This section specifies a resonance mode problem. It is the aim to find pairs , or equivalently satisfying the time-harmonic Maxwell’s equations in a source-free medium.
When passing a two-dimensional grid file
JCMsolve treats the geometry as infinitely extended in the -direction. The computed eigenfields depend harmonically on , that is
The user must fix the longitudinal component of the BlochVector.
When the geometry exhibits a cylinder symmetry with respect to the -axis, it is possible to reduce the eigenmode computation to a two dimensional problem.
Let denote the cylindrical coordinates related to the Cartesian coordinates by
The material distribution is rotational symmetric when the permittivity tensor field and the permeability tensor field satisfy
with the rotation matrix
Hence, the device is fully described by the material distribution within the cross section , so that
JCMsolve expects a two-dimensional grid file
Any eigenfield show up a symmetry with respect to the angular variable . For the electric and field it holds true that:
with an integer value .
Up to a phase factor , 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 or the Poynting vector . Scalar fields such as the electromagnetic field energy density are independent of .
For a resonance mode computation, the user must fix the integer wave number , see BlochVector.