Electromagnetic scattering from a sphere (axisymmetric)

Copyright (C) 2022 Michele Castriotta, Igor Baratta, Jørgen S. Dokken

This demo is implemented in three files: one for the mesh generation with gmsh, one for the calculation of analytical efficiencies, and one for the variational forms and the solver. It illustrates how to:

• Setup and solve Maxwell’s equations for axisymmetric geometries

• Implement (axisymmetric) perfectly matched layers

Equations, problem definition and implementation

Import the modules that will be used:

import sys
from functools import partial

from mpi4py import MPI

import numpy as np
from scipy.special import jv, jvp

try:
    from petsc4py import PETSc

    import dolfinx

    # The time-harmonic Maxwell equation is complex-valued PDE. PETSc
    # must therefore have compiled with complex scalars.
    if not np.issubdtype(PETSc.ScalarType, np.complexfloating):  # type: ignore
        print("Demo can only be executed when PETSc using complex scalars.")
        exit(0)

    scalar_type = PETSc.ScalarType  # type: ignore
    real_type = PETSc.RealType  # type: ignore

    if not dolfinx.has_petsc:
        print("This demo requires DOLFINx to be compiled with PETSc enabled.")
        exit(0)
except ModuleNotFoundError:
    print("This demo requires petsc4py.")
    exit(0)

import ufl
from basix.ufl import element, mixed_element
from dolfinx import fem, io, mesh, plot
from dolfinx.fem.petsc import LinearProblem

try:
    from dolfinx.io import VTXWriter

    has_vtx = True
except ImportError:
    print("VTXWriter not available, solution will not be saved.")
    has_vtx = False

try:
    import gmsh
except ModuleNotFoundError:
    print("This demo requires gmsh to be installed.")
    sys.exit(0)

try:
    import pyvista

    have_pyvista = True
except ModuleNotFoundError:
    print("pyvista and pyvistaqt are required to visualise the solution")
    have_pyvista = False

def generate_mesh_sphere_axis(
    radius_sph: float,
    radius_scatt: float,
    radius_dom: float,
    radius_pml: float,
    in_sph_size: float,
    on_sph_size: float,
    scatt_size: float,
    pml_size: float,
    au_tag: int,
    bkg_tag: int,
    pml_tag: int,
    scatt_tag: int,
):
    gmsh.model.add("geometry")

    gmsh.model.occ.addCircle(0, 0, 0, radius_sph * 0.5, angle1=-np.pi / 2, angle2=np.pi / 2, tag=1)
    gmsh.model.occ.addCircle(0, 0, 0, radius_sph, angle1=-np.pi / 2, angle2=np.pi / 2, tag=2)
    gmsh.model.occ.addCircle(0, 0, 0, radius_scatt, angle1=-np.pi / 2, angle2=np.pi / 2, tag=3)
    gmsh.model.occ.addCircle(0, 0, 0, radius_dom, angle1=-np.pi / 2, angle2=np.pi / 2, tag=4)
    gmsh.model.occ.addCircle(
        0, 0, 0, radius_dom + radius_pml, angle1=-np.pi / 2, angle2=np.pi / 2, tag=5
    )

    gmsh.model.occ.addLine(10, 8, tag=6)
    gmsh.model.occ.addLine(8, 6, tag=7)
    gmsh.model.occ.addLine(6, 4, tag=8)
    gmsh.model.occ.addLine(4, 2, tag=9)
    gmsh.model.occ.addLine(2, 1, tag=10)
    gmsh.model.occ.addLine(1, 3, tag=11)
    gmsh.model.occ.addLine(3, 5, tag=12)
    gmsh.model.occ.addLine(5, 7, tag=13)
    gmsh.model.occ.addLine(7, 9, tag=14)

    gmsh.model.occ.addCurveLoop([10, 1], tag=1)
    gmsh.model.occ.addPlaneSurface([1], tag=1)
    gmsh.model.occ.addCurveLoop([11, 2, 9, -1], tag=2)
    gmsh.model.occ.addPlaneSurface([2], tag=2)
    gmsh.model.occ.addCurveLoop([8, -2, 12, 3], tag=3)
    gmsh.model.occ.addPlaneSurface([3], tag=3)
    gmsh.model.occ.addCurveLoop([13, 4, 7, -3], tag=4)
    gmsh.model.occ.addPlaneSurface([4], tag=4)
    gmsh.model.occ.addCurveLoop([4, -6, -5, -14], tag=5)
    gmsh.model.occ.addPlaneSurface([5], tag=5)

