Light Localization in Nonline- ar Binary Two-Dimensional Lieb Lattices

Abstract

Light localization in photonic lattices (PLs) is a well- known phenomenon which has been in-vestigated dur- ing decades. It has been shown that light localization in the linear regime can be achieved by designing PLs with specifi c geometries, instead of embedding defects or disorder in otherwise periodic lattices [1]. Th ese ge- ometries provide conditions necessary for destructive wave interference, leading to formation of a perfectly fl at (dispersionless) energy band. Eigenvectors as-soci- ated to the fl at-band (FB) eigenfrequencies are entirely degenerate and compact states (FB modes) and any su- perposition of them is nondiff racting. One of the sim- plest FB lattice patterns is the two-dimensional (2D) Lieb lattice [2,3] in which the primitive cell contains three sites. By ap-propriate spatial repetition of this fun- damental block, it is possible to achieve a FB in the en- ergy spectrum. Light confi nement in PLs can also be a consequence of the interplay between nonlinearity and diff raction when these eff ects cancel each other, leading to formation of solitons. Recently, it has been reported that nonlinearity and “binarism” in quasi-one-dimen- sional FB systems can in-crease the range of existence and stability of FB ring modes [4]. We model a 2D binary Lieb lattice with nonlinearity of Kerr type and analyse numerically and analytically the existence, stability and dynamical properties of various localized modes found to emerge in spectrum. From the derived dispersion relation we found that binarism does not aff ect the FB. However, due to the presence of additional periodicity, new gaps occur in the ener- gy spec-trum above and below the FB and their widths depend on the ratio between coupling constants. Like in the uniform Lieb lattice, we found eigenmodes in the form of a staggered four-peak “ring” structure, but only under certain conditions which require a particular re- lation between the fi eld am-plitudes in neighbouring sites. In the nonlinear regime, ring modes survive in the uniform Lieb lat-tice but lose their stability moving away from the FB. On the other hand, nonlinearity de- stroys the existence of ring solutions in the binary Lieb lattice, leading to a new class of stable localized solu- tions which can be found in minigaps. As in previous kagome and ladder binary nonlinear strips [4], it is shown that the binarism increases the existence range of stable nonlinear localized solutions.