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Author SHA1 Message Date
Jan Petykiewicz
03fc9e6d70 deprecate 2020-02-19 19:10:18 -08:00
52 changed files with 2517 additions and 6968 deletions
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10
.gitignore vendored
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@ -52,17 +52,11 @@ coverage.xml
# Django stuff:
*.log
# documentation
doc/
# Sphinx documentation
docs/_build/
# PyBuilder
target/
.idea/
.*.sw[op]
*.svg
*.html

3
MANIFEST.in
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@ -1,3 +0,0 @@
include README.md
include LICENSE.md
include meanas/VERSION

109
README.md
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@ -1,111 +1,52 @@
# meanas
# fdfd_tools
**meanas** is a python package for electromagnetic simulations
** DEPRECATED **
** UNSTABLE / WORK IN PROGRESS **
The functionality in this module is now provided by [meanas](https://mpxd.net/code/jan/meanas).
Formerly known as [fdfd_tools](https://mpxd.net/code/jan/fdfd_tools).
-----------------------
This package is intended for building simulation inputs, analyzing
simulation outputs, and running short simulations on unspecialized hardware.
It is designed to provide tooling and a baseline for other, high-performance
purpose- and hardware-specific solvers.
**fdfd_tools** is a python package containing utilities for
creating and analyzing 2D and 3D finite-difference frequency-domain (FDFD)
electromagnetic simulations.
**Contents**
- Finite difference frequency domain (FDFD)
* Library of sparse matrices for representing the electromagnetic wave
equation in 3D, as well as auxiliary matrices for conversion between fields
* Waveguide mode operators
* Waveguide mode eigensolver
* Stretched-coordinate PML boundaries (SCPML)
* Functional versions of most operators
* Anisotropic media (limited to diagonal elements eps_xx, eps_yy, eps_zz, mu_xx, ...)
* Arbitrary distributions of perfect electric and magnetic conductors (PEC / PMC)
- Finite difference time domain (FDTD)
* Basic Maxwell time-steps
* Poynting vector and energy calculation
* Convolutional PMLs
* Library of sparse matrices for representing the electromagnetic wave
equation in 3D, as well as auxiliary matrices for conversion between fields
* Waveguide mode solver and waveguide mode operators
* Stretched-coordinate PML boundaries (SCPML)
* Functional versions of most operators
* Anisotropic media (eps_xx, eps_yy, eps_zz, mu_xx, ...)
* Arbitrary distributions of perfect electric and magnetic conductors (PEC / PMC)
This package does *not* provide a fast matrix solver, though by default
`meanas.fdfd.solvers.generic(...)` will call
`scipy.sparse.linalg.qmr(...)` to perform a solve.
For 2D FDFD problems this should be fine; likewise, the waveguide mode
```fdfd_tools.solvers.generic(...)``` will call
```scipy.sparse.linalg.qmr(...)``` to perform a solve.
For 2D problems this should be fine; likewise, the waveguide mode
solver uses scipy's eigenvalue solver, with reasonable results.
For solving large (or 3D) FDFD problems, I recommend a GPU-based iterative
solver, such as [opencl_fdfd](https://mpxd.net/code/jan/opencl_fdfd) or
those included in [MAGMA](http://icl.cs.utk.edu/magma/index.html). Your
For solving large (or 3D) problems, I recommend a GPU-based iterative
solver, such as [opencl_fdfd](https://mpxd.net/gogs/jan/opencl_fdfd) or
those included in [MAGMA](http://icl.cs.utk.edu/magma/index.html)). Your
solver will need the ability to solve complex symmetric (non-Hermitian)
linear systems, ideally with double precision.
- [WIP Source repository](https://mpxd.net/code/jan/fdfd_tools/src/branch/wip)
- *TODO* [Source repository](https://mpxd.net/code/jan/meanas)
- PyPI *TBD*
## Installation
**Requirements:**
* python 3 (tests require 3.7)
* python 3 (written and tested with 3.5)
* numpy
* scipy
Install from PyPI with pip:
Install with pip, via git:
```bash
pip3 install 'meanas[test,examples]'
pip install git+https://mpxd.net/gogs/jan/fdfd_tools.git@release
```
### Development install
Install python3.7, virtualenv, and git:
```bash
# This is for Debian/Ubuntu/other-apt-based systems; you may need an alternative command
sudo apt install python3.7 virtualenv build-essential python3.7-dev git
```
If python 3.7 is not your default python3 version, create a virtualenv:
```bash
# Check python3 version:
python3 --version
# output on my system: Python 3.7.5rc1
# If this indicates a version >= 3.7, you can skip all
# the steps involving virtualenv or referencing the venv/ directory
# Create a virtual environment using python3.7 and place it in the directory `venv/`
virtualenv -p python3.7 venv
```
In-place development install:
```bash
# Download using git
git clone --branch wip https://mpxd.net/code/jan/fdfd_tools.git meanas/
# NOTE: In the future this will become
#git clone https://mpxd.net/code/jan/meanas.git
# If you are using a virtualenv, activate it
source venv/bin/activate
# Install in-place (-e, editable) from ./meanas, including testing and example dependencies ([test, examples])
pip3 install --user -e './meanas[test,examples]'
# Run tests
cd meanas
python3 -m pytest -rsxX | tee test_results.txt
```
#### See also:
- [git book](https://git-scm.com/book/en/v2)
- [virtualenv documentation](https://virtualenv.pypa.io/en/stable/userguide/)
- [python language reference](https://docs.python.org/3/reference/index.html)
- [python standard library](https://docs.python.org/3/library/index.html)
## Use
See `examples/` for some simple examples; you may need additional
packages such as [gridlock](https://mpxd.net/code/jan/gridlock)
See examples/test.py for some simple examples; you may need additional
packages such as [gridlock](https://mpxd.net/gogs/jan/gridlock)
to run the examples.

