Beam dynamics inside a symmetrically chirped waveguide array mimicking graded index media

In this article, we explore the beam dynamics within symmetrically chirped nonlinear waveguide arrays, focusing on linear and quadratic chirping schemes. We propose a practical structure for these arrays that enhances control over light propagation. By employing a continuous approximation of the discrete nonlinear Schr"odinger equation, we utilize a semianalytical variational method to analyze beam behavior under waveguide chirping. Our findings indicate that the symmetrically chirped waveguide arrays behave similarly to graded index systems, with varying coupling coefficients analogous to the refractive index in continuous media. We derive a steady-state solution and validate it against numerical simulations, alongside conducting a linear stability analysis to assess the robustness of these solutions. The results reveal that input Gaussian beams in such waveguide arrays follow an oscillatory trajectory akin to that in parabolic index media. Notably, under nonlinear conditions, these beams evolve as discrete solitons. Our rigorous investigation of the propagation characteristics in both linear and nonlinear regimes highlights the intricate dynamics of optical beams within the engineered chirped waveguide arrays, supported by comparisons to comprehensive numerical simulations.