COMSOL 6.2 - The Electromagnetic Waves, Frequency Domain Interface

The interface (

) is used to solve for time-harmonic electromagnetic field distributions. It is found under the branch (

) when adding a physics interface.

For this physics interface, the maximum mesh element size should be limited to a fraction of the wavelength. The domain size that can be simulated thus scales with the amount of available computer memory and the wavelength. The physics interface supports the Frequency Domain, Wavelength Domain, Eigenfrequency, Mode Analysis, and Boundary Mode Analysis study types. The Frequency Domain and Wavelength Domain study types are used for source driven simulations for a single frequency/wavelength or a sequence of frequencies/wavelengths. The Eigenfrequency study type is used to find resonance frequencies and their associated eigenmodes in resonant cavities.

This physics interface solves the time-harmonic wave equation for the electric field.

When this physics interface is added, these default nodes are also added to the — , , and . Then, from the toolbar, add other nodes that implement, for example, boundary conditions. You can also right-click to select physics features from the context menu.

The Mode analysis study type is applicable only for 2D and 2D axisymmetric cross sections of waveguides and transmission lines where it is used to find allowed propagating modes. Boundary mode analysis is used for the same purpose in 2D, 2D axisymmetry, and 3D and applies to boundaries representing waveguide ports.

The physics-controlled mesh is controlled from the window for the node (if the is ). In the table in the section, find the physics interface in the column and select or clear the check box in the column on the same row for enabling (the default) or disabling contributions from the physics interface to the physics-controlled mesh.

When the check box for the physics interface is selected, this invokes a parameter for the maximum mesh element size in free space. The physics-controlled mesh automatically scales the maximum mesh element size as the wavelength changes in different dielectric and magnetic regions. If the model is configured by any periodic conditions, identical meshes are generated on each pair of periodic boundaries. Perfectly matched layers are built with a structured mesh, specifically, a swept mesh in 3D and a mapped mesh in 2D.

When the check box is selected for the physics interface, in the section for the physics interface below the table, choose one of the four options for the — (the default), , , or . When is selected, 1/8 in 2D or 1/5 in 3D of the vacuum wavelength from the highest frequency defined in the study step is used for the maximum mesh element size. For the option , enter a suitable . For example, 1/5 of the vacuum wavelength or smaller. When is selected, enter the highest frequency intended to be used during the simulation. The maximum mesh element size in free space is 1/8 in 2D and 1/5 in 3D of the vacuum wavelength for the entered frequency. For the option, enter the smallest vacuum wavelength intended to be used during the simulation. The maximum mesh element size in free space is 1/8 in 2D and 1/5 in 3D of the entered wavelength.

The maximum mesh element sizes discussed above are used with quadratic shape functions. When linear shape functions are used, 1/2 of the maximum mesh element size for quadratic shape functions are used. Similarly, when cubic shape functions are used, the maximum mesh element size is 2.25 times the maximum mesh element size for quadratic shape functions.

Furthermore, for Port features, the maximum mesh element size can be slightly finer than what is discussed above.

When is selected, the outer boundaries of lossy media domains are meshed with a maximum mesh element size in free space given by the minimum value of half a skin depth and 1/5 of the vacuum wavelength.

The maximum mesh element size in dielectric media is equal to the maximum mesh element size in vacuum divided by the square root of the product of the relative permittivity and permeability.

The material property can be defined by a function, if the function use a single input argument, for example called freq, and this variable is also marked as a Frequency Model input in the material property group node where the function is used.

	In the COMSOL Multiphysics Reference Manual see the Physics-Controlled Mesh section for more information about how to define the physics-controlled mesh.

The is the default physics interface name.

The is used primarily as a scope prefix for variables defined by the physics interface. Refer to such physics interface variables in expressions using the pattern <name>.<variable_name>. In order to distinguish between variables belonging to different physics interfaces, the name string must be unique. Only letters, numbers, and underscores (_) are permitted in the field. The first character must be a letter.

The default (for the first physics interface in the model) is ewfd.

From the list, select whether to solve for the (the default) or the .

For select a according to the following table:

Table 3-1: Background wave type based on component dimension.
Component	Background wave type
2D	User defined (default), Gaussian beam
2D Axisymmetric	User defined (default), Circularly polarized plane wave, Linearly polarized plane wave
3D	User defined (default), Gaussian beam, Linearly polarized plane wave

Enter the component expressions for the Eb (SI unit: V/m). The entered expressions must be differentiable.

	Notice that expressions including coupling operators are not differentiable and cannot be used as background fields.

For select the — (the default) or .

