Excludes:PropagatingMode, Scattering

This section specifies a resonance mode problem. It is the aim to find pairs (\omega, \VField{E}), or equivalently (\omega, \VField{H}), satisfying the time-harmonic Maxwell’s equations in a source-free medium.

Special geometries

2-dimensional, straight

When passing a two-dimensional grid file grid.jcm, JCMsolve treats the geometry as infinitely extended in the z-direction. The computed eigenfields depend harmonically on z, that is

\VField{E}(x, y, z) & = &  \VField{E}(x, y) e^{ik_z z}, \\
\VField{H}(x, y, z) & = &  \VField{H}(x, y) e^{ik_z z} \\

The user must fix the longitudinal component k_z of the BlochVector.

3-dimensional, cylindrical

When the geometry exhibits a cylinder symmetry with respect to the y-axis, it is possible to reduce the eigenmode computation to a two dimensional problem.

Let (r,y,\varphi) denote the cylindrical coordinates related to the Cartesian coordinates (x, y, z) by

(x,y,z) & = & (r \cos(\varphi), y, r \sin(\varphi)).

The material distribution is rotational symmetric when the permittivity tensor field \varepsilon and the permeability tensor field \mu satisfy

\varepsilon(r \cos(\varphi), y, r \sin(\varphi)) & = &\TField{R} \cdot \varepsilon(r, y, 0) \cdot \TField{R}^{\mathrm{T}}, \\
\mu(r \cos(\varphi), y, r \sin(\varphi)) & = &\TField{R} \cdot \mu(r, y, 0) \cdot \TField{R}^{\mathrm{T}},

with the rotation matrix

\TField{R} & = & \left [
\cos(\varphi) & 0 & -\sin(\varphi) \\
0 & 1 & 0 \\
\sin(\varphi) & 0 & \cos(\varphi) \\
\right ].

Hence, the device is fully described by the material distribution within the cross section r\geq0,\, z=0, so that JCMsolve expects a two-dimensional grid file grid.jcm.

Any eigenfield show up a symmetry with respect to the angular variable \varphi. For the electric and field it holds true that:

\VField{E}(r \cos(\varphi), y, r \sin(\varphi)) & = &
\TField{R}\cdot\VField{E}(r, y, 0)e^{i n_\varphi \phi},

with an integer value n_\varphi.

Up to a phase factor e^{i n_\varphi \phi}, the electric field is determined by the values within the cross section. The same holds true for other vector fields, i.e., for the magnetic field \VField{H} or the Poynting vector \VField{S}. Scalar fields such as the electromagnetic field energy density are independent of \varphi.

For a resonance mode computation, the user must fix the integer wave number n_\varphi, see BlochVector.