Abstract: Considering conversion to the third harmonic (TH) wavelength, present concepts are based on two second-order nonlinear processes involving two conversion crystals. Even though corresponding modules are commercially available, there is a demand for simpler solutions. One possibility is direct third-harmonic generation (THG) in a third-order process [1]. However, compared to second-order processes, phase matching imposes a certain challenge here, because the refractive index differences between the fundamental and third harmonics are significantly larger for typical conversion materials. The requirement of phase matching can be met using stacks of dielectric coatings, so-called frequency tripling mirrors (FTM) [2,3]. To generate the third harmonic, materials with low absorption for both, the fundamental and the TH wavelength have to be employed. From numerical calculations for typical resonant layer designs, an upper limit in the low 10{\^}(-6) range for the absorption at the fundamental wavelength can be deduced. In addition, one of the materials must have a high third order susceptibility χ{\^}((3)) to enable efficient conversion. Since near the band gap of a material its χ{\^}((3)) increases strongly [4], the material should be chosen depending on the laser wavelength used. In the present study a Ti:Sa laser with a wavelength of approximately 775 nm was considered as a source for the fundamental wavelength. Accordingly, the TH of this system is 258 nm. A suitable material for THG of 775 nm is HfAlO, since the band gap of this material is not far below the TH wavelength. Like HfO2, HfAlO has the property that its refractive index as well as its absorption strongly depends on the manufacturing parameters. In a parameter study, various single layers of HfAlO were produced with different oxygen additions and different aluminium contents, some with substrate heating and some without. Subsequently, the 1-T-R values at the wavelength of the TH were determined with the help of a VUV spectrometer and coating parameters with almost disappearing absorption were identified. Based on these results, FTMs were produced. We are optimistic that with these improved coating parameters, we are able to or already have produced FTMs with a conversion efficiency of more than 3\%. The measurements are still pending. Our best FTM produced so far, prior to the parameter study, delivered a conversion efficiency of 1.8\%.