    gmsh.model.occ.synchronize()

    gmsh.model.addPhysicalGroup(2, [1, 2], tag=au_tag)
    gmsh.model.addPhysicalGroup(2, [3, 4], tag=bkg_tag)
    gmsh.model.addPhysicalGroup(2, [5], tag=pml_tag)
    gmsh.model.addPhysicalGroup(1, [3], tag=scatt_tag)

    gmsh.model.mesh.setSize([(0, 1)], size=in_sph_size)
    gmsh.model.mesh.setSize([(0, 2)], size=in_sph_size)
    gmsh.model.mesh.setSize([(0, 3)], size=on_sph_size)
    gmsh.model.mesh.setSize([(0, 4)], size=on_sph_size)
    gmsh.model.mesh.setSize([(0, 5)], size=scatt_size)
    gmsh.model.mesh.setSize([(0, 6)], size=scatt_size)
    gmsh.model.mesh.setSize([(0, 7)], size=pml_size)
    gmsh.model.mesh.setSize([(0, 8)], size=pml_size)
    gmsh.model.mesh.setSize([(0, 9)], size=pml_size)
    gmsh.model.mesh.setSize([(0, 10)], size=pml_size)

    gmsh.model.mesh.generate(2)

    return gmsh.model

Problem formulation

Consider a metallic sphere embedded in a background medium (e.g., a vacuum or water) subject to an incident plane wave, with the objective to calculate the scattered electric field. We can simplify the problem by considering the axisymmetric case. To verify this, let’s consider the weak form of the problem with PML:

\[\begin{split} \begin{split} \int_{\Omega_m\cup\Omega_b}-(\nabla \times \mathbf{E}_s) \cdot (\nabla \times \bar{\mathbf{v}})+\varepsilon_{r} k_{0}^{2} \mathbf{E}_s \cdot \bar{\mathbf{v}}+k_{0}^{2}\left(\varepsilon_{r} -\varepsilon_b\right)\mathbf{E}_b \cdot \bar{\mathbf{v}}~\mathrm{d} x\\ +\int_{\Omega_{pml}}\left[\boldsymbol{\mu}^{-1}_{pml} \nabla \times \mathbf{E}_s \right]\cdot \nabla \times \bar{\mathbf{v}}-k_{0}^{2} \left[\boldsymbol{\varepsilon}_{pml} \mathbf{E}_s \right]\cdot \bar{\mathbf{v}}~ d x=0 \end{split} \end{split}\]

Since we want to exploit axisymmetry of the problem, we can use cylindrical coordinates as a more appropriate reference system:

\[\begin{split} \begin{split} \int_{\Omega_{cs}}\int_{0}^{2\pi}-(\nabla \times \mathbf{E}_s) \cdot (\nabla \times \bar{\mathbf{v}})+\varepsilon_{r} k_{0}^{2} \mathbf{E}_s \cdot \bar{\mathbf{v}}+k_{0}^{2}\left(\varepsilon_{r} -\varepsilon_b\right)\mathbf{E}_b \cdot \bar{\mathbf{v}}~ \rho d\rho dz d \phi\\ +\int_{\Omega_{cs}} \int_{0}^{2\pi}\left[\boldsymbol{\mu}^{-1}_{pml} \nabla \times \mathbf{E}_s \right]\cdot \nabla \times \bar{\mathbf{v}}-k_{0}^{2} \left[\boldsymbol{\varepsilon}_{pml} \mathbf{E}_s \right]\cdot \bar{\mathbf{v}}~ \rho d\rho dz d \phi=0 \end{split} \end{split}\]

Expanding \(\mathbf{E}_s\), \(\mathbf{E}_b\) and \(\bar{\mathbf{v}}\) in cylindrical harmonics:

\[\begin{split} \begin{align} \mathbf{E}_s(\rho, z, \phi) &= \sum_m\mathbf{E}^{(m)}_s(\rho, z)e^{-jm\phi} \\ \mathbf{E}_b(\rho, z, \phi) &= \sum_m\mathbf{E}^{(m)}_b(\rho, z)e^{-jm\phi} \\ \bar{\mathbf{v}}(\rho, z, \phi) &= \sum_m\bar{\mathbf{v}}^{(m)}(\rho, z)e^{+jm\phi} \end{align} \end{split}\]

