Frequency-dependent polarization of light scattered near theNa D2resonance line

We report measurements of the degree of linear polarization of scattered light as a function of excitation frequency near the Doppler-broadened $\mathrm{Na} D2$ resonance line. For resonant excitation of the $D2$ line, the hyperfine interaction strongly depolarizes the emitted light due to precession of the electronic angular momentum $\overrightarrow{\mathrm{J}}$ prior to spontaneous emission. However, as the laser frequency is tuned away from resonance, the polarization degree sharply rises to the level predicted using a simple $J{M}_J$ basis, i.e., neglecting any coupling of the electronic angular momentum $\overrightarrow{\mathrm{J}}$ with the nuclear spin $\overrightarrow{\mathrm{I}}$. Roughly speaking, there is a transition from $F$ selection rules ($\overrightarrow{\mathrm{F}}=\overrightarrow{\mathrm{I}}+\overrightarrow{\mathrm{J}}$) for resonant excitation to $J$ selection rules for off-resonant excitation. We present a rigorous calculation of the polarization degree with the use of multipole moments of the density matrix which agrees quantitatively with our data and also predicts complicated interference structure when Doppler broadening is negligible. We show that this structure may shift peaks observed in precision laser-spectroscopy experiments. For Doppler-broadened transitions we discuss a simplified approach (based on an approximation to the Voigt profile) which treats separately the contributions of resonant and off-resonant scattering.