51
examples/bloch.py
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@ -1,44 +1,12 @@
import numpy, scipy, gridlock, meanas
from meanas.fdfd import bloch
import numpy, scipy, gridlock, fdfd_tools
from fdfd_tools import bloch
from numpy.linalg import norm
import logging
from pathlib import Path
logging.basicConfig(level=logging.DEBUG)
logger = logging.getLogger(__name__)
WISDOM_FILEPATH = pathlib.Path.home() / '.local' / 'share' / 'pyfftw' / 'wisdom.pickle'
def pyfftw_save_wisdom(path):
path = pathlib.Path(path)
try:
import pyfftw
import pickle
except ImportError as e:
pass
path.parent.mkdir(parents=True, exist_ok=True)
with open(path, 'wb') as f:
pickle.dump(wisdom, f)
def pyfftw_load_wisdom(path):
path = pathlib.Path(path)
try:
import pyfftw
import pickle
except ImportError as e:
pass
try:
with open(path, 'rb') as f:
wisdom = pickle.load(f)
pyfftw.import_wisdom(wisdom)
except FileNotFoundError as e:
pass
logger.info('Drawing grid...')
dx = 40
x_period = 400
y_period = z_period = 2000
@ -62,13 +30,11 @@ g2.shifts = numpy.zeros((6,3))
g2.grids = [numpy.zeros(g.shape) for _ in range(6)]
epsilon = [g.grids[0],] * 3
reciprocal_lattice = numpy.diag(1000/numpy.array([x_period, y_period, z_period])) #cols are vectors
pyfftw_load_wisdom(WISDOM_FILEPATH)
reciprocal_lattice = numpy.diag(1e6/numpy.array([x_period, y_period, z_period])) #cols are vectors
#print('Finding k at 1550nm')
#k, f = bloch.find_k(frequency=1000/1550,
# tolerance=(1000 * (1/1550 - 1/1551)),
#k, f = bloch.find_k(frequency=1/1550,
# tolerance=(1/1550 - 1/1551),
# direction=[1, 0, 0],
# G_matrix=reciprocal_lattice,
# epsilon=epsilon,
@ -76,15 +42,15 @@ pyfftw_load_wisdom(WISDOM_FILEPATH)
#
#print("k={}, f={}, 1/f={}, k/f={}".format(k, f, 1/f, norm(reciprocal_lattice @ k) / f ))
logger.info('Finding f at [0.25, 0, 0]')
print('Finding f at [0.25, 0, 0]')
for k0x in [.25]:
k0 = numpy.array([k0x, 0, 0])
kmag = norm(reciprocal_lattice @ k0)
tolerance = (1000/1550) * 1e-4/1.5 # df = f * dn_eff / n
tolerance = (1e6/1550) * 1e-4/1.5 # df = f * dn_eff / n
logger.info('tolerance {}'.format(tolerance))
n, v = bloch.eigsolve(4, k0, G_matrix=reciprocal_lattice, epsilon=epsilon, tolerance=tolerance**2)
n, v = bloch.eigsolve(4, k0, G_matrix=reciprocal_lattice, epsilon=epsilon, tolerance=tolerance)
v2e = bloch.hmn_2_exyz(k0, G_matrix=reciprocal_lattice, epsilon=epsilon)
v2h = bloch.hmn_2_hxyz(k0, G_matrix=reciprocal_lattice, epsilon=epsilon)
ki = bloch.generate_kmn(k0, reciprocal_lattice, g.shape)
@ -100,4 +66,3 @@ for k0x in [.25]:
n_eff = norm(reciprocal_lattice @ k0) / f
print('kmag/f = n_eff = {} \n wl = {}\n'.format(n_eff, 1/f ))
pyfftw_save_wisdom(WISDOM_FILEPATH)

90
examples/tcyl.py
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@ -1,90 +0,0 @@
import importlib
import numpy
from numpy.linalg import norm
from meanas import vec, unvec
from meanas.fdfd import waveguide_mode, functional, scpml
from meanas.fdfd.solvers import generic as generic_solver
import gridlock
from matplotlib import pyplot
__author__ = 'Jan Petykiewicz'
def test1(solver=generic_solver):
dx = 20 # discretization (nm/cell)
pml_thickness = 10 # (number of cells)
wl = 1550 # Excitation wavelength
omega = 2 * numpy.pi / wl
# Device design parameters
w = 800
th = 220
center = [0, 0, 0]
r0 = 8e3
# refractive indices
n_wg = numpy.sqrt(12.6) # ~Si
n_air = 1.0 # air
# Half-dimensions of the simulation grid
y_max = 1200
z_max = 900
xyz_max = numpy.array([800, y_max, z_max]) + (pml_thickness + 2) * dx
# Coordinates of the edges of the cells.
half_edge_coords = [numpy.arange(dx/2, m + dx/2, step=dx) for m in xyz_max]
edge_coords = [numpy.hstack((-h[::-1], h)) for h in half_edge_coords]
edge_coords[0] = numpy.array([-dx, dx])
# #### Create the grid and draw the device ####
grid = gridlock.Grid(edge_coords, initial=n_air**2, num_grids=3)
grid.draw_cuboid(center=center, dimensions=[8e3, w, th], eps=n_wg**2)
dxes = [grid.dxyz, grid.autoshifted_dxyz()]
for a in (1, 2):
for p in (-1, 1):
dxes = scmpl.stretch_with_scpml(dxes, omega=omega, axis=a, polarity=p,
thickness=pml_thickness)
wg_args = {
'omega': omega,
'dxes': [(d[1], d[2]) for d in dxes],
'epsilon': vec(g.transpose([1, 2, 0]) for g in grid.grids),
'r0': r0,
}
wg_results = waveguide_mode.solve_waveguide_mode_cylindrical(mode_number=0, **wg_args)
E = wg_results['E']
n_eff = wl / (2 * numpy.pi / wg_results['wavenumber'])
print('n =', n_eff)
print('alpha (um^-1) =', -4 * numpy.pi * numpy.imag(n_eff) / (wl * 1e-3))
'''
Plot results
'''
def pcolor(v):
vmax = numpy.max(numpy.abs(v))
pyplot.pcolor(v.T, cmap='seismic', vmin=-vmax, vmax=vmax)
pyplot.axis('equal')
pyplot.colorbar()
pyplot.figure()
pyplot.subplot(2, 2, 1)
pcolor(numpy.real(E[0][:, :]))
pyplot.subplot(2, 2, 2)
pcolor(numpy.real(E[1][:, :]))
pyplot.subplot(2, 2, 3)
pcolor(numpy.real(E[2][:, :]))
pyplot.subplot(2, 2, 4)
pyplot.show()
if __name__ == '__main__':
test1()

