API

IrreducibleBrillouin{T}

Irreducible Brillouin zone of 2D square lattices. It is a triangle with three vertices: Γ, X, and M.

Fields

• Γ: Center of the Brillouin zone $(0, 0)$
• X: $(\pi / x, 0)$
• M: $(\pi / x, \pi / y)$

SquareLattice{T}

2D Square lattice described by x and y.

(0, y)
     ^ 
     |
     |
     |
     ---------> (x, 0) 

allocate_matries(dh::DofHandler, cst::ConstraintHandler)

Create a sparsity pattern by DofHandler dh and ConstraintHandler cst.

assemble_A(cv::CellValues, dh::DofHandler, A::SparseMatrixCSC, α::Float64)

Assemble the matrix $\mathbf{A}$ which is given by the sesquilinear form

\[\int \nabla u \cdot \nabla \bar{v} - 2i \alpha \partial_{1} u \bar{v} + \alpha^{2} u \bar{v} dx.\]

assemble_B(cv::CellValues, dh::DofHandler, B::SparseMatrixCSC, n::Function)

Assemble the matrix $\mathbf{B}$ which is given by the sesquilinear form

\[\int n(x_{2}, x_{2}) u \bar{v} dx.\]

calc_diagram(cv::CellValues, dh::DofHandler, cst::ConstraintHandler, A::SparseMatrixCSC, B::SparseMatrixCSC, n::Function, bz; nevs::Int=6)

Compute the dispersion diagram for a 2D periodic closed waveguide. The propagation of the wave in this waveguide is described by

\[\Delta u + k^2 n(x_{1}, x_{2})u = 0.\]

To get the dispersion diagram, We need to solve generalized linear eigenvalue problems parametered by $\alpha$ in the Brillouin zone bz which are derived by Finite element method. The generalized linear eigenvalue problems come from

\[\int \nabla u \cdot \nabla \bar{v} - 2i \alpha \partial_{1} u \bar{v} + \alpha^{2} u \bar{v} dx = k^{2} \int n(x_{2}, x_{2}) u \bar{v} dx.\]

Arguments

• cv: CellValues
• dh: DofHandler
• cst: ConstraintHandler
• A: matrix $\mathbf{A}$ generated by assemble_A
• B: matrix $\mathbf{B}$ generated by assemble_B
• n: refractive index which describe the property of the medium
• bz: discrete Brillouin zone
• nevs: the number of eigenvalues

get_discrete_irreducibleBrillouin(ibz::IrreducibleBrillouin{T}, n::Int64) where{T}

Get discrete points on the boundary of the irreducible Brillouin zone (a triangle for square lattices): Γ -> X -> M -> Γ.

             M
            /|   
           / |
         /   |
   d   /     | π_y
     /       |
   /         |
 /           |
--------------
Γ     π_x    X

Arguments

• ibz: the irreducible Brillouin zone
• n: the number of points in the interior of each edge

Output

• dibz: the points along the boundary of the irreducible Brillouin zone
• para: a real number in $[0, 1]$. which is the parameter corresponding to the points in dibz. We need para when we plot the band structure

plot_diagram(bz, μ;period=1.0, title="Dispersion Diagram", xlabel="Brillouin zone")

Plot the disperison diagram.

Arguments

• bz: discrete Brillouin zone
• μ: eigenvalues with respect to $\alpha$ in bz
• period: period of the closed waveguide in the horizontal direction

setup_bdcs(dh::DofHandler; period=1.0)

Impose the periodic boundary condition in the horizontal direction.

Arguments

• dh: DofHandler
• period: the period of the closed waveguide in the horizontal direction

setup_dofs(grid::Grid, ip)

Define the DofHandler according to the mesh grid and the interpolation ip.

setup_fevs(ip)

Set up the quadreture rule and CellValues (for the integral on domain). Here we only consider 2D problem.

Arguments

• ip: the interpolation

setup_grid(;lc=0.05, period=1.0, height=1.0)

Generate the mesh for the periodic cell by using Gmsh.

Arguments

• lc: the mesh size
• period: the period of the periodic closed waveguide
• height: the height of the periodic closed waveguide

setup_grid_squareLattice(;lc=0.05, period=2π)

Generate meshes for periodic cells of square lattices.

Arguments

• lc: the mesh size
• period: the period of the square