In the past decades, fibre laser and amplifier systems have seen a rapid power increase due to the efficient heat dissipation and robust single-mode operation. The fibre lasers and amplifiers have become attractive sources for applications spanning from industry and defence to scientific research. In particular, the fibre lasers and amplifiers have penetrated for material processing industry including marking, cutting, welding and drilling. It had been known that the highest achievable output power in fibre laser and amplifier systems is largely limited by the nonlinear scatterings such as stimulated Brillouin scattering (SBS) and stimulated Raman scattering (SRS). However, a new phenomenon under high power operation, the so-called transverse mode instability (TMI), was experimentally observed in a large mode area (LMA) ytterbium (Yb)-doped fibre amplifier in 2011. The TMI can occur much earlier than the SBS and SRS in average power scaling, hence becoming the first limitation in high-power fibre operations till now. The TMI refers to the output beam degradation, caused by the power transfer from the fundamental mode (FM) to the higher-order mode (HOM), for input pump power above a certain threshold. The origin of TMI is attributed to the stimulated thermal Rayleigh scattering x