105
examples/fdfd.py → examples/test.py
View File

@ -2,19 +2,16 @@ import importlib
import numpy
from numpy.linalg import norm
import meanas
from meanas import fdtd
from meanas.fdmath import vec, unvec
from meanas.fdfd import waveguide_3d, functional, scpml, operators
from meanas.fdfd.solvers import generic as generic_solver
from fdfd_tools import vec, unvec, waveguide_mode
import fdfd_tools
import fdfd_tools.functional
import fdfd_tools.grid
from fdfd_tools.solvers import generic as generic_solver
import gridlock
from matplotlib import pyplot
import logging
logging.basicConfig(level=logging.DEBUG)
__author__ = 'Jan Petykiewicz'
@ -60,24 +57,18 @@ def test0(solver=generic_solver):
dxes = [grid.dxyz, grid.autoshifted_dxyz()]
for a in (0, 1, 2):
for p in (-1, 1):
dxes = meanas.fdfd.scpml.stretch_with_scpml(dxes, axis=a, polarity=p, omega=omega,
thickness=pml_thickness)
dxes = fdfd_tools.grid.stretch_with_scpml(dxes, axis=a, polarity=p, omega=omega,
thickness=pml_thickness)
J = [numpy.zeros_like(grid.grids[0], dtype=complex) for _ in range(3)]
J[1][15, grid.shape[1]//2, grid.shape[2]//2] = 1
J[1][15, grid.shape[1]//2, grid.shape[2]//2] = 1e5
'''
Solve!
'''
sim_args = {
'omega': omega,
'dxes': dxes,
'epsilon': vec(grid.grids),
}
x = solver(J=vec(J), **sim_args)
A = operators.e_full(omega, dxes, vec(grid.grids)).tocsr()
A = fdfd_tools.functional.e_full(omega, dxes, vec(grid.grids)).tocsr()
b = -1j * omega * vec(J)
print('Norm of the residual is ', norm(A @ x - b))
@ -122,26 +113,25 @@ def test1(solver=generic_solver):
dxes = [grid.dxyz, grid.autoshifted_dxyz()]
for a in (0, 1, 2):
for p in (-1, 1):
dxes = scpml.stretch_with_scpml(dxes,omega=omega, axis=a, polarity=p,
thickness=pml_thickness)
dxes = fdfd_tools.grid.stretch_with_scpml(dxes,omega=omega, axis=a, polarity=p,
thickness=pml_thickness)
half_dims = numpy.array([10, 20, 15]) * dx
dims = [-half_dims, half_dims]
dims[1][0] = dims[0][0]
ind_dims = (grid.pos2ind(dims[0], which_shifts=None).astype(int),
grid.pos2ind(dims[1], which_shifts=None).astype(int))
src_axis = 0
wg_args = {
'omega': omega,
'slices': [slice(i, f+1) for i, f in zip(*ind_dims)],
'dxes': dxes,
'axis': src_axis,
'axis': 0,
'polarity': +1,
}
wg_results = waveguide_3d.solve_mode(mode_number=0, omega=omega, epsilon=grid.grids, **wg_args)
J = waveguide_3d.compute_source(E=wg_results['E'], wavenumber=wg_results['wavenumber'],
omega=omega, epsilon=grid.grids, **wg_args)
e_overlap = waveguide_3d.compute_overlap_e(E=wg_results['E'], wavenumber=wg_results['wavenumber'], **wg_args)
wg_results = waveguide_mode.solve_waveguide_mode(mode_number=0, **wg_args, epsilon=grid.grids)
J = waveguide_mode.compute_source(**wg_args, **wg_results)
H_overlap = waveguide_mode.compute_overlap_e(**wg_args, **wg_results)
pecg = gridlock.Grid(edge_coords, initial=0.0, num_grids=3)
# pecg.draw_cuboid(center=[700, 0, 0], dimensions=[80, 1e8, 1e8], eps=1)
@ -151,19 +141,6 @@ def test1(solver=generic_solver):
# pmcg.draw_cuboid(center=[700, 0, 0], dimensions=[80, 1e8, 1e8], eps=1)
# pmcg.visualize_isosurface()
def pcolor(v):
vmax = numpy.max(numpy.abs(v))
pyplot.pcolor(v, cmap='seismic', vmin=-vmax, vmax=vmax)
pyplot.axis('equal')
pyplot.colorbar()
ss = (1, slice(None), J.shape[2]//2+6, slice(None))
# pyplot.figure()
# pcolor(J3[ss].T.imag)
# pyplot.figure()
# pcolor((numpy.abs(J3).sum(axis=2).sum(axis=0) > 0).astype(float).T)
pyplot.show(block=True)
'''
Solve!
'''
@ -178,7 +155,7 @@ def test1(solver=generic_solver):
x = solver(J=vec(J), **sim_args)
b = -1j * omega * vec(J)
A = operators.e_full(**sim_args).tocsr()
A = fdfd_tools.operators.e_full(**sim_args).tocsr()
print('Norm of the residual is ', norm(A @ x - b))
E = unvec(x, grid.shape)
@ -186,45 +163,42 @@ def test1(solver=generic_solver):
'''
Plot results
'''
def pcolor(v):
vmax = numpy.max(numpy.abs(v))
pyplot.pcolor(v, cmap='seismic', vmin=-vmax, vmax=vmax)
pyplot.axis('equal')
pyplot.colorbar()
center = grid.pos2ind([0, 0, 0], None).astype(int)
pyplot.figure()
pyplot.subplot(2, 2, 1)
pcolor(numpy.real(E[1][center[0], :, :]).T)
pcolor(numpy.real(E[1][center[0], :, :]))
pyplot.subplot(2, 2, 2)
pyplot.plot(numpy.log10(numpy.abs(E[1][:, center[1], center[2]]) + 1e-10))
pyplot.grid(alpha=0.6)
pyplot.ylabel('log10 of Ey @ centerline')
pyplot.subplot(2, 2, 3)
pcolor(numpy.real(E[1][:, :, center[2]]).T)
pcolor(numpy.real(E[1][:, :, center[2]]))
pyplot.subplot(2, 2, 4)
def poyntings(E):
H = functional.e2h(omega, dxes)(E)
poynting = fdtd.poynting(e=E, h=H.conj(), dxes=dxes)
cross1 = operators.poynting_e_cross(vec(E), dxes) @ vec(H).conj()
cross2 = operators.poynting_h_cross(vec(H), dxes) @ vec(E).conj() * -1
s1 = 0.5 * unvec(numpy.real(cross1), grid.shape)
s2 = 0.5 * unvec(numpy.real(cross2), grid.shape)
s0 = 0.5 * poynting.real
# s2 = poynting.imag
return s0, s1, s2
e = vec(E)
h = fdfd_tools.operators.e2h(omega, dxes) @ e
cross1 = fdfd_tools.operators.poynting_e_cross(e, dxes) @ h.conj()
cross2 = fdfd_tools.operators.poynting_h_cross(h.conj(), dxes) @ e
s1 = unvec(0.5 * numpy.real(cross1), grid.shape)
s2 = unvec(0.5 * numpy.real(-cross2), grid.shape)
return s1, s2
s0x, s1x, s2x = poyntings(E)
pyplot.plot(s0x[0].sum(axis=2).sum(axis=1), label='s0', marker='.')
pyplot.plot(s1x[0].sum(axis=2).sum(axis=1), label='s1', marker='.')
pyplot.plot(s2x[0].sum(axis=2).sum(axis=1), label='s2', marker='.')
pyplot.plot(E[1][:, center[1], center[2]].real.T, label='Ey', marker='x')
pyplot.grid(alpha=0.6)
pyplot.legend()
s1x, s2x = poyntings(E)
pyplot.plot(s1x[0].sum(axis=2).sum(axis=1))
pyplot.plot(s2x[0].sum(axis=2).sum(axis=1))
pyplot.show()
q = []
for i in range(-5, 30):
e_ovl_rolled = numpy.roll(e_overlap, i, axis=1)
q += [numpy.abs(vec(E) @ vec(e_ovl_rolled).conj())]
H_rolled = [numpy.roll(h, i, axis=0) for h in H_overlap]
q += [numpy.abs(vec(E) @ vec(H_rolled))]
pyplot.figure()
pyplot.plot(q, marker='.')
pyplot.grid(alpha=0.6)
pyplot.plot(q)
pyplot.title('Overlap with mode')
pyplot.show()
print('Average overlap with mode:', sum(q)/len(q))
@ -235,8 +209,7 @@ def module_available(name):
if __name__ == '__main__':
#test0()
# test1()
# test0()
if module_available('opencl_fdfd'):
from opencl_fdfd import cg_solver as opencl_solver

15
examples/fdtd.py → examples/test_fdtd.py
View File

@ -10,7 +10,7 @@ import time
import numpy
import h5py
from meanas import fdtd
from fdfd_tools import fdtd
from masque import Pattern, shapes
import gridlock
import pcgen
@ -20,10 +20,9 @@ def perturbed_l3(a: float, radius: float, **kwargs) -> Pattern:
"""
Generate a masque.Pattern object containing a perturbed L3 cavity.
Args:
a: Lattice constant.
radius: Hole radius, in units of a (lattice constant).
**kwargs: Keyword arguments:
:param a: Lattice constant.
:param radius: Hole radius, in units of a (lattice constant).
:param kwargs: Keyword arguments:
hole_dose, trench_dose, hole_layer, trench_layer: Shape properties for Pattern.
Defaults *_dose=1, hole_layer=0, trench_layer=1.
shifts_a, shifts_r: passed to pcgen.l3_shift; specifies lattice constant (1 -
@ -31,13 +30,11 @@ def perturbed_l3(a: float, radius: float, **kwargs) -> Pattern:
holes adjacent to the defect (same row). Defaults are 0.15 shift for
first hole, 0.075 shift for third hole, and no radius change.
xy_size: [x, y] number of mirror periods in each direction; total size is
`2 * n + 1` holes in each direction. Default `[10, 10]`.
2 * n + 1 holes in each direction. Default [10, 10].
perturbed_radius: radius of holes perturbed to form an upwards-driected beam
(multiplicative factor). Default 1.1.
trench width: Width of the undercut trenches. Default 1.2e3.
Return:
`masque.Pattern` object containing the L3 design
:return: masque.Pattern object containing the L3 design
"""
default_args = {'hole_dose': 1,

