A Frequency comb comprises of mutually phase coherent pulse trains generated from a mode-locked pulsed laser, which translates to equally spaced narrow peaks in the frequency domain. Nowadays, frequency combs are considered essential in the field of frequency metrology. However, their spectral coverage is usually limited to the range extending from the visible to the infrared wavelengths due to the fundamental difficulty of developing stable mode-locked lasers in the ultraviolet region. To overcome this, we employ high-order harmonic generation (HHG) to transfer the spectral properties of an IR frequency comb to the extreme ultraviolet (XUV) wavelengths. This way, optical frequency metrology can be extended into unexplored wavelength regions of XUV. One of the important applications is high-precision spectroscopy of simple atomic targets. The energy levels of hydrogen-like atoms can be precisely described by bound-state quantum electrodynamics (QED). The 1S-2S transition of hydrogen-like He⁺ ion is an important XUV spectroscopy target for testing QED. Due to their charge, He⁺ ions can be held near-motionless in the field-free environment of a Paul trap, providing ideal conditions for high-precision measurement. Interesting higher-order QED corrections scale with large exponents of the nuclear charge, which makes this measurement much more sensitive to these corrections.

In this talk, we show our recent progress on (X)UV frequency comb generation and its applications. Our effort toward high-precision spectroscopy of the He⁺ 1S-2S transition will be presented. We also discuss future prospects to perform (X)UV frequency metrology with a compact and transportable setup.