An analytical analysis of the nonlinear modes of the coupled silicon-on-insulator waveguides

In the present work we analyze the nonlinear modes of silicon-on-insulator (SOI) nanowires and supermodes of the coupled SOI waveguides. A generalized analysis of the nonlinear modes of silicon nanowires is given where we have considered the scalar approximation and its vectorial nature to obtain the analytical profiles. In the scalar approximation, the analytical analysis of the profiles of the transversal modes is based on the solutions of the Helmholtz equation for nonlinear periodic media, where we obtain an integral solution for the intensity which is identified with the help of the elliptic functions. Those modes are characterized by two constants of motion of particular physical significance and in some approximations the solution could become a soliton or cosenoidal type. Therefore, we describe the solutions on terms of the movement and integration constants. This is an important result because defines the nature of the solutions, therein the analysis of the third order polynomials roots of those elliptic functions. The general theoretical model includes the two-photon absorption (TPA) and the nonlinear Kerr effect implicit in the refraction index.