Electromagnetic scattering from a wire with scattering boundary conditions

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

Use complex quantities in FEniCSx
Setup and solve Maxwell’s equations
Implement Scattering Boundary Conditions

Equations, problem definition and implementation

First of all, let’s import the modules that will be used:

import sys
from typing import Tuple

from mpi4py import MPI

from analytical_efficiencies_wire import calculate_analytical_efficiencies
from mesh_wire import generate_mesh_wire

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

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

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

Since we want to solve time-harmonic Maxwell’s equation, we need to solve a complex-valued PDE, and therefore need to use PETSc compiled with complex numbers.

if not np.issubdtype(default_scalar_type, np.complexfloating):
    print("Demo should only be executed with DOLFINx complex mode")
    exit(0)

Now, let’s consider an infinite metallic wire immersed in a background medium (e.g. vacuum or water). Let’s now consider the plane cutting the wire perpendicularly to its axis at a generic point. Such plane \(\Omega=\Omega_{m} \cup\Omega_{b}\) is formed by the cross-section of the wire \(\Omega_m\) and the background medium \(\Omega_{b}\) surrounding the wire. Let’s consider just the portion of this plane delimited by an external circular boundary \(\partial \Omega\). We want to calculate the electric field \(\mathbf{E}_s\) scattered by the wire when a background wave \(\mathbf{E}_b\) impinges on it. We will consider a background plane wave at \(\lambda_0\) wavelength, that can be written analytically as:

\[ \mathbf{E}_b = \exp(\mathbf{k}\cdot\mathbf{r})\hat{\mathbf{u}}_p \]

with \(\mathbf{k} = \frac{2\pi}{\lambda_0}n_b\hat{\mathbf{u}}_k\) being the wavevector of the plane wave, pointing along the propagation direction, with \(\hat{\mathbf{u}}_p\) being the polarization direction, and with \(\mathbf{r}\) being a point in \(\Omega\). We will only consider \(\hat{\mathbf{u}}_k\) and \(\hat{\mathbf{u}}_p\) with components belonging to the \(\Omega\) domain and perpendicular to each other, i.e. \(\hat{\mathbf{u}}_k \perp \hat{\mathbf{u}}_p\) (transversality condition of plane waves). Using a Cartesian coordinate system for \(\Omega\), and by defining \(k_x = n_bk_0\cos\theta\) and \(k_y = n_bk_0\sin\theta\), with \(\theta\) being the angle defined by the propagation direction \(\hat{\mathbf{u}}_k\) and the horizontal axis \(\hat{\mathbf{u}}_x\), we have:

\[ \mathbf{E}_b = -\sin\theta e^{j (k_xx+k_yy)}\hat{\mathbf{u}}_x + \cos\theta e^{j (k_xx+k_yy)}\hat{\mathbf{u}}_y \]

The following class implements this functions. The inputs to the function are the angle \(\theta\), the background refractive index \(n_b\) and the vacuum wavevector \(k_0\).

class BackgroundElectricField:

    def __init__(self, theta: float, n_bkg: float, k0: complex):
        self.theta = theta  # incident angle
        self.k0 = k0  # vacuum wavevector
        self.n_bkg = n_bkg  # background refractive index

    def eval(self, x: np.typing.NDArray[np.float64]) -> Tuple[np.typing.NDArray[np.complex128],
                                                              np.typing.NDArray[np.complex128]]:
        kx = self.n_bkg * self.k0 * np.cos(self.theta)
        ky = self.n_bkg * self.k0 * np.sin(self.theta)
        phi = kx * x[0] + ky * x[1]
        ax, ay = np.sin(self.theta), np.cos(self.theta)
        return (-ax * np.exp(1j * phi), ay * np.exp(1j * phi))

The Maxwell’s equation for scattering problems takes the following form:

\[ -\nabla \times \nabla \times \mathbf{E}_s+\varepsilon_{r} k_{0}^{2} \mathbf{E}_s +k_{0}^{2}\left(\varepsilon_{r}-\varepsilon_{b}\right) \mathbf{E}_{\mathrm{b}}=0 \textrm{ in } \Omega, \]

where \(k_0 = 2\pi/\lambda_0\) is the vacuum wavevector of the background field, \(\varepsilon_b\) is the background relative permittivity and \(\varepsilon_r\) is the relative permittivity as a function of space, i.e.:

\[\begin{split} \varepsilon_r = \begin{cases} \varepsilon_m & \textrm{on }\Omega_m \\ \varepsilon_b & \textrm{on }\Omega_b \end{cases} \end{split}\]

with \(\varepsilon_m\) being the relative permittivity of the metallic wire. As reference values, we will consider \(\lambda_0 = 400\textrm{nm}\) (violet light), \(\varepsilon_b = 1.33^2\) (relative permittivity of water), and \(\varepsilon_m = -1.0782 + 5.8089\textrm{j}\) (relative permittivity of gold at \(400\textrm{nm}\)).

