Adiabatic modes of a smoothly irregular waveguide: zero approximation of vector theory

The paper concerns the construction of specific asymptotic solutions of Maxwell's equations in a two-layer waveguide covered from above and below by semi-infinite subspaces. One of the layers is uniform in thickness, having a constant refraction index. The second layer is non-uniform with another constant refraction index. Tangential components of solutions are continuous across the boundaries and solutions are assumed to be bounded at infinity. The asymptotic solutions are constructed in an adiabatic approximation. The asymptotic parameter is small provided the coefficient of the phase slowness varies insignificantly on the distances of the order of wave length. The problem is, first, reduced to a coupled ODE for the longitudinal components of the electromagnetic field. Then the adiabatic expansion for the longitudinal components is substituted into the equations. In the leading approximation, simple equations for the components are derived which are supplemented by the corresponding boundary conditions. Non-trivial solutions of the problem (adiabatic modes) are constructed by integration of the equations. With the aid of the boundary conditions, a homogeneous linear system of equations for the unknown amplitudes is derived. The system has a nontrivial solution provided its determinant is zero. The authors consider an example of the construction of the modes for a thin film waveguide with the so-called Luneburg lens. The (nonlinear) equations for the phase slowness coefficient are obtained and numerically solved. The two-dimensional rays associated to the problem at hand are studied. The algorithm of calculation of the vertical distribution of the electromagnetic field in the leading adiabatic approximation is discussed for the given example.