Input Port Boundary Condition

The input port boundary condition models the electromagnetic field entering the computational domain from an attached waveguide. The input boundary condition assumes that the injected radiation corresponds to a well-established waveguide mode, with the complex amplitude \(\tilde{\mathbf{E}}^\text{in}\) of the input electric field expressed as

\[\tilde{\mathbf{E}}^\text{in} = \tilde{\mathbf{e}}(\mathbf{r})\, e^{i\beta \mathbf{r} \cdot \hat{\mathbf{n}}},\]

where \(\tilde{\mathbf{e}}(\mathbf{r})\) describes the mode profile, defined in the plane of the port, \(\beta\) is the mode propagation constant, \(\mathbf{r}=(x,y,z)\) is the radius-vector and \(\hat{\mathbf{n}}\) is the unit vector normal to the boundary. In turn, the mode profile \(\tilde{\mathbf{e}}\) and the mode propagation constant \(\beta\) are calculated from the following eigenvalue problem, defined in the plane of the port under the perfect electric conductor boundary conditions:

\[\nabla \times \left( \frac{1}{\mu} \nabla \times \tilde{\mathbf{e}} \right) = \lambda\, \tilde{\mathbf{e}}.\]

The eigenvectors \(\tilde{\mathbf{e}}\) provide the mode profile, while the eigenvalues \(\lambda\) are used to calculate the cutoff wavenumber \(k_c=\sqrt{\lambda}\) which, in turn, determines the mode propagation constant as \(\beta=\sqrt{\mu\varepsilon\omega^2 - k_c^2}\), where \(\omega\) is the angular frequency of the field.

Under the above assumptions the input port boundary condition results in the following equation for the amplitude \(\tilde{\mathbf{E}}\) of the electric field at the port boundary:

\[\hat{\mathbf{n}} \times \left(\frac{1}{\mu} \nabla \times \tilde{\mathbf{E}}\right) = -\frac{i\beta}{\mu} \hat{\mathbf{n}} \times \left(\tilde{\mathbf{E}} \times \hat{\mathbf{n}}\right) +\frac{2i\beta}{\mu} \hat{\mathbf{n}} \times \left(\tilde{\mathbf{E}}^\text{in} \times \hat{\mathbf{n}}\right).\]