To form a well-determined system, we add boundary conditions on \(\partial \Omega\). It is common to use scattering boundary conditions (ref), which make the boundary transparent for \(\mathbf{E}_s\), allowing us to restrict the computational boundary to a finite \(\Omega\) domain. The first-order boundary conditions in the 2D case take the following form:

\[\mathbf{n} \times \nabla \times \mathbf{E}_s+\left(j k_{0}n_b + \frac{1}{2r} \right) \mathbf{n} \times \mathbf{E}_s \times \mathbf{n}=0\quad \textrm{ on } \partial \Omega, \]

with \(n_b = \sqrt{\varepsilon_b}\) being the background refractive index, \(\mathbf{n}\) being the normal vector to \(\partial \Omega\), and \(r = \sqrt{(x-x_s)^2 + (y-y_s)^2}\) being the distance of the \((x, y)\) point on \(\partial\Omega\) from the wire centered in \((x_s, y_s)\). We consider a wired centered at the origin, i.e. \(r =\sqrt{x^2 + y^2}\).

The radial distance function \(r(x)\) and \(\nabla \times\) operator for a 2D vector (in UFL syntax) is defined below.

def radial_distance(x: ufl.SpatialCoordinate):
    """Returns the radial distance from the origin"""
    return ufl.sqrt(x[0]**2 + x[1]**2)


def curl_2d(f: fem.Function):
    """Returns the curl of two 2D vectors as a 3D vector"""
    return ufl.as_vector((0, 0, f[1].dx(0) - f[0].dx(1)))

Next we define some mesh specific parameters. Please notice that the length units are normalized with respect to \(1\mu m\).

pi = np.pi
epsilon_0 = 8.8541878128 * 10**-12
mu_0 = 4 * pi * 10**-7

# Radius of the wire and of the boundary of the domain
radius_wire = 0.050
radius_dom = 1

# The smaller the mesh_factor, the finer is the mesh
mesh_factor = 1.2

# Mesh size inside the wire
in_wire_size = mesh_factor * 7.0e-3

# Mesh size at the boundary of the wire
on_wire_size = mesh_factor * 3.0e-3

# Mesh size in the background
bkg_size = mesh_factor * 60.0e-3

# Mesh size at the boundary
boundary_size = mesh_factor * 30.0e-3

# Tags for the subdomains
au_tag = 1          # gold wire
bkg_tag = 2         # background
boundary_tag = 3    # boundary

We generate the mesh using GMSH and convert it to a dolfinx.mesh.Mesh.

model = None
gmsh.initialize(sys.argv)
if MPI.COMM_WORLD.rank == 0:
    model = generate_mesh_wire(radius_wire, radius_dom, in_wire_size, on_wire_size, bkg_size,
                               boundary_size, au_tag, bkg_tag, boundary_tag)

model = MPI.COMM_WORLD.bcast(model, root=0)
domain, cell_tags, facet_tags = io.gmshio.model_to_mesh(model, MPI.COMM_WORLD, 0, gdim=2)

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

The mesh is visualized with PyVista

if have_pyvista:
    topology, cell_types, geometry = plot.vtk_mesh(domain, 2)
    grid = pyvista.UnstructuredGrid(topology, cell_types, geometry)
    plotter = pyvista.Plotter()
    num_local_cells = domain.topology.index_map(domain.topology.dim).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("wire_mesh.png", window_size=[8000, 8000])

Now we define some other problem specific parameters:

wl0 = 0.4  # Wavelength of the background field
n_bkg = 1.33  # Background refractive index
eps_bkg = n_bkg**2  # Background relative permittivity
k0 = 2 * np.pi / wl0  # Wavevector of the background field
theta = np.pi / 4  # Angle of incidence of the background field