25
fdfd_tools/__init__.py Normal file
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@ -0,0 +1,25 @@
"""
Electromagnetic FDFD simulation tools
Tools for 3D and 2D Electromagnetic Finite Difference Frequency Domain (FDFD)
simulations. These tools handle conversion of fields to/from vector form,
creation of the wave operator matrix, stretched-coordinate PMLs, PECs and PMCs,
field conversion operators, waveguide mode operator, and waveguide mode
solver.
This package only contains a solver for the waveguide mode eigenproblem;
if you want to solve 3D problems you can use your favorite iterative sparse
matrix solver (so long as it can handle complex symmetric [non-Hermitian]
matrices, ideally with double precision).
Dependencies:
- numpy
- scipy
"""
from .vectorization import vec, unvec, field_t, vfield_t
from .grid import dx_lists_t
__author__ = 'Jan Petykiewicz'

513
fdfd_tools/bloch.py Normal file
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@ -0,0 +1,513 @@
'''
Bloch eigenmode solver/operators
This module contains functions for generating and solving the
3D Bloch eigenproblem. The approach is to transform the problem
into the (spatial) fourier domain, transforming the equation
1/mu * curl(1/eps * curl(H)) = (w/c)^2 H
into
conv(1/mu_k, ik x conv(1/eps_k, ik x H_k)) = (w/c)^2 H_k
where:
- the _k subscript denotes a 3D fourier transformed field
- each component of H_k corresponds to a plane wave with wavevector k
- x is the cross product
- conv denotes convolution
Since k and H are orthogonal for each plane wave, we can use each
k to create an orthogonal basis (k, m, n), with k x m = n, and
|m| = |n| = 1. The cross products are then simplified with
k @ h = kx hx + ky hy + kz hz = 0 = hk
h = hk + hm + hn = hm + hn
k = kk + km + kn = kk = |k|
k x h = (ky hz - kz hy,
kz hx - kx hz,
kx hy - ky hx)
= ((k x h) @ k, (k x h) @ m, (k x h) @ n)_kmn
= (0, (m x k) @ h, (n x k) @ h)_kmn # triple product ordering
= (0, kk (-n @ h), kk (m @ h))_kmn # (m x k) = -|k| n, etc.
= |k| (0, -h @ n, h @ m)_kmn
k x h = (km hn - kn hm,
kn hk - kk hn,
kk hm - km hk)_kmn
= (0, -kk hn, kk hm)_kmn
= (-kk hn)(mx, my, mz) + (kk hm)(nx, ny, nz)
= |k| (hm * (nx, ny, nz) - hn * (mx, my, mz))
where h is shorthand for H_k, (...)_kmn deontes the (k, m, n) basis,
and e.g. hm is the component of h in the m direction.
We can also simplify conv(X_k, Y_k) as fftn(X * ifftn(Y_k)).
Using these results and storing H_k as h = (hm, hn), we have
e_xyz = fftn(1/eps * ifftn(|k| (hm * n - hn * m)))
b_mn = |k| (-e_xyz @ n, e_xyz @ m)
h_mn = fftn(1/mu * ifftn(b_m * m + b_n * n))
which forms the operator from the left side of the equation.
We can then use a preconditioned block Rayleigh iteration algorithm, as in
SG Johnson and JD Joannopoulos, Block-iterative frequency-domain methods
for Maxwell's equations in a planewave basis, Optics Express 8, 3, 173-190 (2001)
(similar to that used in MPB) to find the eigenvectors for this operator.
===
Typically you will want to do something like
recip_lattice = numpy.diag(1/numpy.array(epsilon[0].shape * dx))
n, v = bloch.eigsolve(5, k0, recip_lattice, epsilon)
f = numpy.sqrt(-numpy.real(n[0]))
n_eff = norm(recip_lattice @ k0) / f
v2e = bloch.hmn_2_exyz(k0, recip_lattice, epsilon)
e_field = v2e(v[0])
k, f = find_k(frequency=1/1550,
tolerance=(1/1550 - 1/1551),
direction=[1, 0, 0],
G_matrix=recip_lattice,
epsilon=epsilon,
band=0)
'''
from typing import List, Tuple, Callable, Dict
import logging
import numpy
from numpy.fft import fftn, ifftn, fftfreq
import scipy
import scipy.optimize
from scipy.linalg import norm
import scipy.sparse.linalg as spalg
from .eigensolvers import rayleigh_quotient_iteration
from . import field_t
logger = logging.getLogger(__name__)
def generate_kmn(k0: numpy.ndarray,
G_matrix: numpy.ndarray,
shape: numpy.ndarray
) -> Tuple[numpy.ndarray, numpy.ndarray, numpy.ndarray]:
"""
Generate a (k, m, n) orthogonal basis for each k-vector in the simulation grid.
:param k0: [k0x, k0y, k0z], Bloch wavevector, in G basis.
:param G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns.
:param shape: [nx, ny, nz] shape of the simulation grid.
:return: (|k|, m, n) where |k| has shape tuple(shape) + (1,)
and m, n have shape tuple(shape) + (3,).
All are given in the xyz basis (e.g. |k|[0,0,0] = norm(G_matrix @ k0)).
"""
k0 = numpy.array(k0)
Gi_grids = numpy.meshgrid(*(fftfreq(n, 1/n) for n in shape[:3]), indexing='ij')
Gi = numpy.stack(Gi_grids, axis=3)
k_G = k0[None, None, None, :] - Gi
k_xyz = numpy.rollaxis(G_matrix @ numpy.rollaxis(k_G, 3, 2), 3, 2)
m = numpy.broadcast_to([0, 1, 0], tuple(shape[:3]) + (3,)).astype(float)
n = numpy.broadcast_to([0, 0, 1], tuple(shape[:3]) + (3,)).astype(float)
xy_non0 = numpy.any(k_xyz[:, :, :, 0:1] != 0, axis=3)
if numpy.any(xy_non0):
u = numpy.cross(k_xyz[xy_non0], [0, 0, 1])
m[xy_non0, :] = u / norm(u, axis=1)[:, None]
z_non0 = numpy.any(k_xyz != 0, axis=3)
if numpy.any(z_non0):
v = numpy.cross(k_xyz[z_non0], m[z_non0])
n[z_non0, :] = v / norm(v, axis=1)[:, None]
k_mag = norm(k_xyz, axis=3)[:, :, :, None]
return k_mag, m, n
def maxwell_operator(k0: numpy.ndarray,
G_matrix: numpy.ndarray,
epsilon: field_t,
mu: field_t = None
) -> Callable[[numpy.ndarray], numpy.ndarray]:
"""
Generate the Maxwell operator
conv(1/mu_k, ik x conv(1/eps_k, ik x ___))
which is the spatial-frequency-space representation of
1/mu * curl(1/eps * curl(___))
The operator is a function that acts on a vector h_mn of size (2 * epsilon[0].size)
