GroupTheory`
GroupTheory`

GTPhUncoupledBands

GTPhUncoupledBands[gvecs,bands,matrix] finds uncoupled bands from bands if the bands are calculated with the reciprocal lattice vectors gvecs. matrix is a transformation matrix used to express the symmetry to find the uncoupled bands.

Details and Options

  • Not all plane waves incident on a photonic crystal can couple to the bands of the photonic crystal due to symmetry reasons. To figure out, which plane waves can couple to bands of the photonic crystal GTCharProjectionOperator and GTCompatibility can be used, i.e. the discussion uses the compatibility relations.
  • In some cases it is sufficient to observe if an eigenmode of the photonic crystal is symmetric or antisymmetric to a certain symmetry operation (mirror plane) to find the uncoupled bands. This is the task of GTPhUncoupledBands.
  • The following options can be given:
  • GOPlotBandsTrueOutput for a band structure plot.
    GOVerboseFalseControl the output of additional information.
  • See: W. Hergert, M. Geilhufe, Group Theory in Solid State Physics and Photonics. Problem Solving with Mathematica, chapter 11.4

Examples

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Basic Examples  (1)

First load the package:

The photonic band structure for circular rods in a square lattice will be calculated. The structure is defined as:

The reciprocal lattice vecotrs are generated and the "Hamiltonian" is set up. This takes a while.

The band structure along ΓX is calculated. This is also a bit time consuming.

If the uncoupled bands are found by means of GTPhUncoupledBands, a transformation matrix has to be defined, expressing the symmetry properties. In case of the propagation along Γ-X, the plane given by Γ-X and the z-axis is the mirror plane. Such a mirror plane projects a point (x,y,z) in a point (x,-y,z). Thus, the following matrix has to be applied:

Those bands appear in the output newbd, which couple th th external plane wave.

The uncoupled bands are given in blue.

Neat Examples  (1)