We use a function space consisting of degree 3 Nedelec (first kind) elements to represent the electric field

degree = 3
curl_el = element("N1curl", domain.basix_cell(), degree)
V = fem.functionspace(domain, curl_el)

Next, we can interpolate \(\mathbf{E}_b\) into the function space \(V\):

f = BackgroundElectricField(theta, n_bkg, k0)
Eb = fem.Function(V)
Eb.interpolate(f.eval)

x = ufl.SpatialCoordinate(domain)
r = radial_distance(x)

# Create test and trial functions
Es = ufl.TrialFunction(V)
v = ufl.TestFunction(V)

# Definition of 3d fields for cross and curl operations
Es_3d = ufl.as_vector((Es[0], Es[1], 0))
v_3d = ufl.as_vector((v[0], v[1], 0))

# Measures for subdomains
dx = ufl.Measure("dx", domain, subdomain_data=cell_tags)
ds = ufl.Measure("ds", domain, subdomain_data=facet_tags)
dDom = dx((au_tag, bkg_tag))
dsbc = ds(boundary_tag)

# Normal to the boundary
n = ufl.FacetNormal(domain)
n_3d = ufl.as_vector((n[0], n[1], 0))

We turn our focus to the permittivity \(\varepsilon\). First, we define the relative permittivity \(\varepsilon_m\) of the gold wire at \(400nm\). This data can be found in Olmon et al. 2012 or at refractiveindex.info):

eps_au = -1.0782 + 1j * 5.8089

We define a permittivity function \(\varepsilon\) that takes the value of the gold permittivity \(\varepsilon_m\) for cells inside the wire, while it takes the value of the background permittivity otherwise:

D = fem.functionspace(domain, ("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()

Next we derive the weak formulation of the Maxwell’s equation plus with scattering boundary conditions. First, we take the inner products of the equations with a complex test function \(\mathbf{v}\), and integrate the terms over the corresponding domains:

\[\begin{split} & \int_{\Omega}-\nabla \times( \nabla \times \mathbf{E}_s) \cdot \bar{\mathbf{v}}+\varepsilon_{r} k_{0}^{2} \mathbf{E}_s \cdot \bar{\mathbf{v}}+k_{0}^{2}\left(\varepsilon_{r}-\varepsilon_b\right) \mathbf{E}_b \cdot \bar{\mathbf{v}}~\mathrm{d}x \\ +& \int_{\partial \Omega} (\mathbf{n} \times \nabla \times \mathbf{E}_s) \cdot \bar{\mathbf{v}} +\left(j n_bk_{0}+\frac{1}{2r}\right) (\mathbf{n} \times \mathbf{E}_s \times \mathbf{n}) \cdot \bar{\mathbf{v}}~\mathrm{d}s=0 \end{split}\]

By using \((\nabla \times \mathbf{A}) \cdot \mathbf{B}=\mathbf{A} \cdot(\nabla \times \mathbf{B})+\nabla \cdot(\mathbf{A} \times \mathbf{B}),\) we can change the first term into:

\[\begin{split} & \int_{\Omega}-\nabla \cdot(\nabla\times\mathbf{E}_s \times \bar{\mathbf{v}})-\nabla \times \mathbf{E}_s \cdot \nabla \times\bar{\mathbf{v}}+\varepsilon_{r} k_{0}^{2} \mathbf{E}_s \cdot \bar{\mathbf{v}}+k_{0}^{2}\left(\varepsilon_{r}-\varepsilon_b\right) \mathbf{E}_b \cdot \bar{\mathbf{v}}~\mathrm{dx} \\ +&\int_{\partial \Omega} (\mathbf{n} \times \nabla \times \mathbf{E}_s) \cdot \bar{\mathbf{v}} +\left(j n_bk_{0}+\frac{1}{2r}\right) (\mathbf{n} \times \mathbf{E}_s \times \mathbf{n}) \cdot \bar{\mathbf{v}}~\mathrm{d}s=0, \end{split}\]

using the divergence theorem \(\int_\Omega\nabla\cdot\mathbf{F}~\mathrm{d}x = \int_{\partial\Omega} \mathbf{F}\cdot\mathbf{n}~\mathrm{d}s\), we can write:

