3. Maxwell Eigenmodes

This section explains the theory behind the Maxwell eigenmode Solver.

The temporal evolution of electromagnetic fields is governed by Maxwell’s equations, which read as follows in the absence of free charges or currents [Che92, RC01]:

(3.45)\[\nabla \times \mathbf{E} = -\mu \frac{\partial \mathbf{H}}{\partial t},\]

(3.46)\[\nabla \times \mathbf{H} = \varepsilon \frac{\partial \mathbf{E}}{\partial t},\]

(3.47)\[\nabla \cdot (\varepsilon \mathbf{E}) = 0,\]

(3.48)\[\nabla \cdot (\mu \mathbf{H}) = 0,\]

where \(\mathbf E(\mathbf r, t)\) is the electric field and \(\mathbf{H}(\mathbf r, t)\) is the magnetic field intensity (often referred to as magnetic field). Also, \(\varepsilon(\mathbf r)\) and \(\mu(\mathbf r)\) are the permittivity and permeability (assumed isotropic and linear).

If the fields have time-harmonic [1] variation, one can simplify Maxwell’s equations by assuming the following temporal dependence of the field [Che92, Jac99]:

(3.49)\[\mathbf E(\mathbf r, t) = \operatorname{Re}[\mathbf E(\mathbf r) e^{i\omega t}],\]

(3.50)\[\mathbf H(\mathbf r, t) = \operatorname{Re}[\mathbf H(\mathbf r) e^{i\omega t}],\]

which allows writing the frequency-domain form of Eqs. (3.45)–(3.48) as

(3.51)\[\nabla \times \mathbf{E} = -i \omega \mu \mathbf{H},\]

(3.52)\[\nabla \times \mathbf{H} = i\omega \varepsilon \mathbf{E},\]

(3.53)\[\nabla \cdot (\varepsilon \mathbf{E}) = 0,\]

(3.54)\[\nabla \cdot (\mu \mathbf{H}) = 0.\]

In order to obtain a single equation involving only the electric field, we can divide both sides of Eq. (3.51) by \(\mu\) and apply the curl, obtaining

(3.55)\[\nabla \times (\mu^{-1}\nabla \times \mathbf{E}) = - i \omega \nabla \times \mathbf{H}.\]

Substituting Eq. (3.52) then leads to the following equation:

(3.56)\[\nabla \times (\mu^{-1}\nabla \times \mathbf{E}) = \omega^2 \varepsilon \mathbf{E}.\]

In a resonating structure, the resonance frequencies and the corresponding field distributions are the non-trivial solutions of Eq. (3.56), known as eigenmodes.

Boundary conditions

This section outlines the boundary conditions of the Device supported by the Maxwell eigenmode solver.

Perfect electric conductor boundaries (API reference)

On this type of boundaries, the tangential component of the electric field is set to zero,

(3.57)\[\mathbf{\hat n} \times \mathbf{E} = \mathbf{0},\]

where \(\mathbf{\hat n}\) is the unit vector normal to the surface. This boundary condition is used to model perfect electric conductors (PECs), which are ideal conductors with zero resistivity. This boundary condition is appropriate for the superconducting materials used in quantum devices.

Natural boundaries

All boundaries at which no specific boundary condition is specified by the user are called natural boundaries. In the case of the boundaries external to the device in the Maxwell eigenmode solver, such natural boundary conditions correspond to setting the tangential component of the curl of the electric field to zero,

(3.58)\[\mathbf{\hat n} \times (\nabla \times \mathbf{E}) = \mathbf{0},\]

where \(\mathbf{\hat n}\) is the unit vector normal to the surface, pointing outside the simulation domain. In other words, natural boundaries set the component of the magnetic field tangential to the exterior [2] surface to zero.

Inductor boundaries (API reference)

This type of boundary is used to create inductor ports by imposing the surface current density in accordance with the inductance current-voltage relation \(V = i \omega L I\). Here, \(V\) is the voltage drop across the inductor terminals, \(L\) is the inductance, and \(I\) is the current in the inductor.

Fig. 3.8 Inductor boundary (left) and the inductor element represented by the boundary (right).

Specifically, the current density imposed is given by

(3.59)\[\vec{J}_S = \dfrac{1}{i \omega L} \dfrac{l_P}{w_P} (\vec E \cdot \hat t_P) \hat t_P,\]

where \(\hat t_P\) is a unit vector defining the direction of the current flow, \(l_P\) is the length of the boundary (in the direction of \(\hat t_P\)) and \(w_P\) is the width of the boundary.

Note

By neglecting variations of the electric field along the boundary, one can verify that Eq. (3.59) is consistent with the current-voltage relation for an inductor:

(3.60)\[I = \int_{C_w} \vec{J}_S \cdot \hat t_P \ ds = \dfrac{1}{i \omega L} \dfrac{l_P}{w_P} \int_{C_w} \vec E \cdot \hat t_P \ ds = \dfrac{l_P}{i \omega L } \vec E \cdot \hat t_P\]

and

(3.61)\[V = \int_{C_l} \vec{E} \cdot \hat t_P \ ds = l_P \vec{E} \cdot \hat t_P,\]

where the straight integration curves \(C_w\) and \(C_l\) are shown in Fig. 3.8. From Eqs. (3.60) and (3.61), we obtain \(V = i \omega L I\).

[1]

Meaning they oscillate sinusoidally as a function of time.

[2]

Natural boundary conditions may only be considered on the exterior boundaries of the simulation domain. If a natural boundary is embedded inside the device, it will have no effect on the field.