3. Maxwell Eigenmodes

This section explains the theory behind the Maxwell eigenmode Solver.

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

If the fields have time-harmonic temporal 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 (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 (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 (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

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

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

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 surface to zero. If a natural boundary is located inside the device, it will have no effect on the field.