Symmetric and asymmetric localized modes in linear lattices with an embedded pair of $\chi ^{(2)}$-nonlinear sites

We construct families of symmetric, antisymmetric, and asymmetric solitary modes in one-dimensional bichromatic lattices with the second-harmonic-generating ($\chi ^{(2)}$) nonlinearity concentrated at a pair of sites placed at distance $l$. The lattice can be built as an array of optical waveguides. Solutions are obtained in an implicit analytical form, which is made explicit in the case of adjacent nonlinear sites, $l=1$. The stability is analyzed through the computation of eigenvalues for small perturbations, and verified by direct simulations. In the cascading limit, which corresponds to large mismatch $q$, the system becomes tantamount to the recently studied single-component lattice with two embedded sites carrying the cubic nonlinearity. The modes undergo qualitative changes with the variation of $q$. In particular, at $l\geq 2$, the symmetry-breaking bifurcation (SBB), which creates asymmetric states from symmetric ones, is supercritical and subcritical for small and large values of $q$, respectively, while the bifurcation is always supercritical at $l=1$. In the experiment, the corresponding change of the phase transition between the second and first kinds may be implemented by varying the mismatch, via the wavelength of the input beam. The existence threshold (minimum total power) for the symmetric modes vanishes exactly at $q=0$, which suggests a possibility to create the solitary mode using low-power beams. The stability of solution families also changes with $q$.