119 lines
4.3 KiB
Markdown
119 lines
4.3 KiB
Markdown
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*, *ω* space.
|
||
2. Pre-calculating the
|
||
Ewald-summed translation operators.
|
||
3. For each *k*, *ω* 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:
|
||
|
||
```bash
|
||
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*, *ω* 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
|
||
\f$ k = (0, 10^5\,\mathrm{m}^{-1}) \f$, we can generate an input
|
||
running
|
||
```bash
|
||
for ky in $(seq 0 1e3 1e5); do
|
||
echo 0 $ky >> klist
|
||
done
|
||
```
|
||
|
||
It also make sense to pre-generate the list of *ω* values,
|
||
e.g.
|
||
```bash
|
||
seq 6.900 0.002 7.3 | sed -e 's/,/./g' > omegalist
|
||
```
|
||
|
||
|
||
Step 2: Pre-calculating the translation operators
|
||
-------------------------------------------------
|
||
|
||
`ew_gen_kin` currently uses command-line arguments in
|
||
an atrocious way with a hard-coded order:
|
||
```
|
||
ew_gen_kin outfile b1.x b1.y b2.x b2.y lMax scuffomega refindex npart part0.x part0.y [part1.x part1.y [...]]
|
||
```
|
||
where `outfile` specifies the path to the output, `b1` and `b2` are the
|
||
direct lattice vectors, `lMax` is the multipole degree cutoff,
|
||
`scuffomega` is the frequency in the units used by `scuff-tmatrix`
|
||
(TODO specify), `refindex` is the refractive index of the background
|
||
medium, `npart` number of particles in the unit cell, and `partN` are
|
||
the positions of these particles inside the unit cell.
|
||
|
||
Assuming we have the `ew_gen_kin` binary in our `${PATH}`, we can
|
||
now run e.g.
|
||
```bash
|
||
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; this is a stupid but working
|
||
way to do it in bash:
|
||
```bash
|
||
N=4 # number of parallel processes
|
||
for omega in $(cat omegalist); do
|
||
((i=i%N)); ((i++==0)) && wait
|
||
ew_gen_kin $omega 621e-9 0 0 571e-9 3 w_$omega 1.52 1 0 0 < klist
|
||
echo $omega # optional, to follow progress
|
||
done
|
||
```
|
||
|
||
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 w_*
|
||
```
|
||
which would create a directory named `all` containing several
|
||
.npy files.
|
||
|
||
|
||
Steps 3, 4
|
||
----------
|
||
|
||
TODO. For the time being, see e.g. the `SaraRect/dispersions.ipynb` jupyter notebook
|
||
from the `qpms_ipynotebooks` repository
|
||
for the remaining steps.
|
||
|