<h1> Introduction </h1>
-A surface plasmon-polariton (SPP) is a slowly decaying electromagnetic
-wave, confined near a metal-air (or similar) interface. SPP structures on
+A surface plasmon-polariton (SPP) is a slowly decaying electromagnetic wave
+that is confined near a metal-air (or similar) interface. SPP structures on
novel "2D" materials such as graphene, a monoatomic layer of carbon atoms
arranged in a hexagonal lattice, typically have wavelengths much shorter
than the wavelength of the free-space radiation. This scale separation
\\
\frac{\partial}{\partial t} (\varepsilon\mathbf{E}) - \nabla\times(\mu^{-1}\mathbf{H}) &= - \mathbf{J}_a,
\\
- \nabla\cdot(\varepsilon\mathbf{E}) &= \rho_m,
+ \nabla\cdot(\varepsilon\mathbf{E}) &= \rho_m.
@f}
-in which $\nabla\times$ is the curl operator, $\nabla\cdot$ is the divergence operator,
-$\varepsilon$ is the
+Here, $\nabla\times$ is the curl operator, $\nabla\cdot$ is the divergence
+operator, $\varepsilon$ is the
<a href="https://en.wikipedia.org/wiki/Permittivity">electric permittivity</a>,
$\mu$ is the
<a href="https://en.wikipedia.org/wiki/Permeability">magnetic permeability</a>,
such that $\sqrt{\mu_r^{-1}\varepsilon_r}$ is real valued and strictly positive
in $\partial\Omega$.
-$\mathbf{H}(curl;\Omega)$ is space of vector-valued, measurable and square
+$\mathbf{H}(\text{curl};\Omega)$ is space of vector-valued, measurable and square
integrable functions whose weak curl admits a representation by a
square integrable function. Define a Hilbert space
@f[
-X(\Omega) = \{\varphi \in \mathbf{H}(curl;\Omega)\;\;:\;\; \varphi_T|_{\Sigma}
+X(\Omega) = \{\varphi \in \mathbf{H}(\text{curl};\Omega)\;\;:\;\; \varphi_T|_{\Sigma}
\in L^2(\Sigma)^2,\;\varphi_T|_{\partial\Omega} \in L^2(\partial\Omega)^2\}
@f]
equipped with the norm
the discretized variational problem is:
@f[
- \text{Find a unique } \varphi_i \in X_h(\Omega) \text{ such that, for all } \varphi_j \in X_h(\Omega),
+ \text{Find a unique } \mathbf{E}_h = \sum_j U_j\mathbf{\varphi}_j \in
+ X_h(\Omega) \text{ such that}
@f]
@f[
-A_{ij} = F_i
+\sum_jA_{ij}U_j = F_i\qquad\text{for all }i.
@f]
<h3>Perfectly Matched Layer</h3>
// @sect3{Class Template Declarations}
+//
// We begin our actual implementation by declaring all classes with their
// data structures and methods upfront.
// the interface between two materials. If we are at an interface, we assign
// the i^th diagonal element of the tensor to the private sigma_ value.
- // J_a is the strength and orientation of the dipole. As mentioned in the rescaling,
+ // J_a is the strength and orientation of the dipole. As mentioned in the
+ // rescaling,
// @f[
// \mathbf{J}_a = J_0 e_i\delta(x-a)
// @f]
// It is a rank 1 tensor
- // that depends on the private dipole_position_, dipole_radius_,
- // dipole_strength_, dipole_orientation_ variables.
+ // that depends on the private dipole_position, dipole_radius,
+ // dipole_strength, dipole_orientation variables.
template <int dim>
class Parameters : public ParameterAcceptor
}
// @sect4{PerfectlyMatchedLayer Class}
- // The PerfectlyMatchedLayer class inherits ParameterAcceptor,
- // and it modifies our coefficients from Parameters.
- // The radii and the strength of the PML is specified, and the
- // coefficients will be modified using transformation
- // matrices within the PML region. The radii and strength of
- // the PML are editable through a .prm file
- // The rotation function is the $T_{exer}$ mentioned in the
- // perfectly matched layer section of the introduction.
