Quasisymmetric and asymmetric gap solitons in linearly coupled Bragg gratings with a phase shift

We introduce a model including two linearly coupled Bragg gratings, with a mismatch (phase shift theta) between them. The model may be realized as parallel-coupled fiber Bragg gratings (FBGs), or, in the spatial domain, as two parallel planar waveguides carrying diffraction gratings. The phase shift induced by a shear stress may be used to design a different type of FBG sensor. In the absence of the mismatch, the symmetry-breaking bifurcation of gap solitons (GSs) in this model was investigated before. Our objective is to study how mismatch theta affects families of symmetric and asymmetric GSs, and the bifurcation between them. We find that the system's band gap is always filled with solitons (for theta not equal to 0, the gap's width does not depend on coupling constant lambda if it exceeds some minimum value). The largest velocity of the moving soliton, cmax, is found as a function of theta and lambda (cmax grows with theta). The mismatch transforms symmetric GSs into quasisymmetric (QS) ones, in which the two components are not identical, but their peak powers and energies are equal. The mismatch also breaks the spatial symmetry of the GSs. The QS solitons are stable against symmetry-breaking perturbations as long as asymmetric (AS) solutions do not exist. If theta is small, AS solitons emerge from their QS counterparts through a supercritical bifurcation. However, the bifurcation may become subcritical at larger theta. The condition for the stability against oscillatory perturbations (unrelated to the symmetry breaking) is essentially the same as in the ordinary FBG model: both QS and AS solitons are stable if their intrinsic frequency is positive (i.e., in a half of the band gap).