Elliptic Waveguide
Elliptic waveguides are less common than the rectangular and circular geometries. Like other waveguide types, they support both TE and TM modes. Their distinctive characteristic is that they admit even and odd solution pairs formed using even and odd Mathieu functions. Consequently, except for m = 0, each mode designation (m, n) corresponds to two distinct modes with different cutoff frequencies.
Numerical Example
As in previous sections, we first present numerical solutions obtained with Gridap.jl to serve as validation references for the analytical expressions.
Mesh files (.msh) can be found in \src\Assets\mesh and are generated with Gmsh.jl using functions in \src\Assets\example_mesh.jl.
The Hz field (longitudinal magnetic component) for the first TE modes computed numerically:
using GLMakie
name = "../Assets/mesh/elliptic_wg1.msh"
a = 1
b = 0.7
ellipticwg_mesh(a, b; dl = 0.05, name)
model = GmshDiscreteModel(name, renumber = false)
A_te, B_te, U_te, V_te = engeimode_gridap(model, :TE, order = 2)
A_tm, B_tm, U_tm, V_tm = engeimode_gridap(model, :TM, order = 2)
nev = 24
λ_te, ϕ_te = eigs(A_te, B_te; nev=nev, which=:SM, maxiter=5300)
λ_tm, ϕ_tm = eigs(A_tm, B_tm; nev=nev, which=:SM, maxiter=5300)
t = range(0, 2*pi, length=100)
ellip_x_r = @. b * sin(t)
ellip_y_r = @. a * cos(t)
tria = Triangulation(model)
gridap_coords = get_node_coordinates(model)
makie_coords = map(x->Point2f(x.data), gridap_coords)
face_to_nodes = stack(map(x->x, get_cell_node_ids(model)))'
fig = Figure(size = (1300, 910))
plot_ids = Iterators.product(1:5, 1:4) |> collect |> vec
for (idp, (ii, jj)) in enumerate(plot_ids)
uh = FEFunction(V_te, real.(ϕ_te[:, idp+1]), get_dirichlet_dof_values(V_te))
gridap_values = evaluate(uh, gridap_coords)
fz_num = abs.(gridap_values)
fz_num = fz_num / maximum(fz_num)
coord2 = makie_coords[argmin(fz_num)]
axi = Axis(fig[jj,ii])
t = lines!(axi, ellip_y_r, ellip_x_r, linewidth = 3.5, color = :black)
translate!(t, (0, 0, 2))
xlims!(-a*1.1, a*1.1)
ylims!(-a, a)
mesh!(axi, makie_coords, face_to_nodes, color = fz_num, colormap = :jet)
hidedecorations!(axi)
end
Label(fig[0, 2:4], "Numeric TE Modes", fontsize = 20)
fig
The numerical eigenmode solution reveals the complex field patterns characteristic of elliptic geometries.
The Ez field (longitudinal electric component) for the first TM modes:
fig = Figure(size = (1300, 910))
plot_ids = Iterators.product(1:5, 1:4) |> collect |> vec
for (idp, (ii, jj)) in enumerate(plot_ids)
uh = FEFunction(V_tm, real.(ϕ_tm[:, idp]), get_dirichlet_dof_values(V_tm))
gridap_values = evaluate(uh, gridap_coords)
fz_num = abs.(gridap_values)
fz_num = fz_num / maximum(fz_num)
coord2 = makie_coords[argmin(fz_num)]
axi = Axis(fig[jj,ii])
t = lines!(axi, ellip_y_r, ellip_x_r, linewidth = 3.5, color = :black)
translate!(t, (0, 0, 2))
xlims!(-a*1.1, a*1.1)
ylims!(-a, a)
mesh!(axi, makie_coords, face_to_nodes, color = fz_num, colormap = :jet)
hidedecorations!(axi)
end
Label(fig[0, 2:4], "Numeric TM Modes", fontsize = 20)
fig
TM modes in elliptical waveguides exhibit similar complexity, with field patterns governed by the interplay of Mathieu functions in both coordinate directions.
Analytical Solutions
The analytical evaluation of elliptic waveguide modes uses Mathieu functions, which are more computationally intensive than Bessel or trigonometric functions. The field solutions require evaluating both angular Mathieu functions (ceₘ, seₘ for even/odd) and radial Mathieu functions (Ceₘ, Seₘ). To improve computational efficiency, the Mathieu function coefficients can be pre-computed outside the field evaluation calls.
