2D photonic crystal ω-formulation (TE and TM mode) - sampling over quasi-momenta k

Filename

waveprop/PCEVSolve.cc

• Computation of the band structure of 2D photonic crystal (TE or TM mode) and the respective Bloch modes
• ω-formulation, sampling over quasi-momenta k

Call	PCEVSolve.cc -f <FILE>

Requirements

The external libraries SuperLU (sparse direct solver) and Arpack (sparse eigenvalue solver) have to be installed.

Usage

The files for the mesh are expected to have the following specified relation to the input file <FILE> e.g. for the input file example.cig it is searched for:

example_Coord.dat	File with coordinates of the mesh vertices.
example_Elms.dat	File with information of the mesh cells.
example_Attr.dat	File with cell, edge and vertex attributes.
example_EdgRadia.dat	File with radia of circular edges.
example_EdgCorr.dat	File with information about periodic edges.

Furthermore, semi-infinite rectangles can be defined via the file for the input parameter.

Type	Name (default value)	Stands for		Compulsory
string array	InfiniteQuad (empty)	Array of semi-infinite rectangles, e.g. string array InfiniteQuad { 1 "2 1 4", 2 "104 103 4" }		No

The lattice constant a results from the coordinates, i.e. the coordinates will not be scaled inside the programme. However, as the non-absorbing Maxwell's equations scale with the size can interprete the results in a scaling independent of the lattice constant a.

Input Parameter

Type	Name (default value)	Stands for		Compulsory
double array	epsilon (1)	Map from attribute to relative permittivity		Yes
double array	k0 (empty)	Array of quasi-momenta in e1 direction, in [-π/a, π/a]		No
double array	k1 (empty)	Array of quasi-momenta in e2 direction, in [-π/a, π/a]		No
string	samplemode ( "simple")	Sample modes in the reciprocal lattice.		No
		"simple"	Arrays of k0, k1 build one common array (k0, k1)
		"tensor"	Tensor product of arrays of k0, k1, beginning to sample in k0.
		"angle = <ANGLE> "	Array of k0 is abs(k) and angle is read from string.
double	k0 (0)	Single quasi-momentum in e1 direction (taken if array k0 is empty)		No
double	k1 (0)	Single quasi-momentum in e2 direction (taken if array k1 is empty)		No
int	neig (4)	Number of eigenvalues ω		No
string	mode ( "TM")	TE or TM mode.		No

string	dirichl_bdAttrib ( "")	List of attributes of edges with Dirichlet boundary condition, the attributes are separated by spaces, e.g. "1 3 8 7"		No
int	p (1)	Starting polynomial degree, will be set at least to 1.		No
string	refinement ( "")	Further refinement, e.g. ="p2 in 1, h2 in 2, p2|0 in 3".		No
		h -refinement	"h" - once
			"h2" - twice
			"h2|1" - twice in local x1-direction and once in x2-direction.
		p -refinement	"p" - increase polynomial degrees by 1
			"p2" - increase polynomial degrees by 2
			"p2|0" - increase in local x1-direction by 2
		only in some cells	"p2|0 in 3" - refinement as above, but only in cells with attribute 3
		h - or hp -refinement towards vertices or edges	"hp -> v101" - one step of hp -refinement towards the vertex/vertices with attribute 101
			h2 in 3 -> e20" - twofold h -refinement towards the edge(s) with attribute 20, only in cells with attribute 3
bool	trunk ( true)	Use of linear trunk space.		No
double	epsArpack (EPS)	Accuracy of the eigensolver, by default machine precision is taken.		No
double	omega (0)	Shift for the eigensolver.		No

string	outputdirectory (".")	Directory of the output files, by default the current working directory (where the executable is called from), and related to current the working directory.		No
string	outputprefix (<FILEPRE>)	Prefix of the output files, for input file example.cig the prefix is by default example.		No
bool	verbose (false)	Let print more information out.		No

Example

Output

Mesh

Filename	<OUTPUTDIRECTORY>/<OUTPUTPREFIX> _Mesh.m
	e.g. for example.cig by default ./example_Mesh.m
Type	Matlab
Call	plot(x(edgemsh), y(edgemsh), 'k-');

Band structure

Filename	<OUTPUTDIRECTORY>/<OUTPUTPREFIX> _Omega.m
	e.g. for example.cig by default ./example_Omega.m
Type	Matlab
Variables	ndof	Number of degrees of freedom.
	ks_TE or ks_TM	4 x nk-matrix with quasi-momenta and eigen frequencies.

Structure of ks_TE and ks_TM

Row	Values
1	k0-values, not scaled, i.e. as given for example in [-π/a, π/a] with lattice constant of mesh.
2	k1-values, also not scaled.
3	ω^2-values, scaled by square of the vacuum speed of light c
4	ω-values, scaled by the vacuum speed of light c

To get quasi-momenta in the usual scaling independent of the lattice constant a, divide the values by a/(2π). To get the eigenfrequencies ω in the usual scaling independent of a and the vacuum speed of light c, multiply the values by a/(2π).

Eigenmodes

The eigenmodes are written with their coefficient vectors in the FE space into the following files, where <INDEX> goes from 00000 to the number of eigenmodes minus 1.

Filename	<OUTPUTDIRECTORY>=/eigenfuns/=<OUTPUTPREFIX> EF <INDEX>.dat=
	e.g. for example.cig by default ./eigenfuns/example_TE_00000.dat and so on.
Type	Binary

The sub-directory eigenfuns will be created if not existing. Files for graphical output in Matlab can be generated with showSol2D.cc. See here.