The curl operator \(\nabla\times\) in cylindrical coordinates becomes:

\[ \nabla \times \mathbf{a}= \sum_{m}\left(\nabla \times \mathbf{a}^{(m)}\right) e^{-j m \phi} \]

with:

\[\begin{split} \begin{split} \left(\nabla \times \mathbf{a}^{(m)}\right) = \left[\hat{\rho} \left(-\frac{\partial a_{\phi}^{(m)}}{\partial z} -j\frac{m}{\rho} a_{z}^{(m)}\right)+\\ \hat{\phi} \left(\frac{\partial a_{\rho}^{(m)}}{\partial z} -\frac{\partial a_{z}^{(m)}}{\partial \rho}\right) \right.\\ \left.+\hat{z}\left(\frac{a_{\phi}^{(m)}}{\rho} +\frac{\partial a_{\phi}^{(m)}}{\partial \rho} +j \frac{m}{\rho} a_{\rho}^{(m)}\right)\right] \end{split} \end{split}\]

Assuming an axisymmetric permittivity \(\varepsilon(\rho, z)\),

\[\begin{split} \sum_{n, m}\int_{\Omega_{cs}} -(\nabla \times \mathbf{E}^{(m)}_s) \cdot (\nabla \times \bar{\mathbf{v}}^{(m)})+\varepsilon_{r} k_{0}^{2} \mathbf{E}^{(m)}_s \cdot \bar{\mathbf{v}}^{(m)}+k_{0}^{2} \left(\varepsilon_{r} -\varepsilon_b\right)\mathbf{E}^{(m)}_b \cdot \bar{\mathbf{v}}^{(m)}\\ +\left(\boldsymbol{\mu}^{-1}_{pml} \nabla \times \mathbf{E}^{(m)}_s \right)\cdot \nabla \times \bar{\mathbf{v}}^{(m)}-k_{0}^{2} \left(\boldsymbol{\varepsilon}_{pml} \mathbf{E}^{(m)}_s \right)\cdot \bar{\mathbf{v}}^{(m)}~ \rho d\rho dz \int_{0}^{2 \pi} e^{-i(m-n) \phi} d \phi=0 \end{split}\]

For the last integral, we have that:

\[\begin{split} \int_{0}^{2 \pi} e^{-i(m-n) \phi}d \phi= \begin{cases} 2\pi &\textrm{if } m=n\\ 0 &\textrm{if }m\neq n \end{cases} \end{split}\]

We can therefore consider only the case \(m = n\) and simplify the summation in the following way:

\[\begin{split} \begin{split} \sum_{m}\int_{\Omega_{cs}}-(\nabla \times \mathbf{E}^{(m)}_s) \cdot (\nabla \times \bar{\mathbf{v}}^{(m)})+\varepsilon_{r} k_{0}^{2} \mathbf{E}^{(m)}_s \cdot \bar{\mathbf{v}}^{(m)} +k_{0}^{2}\left(\varepsilon_{r} -\varepsilon_b\right)\mathbf{E}^{(m)}_b \cdot \bar{\mathbf{v}}^{(m)}\\ +\left(\boldsymbol{\mu}^{-1}_{pml} \nabla \times \mathbf{E}^{(m)}_s \right)\cdot \nabla \times \bar{\mathbf{v}}^{(m)}-k_{0}^{2} \left(\boldsymbol{\varepsilon}_{pml} \mathbf{E}^{(m)}_s \right)\cdot \bar{\mathbf{v}}^{(m)}~ \rho d\rho dz =0 \end{split} \end{split}\]

What we have just obtained are multiple weak forms corresponding to different cylindrical harmonics propagating independently. Let’s now solve it in DOLFINx. As a first step we can define the function for the \(\nabla\times\) operator in cylindrical coordinates:

def curl_axis(a, m: int, rho):
    curl_r = -a[2].dx(1) - 1j * m / rho * a[1]
    curl_z = a[2] / rho + a[2].dx(0) + 1j * m / rho * a[0]
    curl_p = a[0].dx(1) - a[1].dx(0)
    return ufl.as_vector((curl_r, curl_z, curl_p))

