition—for which analytical solutions for the density matrix are readily obtained. We assume that the transverse thickness of the sample is small, so that the trapping of spontaneously-emitted radiation can be neglected. (This assumption may not be justified in actual experiments with optically thick media [9].) We also neglect mix-