  Immense optical field enhancement was predicted to occur for the Berreman
mode in ultrathin films at frequencies in the vicinity of epsilon near zero
(ENZ). Here, we report the first experimental proof of this prediction in the
mid-infrared by probing the resonantly enhanced second harmonic generation
(SHG) at the longitudinal optic phonon frequency from a deeply
subwavelength-thin aluminum nitride (AlN) film. Employing a transfer matrix
formalism, we show that the field enhancement is completely localized inside
the AlN layer, revealing that the observed SHG signal of the Berreman mode is
solely generated in the AlN film. Our results demonstrate that ENZ Berreman
modes in intrinsically low-loss polar dielectric crystals constitute a
promising platform for nonlinear nanophotonic applications.