Then we need to define the analytical formula for the background field. We will consider a TMz polarized plane wave, that is a plane wave with the magnetic field normal to our reference plane

For a TMz polarization, we can write the cylindrical harmonics \(\mathbf{E}^{(m)}_b\) of the background field in this way (Wait 1955):

\[\begin{split} \begin{split} \mathbf{E}^{(m)}_b = \hat{\rho} \left(E_{0} \cos \theta e^{j k z \cos \theta} j^{-m+1} J_{m}^{\prime}\left(k_{0} \rho \sin \theta\right)\right) +\hat{z} \left(E_{0} \sin \theta e^{j k z \cos \theta}j^{-m} J_{m} \left(k \rho \sin \theta\right)\right)\\ +\hat{\phi} \left(\frac{E_{0} \cos \theta}{k \rho \sin \theta} e^{j k z \cos \theta} mj^{-m} J_{m}\left(k \rho \sin \theta\right)\right) \end{split} \end{split}\]

with \(k = 2\pi n_b/\lambda = k_0n_b\) being the wavevector, \(\theta\) being the angle between \(\mathbf{E}_b\) and \(\hat{\rho}\), and \(J_m\) representing the \(m\)-th order Bessel function of first kind and \(J_{m}^{\prime}\) its derivative. We implement these functions:

def background_field_rz(theta: float, n_bkg: float, k0: float, m: int, x):
    k = k0 * n_bkg
    a_r = (
        np.cos(theta)
        * np.exp(1j * k * x[1] * np.cos(theta))
        * (1j) ** (-m + 1)
        * jvp(m, k * x[0] * np.sin(theta), 1)
    )
    a_z = (
        np.sin(theta)
        * np.exp(1j * k * x[1] * np.cos(theta))
        * (1j) ** -m
        * jv(m, k * x[0] * np.sin(theta))
    )
    return (a_r, a_z)


def background_field_p(theta: float, n_bkg: float, k0: float, m: int, x):
    k = k0 * n_bkg
    a_p = (
        np.cos(theta)
        / (k * x[0] * np.sin(theta))
        * np.exp(1j * k * x[1] * np.cos(theta))
        * m
        * (1j) ** (-m)
        * jv(m, k * x[0] * np.sin(theta))
    )
    return a_p

PML can be implemented in a spherical shell surrounding the background domain. We can use the following complex coordinate transformation for PML:

\[\begin{split} \begin{align} &\rho^{\prime} = \rho\left[1 +j \alpha/k_0 \left(\frac{r - r_{dom}}{r~r_{pml}}\right)\right] \\ &z^{\prime} = z\left[1 +j \alpha/k_0 \left(\frac{r - r_{dom}}{r~r_{pml}}\right)\right] \\ &\phi^{\prime} = \phi \end{align} \end{split}\]

with \(\alpha\) tuning the absorption inside the PML, and \(r = \sqrt{\rho^2 + z^2}\). This coordinate transformation has the following Jacobian:

\[\begin{split} \mathbf{J}=\mathbf{A}^{-1}= \nabla\boldsymbol{\rho}^ \prime(\boldsymbol{\rho}) = \left[\begin{array}{ccc} \frac{\partial \rho^{\prime}}{\partial \rho} & \frac{\partial z^{\prime}}{\partial \rho} & 0 \\ \frac{\partial \rho^{\prime}}{\partial z} & \frac{\partial z^{\prime}}{\partial z} & 0 \\ 0 & 0 & \frac{\rho^\prime}{\rho}\frac{\partial \phi^{\prime}}{\partial \phi} \end{array}\right]=\left[\begin{array}{ccc} J_{11} & J_{12} & 0 \\ J_{21} & J_{22} & 0 \\ 0 & 0 & J_{33} \end{array}\right] \end{split}\]

which we can use to calculate \({\boldsymbol{\varepsilon}_{pml}}\) and \({\boldsymbol{\mu}_{pml}}\):

\[\begin{split} \begin{align} & {\boldsymbol{\varepsilon}_{pml}} = A^{-1} \mathbf{A} {\boldsymbol{\varepsilon}_b}\mathbf{A}^{T}\\ & {\boldsymbol{\mu}_{pml}} = A^{-1} \mathbf{A} {\boldsymbol{\mu}_b}\mathbf{A}^{T} \end{align} \end{split}\]

