Influence of central wavelength of pump source and pump direction on transverse mode instability threshold of fiber laser

•Transverse Mode Instability threshold can be enhanced when the central wavelength of the pump source deviates from 976 nm.•TMI threshold of fiber lasers with bi-pumping structure is in turn greater than that of counter-pumping, co-pumping.•TMI threshold of bi-pumping structure can be greater, equal to, or less than the sum of co-pumping and counter-pumping.•The counter-pump injection power in the TMI threshold of bi-pumping is larger than the TMI threshold of counter-pumping.