When selecting , the Gaussian beam background field is a solution to the paraxial wave equation, which is an approximation to the Helmholtz equation solved for by the interface. The approximation is valid for Gaussian beams that have a beam radius that is much larger than the wavelength. Since the paraxial Gaussian beam background field is an approximation to the Helmholtz equation, for tightly focused beams, you can get a nonzero scattered field solution, even if you do not have any scatterers. The option means that the electric field for the Gaussian beam is approximated by an expansion of the electric field into a number of plane waves. Since each plane wave is a solution to the Helmholtz equation, the plane wave expansion of the electric field is also a solution to the Helmholtz equation. Thus, this option can be used also for tightly focused Gaussian beams.

If the beam spot radius is smaller than the wavelength, evanescent plane waves need to be included in the expansion. The evanescent waves decay exponentially in the propagation direction, why it only makes sense to model such tightly focused beams if the focal plane coincides with the input boundary. If the focal plane is located inside the modeled domain, the field can be dominated by the exponentially decaying evanescent waves. Those waves can have a very high field strength before the focal plane even though they only provide a small contribution to the field at the focal plane.

For select — (the default) or . For also check , to include evanescent waves in the plane wave expansion. For also enter values for the Nk (the default value is 13) and kt,max (SI unit: rad/m, default value is (2*(sqrt(2*log(10))))/ewfd.w0). Use an odd number for the Nk to make sure that a wave vector pointing in the main propagation direction is included in the plane-wave expansion. The Nk specifies the number of wave vectors that will be included per transverse dimension. So for 3D the total number of wave vectors will be Nk·Nk.

	Evanescent waves are included in the plane wave expansion if the Maximum transverse wave number kt,max is larger that the specified Wave number k. When the Wave vector distribution type is set to Automatic, evanescent waves are included in the expansion if the Allow evanescent waves check box is selected.

A plane wave expansion with a finite number of plane waves included will make the field periodic in the plane orthogonal to the main propagation direction. If the separation between the transverse wave vector components, given by 2kt,max/(Nk − 1), is too small, replicas of the Gaussian beam background field can appear. To avoid that, increase the value for the Nk.

The number of plane waves included in the expansion can be quite large, especially for 3D. For instance, using the default settings, 2·13·13 = 338 plane waves will be included (the factor 2 accounts for the two possible polarizations for each wave vector). Thus, initializing the plane-wave expansion for the Gaussian beam background field can take some time in 3D.

For more information about the Gaussian beam theory, see Gaussian Beams as Background Fields and Input Fields.

Define the Gaussian beam background field using the parameters below:

•

Enter the component expressions for the ETbg0 (SI unit: V/m) if the is . Notice that this is the transverse Gaussian beam amplitude in the focal plane. When the is set to the background field is always orthogonal (transverse) to . However, when the is set to , the background field amplitude can also have a component in the propagation direction. Specify here only the field amplitude components that are orthogonal to the propagation direction. COMSOL computes automatically the component in the propagation direction, if needed.

•

If the is set to , enter the (SI unit: W in 2D axisymmetry and 3D and W/m in 2D) and the component expressions for the ETbg0 (SI unit: V/m).

•

Enter a k (SI unit: rad/m). The default is ewfd.k0 rad/m. The wave number must evaluate to a value that is the same for all the domains the scattered field is applied to. Setting the k to a positive value, means that the wave is propagating in the positive x-, y-, or z-axis direction, whereas setting the k to a negative value means that the wave is propagating in the negative x-, y-, or z-axis direction.

	Nanorods: Application Library path Wave_Optics_Module/Optical_Scattering/nanorods demonstrates how to set up the Gaussian background field, based on the plane-wave expansion.

The initial background wave is predefined as E0 = exp(−jkxx)z. This field is transformed by three successive rotations along the roll, pitch, and yaw angles, in that order. For a graphic representation of the initial background field and the definition of the three rotations compare with Figure 3-1 below.

•

Enter a (SI unit: rad), which is a right-handed rotation with respect to the +y direction. The default is 0 rad, corresponding to the initial direction of propagation pointing in the +x direction.

Figure 3-1: Schematic of the directions for the wave vector k, the electric field E0, and the roll, pitch, and yaw rotations. The top image represents an initial wave propagating in the x direction with a polarization along the z direction.

The background wave is defined as

m is the azimuthal mode number (+1 or −1) varying depending on the and settings, and

and

are the unit vectors in the r and φ directions, respectively.

