  We study the 1S-3S two-photon transition of hydrogen in a thermal atomic
beam, using a homemade cw laser source at 205 nm. The experimental method is
described, leading in 2017 to the measurement of the 1S-3S transition frequency
in hydrogen atom with a relative uncertainty of $9 \times 10^{-13}$. This
result contributes to the "proton puzzle" resolution but is in disagreement
with the ones of some others experiments. We have recently improved our setup
with the aim of carrying out the same measurement in deuterium. With the
improved detection system, we have observed a broadened fluorescence signal,
superimposed on the narrow signal studied so far, and due to the stray
accumulation of atoms in the vacuum chamber. The possible resulting systematic
effect is discussed.