See the module-level docstring for more information.
:param k0: Bloch wavevector, [k0x, k0y, k0z].
:param G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns.
:param epsilon: Dielectric constant distribution for the simulation.
All fields are sampled at cell centers (i.e., NOT Yee-gridded)
:param mu: Magnetic permability distribution for the simulation.
Default None (1 everywhere).
:return: Function which applies the maxwell operator to h_mn.
"""
shape = epsilon[0].shape + (1,)
k_mag, m, n = generate_kmn(k0, G_matrix, shape)
epsilon = numpy.stack(epsilon, 3)
if mu is not None:
mu = numpy.stack(mu, 3)
def operator(h: numpy.ndarray):
"""
Maxwell operator for Bloch eigenmode simulation.
h is complex 2-field in (m, n) basis, vectorized
:param h: Raveled h_mn; size (2 * epsilon[0].size).
:return: Raveled conv(1/mu_k, ik x conv(1/eps_k, ik x h_mn)).
"""
hin_m, hin_n = [hi.reshape(shape) for hi in numpy.split(h, 2)]
#{d,e,h}_xyz fields are complex 3-fields in (1/x, 1/y, 1/z) basis
# cross product and transform into xyz basis
d_xyz = (n * hin_m -
m * hin_n) * k_mag
# divide by epsilon
e_xyz = fftn(ifftn(d_xyz, axes=range(3)) / epsilon, axes=range(3))
# cross product and transform into mn basis
b_m = numpy.sum(e_xyz * n, axis=3)[:, :, :, None] * -k_mag
b_n = numpy.sum(e_xyz * m, axis=3)[:, :, :, None] * +k_mag
if mu is None:
h_m, h_n = b_m, b_n
else:
# transform from mn to xyz
b_xyz = (m * b_m[:, :, :, None] +
n * b_n[:, :, :, None])
# divide by mu
h_xyz = fftn(ifftn(b_xyz, axes=range(3)) / mu, axes=range(3))
# transform back to mn
h_m = numpy.sum(h_xyz * m, axis=3)
h_n = numpy.sum(h_xyz * n, axis=3)
return numpy.hstack((h_m.ravel(), h_n.ravel()))
return operator
def hmn_2_exyz(k0: numpy.ndarray,
G_matrix: numpy.ndarray,
epsilon: field_t,
) -> Callable[[numpy.ndarray], field_t]:
"""
Generate an operator which converts a vectorized spatial-frequency-space
h_mn into an E-field distribution, i.e.
ifft(conv(1/eps_k, ik x h_mn))
The operator is a function that acts on a vector h_mn of size (2 * epsilon[0].size)
See the module-level docstring for more information.
:param k0: Bloch wavevector, [k0x, k0y, k0z].
:param G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns.
:param epsilon: Dielectric constant distribution for the simulation.
All fields are sampled at cell centers (i.e., NOT Yee-gridded)
:return: Function for converting h_mn into E_xyz
"""
shape = epsilon[0].shape + (1,)
epsilon = numpy.stack(epsilon, 3)
k_mag, m, n = generate_kmn(k0, G_matrix, shape)
def operator(h: numpy.ndarray) -> field_t:
hin_m, hin_n = [hi.reshape(shape) for hi in numpy.split(h, 2)]
d_xyz = (n * hin_m -
m * hin_n) * k_mag
# divide by epsilon
return [ei for ei in numpy.rollaxis(ifftn(d_xyz, axes=range(3)) / epsilon, 3)]
return operator
def hmn_2_hxyz(k0: numpy.ndarray,
G_matrix: numpy.ndarray,
epsilon: field_t
) -> Callable[[numpy.ndarray], field_t]:
"""
Generate an operator which converts a vectorized spatial-frequency-space
h_mn into an H-field distribution, i.e.
ifft(h_mn)
The operator is a function that acts on a vector h_mn of size (2 * epsilon[0].size)
See the module-level docstring for more information.
:param k0: Bloch wavevector, [k0x, k0y, k0z].
:param G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns.
:param epsilon: Dielectric constant distribution for the simulation.
Only epsilon[0].shape is used.
:return: Function for converting h_mn into H_xyz
"""
shape = epsilon[0].shape + (1,)
k_mag, m, n = generate_kmn(k0, G_matrix, shape)
def operator(h: numpy.ndarray):
hin_m, hin_n = [hi.reshape(shape) for hi in numpy.split(h, 2)]
h_xyz = (m * hin_m +
n * hin_n)
return [ifftn(hi) for hi in numpy.rollaxis(h_xyz, 3)]
return operator
def inverse_maxwell_operator_approx(k0: numpy.ndarray,
G_matrix: numpy.ndarray,
epsilon: field_t,
mu: field_t = None
) -> Callable[[numpy.ndarray], numpy.ndarray]:
"""
Generate an approximate inverse of the Maxwell operator,
ik x conv(eps_k, ik x conv(mu_k, ___))
which can be used to improve the speed of ARPACK in shift-invert mode.
See the module-level docstring for more information.
:param k0: Bloch wavevector, [k0x, k0y, k0z].
:param G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns.
:param epsilon: Dielectric constant distribution for the simulation.
All fields are sampled at cell centers (i.e., NOT Yee-gridded)
:param mu: Magnetic permability distribution for the simulation.
Default None (1 everywhere).
:return: Function which applies the approximate inverse of the maxwell operator to h_mn.
"""
shape = epsilon[0].shape + (1,)
epsilon = numpy.stack(epsilon, 3)
k_mag, m, n = generate_kmn(k0, G_matrix, shape)
if mu is not None:
mu = numpy.stack(mu, 3)
def operator(h: numpy.ndarray):
"""
Approximate inverse Maxwell operator for Bloch eigenmode simulation.
h is complex 2-field in (m, n) basis, vectorized
:param h: Raveled h_mn; size (2 * epsilon[0].size).
:return: Raveled ik x conv(eps_k, ik x conv(mu_k, h_mn))
"""
hin_m, hin_n = [hi.reshape(shape) for hi in numpy.split(h, 2)]
#{d,e,h}_xyz fields are complex 3-fields in (1/x, 1/y, 1/z) basis
if mu is None:
b_m, b_n = hin_m, hin_n
else:
# transform from mn to xyz
h_xyz = (m * hin_m[:, :, :, None] +
n * hin_n[:, :, :, None])
# multiply by mu
b_xyz = fftn(ifftn(h_xyz, axes=range(3)) * mu, axes=range(3))
# transform back to mn
b_m = numpy.sum(b_xyz * m, axis=3)
b_n = numpy.sum(b_xyz * n, axis=3)
# cross product and transform into xyz basis
e_xyz = (n * b_m -
m * b_n) / k_mag
# multiply by epsilon
d_xyz = fftn(ifftn(e_xyz, axes=range(3)) * epsilon, axes=range(3))
