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Spheroidal Modes

This section exposes the spheroidal vector-wave API currently available in AnalyticEMModes.jl.

Scope and convention

  • Coordinates are local spheroidal (ξ, η, ϕ) (prolate or oblate).
  • Base vectors are returned as (Mξ, Mη, Mϕ, Nξ, Nη, Nϕ).
  • The low-level M/N vectors are provided without enforcing TE/TM physical power normalization.
  • kc_spheroidal() is currently a placeholder and returns 0.0.

Single mode evaluation

using AnalyticEMModes

k = 2π
mode = SpheroidalB(1, 1, 2.4)  # prolate; for oblate use complex c

ξ, η, ϕ = 1.4, 0.2, 0.8
Mξ, Mη, Mϕ, Nξ, Nη, Nϕ = mn_spheroidal_vector(ξ, η, ϕ, mode, k; family=:z, even=true, radial=4)

Batch evaluation on points and basis

using AnalyticEMModes, StaticArrays

k = 2π
points = [SVector(1.3, -0.3, 0.1), SVector(1.5, 0.1, 1.2), SVector(1.9, 0.7, 2.0)]
basis = ProlateSpheroidalBasis(2, 3, 2.4)

A = mn_spheroidal_vectors(points, basis, k; family=:z, even=true, radial=4)
# A[p, i] = (Mξ, Mη, Mϕ, Nξ, Nη, Nϕ) for point p and mode i

Build basis directly from m/n limits

using AnalyticEMModes, StaticArrays

k = 2π
points = [SVector(1.3, -0.2, 0.4), SVector(1.8, 0.4, 1.1)]

A = mn_spheroidal_vectors_mnmax(points, 2, 3, 2.4, k; oblate=false, family=:r, even=false, radial=4)

Notes for modal normalization

For spheroidal geometry, the recommended current workflow is:

  1. Compute raw M/N vectors with mn_spheroidal_vector(s) for a chosen family/parity/radial type.
  2. Build the TE/TM convention at the field level (E/H) on top of those vectors.
  3. Normalize physically by flux/power on the chosen surface.

This keeps the curl(M) = kN style relations explicit at the raw-basis level.

Visualization on Spheroidal Mesh

f = 10e9
μr, εr = 1.0, 1.0
k = wavenumber(f, μr, εr)
name = "../Assets/mesh/spheroidal_prolate.msh"

coord, conn = mesh_data(name)
coords = coord[:, 1:maximum(conn)]
xcoords = coords[1, :]
ycoords = coords[2, :]
zcoords = coords[3, :]

# Estimate spheroidal focal parameter a from mesh axes (z-major spheroid).
major_axis = maximum(zcoords) - minimum(zcoords)
minor_axis = 2 * maximum(hypot.(xcoords, ycoords))
a = spheroidal_parameter(major_axis, minor_axis)

ξηϕ = map(eachindex(xcoords)) do i
    cart2pro(a, xcoords[i], ycoords[i], zcoords[i])
end
ξs = map(t -> t[1], ξηϕ)
ηs = map(t -> t[2], ξηϕ)
ϕs = map(t -> t[3], ξηϕ)
points = to_svector(ξs, ηs, ϕs)

basis = ProlateSpheroidalBasis(4, 7, 2.4)
mn = mn_spheroidal_vectors(points, basis, k; family=:z, even=true, radial=4)

mode_idx = 2
Mcart = map(eachindex(points)) do i
    ξ, η, ϕ = points[i]
    Mξ, Mη, Mϕ, Nξ, Nη, Nϕ = mn[i, mode_idx]
    Mx, My, Mz = prolate_vector_to_cartesian(a, ξ, η, ϕ, Mξ, Mη, Mϕ)
    (Mx, My, Mz, Nξ)
end

Mx = map(v -> real(v[1]), Mcart)
My = map(v -> real(v[2]), Mcart)
Mz = map(v -> real(v[3]), Mcart)
Nξ = map(v -> real(v[4]), Mcart)
Mf = map((x, y, z) -> hypot(x, y, z), Mx, My, Mz)
Mf_safe = map(v -> v > 0 ? v : 1.0, Mf)

Mx ./= Mf_safe
My ./= Mf_safe
Mz ./= Mf_safe

fig = Figure(size = (900, 420))
ga = fig[1, 1] = GridLayout()
Label(ga[0, 1:2], "Spheroidal M/N example", fontsize = 28, tellwidth = false)

ax1 = LScene(ga[1, 1], show_axis = false)
mesh!(ax1, coords, conn, color = :gray)
arrows3d!(ax1, xcoords, ycoords, zcoords, 0.08 .* Mx, 0.08 .* My, 0.08 .* Mz, color = Mf, colormap = :turbo)

ax2 = LScene(ga[1, 2], show_axis = false)
mesh!(ax2, coords, conn, color = Nξ, colormap = :balance)

fig
Example block output