As part of the research on thermal noise reduction in gravitational-wave detectors, we experimentally demonstrate the conversion of a fundamental TEM00 laser mode at 1064 nm to higher-order Hermite-Gaussian modes (HG) of arbitrary order via a commercially available liquid crystal spatial light modulator. We particularly studied the HG5,5/HG10,10/HG15,15 modes. A two-mirror plano-spherical cavity filters the higher-order modes spatially. We analyze the cleaned modes via a three-mirror diagnosis cavity and measure a mode purity of 96/93/78% and a conversion efficiency of 6.6%/3.7%/1.7%, respectively. A full set of simulations and mathematical proofs are also presented which shows that (i) Hermite-Gauss modes resonate in a two-mirror cavity provided mirrors are properly angled with respect to the impinging mode, and (ii) Hermite-Gauss modes resonate in triangular cavities. Hence, higher-order Hermite-Gauss modes are compatible with ground-based gravitational-wave detectors’ architecture and can be employed for the mitigation of mirror thermal noise for the third generation Einstein Telescope or Cosmic Explorer.