Low-order modes of two-scale nonuniform waveguide

It is well known that the waves of the lowest orders in an nonuniform waveguide can be described by the parabolic equation if the following conditions are fulfilled: κ=kB≫1, v=B/L≫1, κv=σ=kB ²/L~1. Here k=2π/λ is the wave number, B is the characteristic width of the longitudinal nonuniformity. Here, it is implied the range D to be of the order of L. The authors derive an approximate description of lower-order modes of a two-scale nonuniform waveguide. They are submitted to a parabolic equation containing an additional term that gives an integral correction to the wave phase of the order 1/v at great distances X~D. This correction determined by the condition of removing secular (growing in X) terms in the asymptotic expansion can be evaluated by two-scale perturbation theory