1：Experimental Design and Method Selection:

Utilized an internally-developed, all-fibre 200 W-class multimode 1907 nm pump laser. Five 45 W single mode all-fiber FBG-locked Tm lasers are multiplexed in a tapered fibre combiner to provide > 200 W in a 105/125 μm

2：22 NA pump delivery fibre. This fiber is spliced onto a low-loss taper that transfers the pump power into the glass guidance of a capillary fibre. The laser cavity is formed around an end-pumped 5 at. % doped Ho:

YAG crystal of 5 mm diameter and 40 mm length, and between the pump input dichroic (laser high-reflector) and a 10% laser output coupler. The annular pump light is relay-imaged from the capillary facet into the Ho:YAG crystal by f =

3：5 and 200 mm lenses, resulting in a 6 mm beam diameter in the crystal. An intra-cavity f = 50 mm lens is utilised to control the size of the cavity modes, and therefore their overlap with the spatially-distributed gain, allowing selection of preferred laser modes. Sample Selection and Data Sources:

Ho:YAG crystal of 5 mm diameter and 40 mm length.

4：List of Experimental Equipment and Materials:

All-fibre 200 W-class multimode 1907 nm pump laser, five 45 W single mode all-fiber FBG-locked Tm lasers, tapered fibre combiner, 105/125 μm

5：22 NA pump delivery fibre, capillary fibre (200μm diameter & 100μm hole with 220μm fluorine-doped 22NA glass cladding), Ho:

YAG crystal, pump input dichroic, 10% laser output coupler, f =

6：5 and 200 mm lenses, intra-cavity f = 50 mm lens. Experimental Procedures and Operational Workflow:

The pump light is relay-imaged into the Ho:YAG crystal, creating a ring-shaped inversion profile that gives preferential gain to the set of LG0n modes. The intra-cavity lens controls the size of the cavity modes for mode selection.

7：Data Analysis Methods:

Output power and polarisation extinction ratio were measured.