- // Moreover, the matrices A, B and C are defined as mentioned
+ // The PerfectlyMatchedLayer class inherits ParameterAcceptor as well. It
+ // implements the transformation matrices used to modify the permittivity
+ // and permeability tensors supplied from the Parameters class. The
+ // actual transformation of the material tensors will be done in the
+ // assembly loop. The radii and the strength of the PML is specified, and
+ // the coefficients will be modified using transformation matrices within
+ // the PML region. The radii and strength of the PML are editable through
+ // a .prm file. The rotation function $T_{exer}$ is the same as
+ // introduced in the perfectly matched layer section of the introduction.
+ // Similarly, the matrices A, B and C are defined as follows
// @f[
// A = T_{e_xe_r}^{-1}
// \text{diag}\left(\frac{1}{\bar{d}^2},\frac{1}{d\bar{d}}\right)T_{e_xe_r},\qquad
// @sect4{Maxwell Class}
- // At this point we are ready to instantiate all the major functions of
- // the finite element program and also a list of variables. Most of these
- // an exact copy of the functions in the tutorial programs. In addition,
- // we instantiate the parameters and the perfectly matched layer. The
- // default values of these parameters are set to show us a standing wave
- // with absorbing boundary conditions and a PML.
+ // At this point we are ready to declare all the major building blocks of
+ // the finite element program which consists of the usual setup and
+ // assembly routines. Most of the structure has already been introduced
+ // in previous tutorial programs. The Maxwell class also holds private
+ // instances of the Parameters and PerfectlyMatchedLayers classes
+ // introduced above. The default values of these parameters are set to
+ // show us a standing wave with absorbing boundary conditions and a PML.
template <int dim>
class Maxwell : public ParameterAcceptor
Vector<double> system_rhs;
};
-
+ // @sect3{Class Template Definitions and Implementation}
+ //
// @sect4{The Constructor}
- // The Constructor simply consists specifications for the mesh
- // and the order of the finite elements. These are editable through
- // the .prm file. The absorbing_boundary boolean can be modified to
- // remove the absorbing boundary conditions (in which case our boundary
- // would be perfectly conducting).
+ // The Constructor simply consists of default initialization a number of
+ // discretization parameters (such as the domain size, mesh refinement,
+ // and the order of finite elements and quadrature) and declaring a
+ // corresponding entry via ParameterAcceptor::add_parameter(). All of
+ // these can be modified by editing the .prm file. Absorbing boundary
+ // conditions can be controlled with the absorbing_boundary boolean. If
+ // absorbing boundary conditions are disabled we simply enforce
+ // homogeneous Dirichlet conditions on the tangential component of the
+ // electric field. In the context of time-harmonic Maxwell's equations
+ // these are also known as \emph{perfectly conducting boundary
+ // conditions}.
template <int dim>
Maxwell<dim>::Maxwell()
fe = std::make_unique<FESystem<dim>>(FE_NedelecSZ<dim>(fe_order), 2);
}
- // Make the mesh for the domain and generate the triangulation required.
- // Additionally, there is an interface added here to visualize
- // a standing wave. To generate a solution without any interface,
- // comment out lines 455-459.
+ // The Maxwell::make_grid() routine creates the mesh for the
+ // computational domain which in our case is a scaled square domain.
+ // Additionally, a material interface is introduced by setting the
+ // material id of the upper half ($y>0$) to 1 and of the lower half
+ // ($y<0$) of the computational domain to 2.
template <int dim>
void Maxwell<dim>::make_grid()
<< std::endl;
}
- // Enumerate all the degrees of freedom and set up matrix and vector
- // objects to hold the system data. Enumerating is done by using
+ // The Maxwell::setup_system() routine follows the usual routine of
+ // enumerating all the degrees of freedom and setting up the matrix and
+ // vector objects to hold the system data. Enumerating is done by using
// DoFHandler::distribute_dofs().
template <int dim>
// Now we assemble the face and the boundary. The following loops will
// assemble
- // //\f{align*}{
+ // \f{align*}{
// - i\int_\Sigma (\sigma_r^{\Sigma}(\varphi_i)_T) \cdot
// (\bar{\varphi}_j)_T\text{do}x \f} and \f{align}{
// - i\int_{\partial\Omega} (\sqrt{\mu_r^{-1}\varepsilon}(\varphi_i)_T)
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