Unveiling Quasiperiodicity through Nonlinear Wave Mixing in Periodic Media

Quasiperiodicity is the concept of order without translation symmetry. The discovery of quasiperiodic order in natural materials transformed the way scientists examine and define ordered structure. We show and verify experimentally that quasiperiodicity can be observed by scattering processes from a periodic structure, provided the interaction area is of finite width. This is made through a momentum conservation condition, physically realizing a geometrical method used to model quasiperiodic structures by projecting a periodic structure of a higher dimension. The outcome of scattering processes by periodic media is diffraction —the generation of new waves at particular directions. There are numerous different kinds of such processes, e.g., electromagnetic optical waves diffracted from gratings, electrons diffracted from atomic crystals, etc. In all of these processes the condition for efficient generation of the scattered wave (diffraction condition) can be formulated in the form of momentum conservation: it needs to be satisfied only up to a reciprocal lattice vector (RLV) of the structure. Each such RLV can give rise to a different diffraction condition. We show that for a restricted-width interaction in which all beams propagate in the same direction within a periodic structure, momentum conservation can also be satisfied up to a projection of an RLV onto the direction of propagation. The real space description of this phenomenon entails projecting part of the structure defining lattice onto a line, but such a projection of a periodic lattice onto a subspace is a well-known scheme for the creation of quasicrystal models [1‐3] (ordered but not periodic structures). As such, for certain propagation directions, the set of processes for which momentum conservation is satisfied exhibit quasiperiodic relations even though they take place within a periodic structure. We experimentally demonstrate this phenomenon using nonlinear wave mixing in a material with planar periodic modulation of � � 2� —its second-order nonlinear susceptibility, but it should be observable for any scattering process in which momentum conservation can be satisfied up to an RLV of some periodic lattice.