The following code evaluates the analytical TE mode fields using te_ewg_fields. The parameter even determines whether even (true) or odd (false) Mathieu functions are used, corresponding to different mode symmetries and cutoff frequencies.
using GLMakie
name = "../Assets/mesh/elliptic_wg1.msh"
a = 1
b = 0.7
#ellipticwg_mesh(a, b; dl = 0.05, name)
coord, conn = mesh_data(name)
coords = coord[:, 1:maximum(conn)]
xcoords = coords[1, :]
ycoords = coords[2, :]
ellipcoords = map((x, y) -> cart2elliptic(x, y, a, b), xcoords, ycoords)
ξcoords = getindex.(ellipcoords, 1)
ηcoords = getindex.(ellipcoords, 2)
t = range(0, 2*pi, length=100)
ellip_x_r = @. b * sin(t)
ellip_y_r = @. a * cos(t)
modekind = [(1, 1, true), (1, 1, false), (2, 1, true), (2, 1, false), (3, 1, true),
(3, 1, false), (0, 1, true), (4, 1, true), (4, 1, false), (1, 2, true), (1, 2, false),
(5, 1, true), (5, 1, false), (2, 2, true), (2, 2, false), (6, 1, true), (6, 1, false),
(3, 2, true), (0, 2, true), (3, 2, false)]
plot_ids = Iterators.product(1:5, 1:4) |> collect |> vec
fig = Figure(size = (1300, 910))
for (idplot, (m, n, even)) in enumerate(modekind)
ii,jj = plot_ids[idplot]
fields = te_ewg_fields(ξcoords, ηcoords, a, b, m, n, even, 100e9, 1.0, 1.0)
fz = getindex.(fields, 6)
stitle = even ? L"TE_{c%$m%$n}" : L"TE_{e%$m%$n}"
axi = Axis(fig[jj,ii], title = stitle, titlesize = 20)
hidedecorations!(axi)
t = lines!(axi, ellip_y_r, ellip_x_r, linewidth = 3.5, color = :black)
translate!(t, (0, 0, 2))
mesh!(axi, coords, conn, color = abs.(fz), colormap = :jet)
xlims!(-a*1.1, a*1.1)
ylims!(-a, a)
end
fig
The visualization shows 20 TE modes with labels indicating even (c) or odd (e) character. Note that for m > 0, both even and odd modes exist with different cutoff frequencies and patterns.
TM modes for the same elliptic geometry:
modekind = [(0, 1, true),(1, 1, true),(1, 1, false),(2, 1, true),(2, 1, false),(3, 1, true),
(0, 2, true),(3, 1, false),(1, 2, true),(4, 1, true),(4, 1, false),(1, 2, false),
(2, 2, true),(5, 1, true),(5, 1, false),(2, 2, false),(3, 2, true),(6, 1, true),
(6, 1, false),(0, 3, true)]
fig = Figure(size = (1300, 910))
for (idplot, (m, n, even)) in enumerate(modekind)
ii,jj = plot_ids[idplot]
fields = tm_ewg_fields(ξcoords, ηcoords, a, b, m, n, even, 100e9, 1.0, 1.0)
fz = getindex.(fields, 3)
stitle = even ? L"TM_{c%$m%$n}" : L"TM_{e%$m%$n}"
axi = Axis(fig[jj,ii], title = stitle, titlesize = 20)
hidedecorations!(axi)
t = lines!(axi, ellip_y_r, ellip_x_r, linewidth = 3.5, color = :black)
translate!(t, (0, 0, 2))
mesh!(axi, coords, conn, color = abs.(fz), colormap = :jet)
xlims!(-a*1.1, a*1.1)
ylims!(-a, a)
end
fig
The even/odd distinction is clearly visible in the field symmetries.
Transverse Field Components
For elliptic coordinates, the transverse field components require coordinate transformation from the native elliptic system (ξ, η) to Cartesian coordinates (x, y). The metric_and_unit_elliptic function provides the necessary metric coefficients and unit vector components for this transformation.
The following example visualizes the transverse magnetic field for selected TM modes:
ρ = sqrt(a^2-b^2)
modekind = [(0, 1, true),(1, 1, true),(1, 1, false),(2, 1, true),(2, 1, false),(3, 1, true)]
plot_ids = Iterators.product(1:3, 1:2) |> collect |> vec
fig = Figure(size = (1300, 910))
for (idplot, (m, n, even)) in enumerate(modekind)
ii,jj = plot_ids[idplot]
fields = tm_ewg_fields(ξcoords, ηcoords, a, b, m, n, even, 100e9, 1.0, 1.0)
fz = map(fields, ξcoords, ηcoords) do ehfields, ξi, ηi
h, exi_x, exi_y, eeta_x, eeta_y = metric_and_unit_elliptic(ρ, ξi, ηi)
Fx = ehfields[4] * exi_x + ehfields[5] * eeta_x
Fy = ehfields[4] * exi_y + ehfields[5] * eeta_y
return (Fx, Fy)
end
stitle = even ? L"TM_{c%$m%$n}" : L"TM_{e%$m%$n}"
axi = Axis(fig[jj,ii], title = stitle, titlesize = 20)
hidedecorations!(axi)
t = lines!(axi, ellip_y_r, ellip_x_r, linewidth = 3.5, color = :black)
translate!(t, (0, 0, 2))
fu = real.(getindex.(fz, 1))
fv = real.(getindex.(fz, 2))
ft = map(x->hypot(x...), fz) * 35
vm = maximum(ft)*10
fu = fu ./ ft
fv = fv ./ ft
arrows2d!(axi, xcoords, ycoords, fu, fv, color = ft, colormap = :jet)
xlims!(-a*1.1, a*1.1)
ylims!(-a, a)
end
fig