GroupTheory`
GroupTheory`

GTPhMPBBands

GTPhMPBBands[file, symmetry points,basis]

is used to read a photonic band structure from a file calculated by MPB. The band structure is calculated along lines defined by the symmetry points. basis contains the reciprocal basis used in the MPB calculation.

Details and Options

  • The results of a photonic bandstructure calculation by means of MPB is stored in the output of MPB which cannot be used inside GTPack. The scripts GSMPBBands2D and GSMPBBands3D extract the information for the bands in a readable form. In the two-dimensional case filles are usually generated for TE and TM polarization. The constructed data set can be used by the GTBandsPlot command. Use of this script:
  • GSMPBBands2D output_mpb
  • Output of this script are the files bands2D_TE and bands2D_TM containing the data in the appropriate form. The scripts are located in the datasets/MPB directory.
  • basis has to be the basis used in the MPB calculation. symmetry points are the symmetry points are expressed not in Cartesian coordinates, but in basis.
  • The following option can be given:
  • GOPlotBands TrueControls the plot of the band structure in the Brillouin zone
    GOVerbose TrueControls the output of additional information
  • See: W. Hergert, M. Geilhufe, Group Theory in Solid State Physics and Photonics. Problem Solving with Mathematica, chapter 11.1
  • J. D. Joannopoulos, S. G. Johnson, J. N. Winn, R. D. Meade, Photonic Crystals. Molding the flow of light, Princton 2008, p.68
  • K. Sakoda, Optiical Properties of Photonic Crystals, Springer, Berlin 2001

Examples

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Basic Examples  (3)

First load the package:

Square Lattice  (1)

The example of a square array of dielectric columns (see Joannopoulos et al., p. 68) with r=0.2 a is used. The permittivity is set to ϵ = 8.9 (alumina). The band structure is calculated along the lines Γ-X-M-Γ.

The reciprocal basis for this example is given by:

It will be checked, if the file exists. First the calculated TM modes will be plotted.

Now the calculated TE modes are plotted

The TM modes (blue) and TE modes (red) shown together.

Hexagonal lattice  (1)

We calculated the TM band of Sakoda's example (see his book, p. 57ff) by means of GTPack.

This is the reciprocal basis as given in Sakoda'S example.

Construction of reciprocal lattice vectors:

Path in the BZ.

The Brillouinzone the basis vectors and the vectors to the symmetry points M and K are presented:

Γ, M, K expressed in terms of the basis vectors:

Γ, M, K in therms of the Cartesian coordinate system:

In MPB k-vectors are expressed in terms of the reciprocal basis vectors. Thus, we provide the basis used in the MPB calculation and the symmetry points defined as multiples of those basis vectors.

The dielectric contrast is low in this case, i.e. we get a good agreement with the MPB result with relatively small number of plane waves.

The input file for the calculation of the hexagonal example is given as hexagonal.ctl in the datatasets/MPB directory.

Options  (2)

GOPlotBands  (1)

Data for the calculation of then density of states are read from a file (square lattice example) . The argument kpoints is not used in this case, but the basis has to be given.

Data are used to calculate the density of states for TM polarisation (dielectric rods in air, ϵ=8.9, r=0.2 a)

GOVerbose  (1)

Information not provided.