Broadening and intensity redistribution in the Na($3p$) hyperfine   excitation spectra due to optical pumping in the weak excitation limit
  Detailed analysis of spectral line broadening and variations in relative intensities of hyperfine spectral components due to optical pumping is presented. Hyperfine levels of sodium $3p_{1/2}$ and $3p_{3/2}$ levels are selectively excited in a supersonic beam at various laser intensities under the conditions when optical pumping time is shorter than transit time of atoms through the laser beam. The excitation spectra exhibit significant line broadening at laser intensities well below the saturation intensity, and redistribution of intensities of hyperfine spectral components is observed, which in some cases is contradicting with intuitive expectations. Theoretical analysis of the dynamics of optical pumping shows that spectral line broadening depends sensitively on branching coefficient of the laser-driven transition. Analytical expressions for branching ratio dependent critical Rabi frequency and critical laser intensity are derived, which give the threshold for onset of noticeable line broadening by optical pumping. Transitions with larger and smaller branching coefficients are relatively less affected. The theoretical excitation spectra were calculated numerically by solving density matrix equations of motion using the split propagation technique, and they well reproduce the observed effects of line broadening and peak intensity variations. The calculations also show that presence of dark (i.e., not laser- coupled) Zeeeman sublevels in the lower state results in effective branching coefficients which vary with laser intensity and differ from those implied by the sum rules, and this can lead to peculiar changes in peak ratios of hyperfine components of the spectra. 
I. Sydoryk, N.N. Bezuglov, I.I. Beterov, K. Miculis, E. Saks, A.   Janovs, P. Spels, A. Ekers