For doing these calculations, we define the pml_coordinate and create_mu_eps functions:

def pml_coordinate(x, r, alpha: float, k0: float, radius_dom: float, radius_pml: float):
    return x + 1j * alpha / k0 * x * (r - radius_dom) / (radius_pml * r)


def create_eps_mu(pml, rho, eps_bkg, mu_bkg):
    J = ufl.grad(pml)

    # Transform the 2x2 Jacobian into a 3x3 matrix.
    J = ufl.as_matrix(((J[0, 0], J[0, 1], 0), (J[1, 0], J[1, 1], 0), (0, 0, pml[0] / rho)))

    A = ufl.inv(J)
    eps_pml = ufl.det(J) * A * eps_bkg * ufl.transpose(A)
    mu_pml = ufl.det(J) * A * mu_bkg * ufl.transpose(A)
    return eps_pml, mu_pml

We can now define some constants and geometrical parameters, and then we can generate the mesh with Gmsh, by using the function generate_mesh_sphere_axis in mesh_sphere_axis.py:

# Constants
epsilon_0 = 8.8541878128 * 10**-12  # Vacuum permittivity
mu_0 = 4 * np.pi * 10**-7  # Vacuum permeability
Z0 = np.sqrt(mu_0 / epsilon_0)  # Vacuum impedance

# Radius of the sphere
radius_sph = 0.025

# Radius of the domain
radius_dom = 1

# Radius of the boundary where scattering efficiency is calculated
radius_scatt = 0.4 * radius_dom

# Radius of the PML shell
radius_pml = 0.25

# Mesh sizes
mesh_factor = 1
in_sph_size = mesh_factor * 2.0e-3
on_sph_size = mesh_factor * 2.0e-3
scatt_size = mesh_factor * 60.0e-3
pml_size = mesh_factor * 40.0e-3

# Tags for the subdomains
au_tag = 1
bkg_tag = 2
pml_tag = 3
scatt_tag = 4

model = None
gmsh.initialize(sys.argv)
if MPI.COMM_WORLD.rank == 0:
    # Mesh generation
    model = generate_mesh_sphere_axis(
        radius_sph,
        radius_scatt,
        radius_dom,
        radius_pml,
        in_sph_size,
        on_sph_size,
        scatt_size,
        pml_size,
        au_tag,
        bkg_tag,
        pml_tag,
        scatt_tag,
    )

model = MPI.COMM_WORLD.bcast(model, root=0)
partitioner = dolfinx.cpp.mesh.create_cell_partitioner(dolfinx.mesh.GhostMode.shared_facet)
msh, cell_tags, facet_tags = io.gmshio.model_to_mesh(
    model, MPI.COMM_WORLD, 0, gdim=2, partitioner=partitioner
)

gmsh.finalize()
MPI.COMM_WORLD.barrier()

Visually check of the mesh and of the subdomains using PyVista:

tdim = msh.topology.dim
if have_pyvista:
    topology, cell_types, geometry = plot.vtk_mesh(msh, 2)
    grid = pyvista.UnstructuredGrid(topology, cell_types, geometry)
    plotter = pyvista.Plotter()
    num_local_cells = msh.topology.index_map(tdim).size_local
    grid.cell_data["Marker"] = cell_tags.values[cell_tags.indices < num_local_cells]
    grid.set_active_scalars("Marker")
    plotter.add_mesh(grid, show_edges=True)
    plotter.view_xy()
    if not pyvista.OFF_SCREEN:
        plotter.show()
    else:
        pyvista.start_xvfb()
        figure = plotter.screenshot("sphere_axis_mesh.png", window_size=[500, 500])

For the \(\hat{\rho}\) and \(\hat{z}\) components of the electric field, we will use Nedelec elements, while for the \(\hat{\phi}\) components we will use Lagrange elements:

degree = 3
curl_el = element("N1curl", msh.basix_cell(), degree, dtype=real_type)
lagr_el = element("Lagrange", msh.basix_cell(), degree, dtype=real_type)
V = fem.functionspace(msh, mixed_element([curl_el, lagr_el]))

The integration domains of our problem are the following:

