This topic compares the analytical solutions and results simulated with MODE for a hollow metal waveguide:

- We first calculate the wavenumbers for 3 TE modes from 5GHz to 25GHz and compare the results to the analytical solutions.
- We also examine the error in the calculated wavenumbers as a function of the number of grid points.

## Simulation setup

This waveguide is created using a simulation box of dimensions 10nm x 20nm with METAL boundary conditions on all sides. The region is discretized such that the grid spacing is the same in both the x and y directions.

## Analysis

Analytical Solutions

The analytical solutions for the TE modes are given by the following equation:

$$\operatorname{TEmn}=\sqrt{\left(\left(\frac{\omega}{c}\right)^{2}-\pi^{2}\left(\left(\frac{m}{a}\right)^{2}+\left(\frac{n}{b}\right)^{2}\right)\right.}$$

where a and b are the dimensions of the waveguide. This will be used to compare with the MODE results for modes TE10, TE01 and TE11.

## Results

### TE modes

The script* hollow_metal_wg.lsf* first performs a frequency sweep from 5GHz to 25GHz and selects the 3 modes that best overlap with the desired TE modes.

The following figure (generated using the Matlab interface) shows the propagation wavevector as a function of frequency for the 3 modes.

(Left) Propagation wavevector as a function of frequency for hollow metal waveguide. Dispersive characteristics of the first three modes are shown for frequencies ranging from 5 to 25 GHz. The solid lines show the analytic response, while the symbols (o) show the results calculated with MODE. (Right) The same figure generated using only built-in MODE functions (no Matlab interface).

Error amplitude as a function of grid points

The second part of *hollow_metal_wg.lsf* performs a systematic increase in the number of grid points and observes the rate of change of the error amplitude as a function of this decrease.

(Left) Error amplitude of MODE calculation for hollow metal waveguide modes at a frequency of 20 GHz compared to the analytic response. The x-axis shows the number of grid points along the long side of the hollow metal waveguide. (Right)The same figure generated using only built-in MODE functions.