Time-Harmonic Maxwell Model¶

Introduction¶

The time-harmonic Maxwell model describes the behavior of electromagnetic field in terms of its complex amplitude, corresponding to a single frequency within the field's spectrum. This model is applicable in scenarios where the electromagnetic field has achieved a steady-state oscillation, meaning they vary periodically at a constant frequency without changing over time. By eliminating the time dependence of the fields, the model focuses solely on the spatial variations of the field's amplitude and phase. The time-harmonic Maxwell model is used to analyze time-harmonic fields in various applications, such as waveguides, antennas, and optical fibers. It is also relevant in electromagnetic scattering problems, making it valuable for the design of radio frequency (RF) and microwave devices. Expressed in terms of the electric field's complex amplitude, the time-harmonic Maxwell model solves the following equation:

\[ \begin{align} \nabla \times \left( \frac{1}{\mu} \nabla \times \tilde{\mathbf{E}} \right) - \omega^2 \varepsilon \tilde{\mathbf{E}} = -j \omega \tilde{\mathbf{J}}, \end{align} \]

where $\tilde{\mathbf{E}}$ is the complex amplitude of electric field at frequency $f$ (with $\omega=2\pi f$ being the angular frequency), $\mu$ is the magnetic permeability, $\varepsilon$ is the electric permittivity and $\tilde{\mathbf{J}}$ is the complex amplitude of the current density. In a conductive material, the current density can be expressed as $\tilde{\mathbf{J}} = \sigma \tilde{\mathbf{E}}$, where $\sigma$ represents the electric conductivity.

Materials¶

In the most general case, the material parameters $\mu$ and $\varepsilon$ in Eq. (1), as well as the electric conductivity $\sigma$, can be complex, frequency-dependent tensors that vary spatially and may also be depended on other physical properties, such as temperature. To provide a unified interface for specifying these material parameters, we introduce the concept of a material. A material is defined by a set of functions that represent the material parameters as functions of frequency, spatial coordinates, and other physical properties. The material functions can be defined using analytical expressions, tabulated data, or numerical models. For further details, please refer to General Material.

To simplify the setup of the most common materials, we offer a set of predefined material functions, listed below:

Predefined time-harmonic Maxwell materials
Name	Description
Constant	Creates a material with a user-defined constant relative magnetic permeability $\mu_\text{usr}$, relative electric permittivity $\varepsilon_\text{usr}$, and electric conductivity $\sigma_\text{usr}$: $$ \begin{align} \mu &= \mu_0\, \mu_\text{usr}, \\ \varepsilon &= \varepsilon_0\, \varepsilon_\text{usr}, \\ \sigma &= \sigma_\text{usr}, \end{align} $$ where $\mu_0$ is the vacuum magnetic permeability and $\varepsilon_0$ is the vacuum electric permittivity. For example, with $\mu_\text{usr}=1$, $\varepsilon_\text{usr}=1$, and $\sigma_\text{usr}=0$ this material can be used to define vacuum or air.

Conditions¶

Equation (1) is solved within a specific domain and must be accompanied by appropriate boundary conditions. The boundary conditions can be categorized into the following types:

List of supported conditions
Name	Supported Entities	Description
Absorbing	Boundary	Generally applied to the outer boundary of the computational domain, this condition effectively absorbs incoming radiation, preventing reflections and simulating an enclosure within infinite free space.
Lumped Port	Internal interface	Models an electromagnetic field source located within the computational domain, representing an attached transmission line or a voltage or current source applied between electrodes.
Perfect Electric Conductor (PEC)	Boundary	Applied at boundaries with conducting materials whose conductivity is assumed to be infinite.
Perfect Magnetic Conductor (PMC)	Boundary	An idealized boundary condition often used as a symmetry condition or as the dual counterpart of a perfect electric conductor condition.
Radiation	Boundary	Serves to model the irradiation of the entire computational domain by an external plane wave.
Surface Impedance	Boundary	Applied at the boundaries of well-conducting materials, whose conductivity, while high, cannot be considered infinite, and where the field penetrates the material to a non-negligible skin depth.
Waveguide Input Port	Boundary	Models the electromagnetic field entering the computational domain from an attached waveguide.
Waveguide Output Port	Boundary	Models the electromagnetic field exiting the computational domain through an attached waveguide.

Reports¶

The Time-harmonic Maxwell model provides the following reports that can be used to analyze and visualize the model data:

List of reports
Name	Type	Description
S-parameters	Complex Matrix	Returns a matrix of s-parameters for a given number of waveguide modes.
Far-Field Radiation Sensor	Sensor Object	Returns a sensor object that can be queried for various properties of far-field radiation.

Coefficients¶

The following functions are available in the time-harmonic Maxwell model for visualization or querying:

List of functions
Name	Field Type	Description
Electric Field-Real	Vector	The real part of the complex amplitude of electric field: $~ \mathrm{Re}(\tilde{\mathbf{E}}).$
Electric Field-Imag	Vector	The imaginary part of the complex amplitude of electric field: $~ \mathrm{Im}(\tilde{\mathbf{E}}).$
Magnetic Field-Real	Vector	The real part of the complex amplitude of magnetic field: $~ \mathrm{Re}(\tilde{\mathbf{H}}).$
Magnetic Field-Imag	Vector	The imaginary part of the complex amplitude of magnetic field: $~ \mathrm{Im}(\tilde{\mathbf{H}}).$
Poynting Vector	Vector	Time-averaged Poynting vector $\mathbf{S} = \frac{1}{2} \mathrm{Re}(\tilde{\mathbf{E}} \times \tilde{\mathbf{H}}^),$ where $^$ denotes the complex conjugate.