Field localization boosts nonlinear optical processes at the hot spots of
metal nanostructures. Fano resonances can further enhance these "local"
processes taking place at the hot spots. However, in conventional nonlinear
materials, the frequency conversion takes place along the entire crystal body.
That is, the conversion process is "unlocalized". The path interference (Fano
resonance) schemes developed for localized processes become useless in such
materials. Here, we develop Fano enhancement schemes for unlocalized nonlinear
optical processes. We show that 3 orders of magnitude Fano enhancement multiply
the enhancements achieved via field trapping techniques, e.g., in
epsilon-near-zero~(ENZ) materials. We demonstrate the phenomenon both
analytically and by numerical solutions of Maxwell's equations. The match
between the two solutions is impressive. We observe that the interference
scheme for unlocalized processes is richer than the one for the local
processes. The method can be employed for any kind of nonlinear optical
conversion. Moreover, the Fano enhancement can be continuously controlled by an
applied voltage.Comment: 10 pages, 5 figure
