  The nonlinear optical response of materials is the foundation upon which
applications such as frequency conversion, all-optical signal processing,
molecular spectroscopy, and nonlinear microscopy are built. However, the
utility of all such parametric nonlinear optical processes is hampered by
phase-matching requirements. Quasi-phase-matching, birefringent phase matching,
and higher-order-mode phase matching have all been developed to address this
constraint, but the methods demonstrated to date suffer from the inconvenience
of only being phase-matched for a single, specific arrangement of beams,
typically co-propagating, resulting in cumbersome experimental configurations
and large footprints for integrated devices. Here, we experimentally
demonstrate that these phase-matching requirements may be satisfied in a
parametric nonlinear optical process for multiple, if not all, configurations
of input and output beams when using low-index media. Our measurement
constitutes the first experimental observation of direction-independent phase
matching for a medium sufficiently long for phase matching concerns to be
relevant. We demonstrate four-wave mixing from spectrally distinct co- and
counter-propagating pump and probe beams, the backward-generation of a
nonlinear signal, and excitation by an out-of-plane probe beam. These results
explicitly show that the unique properties of low-index media relax traditional
phase-matching constraints, which can be exploited to facilitate nonlinear
interactions and miniaturize nonlinear devices, thus adding to the established
exceptional properties of low-index materials.
