Wave Equation, Electric
Wave Equation, Electric is the main feature node for this physics interface. The governing equation can be written in the form
for the time-harmonic and eigenfrequency problems. The wave number of free space k0 is defined as
where c0 is the speed of light in vacuum.
In 2D the electric field varies with the out-of-plane wave number kz as
.
The wave equation is thereby rewritten as
,
where z is the unit vector in the out-of-plane z direction.
Notice that the ansatz above just explains how the wave equation is modified when the out-of-plane wave vector component kz is not zero. As an example, for a plane wave with a nonzero out-of-plane wave vector component, the electric field is of course given by
,
where A is a constant amplitude and kx, ky, and kz are the wave vector components.
In 2D axisymmetry, the electric field varies with the azimuthal mode number m as
.
For this case, the wave equation is rewritten as
,
where is the unit vector in the out-of-plane  direction.
When solving the equations as an eigenfrequency problem the eigenvalue is the complex eigenfrequency λ = −jω + δ, where δ is the damping of the solution. The Q-factor is given from the eigenvalue by the formula
Using the relation εr = n2, where n is the refractive index, the equation can alternatively be written
When the equation is written using the refractive index, the assumption is that μr = 1 and σ = 0 and only the constitutive relations for linear materials are available. When solving for the scattered field the same equations are used but E = Esc + Ei and Esc is the dependent variable.
The Divergence Constraint subnode is available from the context menu (right-click the parent node) or from the Physics toolbar, Attributes menu.
Electric Displacement Field
Select an Electric displacement field model — Relative permittivity (the default), Refractive index, Loss tangent, loss angle, Loss tangent, dissipation factor, Dielectric loss, Drude-Lorentz dispersion model, or Debye dispersion model.
Relative Permittivity
When Relative permittivity is selected, the default Relative permittivity εr takes values From material. Select Effective medium to add an Effective Medium subnode, or for User defined select Isotropic, Diagonal, Symmetric, or Anisotropic and enter values or expressions in the field or matrix.
Refractive Index
When Refractive index is selected, the default Refractive index n and Refractive index, imaginary part k take the values From material. To specify the real and imaginary parts of the refractive index and assume a relative permeability of unity and zero conductivity, for one or both of the options, select User defined then choose Isotropic, Diagonal, Symmetric, or Anisotropic. Enter values or expressions in the field or matrix.
The diagonal components of the input refractive index matrix correspond to the semi-axes of the so called index ellipsoid. You can orient the index ellipsoid by first creating a suitably oriented coordinate system below the Definitions node for the model component. Then select the created coordinate system in the Coordinate system setting in the Coordinate System Selection section in the settings for the Wave Equation, Electric feature.
Beware of the time-harmonic sign convention requiring a lossy material having a negative imaginary part of the refractive index (see Introducing Losses in the Frequency Domain).
Loss Tangent, Loss Angle
When Loss tangent, loss angle is selected, the default Relative permittivity (real part) ε′ and Loss tangent, loss angle δ take values From material. For User defined select Isotropic, Diagonal, Symmetric, or Anisotropic and enter values or expressions in the field or matrix. Then if User defined is selected for Loss tangent, loss angle δ, enter a value to specify a loss angle for dielectric losses. This assumes zero conductivity.
Loss Tangent, Dissipation Factor
When Loss tangent, dissipation factor is selected, the default Relative permittivity (real part) ε′ and Loss tangent, dissipation factor tanδ take values From material. For User defined select Isotropic, Diagonal, Symmetric, or Anisotropic and enter values or expressions in the field or matrix. Then if User defined is selected for Loss tangent, dissipation factor tanδ, enter a value to specify a dissipation for dielectric losses. This assumes zero conductivity.
Dielectric Loss
When Dielectric loss is selected, the default Relative permittivity ε′ and Relative permittivity (imaginary part) ε″ take values From material. For User defined select Isotropic, Diagonal, Symmetric, or Anisotropic and enter values or expressions in the field or matrix.
Beware of the time-harmonic sign convention requiring a lossy material having a negative imaginary part of the relative permittivity (see Introducing Losses in the Frequency Domain).
Drude-Lorentz Dispersion Model
The Drude-Lorentz dispersion model is defined by the equation
where ε∞ is the high-frequency contribution to the relative permittivity, ωP is the plasma frequency, fj is the oscillator strength, ω0j is the resonance frequency, and Γj is the damping coefficient.
When Drude-Lorentz dispersion model is selected, the default Relative permittivity, high frequency ε∞ (dimensionless) takes its value From material. For User defined select Isotropic, Diagonal, Symmetric, or Anisotropic and enter a value or expression in the field or matrix.
Enter a Plasma frequency ω∞ (SI unit: rad/s). The default is 0 rad/s.
In the table, enter values or expressions in the columns for the Oscillator strength, Resonance frequency (rad/s), and Damping in time (rad/s).
Debye Dispersion Model
The Debye dispersion model is given by
where ε∞ is the high-frequency contribution to the relative permittivity, Δεk is the contribution to the relative permittivity, and τk is the relaxation time.
When Debye dispersion model is selected, the default Relative permittivity, high frequency ε∞ (dimensionless) takes its value From material. For User defined select Isotropic, Diagonal, Symmetric, or Anisotropic and enter a value or expression in the field or matrix.
In the table, enter values or expressions in the columns for the Relative permittivity contribution and Relaxation time (s).
Magnetic Field
Select the Constitutive relation — Relative permeability (the default) or Magnetic losses.
For magnetic losses, beware of the time-harmonic sign convention requiring a lossy material having a negative imaginary part of the relative permeability (see Introducing Losses in the Frequency Domain).
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For Relative permeability the relative permeability μr uses values From material. For User defined select Isotropic, Diagonal, Symmetric, or Anisotropic based on the characteristics of the magnetic field, and then enter values or expressions in the field or matrix. If Effective medium is selected, the Effective Medium subnode is available from the context menu (right-click the parent node) or from the Physics toolbar, Attributes menu.
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For Magnetic losses the default values for Relative permeability (real part) μ′ and Relative permeability (imaginary part) μ″ are taken From material. For User defined enter different values.
Conduction Current
By default, the Electrical conductivity σ (SI unit: S/m) uses values From material.
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For User defined select Isotropic, Diagonal, Symmetric, or Anisotropic based on the characteristics of the current and enter values or expressions in the field or matrix.
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For Linearized resistivity the default values for the Reference temperature Tref (SI unit: K), Resistivity temperature coefficient α (SI unit: 1/K), and Reference resistivity ρ0 (SI unit: Ω⋅m) are taken From material. For User defined enter other values or expressions for any of these variables.
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When Effective medium is selected, the Effective Medium subnode is available from the context menu (right-click the parent node) or from the Physics toolbar, Attributes menu.
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When Archie’s law is selected, the Archie’s Law subnode is available from the context menu (right-click the parent node) or from the Physics toolbar, Attributes menu.