Hermite-Gaussian (HG) beams have many important applications in the optical frontier, and the limited output power of the high-purity HG beams is partly due to the small gain volume of the mode. The commonly used off-axis end-pumped scheme offers a narrow gain volume whose diameter is about a hundred microns. In this work, a new method of generating the HG beams based on a slab resonator that has a large mode volume is proposed and experimentally demonstrated. According to the optical resonator theory, the intra-cavity modes in thickness and width direction of the slab resonator are restricted by inserting two size-adjustable apertures, respectively. The one-dimensional HG beam generation is mainly guaranteed by the size of the aperture along the thickness direction of the slab, which matches the diameter of the fundamental mode. The different order one-dimensional HG beams are obtained by refined intra-cavity mode modulation. Since the higher-order modes are less sensitive to the misalignment of the cavity mirror than the lower-order modes, and the manipulation of the modes-loss at different orders is achieved by combining the tilt control of the coupled output mirror and the size control of intra-cavity apertures. By adjusting the optical gain and loss in the resonant cavity, the single mode wins the competition of laser modes. Therefore, high-purity one-dimensional HG beams with 0 to 9 orders (HG ₀₀to HG ₀₉) are generated. The pump module is comprised of a two-dimensional laser diode array which offers face-pumping to the large surface of the slab, therefore the width of the mode volume is extended to several millimeters. By further incorporating the 100mm-level long slab, the total gain volume is much larger than the counterpart in the off-axis pumping scheme. In this work, the output power of the highest order HG ₀₉mode increases up to 244 mW. Owing to the large gain volume and uniform gain distribution caused by the face-pumped slab, the purity of high order HG modes is quite good. The correlation coefficient $ \rho $ between the measured intensity distribution and the theoretical value is larger than 0.95. The beam quality factor $ {M}^{2} $ is also in good agreement with the theoretical one. Finally, a conversion from Hermite-Gaussian beams to the donut-shaped Laguerre-Gaussian beams is realized by using an astigmatic mode converter. Hopefully, power scaling of the HG beam output is also expected by employing cascaded slab amplifiers, and the approach in this paper provides a novel solution for generation of high power HG beams.