# Measures for subdomains
dx = ufl.Measure("dx", msh, subdomain_data=cell_tags, metadata={"quadrature_degree": 5})
dDom = dx((au_tag, bkg_tag))
dPml = dx(pml_tag)

The \(\varepsilon_r\) function:

n_bkg = 1  # Background refractive index
eps_bkg = n_bkg**2  # Background relative permittivity
eps_au = -1.0782 + 1j * 5.8089

D = fem.functionspace(msh, ("DG", 0))
eps = fem.Function(D)
au_cells = cell_tags.find(au_tag)
bkg_cells = cell_tags.find(bkg_tag)
eps.x.array[au_cells] = np.full_like(au_cells, eps_au, dtype=eps.x.array.dtype)
eps.x.array[bkg_cells] = np.full_like(bkg_cells, eps_bkg, dtype=eps.x.array.dtype)
eps.x.scatter_forward()

For the background field, we need to specify the wavelength (and the wavevector), angle of incidence, harmonic numbers and the intensity:

wl0 = 0.4  # Wavelength of the background field
k0 = 2 * np.pi / wl0  # Wavevector of the background field
theta = np.pi / 4  # Angle of incidence of the background field
m_list = [0, 1]  # list of harmonics
I0 = 0.5 * n_bkg / Z0  # Intensity

where m_list contains the harmonic numbers we want to solve for. In general, we would need to solve for \(m\in \mathbb{Z}\). However, for sub-wavelength structure (as our sphere), we can limit the calculation to few harmonic numbers, e.g., \(m = -1, 0, 1\). Besides, we have that:

\[\begin{split} \begin{align} &J_{-m}=(-1)^m J_m \\ &J_{-m}^{\prime}=(-1)^m J_m^{\prime} \\ &j^{-m}=(-1)^m j^m \end{align} \end{split}\]

and therefore:

\[\begin{split} \begin{align} &E_{b, \rho}^{(m)}=E_{b, \rho}^{(-m)} \\ &E_{b, \phi}^{(m)}=-E_{b, \phi}^{(-m)} \\ &E_{b, z}^{(m)}=E_{b, z}^{(-m)} \end{align} \end{split}\]

In light of this, we can solve the problem for \(m\geq 0\).

We now now define eps_pml and mu_pml:

rho, z = ufl.SpatialCoordinate(msh)
alpha = 5
r = ufl.sqrt(rho**2 + z**2)

pml_coords = ufl.as_vector(
    (
        pml_coordinate(rho, r, alpha, k0, radius_dom, radius_pml),
        pml_coordinate(z, r, alpha, k0, radius_dom, radius_pml),
    )
)

eps_pml, mu_pml = create_eps_mu(pml_coords, rho, eps_bkg, 1)

Define other objects that will be used inside our solver loop:

# Total field
Eh_m = fem.Function(V)
Esh = fem.Function(V)

n = ufl.FacetNormal(msh)
n_3d = ufl.as_vector((n[0], n[1], 0))

# Geometrical cross section of the sphere, for efficiency calculation
gcs = np.pi * radius_sph**2

# Marker functions for the scattering efficiency integral
marker = fem.Function(D)
scatt_facets = facet_tags.find(scatt_tag)
incident_cells = mesh.compute_incident_entities(msh.topology, scatt_facets, tdim - 1, tdim)
msh.topology.create_connectivity(tdim, tdim)
midpoints = mesh.compute_midpoints(msh, tdim, incident_cells)
inner_cells = incident_cells[(midpoints[:, 0] ** 2 + midpoints[:, 1] ** 2) < (radius_scatt) ** 2]
marker.x.array[inner_cells] = 1

# Define integration domain for the gold sphere
dAu = dx(au_tag)

# Define integration facet for the scattering efficiency
dS = ufl.Measure("dS", msh, subdomain_data=facet_tags)

We also specify a variable phi, corresponding to the \(\phi\) angle of the cylindrical coordinate system that we will use to post-process the field and save it along the plane at \(\phi = \pi/4\). In particular, the scattered field needs to be transformed for \(m\neq 0\) in the following way:

\[\begin{split} \begin{align} &E_{s, \rho}^{(m)}(\phi)=E_{s, \rho}^{(m)}(e^{-jm\phi}+e^{jm\phi}) \\ &E_{s, \phi}^{(m)}(\phi)=E_{s, \phi}^{(m)}(e^{-jm\phi}-e^{jm\phi}) \\ &E_{s, z}^{(m)}(\phi)=E_{s, z}^{(m)}(e^{-jm\phi}+e^{jm\phi}) \end{align} \end{split}\]

