Radiation from a cut-off point in a two layer nonlinear TE mode waveguide

In this work, the propagation of a nonlinear transverse electric (TE) mode in an optical two layer waveguide is considered for the case in which the layers are slowly varying. For a semi-infinite straight boundary between the layers, it is known that trapped modes exist which travel close to the interface. In the present work the upper layer light channel is taken to be of finite extent, while the lower layer is taken to be semi-infinite. The lateral stratification causes trapped modes to cut-off, so that energy is then beamed into the lower layer. In the present work a canonical nonlinear Schrodinger (NLS) equation is obtained which describes, together with an appropriate boundary condition, the radiation beamed into the lower light channel (material layer). It is found from numerical solutions that the radiating mode in the lower layer propagates as a soliton. Approximate solutions for this radiation are found using two methods. The first assumes that the radiating mode is a soliton whose amplitude and width are constant, but whose velocity can vary. The equation governing the soliton velocity is derived using conservation of energy. The second method allows the amplitude, width and velocity all to vary and the equations governing these parameters are obtained from an averaged Lagrangian for the NLS equation. Solutions obtained from the second approximate method are in much better agreement with numerical solutions since the amplitude of the soliton undergoes significant variation in the lower layer (light channel). Since the equation is canonical, it is apparent that nonlinearity induces coherent propagation in the wave radiated into the lower layer (light channel). (C) 2003 Elsevier Science B.V. All rights reserved.