Tutorial

In this tutorial, we will present some basic knowledge of dispersion digrams and use an example to show how ClosedWaveguideDispersion.jl works.

Problem

Let $\Omega = \mathbb{R} \times (0, 1) \subset \mathbb{R}^{2}$ be a closed waveguide with the boundary

\[\partial \Omega = \{x = (x_{1}, x_{2}) \in \mathbb{R}^{2} : x_{2} = 0 \text{ or } x_{2} = 1\}.\]

Suppose $\Omega$ is filled by periodic medium with period $p$. The medium can be characterized by the real-valued refactive index $q(x_{1}, x_{2})$ satisfying

\[q(x_{1} + p, x_{2}) = q(x_{1}, x_{2}), q \geqslant c > 0, \forall (x_{1}, x_{2}) \in \Omega.\]

Moreover, we assume that $q(x_{1}, x_{2}) \in L^{\infty}(\Omega)$.

The wave propagation in the periodic closed waveguide is modeled by the following boundary value problem

\[\begin{align*} &\Delta u + k^{2}q(x_{1}, x_{2})u = f \text{ in } \Omega, \\ &\frac{\partial u}{\partial x_{2}} = 0 \text{ on } \partial \Omega, \end{align*}\]

where $f$ is a function in $L^{2}(\Omega)$ with compact support, and $k > 0$ is the real wavenumber.

Floquet-Bloch theory

For simplicity, we only present necessary notions for computation. For more theoretical details, we refer to [1] and [2].

The parameter $\alpha$ is introduced by the Floquet-Bloch transform. And we always consider $\alpha$ in the Brillouin zone $(-\pi/p, \pi/p)$.

Transfering the dependence on $\alpha$ from the function space to the PDE, we have the following boundary value problem

\[\begin{align*} &\Delta v + 2i\alpha \partial_{1}v + (k^{2}q(x_{1}, x_{2}) - \alpha^{2}) v = 0 \text{ in } \Omega_{0}, \\ &\frac{\partial v}{\partial x_{2}} = 0 \text{ on } \partial \Omega_{0}, \\ &v \text{ is periodic with respect to } x_{1}. \end{align*}\]

Variational formulation

In this section, we present the variational formulation of the boundary value problem: Find $v \in H_{per}^{1}(\Omega_{0})$ satisfying

\[\int_{\Omega_{0}} \nabla v \cdot \nabla \bar{\phi} - 2i\alpha \partial_{1} v \bar{\phi} - (k^{2}q(x_{1}, x_{2}) - \alpha^{2}) v \bar{\phi} dx = 0.\]

After finite element discretization, we can obtain a generalized linear eigenvalue problem

\[\mathbf{A}_{\alpha} \mathbf{v} = k^{2} \mathbf{B} \mathbf{v},\]

where $\mathbf{A}_{\alpha}$ comes from

\[\int_{\Omega_{0}} \nabla v \cdot \nabla \bar{\phi} - 2i\alpha \partial_{1} v \bar{\phi} + \alpha^{2} v \bar{\phi} dx \]

and $\mathbf{B}$ comes from

\[\int_{\Omega_{0}} q(x_{1}, x_{2}) v \bar{\phi} dx.\]