# cross product and transform into mn basis crossinv_t2c
h_m = numpy.sum(e_xyz * n, axis=3)[:, :, :, None] / +k_mag
h_n = numpy.sum(e_xyz * m, axis=3)[:, :, :, None] / -k_mag
return numpy.hstack((h_m.ravel(), h_n.ravel()))
return operator
def eigsolve(num_modes: int,
k0: numpy.ndarray,
G_matrix: numpy.ndarray,
epsilon: field_t,
mu: field_t = None,
tolerance = 1e-8,
) -> Tuple[numpy.ndarray, numpy.ndarray]:
"""
Find the first (lowest-frequency) num_modes eigenmodes with Bloch wavevector
k0 of the specified structure.
:param k0: Bloch wavevector, [k0x, k0y, k0z].
:param G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns.
:param epsilon: Dielectric constant distribution for the simulation.
All fields are sampled at cell centers (i.e., NOT Yee-gridded)
:param mu: Magnetic permability distribution for the simulation.
Default None (1 everywhere).
:return: (eigenvalues, eigenvectors) where eigenvalues[i] corresponds to the
vector eigenvectors[i, :]
"""
h_size = 2 * epsilon[0].size
kmag = norm(G_matrix @ k0)
'''
Generate the operators
'''
mop = maxwell_operator(k0=k0, G_matrix=G_matrix, epsilon=epsilon, mu=mu)
imop = inverse_maxwell_operator_approx(k0=k0, G_matrix=G_matrix, epsilon=epsilon, mu=mu)
scipy_op = spalg.LinearOperator(dtype=complex, shape=(h_size, h_size), matvec=mop)
scipy_iop = spalg.LinearOperator(dtype=complex, shape=(h_size, h_size), matvec=imop)
y_shape = (h_size, num_modes)
def rayleigh_quotient(Z: numpy.ndarray, approx_grad: bool = True):
"""
Absolute value of the block Rayleigh quotient, and the associated gradient.
See Johnson and Joannopoulos, Opt. Expr. 8, 3 (2001) for details (full
citation in module docstring).
===
Notes on my understanding of the procedure:
Minimize f(Y) = |trace((Y.H @ A @ Y)|, making use of Y = Z @ inv(Z.H @ Z)^(1/2)
(a polar orthogonalization of Y). This gives f(Z) = |trace(Z.H @ A @ Z @ U)|,
where U = inv(Z.H @ Z). We minimize the absolute value to find the eigenvalues
with smallest magnitude.
The gradient is P @ (A @ Z @ U), where P = (1 - Z @ U @ Z.H) is a projection
onto the space orthonormal to Z. If approx_grad is True, the approximate
inverse of the maxwell operator is used to precondition the gradient.
"""
z = Z.view(dtype=complex).reshape(y_shape)
U = numpy.linalg.inv(z.conj().T @ z)
zU = z @ U
AzU = scipy_op @ zU
zTAzU = z.conj().T @ AzU
f = numpy.real(numpy.trace(zTAzU))
if approx_grad:
df_dy = scipy_iop @ (AzU - zU @ zTAzU)
else:
df_dy = (AzU - zU @ zTAzU)
df_dy_flat = df_dy.view(dtype=float).ravel()
return numpy.abs(f), numpy.sign(f) * df_dy_flat
'''
Use the conjugate gradient method and the approximate gradient calculation to
quickly find approximate eigenvectors.
'''
result = scipy.optimize.minimize(rayleigh_quotient,
numpy.random.rand(*y_shape, 2),
jac=True,
method='L-BFGS-B',
tol=1e-20,
options={'maxiter': 2000, 'gtol':0, 'ftol':1e-20 , 'disp':True})#, 'maxls':80, 'm':30})
result = scipy.optimize.minimize(lambda y: rayleigh_quotient(y, True),
result.x,
jac=True,
method='L-BFGS-B',
tol=1e-20,
options={'maxiter': 2000, 'gtol':0, 'disp':True})
result = scipy.optimize.minimize(lambda y: rayleigh_quotient(y, False),
result.x,
jac=True,
method='L-BFGS-B',
tol=1e-20,
options={'maxiter': 2000, 'gtol':0, 'disp':True})
for i in range(20):
result = scipy.optimize.minimize(lambda y: rayleigh_quotient(y, False),
result.x,
jac=True,
method='L-BFGS-B',
tol=1e-20,
options={'maxiter': 70, 'gtol':0, 'disp':True})
if result.nit == 0:
# We took 0 steps, so re-running won't help
break
z = result.x.view(dtype=complex).reshape(y_shape)
'''
Recover eigenvectors from Z
'''
U = numpy.linalg.inv(z.conj().T @ z)
y = z @ scipy.linalg.sqrtm(U)
w = y.conj().T @ (scipy_op @ y)
eigvals, w_eigvecs = numpy.linalg.eig(w)
eigvecs = y @ w_eigvecs
for i in range(len(eigvals)):
v = eigvecs[:, i]
n = eigvals[i]
v /= norm(v)
eigness = norm(scipy_op @ v - (v.conj() @ (scipy_op @ v)) * v )
f = numpy.sqrt(-numpy.real(n))
df = numpy.sqrt(-numpy.real(n + eigness))
neff_err = kmag * (1/df - 1/f)
logger.info('eigness {}: {}\n neff_err: {}'.format(i, eigness, neff_err))
order = numpy.argsort(numpy.abs(eigvals))
return eigvals[order], eigvecs.T[order]
def find_k(frequency: float,
tolerance: float,
direction: numpy.ndarray,
G_matrix: numpy.ndarray,
epsilon: field_t,
mu: field_t = None,
band: int = 0,
k_min: float = 0,
k_max: float = 0.5,
) -> Tuple[numpy.ndarray, float]:
"""
Search for a bloch vector that has a given frequency.
:param frequency: Target frequency.
:param tolerance: Target frequency tolerance.
:param direction: k-vector direction to search along.
:param G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns.
:param epsilon: Dielectric constant distribution for the simulation.
All fields are sampled at cell centers (i.e., NOT Yee-gridded)
:param mu: Magnetic permability distribution for the simulation.
Default None (1 everywhere).
:param band: Which band to search in. Default 0 (lowest frequency).
return: (k, actual_frequency) The found k-vector and its frequency
"""
direction = numpy.array(direction) / norm(direction)
def get_f(k0_mag: float, band: int = 0):
k0 = direction * k0_mag
n, _v = eigsolve(band + 1, k0, G_matrix=G_matrix, epsilon=epsilon)
f = numpy.sqrt(numpy.abs(numpy.real(n[band])))
return f
res = scipy.optimize.minimize_scalar(lambda x: abs(get_f(x, band) - frequency),
(k_min + k_max) / 2,
method='Bounded',
bounds=(k_min, k_max),
options={'xatol': abs(tolerance)})
return res.x * direction, res.fun + frequency

