Waveguide port boundary conditions
Here we introduce the math necessary for waveguide port boundary conditions, which launch a certain mode into the waveguide and absorb the reflections[1].
Eigenmodes
Let’s start with the simplest case: a parallel-plate waveguide with a width of \(b\). In this case, the fields of the modes can simply be described by
\[\begin{split}
h_m(y) = \sqrt{\frac{v_m}{b}}\cos\frac{m\pi y}{b},
\quad
v_m =
\begin{cases}
1, &m=0 \\
2, &m \neq 0
\end{cases}
\end{split}\]
and the propagation constants are
\[
\gamma_m
=
\sqrt{
\left(
\frac{m \pi}{b}
\right)^2
-
k_0^2
}
\]
leading to the propagating modes being described as
\[
H_{z,m}(x,y) = h_m(y)\mathrm{e}^{\gamma_m x}\,.
\]
As these modes are orthogonal to each other, they fullfill the orthogonallity relation
\[
\int_0^b h_m(y)h_{m'}(y)\mathrm{d}y = \delta_{m,m'}
\]
Field at the interface
The field at the interface can be described using these modes as
\[
H_z(x,y)
=
H_z^{inc}(x,y)
+
\sum_{m=0}^\infty
a_m h_m(y) \mathrm{e}^{\gamma_m x},
\]
where the expansion coefficients \(a_m\) can be determined using
\[
a_m
=
\mathrm{e}^{-\gamma_m x_1}
\int_0^b \left[ H_z(x_1,y)-H_z^{inc}(x_1,y) \right] h_m(y) \mathrm{d}y \,.
\]
This way we get for \(H_z\) the expression
\[
H_z(x,y)
=
H_z^{inc}(x,y)
+
\sum_{m=0}^\infty
\mathrm{e}^{\gamma_m (x-x_1)} h_m(y)
\int_0^b \left[ H_z(x_1,y')-H_z^{inc}(x_1,y') \right] h_m(y') \mathrm{d}y'
,
\]
and for the derivative with respect to \(x\)
\[
\frac{
\partial H_z
}{
\partial x
}
=
\frac{
\partial H_z^{inc}
}{
\partial x
}
+
\sum_{m=0}^\infty
\gamma_m\mathrm{e}^{\gamma_m (x-x_1)} h_m(y)
\int_0^b \left[ H_z(x_1,y')-H_z^{inc}(x_1,y') \right] h_m(y') \mathrm{d}y'
,
\]
and at the interface \(x=x_1\)
\[
\left.
\frac{
\partial H_z
}{
\partial x
}
\right|_{x=x_1}
=
\left.
\frac{
\partial H_z^{inc}
}{
\partial x
}
\right|_{x=x_1}
+
\sum_{m=0}^\infty
\gamma_m
\int_0^b \left[ H_z(x_1,y')-H_z^{inc}(x_1,y') \right] h_m(y') \mathrm{d}y'
,
\]
Boundary condition
Using this, we can write it in the form of a generalized boundary condition:
\[
\frac{
\partial H_z
}{
\partial \vec{n}
}
+
\gamma(H_z) = q \,,
\]
where \(\vec{n}\) is the vector orthogonal to the interface. the boundary operator \(\gamma\) is given by
\[
\gamma(H_z) = \sum_{m=0}^\infty \gamma_m h_m(y) \int_0^b H_z(x_1,y') h_m(y') \mathrm{d}y'
\]
and \(q\) is defined as
\[
q
=
\left.
\frac{
\partial H_z^{inc}
}{
\partial \vec{n}
}
\right|_{x=x_1}
+
\sum_{m=0}^\infty \gamma_m h_m(y) \int_0^b H_z^{inc}(x_1,y') h_m(y') \mathrm{d}y'
\]
and simplifies for single-mode incidence of mode \(n\) to
\[
q = 2 \gamma_n H_0 h_n(y) \mathrm{e}^{-\gamma_n x_1} \,,
\]
where \(H_0\) is the magnitude of the incident field and \(n\) is the number of the incident mode.
By setting \(H_0=0\) and thus setting the right-hand side of the boundary condition to zero, this kind of boundary condition can be used as an absorbing boundary condition.
Functional defining the finite-element simulation
Adding this boundary condition to the functional defining the simulation leads to
\[\begin{split}
\begin{aligned}
F(H_z)
=
&\frac{1}{2}
\int_\Omega
\left[
\frac{
\left(
\frac{\partial H_z}{\partial x}
\right)^2
+
\left(
\frac{\partial H_z}{\partial y}
\right)^2
}{\epsilon_r}
-
k_0^2 \mu_r H_z^2
\right]
\mathrm{d}\Omega
\\
&+
\sum_\sigma
\int_\sigma
\left[
\frac{1}{2} H_z \gamma(H_z) - q H_z
\right]
\mathrm{d}y
\end{aligned}
\end{split}\]
where \(\Omega\) is the simulation domain and the \(\sigma\) are the boundaries of \(\Omega\), where the waveguide port condition are applied to.
As \(\lim_{m\to\infty} h_m=0\), the functional converges and only needs a limited amount of summands.
Bibliography
[1]
Jian-Ming Jin. The Finite Element Method in Electromagnetics. John Wiley & Sons, New York, edition, 2015. ISBN 978-1-118-84202-7.
