# Accurate Modeling of Conductor Rough Surfaces in Waveguide Devices
## 2 Theories  
### Conductivity Gradient Model for a Conductor Plane with Surface Roughness  
The surface roughness profiles of conductors can be assumed to be irregular and random. The roughness profile of a plane can be measured using a stylus profilometer or by SEM. The roughness is usually characterized by two parameters: \(R_{q}\), RMS average of the departure of the roughness profile from the mean surface line, and \(\Lambda_{r}\) the correlation length measuring the mean distance between consecutive peaks and valleys on the surface. The characteristics of the roughness parameters are dependent on the manufacturing process. In waveguide structure fabrication, the RMS surface roughness parameter (\(R_{q}\)) of the inner surface is typically in the order of \(1\upmu\)m. For the frequencies of interest, for example at 5 GHz, the skin depth in copper is \(0.93\upmu\)m. As this skin depth is comparable to the surface roughness parameter, the roughness effects on signal propagation in waveguides cannot be neglected.  
The sizes of waveguide components are usually comparable with the signal wavelength, and the components are operated on the resonance region corresponding to the signal frequency. On the other hand, the sizes of the surface roughness are in the microscopic scale and much smaller than the component sizes. If the conductor surfaces on the component are modeled accurately by fine discretization, the numerical technique will inherently handle the fields above and in the conductors but would make the computing effort so large that it would be almost impossible to solve these multi-scale structures. In fact, since the wavelength is much greater than the scale of the RMS surface roughness \(R_{q}\) and the correlation length \(\Lambda_{r}\), it is not necessary to model the microscopic features of surface roughness directly.  
For an air-metal plane interface, the conductor occupies the top half space, as shown in Figure 1. Considering that the conductor and air are separated by the conductor surface profile, the conductivity around the interface is non-homogeneous. Thus, the transition from air to conductor can be quantitatively modeled as a conductivity gradient perpendicular to the surface along the y-axis direction (See Figure 1). For a random surface roughness, the surface profile height can be characterized with a probability density function (PDF), where the conductivity distribution can be expressed through the cumulative distribution function (CDF) as follows:  
\[\sigma(y)=\sigma_{bulk}\cdot CDF(y)=\sigma_{bulk}\int\limits_{-\infty}^{y}PDF( u)du, \tag{2}\]  
where \(\sigma_{bulk}\) is the metal bulk conductivity and \(PDF()\) is the probability density function of the surface profile height. Usually the PDF has a normal distribution due to the manufacturing process and can be obtained by measurement. Naturally, the normal distribution can be expressed with a mean value and standard deviation. With the RMS value \(R_{q}\) for the conductor surface roughness, the conductivity distribution in space based on this normal distribution can be derived as  
\[\sigma(y)=\sigma_{bulk}\cdot\frac{1}{R_{q}\sqrt{2\pi}}\int\limits_{-\infty}^{y}e ^{-\frac{\omega^{2}}{2R_{q}^{2}}}du. \tag{3}\]  
In addition, surface profiles other than the normal distribution also are suitable to be modeled with the CDF directly. With the conductivity gradient model, we transfer the conductor surface roughness topology to a location dependent conductivity filled in space. Then, the resulting problem involves solving [PERSON]'s equations for the model with an inhomogeneous conductivity distribution.