53
meanas/eigensolvers.py → fdfd_tools/eigensolvers.py
View File

@ -15,13 +15,10 @@ def power_iteration(operator: sparse.spmatrix,
"""
Use power iteration to estimate the dominant eigenvector of a matrix.
Args:
operator: Matrix to analyze.
guess_vector: Starting point for the eigenvector. Default is a randomly chosen vector.
iterations: Number of iterations to perform. Default 20.
Returns:
(Largest-magnitude eigenvalue, Corresponding eigenvector estimate)
:param operator: Matrix to analyze.
:param guess_vector: Starting point for the eigenvector. Default is a randomly chosen vector.
:param iterations: Number of iterations to perform. Default 20.
:return: (Largest-magnitude eigenvalue, Corresponding eigenvector estimate)
"""
if numpy.any(numpy.equal(guess_vector, None)):
v = numpy.random.rand(operator.shape[0])
@ -40,23 +37,20 @@ def rayleigh_quotient_iteration(operator: sparse.spmatrix or spalg.LinearOperato
guess_vector: numpy.ndarray,
iterations: int = 40,
tolerance: float = 1e-13,
solver = None,
solver=None,
) -> Tuple[complex, numpy.ndarray]:
"""
Use Rayleigh quotient iteration to refine an eigenvector guess.
Args:
operator: Matrix to analyze.
guess_vector: Eigenvector to refine.
iterations: Maximum number of iterations to perform. Default 40.
tolerance: Stop iteration if `(A - I*eigenvalue) @ v < num_vectors * tolerance`,
Default 1e-13.
solver: Solver function of the form `x = solver(A, b)`.
By default, use scipy.sparse.spsolve for sparse matrices and
scipy.sparse.bicgstab for general LinearOperator instances.
Returns:
(eigenvalues, eigenvectors)
:param operator: Matrix to analyze.
:param guess_vector: Eigenvector to refine.
:param iterations: Maximum number of iterations to perform. Default 40.
:param tolerance: Stop iteration if (A - I*eigenvalue) @ v < tolerance.
Default 1e-13.
:param solver: Solver function of the form x = solver(A, b).
By default, use scipy.sparse.spsolve for sparse matrices and
scipy.sparse.bicgstab for general LinearOperator instances.
:return: (eigenvalue, eigenvector)
"""
try:
_test = operator - sparse.eye(operator.shape[0])
@ -70,7 +64,7 @@ def rayleigh_quotient_iteration(operator: sparse.spmatrix or spalg.LinearOperato
if solver is None:
solver = lambda A, b: spalg.bicgstab(A, b)[0]
v = numpy.squeeze(guess_vector)
v = guess_vector
v /= norm(v)
for _ in range(iterations):
eigval = v.conj() @ (operator @ v)
@ -91,15 +85,12 @@ def signed_eigensolve(operator: sparse.spmatrix or spalg.LinearOperator,
Find the largest-magnitude positive-only (or negative-only) eigenvalues and
eigenvectors of the provided matrix.
Args:
operator: Matrix to analyze.
how_many: How many eigenvalues to find.
negative: Whether to find negative-only eigenvalues.
Default False (positive only).
Returns:
(sorted list of eigenvalues, 2D ndarray of corresponding eigenvectors)
`eigenvectors[:, k]` corresponds to the k-th eigenvalue
:param operator: Matrix to analyze.
:param how_many: How many eigenvalues to find.
:param negative: Whether to find negative-only eigenvalues.
Default False (positive only).
:return: (sorted list of eigenvalues, 2D ndarray of corresponding eigenvectors)
eigenvectors[:, k] corresponds to the k-th eigenvalue
"""
# Use power iteration to estimate the dominant eigenvector
lm_eigval, _ = power_iteration(operator)
@ -108,7 +99,7 @@ def signed_eigensolve(operator: sparse.spmatrix or spalg.LinearOperator,
Shift by the absolute value of the largest eigenvalue, then find a few of the
largest-magnitude (shifted) eigenvalues. A positive shift ensures that we find the
largest _positive_ eigenvalues, since any negative eigenvalues will be shifted to the
range `0 >= neg_eigval + abs(lm_eigval) > abs(lm_eigval)`
range 0 >= neg_eigval + abs(lm_eigval) > abs(lm_eigval)
'''
shift = numpy.abs(lm_eigval)
if negative:

100
meanas/fdfd/farfield.py → fdfd_tools/farfield.py
View File

@ -1,51 +1,45 @@
"""
Functions for performing near-to-farfield transformation (and the reverse).
"""
from typing import Dict, List, Any
from typing import Dict, List
import numpy
from numpy.fft import fft2, fftshift, fftfreq, ifft2, ifftshift
from numpy import pi
from ..fdmath import fdfield_t
def near_to_farfield(E_near: fdfield_t,
H_near: fdfield_t,
def near_to_farfield(E_near: List[numpy.ndarray],
H_near: List[numpy.ndarray],
dx: float,
dy: float,
padded_size: List[int] = None
) -> Dict[str, Any]:
) -> Dict[str]:
"""
Compute the farfield, i.e. the distribution of the fields after propagation
through several wavelengths of uniform medium.
The input fields should be complex phasors.
Args:
E_near: List of 2 ndarrays containing the 2D phasor field slices for the transverse
E fields (e.g. [Ex, Ey] for calculating the farfield toward the z-direction).
H_near: List of 2 ndarrays containing the 2D phasor field slices for the transverse
H fields (e.g. [Hx, hy] for calculating the farfield towrad the z-direction).
dx: Cell size along x-dimension, in units of wavelength.
dy: Cell size along y-dimension, in units of wavelength.
padded_size: Shape of the output. A single integer `n` will be expanded to `(n, n)`.
Powers of 2 are most efficient for FFT computation.
Default is the smallest power of 2 larger than the input, for each axis.
Returns:
Dict with keys
- `E_far`: Normalized E-field farfield; multiply by
(i k exp(-i k r) / (4 pi r)) to get the actual field value.
- `H_far`: Normalized H-field farfield; multiply by
(i k exp(-i k r) / (4 pi r)) to get the actual field value.
- `kx`, `ky`: Wavevector values corresponding to the x- and y- axes in E_far and H_far,
normalized to wavelength (dimensionless).
- `dkx`, `dky`: step size for kx and ky, normalized to wavelength.
- `theta`: arctan2(ky, kx) corresponding to each (kx, ky).
This is the angle in the x-y plane, counterclockwise from above, starting from +x.
- `phi`: arccos(kz / k) corresponding to each (kx, ky).
This is the angle away from +z.
:param E_near: List of 2 ndarrays containing the 2D phasor field slices for the transverse
E fields (e.g. [Ex, Ey] for calculating the farfield toward the z-direction).
:param H_near: List of 2 ndarrays containing the 2D phasor field slices for the transverse
H fields (e.g. [Hx, hy] for calculating the farfield towrad the z-direction).
:param dx: Cell size along x-dimension, in units of wavelength.
:param dy: Cell size along y-dimension, in units of wavelength.
:param padded_size: Shape of the output. A single integer `n` will be expanded to `(n, n)`.
Powers of 2 are most efficient for FFT computation.
Default is the smallest power of 2 larger than the input, for each axis.
:returns: Dict with keys
'E_far': Normalized E-field farfield; multiply by
(i k exp(-i k r) / (4 pi r)) to get the actual field value.
'H_far': Normalized H-field farfield; multiply by
(i k exp(-i k r) / (4 pi r)) to get the actual field value.
'kx', 'ky': Wavevector values corresponding to the x- and y- axes in E_far and H_far,
normalized to wavelength (dimensionless).
'dkx', 'dky': step size for kx and ky, normalized to wavelength.
'theta': arctan2(ky, kx) corresponding to each (kx, ky).
This is the angle in the x-y plane, counterclockwise from above, starting from +x.
'phi': arccos(kz / k) corresponding to each (kx, ky).
This is the angle away from +z.
"""
if not len(E_near) == 2:
@ -121,39 +115,35 @@ def near_to_farfield(E_near: fdfield_t,
def far_to_nearfield(E_far: fdfield_t,
H_far: fdfield_t,
def far_to_nearfield(E_far: List[numpy.ndarray],
H_far: List[numpy.ndarray],
dkx: float,
dky: float,
padded_size: List[int] = None
) -> Dict[str, Any]:
) -> Dict[str]:
"""
Compute the farfield, i.e. the distribution of the fields after propagation
through several wavelengths of uniform medium.
The input fields should be complex phasors.
Args:
E_far: List of 2 ndarrays containing the 2D phasor field slices for the transverse
E fields (e.g. [Ex, Ey] for calculating the nearfield toward the z-direction).
Fields should be normalized so that
E_far = E_far_actual / (i k exp(-i k r) / (4 pi r))
H_far: List of 2 ndarrays containing the 2D phasor field slices for the transverse
H fields (e.g. [Hx, hy] for calculating the nearfield toward the z-direction).
Fields should be normalized so that
H_far = H_far_actual / (i k exp(-i k r) / (4 pi r))
dkx: kx discretization, in units of wavelength.
dky: ky discretization, in units of wavelength.
padded_size: Shape of the output. A single integer `n` will be expanded to `(n, n)`.
Powers of 2 are most efficient for FFT computation.
Default is the smallest power of 2 larger than the input, for each axis.
Returns:
Dict with keys
- `E`: E-field nearfield
- `H`: H-field nearfield
- `dx`, `dy`: spatial discretization, normalized to wavelength (dimensionless)
:param E_far: List of 2 ndarrays containing the 2D phasor field slices for the transverse
E fields (e.g. [Ex, Ey] for calculating the nearfield toward the z-direction).
Fields should be normalized so that
E_far = E_far_actual / (i k exp(-i k r) / (4 pi r))
:param H_far: List of 2 ndarrays containing the 2D phasor field slices for the transverse
H fields (e.g. [Hx, hy] for calculating the nearfield toward the z-direction).
Fields should be normalized so that
H_far = H_far_actual / (i k exp(-i k r) / (4 pi r))
:param dkx: kx discretization, in units of wavelength.
:param dky: ky discretization, in units of wavelength.
:param padded_size: Shape of the output. A single integer `n` will be expanded to `(n, n)`.
Powers of 2 are most efficient for FFT computation.
Default is the smallest power of 2 larger than the input, for each axis.
:returns: Dict with keys
'E': E-field nearfield
'H': H-field nearfield
'dx', 'dy': spatial discretization, normalized to wavelength (dimensionless)
"""
if not len(E_far) == 2:

