Square-Gradient Mechanism of Surface Scattering in Waveguides with Random Rough Surfaces

Abstract

In spite of the extensive research devoted to the subject of wave scattering in surface-random-corrugated guiding systems (see, e.g., [1,2]), there is a number of open questions to be resolved. Experimental and numerical works have revealed transport properties of such waveguides which challenge the possibility of an analytical treatment [3]. Thus, it is important to analyze what are specific mechanisms ofthe surface scattering.The goal of this contribution is to study competing mechanisms of surface scattering and their manifestation in the wave attenuation length Ln (also known as the scattering length or the total mean-free-path corresponding to specific nth propagating mode) of multimode waveguides or electron conducting wires. We focus our attention on quasi-i D waveguides in which the total number Nd of propagating modes (or conducting channels) is large Nd >> 1 The roughness of the lower and upper boundaries of such waveguides are described, respectively, by the random functions 5, (x) and $, (x). The relationship between those functions defines the waveguide's profile or configuration. Here four profiles are of our particular interest:The waveguide with one rough boundary, ;, (x) = 0The waveguide with the uncorrelated boundaries, 5, (x) and 4 (x).The waveguide with the antisymmetric boundaries, J (x) = T (x) = 5(X)The waveguide with symmetric boundaries, -(x)= t (x) = 4X) We assume that due to the multiple scattering of a traveling wave from the rough boundaries, the longitudinal wave number of an nth propagating mode can be written as kn + kn a where kn is its unperturbed value and in = Yn + i(2L ) *.(1)The real part Y,, is responsible for a roughness-induced correction to the phase velocity of a given mode. As is known, the shift Yn does not change the transport properties of a disordered system. Our model is the open waveguide of the average width d, stretched along the x axis. The lower and upper surfaces of the waveguide are assumed to be described, respectively, by the boundaries z = ur, (x), and z = d + 4~( x). Here o-is the root-mean-square roughness height that is assumed to be identical for both boundaries. In other words, the waveguide occupies the region -oo < x so a:; (x)< z < d + or, (x) of the (x, z) -plane. The fluctuating width is defined by w(x) = d + a[CL (x) -4, (x)] with w(4x) = d We emphasize that wherever being possible, we will develop our approach in a general form in order to include the four waveguide's profiles defined above. Then, the expressions of the attenuation length will be given for each profile.