Time-Harmonic Maxwell Model

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 applies in situations where the electromagnetic fields have reached a steady-state oscillation, meaning they vary periodically at a constant frequency without changing over time. The time-harmonic Maxwell model eliminates the time dependence of the fields, focusing solely on the spatial variations of the field’s amplitude and phase. The time-harmonic Maxwell model can be used to analyze time-harmonic fields in waveguides, antennas, and optical fibers, as well as in electromagnetic scattering problems, and thus can be applied in 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:

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

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 written as \(\tilde{\mathbf{J}} = \sigma \tilde{\mathbf{E}}\), where \(\sigma\) is 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, and spatially varying tensors, and may also depend on other physical properties, such as temperature. In order to provide a unified interface for specifying such material parameters, we introduce the concept of a material. A material is defined by a set of functions that introduce the material parameters as a function of frequency, space, and other physical properties. The material functions can be defined in the form of analytical expressions, tabulated data, or numerical models. For more details, please refer to Time-Harmonic Maxwell Material.

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

Predefined time-harmonic Maxwell materials
Name	Description
Constant	Creates a material with constant relative magnetic permeability \(\mu_\text{usr}\), relative electric permittivity \(\varepsilon_\text{usr}\), and electric conductivity \(\sigma_\text{usr}\): \[\begin{split}\begin{aligned} \mu &= \mu_0 \mu_\text{usr}, \\ \varepsilon &= \varepsilon_0 \varepsilon_\text{usr}, \\ \sigma &= \sigma_\text{usr}, \end{aligned}\end{split}\] 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 of the following types:

List of supported conditions
Name	Supported Entities	Description
Perfect electric conductor (PEC)	Boundary	Under PEC boundary condition the tangential component of the electric field amplitude at the boundary is set to zero: \[\tilde{\mathbf{E}} \times \hat{\mathbf{n}} = 0,\] where \(\hat{\mathbf{n}}\) is the unit vector normal to the boundary.
Surface Impedance	Boundary	Under surface impedance boundary condition the amplitude of electric field at the boundary satisfies the following equation: \[\hat{\mathbf{n}} \times \left(\frac{1}{\mu} \nabla \times \tilde{\mathbf{E}}\right) = -i \frac{\omega}{Z_s} \left[\hat{\mathbf{n}} \times \left(\tilde{\mathbf{E}} \times \hat{\mathbf{n}}\right)\right],\] where \(\hat{\mathbf{n}}\) is the unit vector normal to the boundary and \(Z_s = \sqrt{\mu_s/(\varepsilon_s + i\sigma_s/\omega)}\) is the surface impedance with \(\mu_s\), \(\varepsilon_s\), and \(\sigma_s\) being the permeability, permittivity, and conductivity of the boundary material.
Input port	Boundary	Under the input port boundary condition the amplitude of electric field at the boundary satisfies the following equation: \[\hat{\mathbf{n}} \times \left(\frac{1}{\mu} \nabla \times \tilde{\mathbf{E}}\right) = -\frac{i\beta}{\mu} \hat{\mathbf{n}} \times \left(\tilde{\mathbf{E}} \times \hat{\mathbf{n}}\right) +\frac{2i\beta}{\mu} \hat{\mathbf{n}} \times \left(\tilde{\mathbf{E}}^\text{in} \times \hat{\mathbf{n}}\right),\] where \(\hat{\mathbf{n}}\) is the unit vector normal to the boundary, \(\tilde{\mathbf{E}}^\text{in}\) is the complex amplitude of the input electric field being in a waveguide mode configuration with the mode propagation constant \(\beta\).
Output port	Boundary	Under the output port boundary condition the amplitude of electric field at the boundary satisfies the following equation: \[\hat{\mathbf{n}} \times \left(\frac{1}{\mu} \nabla \times \tilde{\mathbf{E}}\right) = -\frac{i\beta}{\mu} \hat{\mathbf{n}} \times \left(\tilde{\mathbf{E}} \times \hat{\mathbf{n}}\right),\] where \(\hat{\mathbf{n}}\) is the unit vector normal to the boundary and \(\beta\) is the propagation constant of the propagating waveguide mode.

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 Report	Complex Matrix	Returns a matrix of s-parameters for a given number of waveguide modes

Coefficient Functions

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

List of functions
Name	Type	Description
Electric Field Real	Vector Field	The real part of the complex amplitude of electric field: \[\mathrm{Re}\left\{\tilde{\mathbf{E}}\right\}\]
Electric Field Imag	Vector Field	The imaginary part of the complex amplitude of electric field: \[\mathrm{Im}\left\{\tilde{\mathbf{E}}\right\}\]
Magnetic Field Real	Vector Field	The real part of the complex amplitude of magnetic field: \[\mathrm{Re}\left\{\tilde{\mathbf{H}}\right\}\]
Magnetic Field Imag	Vector Field	The imaginary part of the complex amplitude of magnetic field: \[\mathrm{Im}\left\{\tilde{\mathbf{H}}\right\}\]
Curl Electric Field Real	Vector Field	The real part of the curl of the complex amplitude of electric field: \[\mathrm{Re}\left\{\nabla \times \tilde{\mathbf{E}}\right\}\]
Curl Electric Field Imag	Vector Field	The imaginary part of the curl of the complex amplitude of electric field: \[\mathrm{Im}\left\{\nabla \times \tilde{\mathbf{E}}\right\}\]
Poynting Vector	Vector Field	Time-averaged Poynting vector: \[S = \frac{1}{2} \mathrm{Re}\left\{\tilde{\mathbf{E}} \times \tilde{\mathbf{H}}^\right\},\] where \(^\) denotes the complex conjugate.
Electromagnetic Energy Density	Scalar Field	Electromagnetic energy density: \[u = \frac{1}{4} \left(\varepsilon \|\tilde{\mathbf{E}}\|^2 + \mu \|\tilde{\mathbf{H}}\|^2\right)\]
Complex Magnetic Permeability	Scalar Field	Complex magnetic permeability: \[\mu\]
Complex Magnetic Reluctivity	Scalar Field	Complex magnetic reluctivity: \[\nu = \frac{1}{\mu}\]
Complex Electric Permittivity	Scalar Field	Complex electric permittivity: \[\varepsilon\]
Complex Electric Conductivity	Scalar Field	Complex electric conductivity: \[\sigma\]