\[\begin{split} & \int_{\Omega}-(\nabla \times \mathbf{E}_s) \cdot (\nabla \times \bar{\mathbf{v}})+\varepsilon_{r} k_{0}^{2} \mathbf{E}_s \cdot \bar{\mathbf{v}}+k_{0}^{2}\left(\varepsilon_{r}-\varepsilon_b\right) \mathbf{E}_b \cdot \bar{\mathbf{v}}~\mathrm{d}x \\ +&\int_{\partial \Omega} -(\nabla\times\mathbf{E}_s \times \bar{\mathbf{v}})\cdot\mathbf{n} + (\mathbf{n} \times \nabla \times \mathbf{E}_s) \cdot \bar{\mathbf{v}} +\left(j n_bk_{0}+\frac{1}{2r}\right) (\mathbf{n} \times \mathbf{E}_s \times \mathbf{n}) \cdot \bar{\mathbf{v}}~\mathrm{d}s=0. \end{split}\]

Cancelling \(-(\nabla\times\mathbf{E}_s \times \bar{\mathbf{V}}) \cdot\mathbf{n}\) and \(\mathbf{n} \times \nabla \times \mathbf{E}_s \cdot \bar{\mathbf{V}}\) using the triple product rule \(\mathbf{A} \cdot(\mathbf{B} \times \mathbf{C})=\mathbf{B} \cdot(\mathbf{C} \times \mathbf{A})=\mathbf{C} \cdot(\mathbf{A} \times \mathbf{B})\), we get:

\[\begin{split} & \int_{\Omega}-(\nabla \times \mathbf{E}_s) \cdot (\nabla \times \bar{\mathbf{v}})+\varepsilon_{r} k_{0}^{2} \mathbf{E}_s \cdot \bar{\mathbf{v}}+k_{0}^{2}\left(\varepsilon_{r}-\varepsilon_b\right) \mathbf{E}_b \cdot \bar{\mathbf{v}}~\mathrm{d}x \\ +&\int_{\partial \Omega} \left(j n_bk_{0}+\frac{1}{2r}\right)( \mathbf{n} \times \mathbf{E}_s \times \mathbf{n}) \cdot \bar{\mathbf{v}} ~\mathrm{d} s = 0. \end{split}\]

We use the UFL to implement the residual

# Weak form
F = - ufl.inner(ufl.curl(Es), ufl.curl(v)) * dDom \
    + eps * (k0**2) * ufl.inner(Es, v) * dDom \
    + (k0**2) * (eps - eps_bkg) * ufl.inner(Eb, v) * dDom \
    + (1j * k0 * n_bkg + 1 / (2 * r)) \
    * ufl.inner(ufl.cross(Es_3d, n_3d), ufl.cross(v_3d, n_3d)) * dsbc

We split the residual into a sesquilinear (lhs) and linear (rhs) form and solve the problem. We store the scattered field \(\mathbf{E}_s\) as Esh:

a, L = ufl.lhs(F), ufl.rhs(F)
problem = LinearProblem(a, L, bcs=[], petsc_options={"ksp_type": "preonly", "pc_type": "lu"})
Esh = problem.solve()

We save the solution as an ADIOS2 bp folder. In order to do so, we need to interpolate our solution discretized with Nedelec elements into a suitable discontinuous Lagrange space.

gdim = domain.geometry.dim
V_dg = fem.functionspace(domain, ("Discontinuous Lagrange", degree, (gdim,)))
Esh_dg = fem.Function(V_dg)
Esh_dg.interpolate(Esh)

with io.VTXWriter(domain.comm, "Esh.bp", Esh_dg) as vtx:
    vtx.write(0.0)

We visualize the solution using PyVista. For more information about saving and visualizing vector fields discretized with Nedelec elements, check this DOLFINx demo.

if have_pyvista:
    V_cells, V_types, V_x = plot.vtk_mesh(V_dg)
    V_grid = pyvista.UnstructuredGrid(V_cells, V_types, V_x)
    Esh_values = np.zeros((V_x.shape[0], 3), dtype=np.float64)
    Esh_values[:, : domain.topology.dim] = Esh_dg.x.array.reshape(V_x.shape[0], domain.topology.dim).real

    V_grid.point_data["u"] = Esh_values

    plotter = pyvista.Plotter()
    plotter.add_text("magnitude", font_size=12, color="black")
    plotter.add_mesh(V_grid.copy(), show_edges=True)
    plotter.view_xy()
    plotter.link_views()
    if not pyvista.OFF_SCREEN:
        plotter.show()
    else:
        pyvista.start_xvfb()
        plotter.screenshot("Esh.png", window_size=[800, 800])

Next we can calculate the total electric field \(\mathbf{E}=\mathbf{E}_s+\mathbf{E}_b\) and save it.

