Non-Linear Modes in Lithium Niobate Ferroelectrics with niobium antisite defects

Ferroelectric belongs to an important class of materials showing very interesting nonlinear optical properties that have a variety of application in photonic devices In a discrete Hamiltonian, if we consider the space and time dependence of polarization vectors with an interaction term between two polarization domains, it gives rise to a nonlinear Klein-Gordon (K-G) equation that has been shown in some systems [1]. This equation as a governing equation enables multiple time-scale analysis (MTSA) to be performed on a real ferroelectric material that reveals new type of information on both linear (known as soft modes) and the nonlinear parts of frequencies and amplitudes of oscillations. MTSA was described in a K-G system based on staggered polarization where more emphasis was given to reveal the presence of intrinsic localized modes (ILM). Here, detailed results for the first three linear modes and their frequencies as a 2 function of a wide range of niobium antisite defects or, impurities in lithium niobate ferroelectric are presented. It is found that when an external excitation (i.e. the input) is only harmonic in nature, it could not be ascertained whether the even modes of the solitonic waves (i.e. the output) will have phase-locking with this type of harmonic external force.