ResonanceMode¶

Type:	section
Appearance:	simple
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

$\begin{eqnarray*} \VField{E}(x, y, z) & = & \VField{E}(x, y) e^{ik_z z}, \\ \VField{H}(x, y, z) & = & \VField{H}(x, y) e^{ik_z z} \\ \end{eqnarray*}$

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

$\begin{eqnarray*} (x,y,z) & = & (r \cos(\varphi), y, r \sin(\varphi)). \end{eqnarray*}$

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

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

with the rotation matrix

$\begin{eqnarray*} \TField{R} & = & \left [ \begin{array}{ccc} \cos(\varphi) & 0 & -\sin(\varphi) \\ 0 & 1 & 0 \\ \sin(\varphi) & 0 & \cos(\varphi) \\ \end{array} \right ]. \end{eqnarray*}$

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:

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

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.