For this reason, we also add a phase constant for the above phase term:

phi = np.pi / 4

# Initialize phase term
phase = fem.Constant(msh, scalar_type(np.exp(1j * 0 * phi)))

We now solve the problem:

for m in m_list:
    # Definition of Trial and Test functions
    Es_m = ufl.TrialFunction(V)
    v_m = ufl.TestFunction(V)

    # Background field
    Eb_m = fem.Function(V)
    f_rz = partial(background_field_rz, theta, n_bkg, k0, m)
    f_p = partial(background_field_p, theta, n_bkg, k0, m)
    Eb_m.sub(0).interpolate(f_rz)
    Eb_m.sub(1).interpolate(f_p)

    curl_Es_m = curl_axis(Es_m, m, rho)
    curl_v_m = curl_axis(v_m, m, rho)

    F = (
        -ufl.inner(curl_Es_m, curl_v_m) * rho * dDom
        + eps * k0**2 * ufl.inner(Es_m, v_m) * rho * dDom
        + k0**2 * (eps - eps_bkg) * ufl.inner(Eb_m, v_m) * rho * dDom
        - ufl.inner(ufl.inv(mu_pml) * curl_Es_m, curl_v_m) * rho * dPml
        + k0**2 * ufl.inner(eps_pml * Es_m, v_m) * rho * dPml
    )
    a, L = ufl.lhs(F), ufl.rhs(F)
    sys = PETSc.Sys()  # type: ignore
    use_superlu = PETSc.IntType == np.int64
    if sys.hasExternalPackage("mumps") and not use_superlu:  # type: ignore
        mat_factor_backend = "mumps"
    elif sys.hasExternalPackage("superlu_dist"):  # type: ignore
        mat_factor_backend = "superlu_dist"
    else:
        if msh.comm.size > 1:
            raise RuntimeError("This demo requires a parallel linear algebra backend.")
        else:
            mat_factor_backend = "petsc"
    problem = LinearProblem(
        a,
        L,
        bcs=[],
        petsc_options={
            "ksp_type": "preonly",
            "pc_type": "lu",
            "pc_factor_mat_solver_type": mat_factor_backend,
        },
    )
    Esh_m = problem.solve()
    assert problem.solver.getConvergedReason() > 0, "Solver did not converge!"

    # Scattered magnetic field
    Hsh_m = -1j * curl_axis(Esh_m, m, rho) / (Z0 * k0)

    # Total electric field
    Eh_m.x.array[:] = Eb_m.x.array[:] + Esh_m.x.array[:]

    # We can add our solution to the total scattered field, which also
    # includes the transformation to the $\phi$ plane:

    phase.value = np.exp(-1j * m * phi)
    rotate_to_phi = ufl.as_vector(
        (phase + ufl.conj(phase), phase + ufl.conj(phase), phase - ufl.conj(phase))
    )

    if m == 0:  # initialize and do not transform
        Esh = Esh_m
    elif m == m_list[0]:  # initialize and transform
        Esh = ufl.elem_mult(Esh_m, rotate_to_phi)
    else:  # transform
        Esh += ufl.elem_mult(Esh_m, rotate_to_phi)

    # To check that  our calculations are correct, we can use as reference
    # quantities the absorption and scattering efficiencies:

