  Transverse Mode Instability (TMI) which results from dynamic nonlinear
thermo-optical scattering is the primary limitation to power scaling in
high-power fiber lasers and amplifiers. It has been proposed that TMI can be
suppressed by exciting multiple modes in a highly multimode fiber. We derive a
semi-analytic frequency-domain theory of the threshold for the onset of TMI
under arbitrary multimode input excitation for general fiber geometries. We
show that TMI results from exponential growth of noise in all the modes at
downshifted frequencies due to the thermo-optical coupling. The noise growth
rate in each mode is given by the sum of signal powers in various modes
weighted by pairwise thermo-optical coupling coefficients. We calculate
thermo-optical coupling coefficients for all $\sim$$10^4$ pairs of modes in a
standard circular multimode fiber and show that modes with large transverse
spatial frequency mismatch are weakly coupled resulting in a banded coupling
matrix. This short-range behavior is due to the diffusive nature of the heat
propagation which mediates the coupling and leads to a lower noise growth rate
upon multimode excitation compared to single mode, resulting in significant TMI
suppression. We find that the TMI threshold increases linearly with the number
of modes that are excited, leading to more than an order of magnitude increase
in the TMI threshold in a 82-mode fiber amplifier. Using our theory, we also
calculate TMI threshold in fibers with non-circular geometries upon multimode
excitation and show the linear scaling of TMI threshold to be a universal
property of different fibers.
