qpms/lattices.md

Using QPMS library for finding modes of 2D-periodic systems

Calculating modes of infinite 2D arrays is now done in several steps (assuming the T-matrices have already been obtained using scuff-tmatrix or can be obtained from Lorenz-Mie solution (spherical particles)):

1. Sampling the k, \omega space.
2. Pre-calculating the Ewald-summed translation operators.
3. For each k, \omega pair, build the LHS operator for the scattering problem (TODO reference), optionally decomposed into suitable irreducible representation subspaces.
4. Evaluating the singular values and finding their minima.

The steps above may (and will) change as more user-friendly interface will be developed.

Preparation: compile the ew_gen_kin utility

This will change, but at this point, the lattice-summed translation operators are computed using the ew_gen_kin utility located in the qpms/apps directory. It has to be built manually like this:

  cd qpms/apps
  c99 -o ew_gen_kin -Wall -I ../.. -I ../../amos/ -O2 -ggdb -DQPMS_VECTORS_NICE_TRANSFORMATIONS -DLATTICESUMS32 2dlattice_ewald.c ../translations.c ../ewald.c ../ewaldsf.c ../gaunt.c ../lattices2d.c ../latticegens.c ../bessel.c -lgsl -lm -lblas ../../amos/libamos.a -lgfortran ../error.c

Step 1: Sampling the k, \omega space

ew_gen_kin expects a list of $(k_x, k_y)$ pairs on standard input (separated by whitespaces), the rest is specified via command line arguments.

So if we want to examine the line between the Г point and the point k = (0, 1\cdot10^5\mathrm{m}^{-1}), we can generate an input running

  for ky in $(seq 0 1e3 1e5); do
    echo 0 $ky >> klist
  done

It also make sense to pre-generate the list of \omega values, e.g.

  seq 6.900 0.002 7.3 | sed -e 's/,/./g' > omegalist

Step 2: Pre-calculating the translation operators

ew_gen_kin currently uses command-line arguments in an atrocious way with a hard-coded order:

  ew_gen_kin outfile b1.x b1.y b2.x b2.y lMax scuffomega refindex npart part0.x part0.y [part1.x part1.y [...]]

where outfile specifies the path to the output, b1 and b2 are the direct lattice vectors, lMax is the multipole degree cutoff, scuffomega is the frequency in the units used by scuff-tmatrix (TODO specify), refindex is the refractive index of the background medium, npart number of particles in the unit cell, and partN are the positions of these particles inside the unit cell.

Assuming we have the ew_gen_in binary in our ${PATH}, we can now run e.g.

  for omega in $(cat omegalist); do
    ew_gen_kin $omega 621e-9 0 0 571e-9 3 w_$omega 1.52 1 0 0 < klist
  done

This pre-calculates the translation operators for a simple (one particle per unit cell) 621 nm × 571 nm rectangular lattice inside a medium with refractive index 1.52, up to the octupole (lMax = 3) order, yielding one file per frequency. This can take some time and it makes sense to run a parallelised for-loop instead.

When this is done, we convert all the text output files into numpy's binary format in order to speed up loading in the following steps. This is done using the processWfiles_sortnames.py script located in the misc directory. Its usage pattern is

  processWfiles_sortnames.py npart dest src1 [src2 ...]

where npart is the number of particles in the unit cell, dest is the destination path for the converted data (this will be a directory), and the remaining arguments are paths to the files generated by ew_gen_kin. In the case above, one could use

  processWfiles_sortnames.py 1 all s_*

which would create a directory named all containing several .npy files.

Steps 3, 4

TODO. For the time being, see e.g. the [TODO SPECIFY] jupyter notebook for the remaining steps.