Linearly polarized plane wave can be written as the sum of an infinite number of azimuthal modes in cylindrical coordinates. Thus, the option available for 2D axisymmetric components enables fast simulation of scattering problem for body-of-revolution geometries. For a graphic representation of the initial background field and the definition of the three rotations have a look at Figure 3-2 below. To define a linearly polarized plane wave with arbitrary angle of incidence θ and polarization α:

•

Click the button. This button creates a parameter modeNum, which will be used as the azimuthal mode number, and a parameter highestMode, which is the highest mode number used in the expansion. Then the auxiliary sweep in the first frequency domain study step under the first study will be enabled and a sweep over modeNum will be added.

Figure 3-2: Schematic of the directions for the wave vector k, the electric field E0, the polarization angle α, and the incident angle θ. The top image represents an initial wave propagating in the -z direction with a polarization along the r direction.

	Cloaking of a Cylindrical Scatterer with Graphene: Application Library path Wave_Optics_Module/Optical_Scattering/cylinder_graphene_cloak demonstrates how to set up the Linearly polarized plane wave background field.

This section is available for 2D and 2D axisymmetric components.

Select the — , , or . Select:

This section is available for 2D and 2D axisymmetric components, when solving for or.

For 2D components, assign a wave vector component to the field. For 2D axisymmetric components, assign an integer constant or an integer parameter expression to the field.

Select the check box to enable the port sweep. When selected, this invokes a parametric sweep over the ports in addition to the frequency/wavelength sweep already added. The generated lumped parameters are in the form of an S-parameter matrix.

For enter a to assign a specific name to the parameter that controls the port number solved for during the sweep. Before making the port sweep, the parameter must also have been added to the list of parameters in the section of the node under the node. This process can be automated by clicking the button. The button helps add a necessary port sweep parameter and a study step in the last study node. If there is already a study step, the sweep settings are adjusted for the port sweep.

	In the COMSOL Multiphysics Reference Manual see the Frequency Domain Source Sweep section for a discussion of how to use the Frequency Domain Source Sweep study type to perform efficient port sweeps.

Select and the S-parameters are subject to . Click to locate the file, or enter a filename and path. Select an : , , or .

Enter a Zref (SI unit: Ω) that is used only for the header in the exported Touchstone file. The default is 50 Ω.

	It is also possible to export Touchstone files during postprocessing under the Results node. In the COMSOL Multiphysics Reference Manual, see the Touchstone section for more information on how to export Touchstone files as a postprocessing step.

To display this section, click the button (

) and select in the dialog box.

The electric field on port boundaries is expanded as

where E, r, Einc, Si, Ei, αi, n, and r0 are, respectively, the electric field on the port boundary, the position vector, the incident electric field, the expansion coefficient (or S-parameter), the mode field, the propagation constant for the mode, the normal vector, and a position on the port boundary.

Select (the default value) from the list. In this formulation, the expansion coefficients (or S-parameters) are calculated by adding a scalar dependent variable for each coefficient. The S-parameter and the tangential electric field on the port boundary are solved for by adding the following weak expression for each port

where Js,i is the surface current density for the port

ET is the tangential electric field (the dependent variable) on the port boundary, δij is the Kronecker delta

Above, Hi is the magnetic mode field for the port, Ebnd is the expansion of the electric field on the port boundary in terms of the electric mode fields, Ei,

and test() is the test operator.

	In the COMSOL Multiphysics Reference Manual, see the Built-In Operators section for more information about the test operator.

When is selected from the list, the expansion coefficients (or S-parameters) are calculated by adding a scalar dependent variable for each coefficient and then adding a constraint to enforce the series expansion above.

This section is available in 2D axisymmetry. To display this section, click the button (

) and select in the dialog box.

Select (the default) or for the . When is selected, the average radius is defined as the average radial position of the geometry bounding box. Select to enter a value for the r0 (SI unit: m). The default value is 1 m.

The average radius is used in Mode analysis in 2D axisymmetry, where the relation between the eigenvalue and the effective mode index and out-of-plane wave number is given by

respectively. Here, β is the out-of-plane wave number, k0 is the vacuum wave number, neff is the effective mode index, and λ is the eigenvalue.

The dependent variables (field variables) are for the E and its components (in the fields). The name can be changed but the names of fields and dependent variables must be unique within a model.

	In 2D axisymmetry, for all studies except Mode Analysis and Boundary Mode Analysis, the Covariant formulation is used. Thereby, Ephi is not a dependent variable. To get the correct out-of-plane electric field component, the variable ewfd.Ephi should be used. Here, it is assumed that the dependent variable has been given the name E and that the tag for the physics interface is ewfd.

Select the shape order for the dependent variable — , , (the default), , , , , , , , , , , or . For more information about the section, see Settings for the Discretization Sections in the COMSOL Multiphysics Reference Manual.