# Accurate Modeling of Conductor Rough Surfaces in Waveguide Devices
## 1 Introduction
In their study, the models of the rough interface between the dielectric and conductive mediums and the rough parallel plate waveguide were investigated using an analytical second-order small perturbation method (SPM2), as well as numerical methods, such as the method of moments (MoM) and the transfer matrix (T-matrix) method. Unfortunately, these models cannot be extended straightforwardly for the models of non-TEM waveguide structures with inner surface roughness. For waveguide structures, [PERSON] et al. [10] investigated the field distribution and cutoff frequency of a rectangular waveguide with surface roughness by using the method of moment rigorously, while the waveguide walls were modeled as perfect conductors. [PERSON] et al. [11] employed transformation-based media for analyzing waveguides with grooves or rough surfaces, thereby improving the computational performance of the finite element method by eliminating the need for mesh generation at each realization corresponding to a different shape of the surface when using a Monte Carlo technique. However, the finite conductivity boundary in the transformation electromagnetics method has not yet been developed to account for the effects of rough surfaces on conductor loss. [PERSON] et al. [12] studied the effects of periodic 3-D surface roughness on the complex propagation constant of rectangular \(\mu\)-coaxial lines using the commercial finite-element method software, HFSS\({}^{\text{TM}}\). Importantly, direct solving for the field inside the rough conductor in the realistic recta-coax line is impractical due to the excessive memory requirements needed to accurately mesh the interior of the metal walls. If surface roughness is modeled at a microscopic level using several intrinsic parameters (e.g., the shape of the assumed structures), it will be expected that the modeling will still differ from reality as the model assumes regular structures rather than a random and irregular surface topology. In addition, it is very likely that significant computational effort will be required for co-simulation of multi-scale structures. Recently [PERSON] et al [13; 14; 15; 16] proposed the conductivity gradient model to model the influence of rough surfaces in PCB transmission lines and waveguides. A frequency-dependent effective conductivity was introduced in a smooth surface conductor to account for surface roughness effects, and this conductivity could easily be integrated into 2D/3D field solvers as a surface impedance boundary condition. In this paper, the effective conductivity concept is employed to analyze metal-based waveguide structures and the effects of inner surface roughness. In Section 2, the conductivity gradient model for rough surfaces is described. Then, the associated Maxwell equations are solved with an inhomogeneous distribution of conductivity filled in the space. Thus, the frequency dependent effective conductivity is derived with the equivalent of conductor loss and can then be straightforwardly integrated into the surface impedance boundary condition and analyzed using the commercial 3-D software, HFSS(tm). In Section 3, three typical and widely used models (rectangular waveguide, cylindrical resonator, and a four pole aperture couple filter) with inner rough surfaces are investigated at millimeter wave frequencies, and the numerical results are compared with models that have ideally smooth surfaces. Finally, the conclusions from the analysis are presented in Section 4.