Metrology of Hydrogen Atom:. Determination of the Rydberg Constant and Lamb Shifts

We present the most recent optical frequency measurements in hydrogen and deuterium. From an analysis taking into account the scaling law of the Lamb shifts, we show that the optical frequency measurements have superseded the microwave determination of the 2S Lamb shift and we deduce optimized values for the Rydberg constant and for the 1S and 2S Lamb shifts. We report also the recent improvement of our 1S-3S experiment and the development of a new method to deduce the second order Doppler effect.