Optical waveguide Hamiltonians leading to step-2 difference equations

We examine the evolution of an N-point signal produced and sensed at finite arrays of points transverse to a planar waveguide, within the framework of the finite quantization of geometric optics. In contradistinction to the common mechanical Hamiltonians (kinetic plus potential energy terms) the classical waveguide Hamiltonian is the square root of a difference of squares of the refractive index profile minus the optical momentum. The finitely quantized model requires the solution of the square eigenvalue and eigenfunction problem which leads to a step-two difference equation that contains two solutions and two signs of energy. We find the proper linear combinations to fit the Kravchuk functions of the finite oscillator model.