A Coherent Frequency Comb in the Extreme Ultraviolet
b"In the course of this work, a system was designed and developed to nonlinearily convert a femtosecond frequency comb laser into the extreme ultraviolet (XUV) spectral range (120-30 nm).\\n\\nThe optical frequency comb, for which the nobel prize 2005 was awarded to John Hall and Theodor W. H\\xe4nsch, has become an indispensable tool for high precision spectroscopy. With the aid of a mode locked femtosecond laser it is possible to directly and phase coherently link the radio frequency domain and the frequency range of visible light. Today's most accurate time standard, the cesium atomic clock operates in the former and therefore it became possible for the first time to compare arbitrary optical frequencies with our primary time standard and measure them with 15 digits of accuracy. Among other things, this method allowed one of the most accurate test of quantum electrodynamics (QED) today in the course of the determination of the 1S-2S transition frequency of atomic hydrogen that is carried out in one of our labs. But also experiments in the field of ultrafast physics rely on the frequency comb technique to generate precisely controlled optical waveforms.\\n \\nAn especially intriguing possibility is to exploit the unique combination of high peak power in the megawatt range and the high spectral quality (on the order of 10^14) of single comb modes of a femtosecond frequency comb. To this end, in the method presented in this thesis, the femtosecond pulse train is coupled to an optical resonator of high finesse. With this trick, the field strength inside the resonator exceeds the driving lasers field by almost an order of magnitude. Enough to efficiently drive a nonlinear process of high order inside a medium of xenon atoms. As a result harmonics of the driving frequency comb up to 15\\\\nth order are generated. The obtained field contains photons with energies exceeding 20~eV, a spectral region which is not or only hard to access by conventional continuous laser source. Therefore the presented XUV frequency comb source brings direct frequency measurements at such high photon energies into the realm of possibility for the first time.\\n\\nIn particular, an improved version of the demonstrated source will be used to take the next step in an experiment with a long tradition in our group, the 1S-2S spectroscopy of atomic hydrogen. The generated frequency comb in the vicinity of 60~nm wave length will be used to probe the 1S-2S transition in singly charged helium, a hydrogen like system with larger nuclear charge. From such a measurement it can be expected that, compared to hydrogen, relativistic corrections from the QED theory become more important as the system has higher energies in general. For this reason this could lead to a test of QED with increased sensitivity.\\n\\nOther applications of such a compact and relatively simple coherent source of XUV radiation could be high resolution spectroscopy, XUV holography, but could also lie in the research area of ultrafast physics."