E = fem.Function(V)
E.x.array[:] = Eb.x.array[:] + Esh.x.array[:]
E_dg = fem.Function(V_dg)
E_dg.interpolate(E)
with io.VTXWriter(domain.comm, "E.bp", E_dg) as vtx:
    vtx.write(0.0)

We validate our numerical solution by computing the absorption, scattering and extinction efficiencies, which are quantities that define how much light is absorbed and scattered by the wire. First of all, we calculate the analytical efficiencies with the calculate_analytical_efficiencies function defined in a separate file:

# Calculation of analytical efficiencies
q_abs_analyt, q_sca_analyt, q_ext_analyt = calculate_analytical_efficiencies(
    eps_au, n_bkg, wl0, radius_wire)

Now we can calculate the numerical efficiencies. The formula for the absorption, scattering and extinction are:

\[\begin{split} & Q_{abs} = \operatorname{Re}\left(\int_{\Omega_{m}} \frac{1}{2} \frac{\operatorname{Im}(\varepsilon_m)k_0}{Z_0n_b} \mathbf{E}\cdot\hat{\mathbf{E}}dx\right) \\ & Q_{sca} = \operatorname{Re}\left(\int_{\partial\Omega} \frac{1}{2} \left(\mathbf{E}_s\times\bar{\mathbf{H}}_s\right) \cdot\mathbf{n}ds\right)\\ \\ & Q_{ext} = Q_{abs} + Q_{sca}, \end{split}\]

with \(Z_0 = \sqrt{\frac{\mu_0}{\varepsilon_0}}\) being the vacuum impedance, and \(\mathbf{H}_s = -j\frac{1}{Z_0k_0n_b}\nabla\times\mathbf{E}_s\) being the scattered magnetic field. We can then normalize these values over the intensity of the electromagnetic field \(I_0\) and the geometrical cross section of the wire, \(\sigma_{gcs} = 2r_w\):

\[\begin{split} & q_{abs} = \frac{Q_{abs}}{I_0\sigma_{gcs}} \\ & q_{sca} = \frac{Q_{sca}}{I_0\sigma_{gcs}} \\ & q_{ext} = q_{abs} + q_{sca}. \end{split}\]

We can calculate these values in the following way:

# Vacuum impedance
Z0 = np.sqrt(mu_0 / epsilon_0)

# Magnetic field H
Hsh_3d = -1j * curl_2d(Esh) / (Z0 * k0 * n_bkg)

Esh_3d = ufl.as_vector((Esh[0], Esh[1], 0))
E_3d = ufl.as_vector((E[0], E[1], 0))

# Intensity of the electromagnetic fields I0 = 0.5*E0**2/Z0 E0 =
# np.sqrt(ax**2 + ay**2) = 1, see background_electric_field
I0 = 0.5 / Z0

# Geometrical cross section of the wire
gcs = 2 * radius_wire

# Quantities for the calculation of efficiencies
P = 0.5 * ufl.inner(ufl.cross(Esh_3d, ufl.conj(Hsh_3d)), n_3d)
Q = 0.5 * np.imag(eps_au) * k0 * (ufl.inner(E_3d, E_3d)) / Z0 / n_bkg

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

# Normalized absorption efficiency
q_abs_fenics_proc = (fem.assemble_scalar(fem.form(Q * dAu)) / gcs / I0).real
q_abs_fenics = domain.comm.allreduce(q_abs_fenics_proc, op=MPI.SUM)

# Normalized scattering efficiency
q_sca_fenics_proc = (fem.assemble_scalar(fem.form(P * dsbc)) / gcs / I0).real
q_sca_fenics = domain.comm.allreduce(q_sca_fenics_proc, op=MPI.SUM)

# Extinction efficiency
q_ext_fenics = q_abs_fenics + q_sca_fenics

# 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

# Check if errors are smaller than 1%
assert err_abs < 0.01
assert err_sca < 0.01
assert err_ext < 0.01

if domain.comm.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}%")