    # +
    # Efficiencies calculation
    if m == 0:  # initialize and do not add 2 factor
        P = np.pi * ufl.inner(-ufl.cross(Esh_m, ufl.conj(Hsh_m)), n_3d) * marker
        Q = np.pi * eps_au.imag * k0 * (ufl.inner(Eh_m, Eh_m)) / Z0
        q_abs_fenics_proc = (fem.assemble_scalar(fem.form(Q * rho * dAu)) / gcs / I0).real
        q_sca_fenics_proc = (
            fem.assemble_scalar(fem.form((P("+") + P("-")) * rho * dS(scatt_tag))) / gcs / I0
        ).real
        q_abs_fenics = msh.comm.allreduce(q_abs_fenics_proc, op=MPI.SUM)
        q_sca_fenics = msh.comm.allreduce(q_sca_fenics_proc, op=MPI.SUM)
    elif m == m_list[0]:  # initialize and add 2 factor
        P = 2 * np.pi * ufl.inner(-ufl.cross(Esh_m, ufl.conj(Hsh_m)), n_3d) * marker
        Q = 2 * np.pi * eps_au.imag * k0 * (ufl.inner(Eh_m, Eh_m)) / Z0 / n_bkg
        q_abs_fenics_proc = (fem.assemble_scalar(fem.form(Q * rho * dAu)) / gcs / I0).real
        q_sca_fenics_proc = (
            fem.assemble_scalar(fem.form((P("+") + P("-")) * rho * dS(scatt_tag))) / gcs / I0
        ).real
        q_abs_fenics = msh.comm.allreduce(q_abs_fenics_proc, op=MPI.SUM)
        q_sca_fenics = msh.comm.allreduce(q_sca_fenics_proc, op=MPI.SUM)
    else:  # do not initialize and add 2 factor
        P = 2 * np.pi * ufl.inner(-ufl.cross(Esh_m, ufl.conj(Hsh_m)), n_3d) * marker
        Q = 2 * np.pi * eps_au.imag * k0 * (ufl.inner(Eh_m, Eh_m)) / Z0 / n_bkg
        q_abs_fenics_proc = (fem.assemble_scalar(fem.form(Q * rho * dAu)) / gcs / I0).real
        q_sca_fenics_proc = (
            fem.assemble_scalar(fem.form((P("+") + P("-")) * rho * dS(scatt_tag))) / gcs / I0
        ).real
        q_abs_fenics += msh.comm.allreduce(q_abs_fenics_proc, op=MPI.SUM)
        q_sca_fenics += msh.comm.allreduce(q_sca_fenics_proc, op=MPI.SUM)

q_ext_fenics = q_abs_fenics + q_sca_fenics
# -

The quantities P and Q have an additional 2 factor for m != 0 due to parity.

We now compare the numerical and analytical efficiencies (he latter were obtained with the following routine provided by the scattnlay library):

from scattnlay import scattnlay

m = np.sqrt(eps_au)/n_bkg
x = 2*np.pi*radius_sph/wl0*n_bkg

q_sca_analyt, q_abs_analyt = scattnlay(np.array([x], dtype=np.complex128),
                                       np.array([m], dtype=np.complex128))[2:4]

The numerical values are reported here below:

q_abs_analyt = 0.9622728008329892
q_sca_analyt = 0.07770397394691526
q_ext_analyt = q_abs_analyt + q_sca_analyt

Finally, we can calculate the error and save our total scattered field:

# Error calculation
err_abs = np.abs(q_abs_analyt - q_abs_fenics) / q_abs_analyt
err_sca = np.abs(q_sca_analyt - q_sca_fenics) / q_sca_analyt
err_ext = np.abs(q_ext_analyt - q_ext_fenics) / q_ext_analyt

if MPI.COMM_WORLD.rank == 0:
    print()
    print(f"The analytical absorption efficiency is {q_abs_analyt}")
    print(f"The numerical absorption efficiency is {q_abs_fenics}")
    print(f"The error is {err_abs * 100}%")
    print()
    print(f"The analytical scattering efficiency is {q_sca_analyt}")
    print(f"The numerical scattering efficiency is {q_sca_fenics}")
    print(f"The error is {err_sca * 100}%")
    print()
    print(f"The analytical extinction efficiency is {q_ext_analyt}")
    print(f"The numerical extinction efficiency is {q_ext_fenics}")
    print(f"The error is {err_ext * 100}%")

# Check whether the geometrical and optical parameters are correct
# assert radius_sph / wl0 == 0.025 / 0.4
# assert eps_au == -1.0782 + 1j * 5.8089
# assert err_abs < 0.01
# assert err_sca < 0.01
# assert err_ext < 0.01

if has_vtx:
    v_dg_el = element("DG", msh.basix_cell(), degree, shape=(3,), dtype=real_type)
    W = fem.functionspace(msh, v_dg_el)
    Es_dg = fem.Function(W)
    Es_expr = fem.Expression(Esh, W.element.interpolation_points())
    Es_dg.interpolate(Es_expr)
    with VTXWriter(msh.comm, "sols/Es.bp", Es_dg) as f:
        f.write(0.0)