The fine-structure splitting of the
4F state of atomic sodium is determined using the technique of two-photon spectroscopy with a resonant intermediate state. The sensitivity of this Doppler-free technique allows observation of the electric-dipole-forbidden $3S\ensuremath{\rightarrow}4F$ two-photon transition. The
4F fine-structure splitting is found to be 229\ifmmode\pm\else\textpm\fi{}4 MHz. Its sign and magnitude are hydrogenic, in complete contrast to the
D levels of Na.
We have demonstrated the feasibility of Doppler-free two-photon spectroscopy with a train of picosecond standing-wave light pulses from a synchronously pumped mode-locked cw dye laser. The actively controlled mode spectrum provides a means for accurate measurements of large frequency intervals. From a multipulse spectrum of the sodium $3s\ensuremath{-}4d$ transition we have determined a new value of the
4d fine-structure splitting, 1028\ifmmode\pm\else\textpm\fi{}0.4 MHz.