  Resonant modes in metamaterials have been widely utilized to amplify the
optical response of 2D materials for practical device applications. However,
the high loss at the resonant mode severely hinders metamaterial applications.
Here, we introduce a field-enhancement gain (FEG) factor to find the FEG mode
for significantly improving light-matter interaction. As a demonstration, we
experimentally compared the second harmonic generation enhancement of monolayer
MoS2 induced by the optimal FEG and resonant modes in hyperbolic
meta-structures. With the optimal FEG mode, we obtained an enhancement of
22145-fold and a conversion efficiency of 1.1*10-6 W-1, which are respectively
one and two orders of magnitude higher than that previously reported of
monolayer MoS2. A broadband high-FEG region over ~80 nm where the nonlinear
enhancement is larger than that induced by the resonant mode is achieved. The
concept of FEG factor is general to metamaterials, opening a new way for
advancing their applications.