239
fdfd_tools/fdtd.py Normal file
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@ -0,0 +1,239 @@
from typing import List, Callable, Tuple, Dict
import numpy
from . import dx_lists_t, field_t
__author__ = 'Jan Petykiewicz'
functional_matrix = Callable[[field_t], field_t]
def curl_h(dxes: dx_lists_t = None) -> functional_matrix:
"""
Curl operator for use with the H field.
:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header
:return: Function for taking the discretized curl of the H-field, F(H) -> curlH
"""
if dxes:
dxyz_b = numpy.meshgrid(*dxes[1], indexing='ij')
def dh(f, ax):
return (f - numpy.roll(f, 1, axis=ax)) / dxyz_b[ax]
else:
def dh(f, ax):
return f - numpy.roll(f, 1, axis=ax)
def ch_fun(h: field_t) -> field_t:
e = [dh(h[2], 1) - dh(h[1], 2),
dh(h[0], 2) - dh(h[2], 0),
dh(h[1], 0) - dh(h[0], 1)]
return e
return ch_fun
def curl_e(dxes: dx_lists_t = None) -> functional_matrix:
"""
Curl operator for use with the E field.
:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header
:return: Function for taking the discretized curl of the E-field, F(E) -> curlE
"""
if dxes is not None:
dxyz_a = numpy.meshgrid(*dxes[0], indexing='ij')
def de(f, ax):
return (numpy.roll(f, -1, axis=ax) - f) / dxyz_a[ax]
else:
def de(f, ax):
return numpy.roll(f, -1, axis=ax) - f
def ce_fun(e: field_t) -> field_t:
h = [de(e[2], 1) - de(e[1], 2),
de(e[0], 2) - de(e[2], 0),
de(e[1], 0) - de(e[0], 1)]
return h
return ce_fun
def maxwell_e(dt: float, dxes: dx_lists_t = None) -> functional_matrix:
curl_h_fun = curl_h(dxes)
def me_fun(e: field_t, h: field_t, epsilon: field_t):
ch = curl_h_fun(h)
for ei, ci, epsi in zip(e, ch, epsilon):
ei += dt * ci / epsi
return e
return me_fun
def maxwell_h(dt: float, dxes: dx_lists_t = None) -> functional_matrix:
curl_e_fun = curl_e(dxes)
def mh_fun(e: field_t, h: field_t):
ce = curl_e_fun(e)
for hi, ci in zip(h, ce):
hi -= dt * ci
return h
return mh_fun
def conducting_boundary(direction: int,
polarity: int
) -> Tuple[functional_matrix, functional_matrix]:
dirs = [0, 1, 2]
if direction not in dirs:
raise Exception('Invalid direction: {}'.format(direction))
dirs.remove(direction)
u, v = dirs
if polarity < 0:
boundary_slice = [slice(None)] * 3
shifted1_slice = [slice(None)] * 3
boundary_slice[direction] = 0
shifted1_slice[direction] = 1
def en(e: field_t):
e[direction][boundary_slice] = 0
e[u][boundary_slice] = e[u][shifted1_slice]
e[v][boundary_slice] = e[v][shifted1_slice]
return e
def hn(h: field_t):
h[direction][boundary_slice] = h[direction][shifted1_slice]
h[u][boundary_slice] = 0
h[v][boundary_slice] = 0
return h
return en, hn
elif polarity > 0:
boundary_slice = [slice(None)] * 3
shifted1_slice = [slice(None)] * 3
shifted2_slice = [slice(None)] * 3
boundary_slice[direction] = -1
shifted1_slice[direction] = -2
shifted2_slice[direction] = -3
def ep(e: field_t):
e[direction][boundary_slice] = -e[direction][shifted2_slice]
e[direction][shifted1_slice] = 0
e[u][boundary_slice] = e[u][shifted1_slice]
e[v][boundary_slice] = e[v][shifted1_slice]
return e
def hp(h: field_t):
h[direction][boundary_slice] = h[direction][shifted1_slice]
h[u][boundary_slice] = -h[u][shifted2_slice]
h[u][shifted1_slice] = 0
h[v][boundary_slice] = -h[v][shifted2_slice]
h[v][shifted1_slice] = 0
return h
return ep, hp
else:
raise Exception('Bad polarity: {}'.format(polarity))
def cpml(direction:int,
polarity: int,
dt: float,
epsilon: field_t,
thickness: int = 8,
epsilon_eff: float = 1,
dtype: numpy.dtype = numpy.float32,
) -> Tuple[Callable, Callable, Dict[str, field_t]]:
if direction not in range(3):
raise Exception('Invalid direction: {}'.format(direction))
if polarity not in (-1, 1):
raise Exception('Invalid polarity: {}'.format(polarity))
if thickness <= 2:
raise Exception('It would be wise to have a pml with 4+ cells of thickness')
if epsilon_eff <= 0:
raise Exception('epsilon_eff must be positive')
m = (3.5, 1)
sigma_max = 0.8 * (m[0] + 1) / numpy.sqrt(epsilon_eff)
alpha_max = 0 # TODO: Decide what to do about non-zero alpha
transverse = numpy.delete(range(3), direction)
u, v = transverse
xe = numpy.arange(1, thickness+1, dtype=float)
xh = numpy.arange(1, thickness+1, dtype=float)
if polarity > 0:
xe -= 0.5
elif polarity < 0:
xh -= 0.5
xe = xe[::-1]
xh = xh[::-1]
else:
raise Exception('Bad polarity!')
expand_slice = [None] * 3
expand_slice[direction] = slice(None)
def par(x):
sigma = ((x / thickness) ** m[0]) * sigma_max
alpha = ((1 - x / thickness) ** m[1]) * alpha_max
p0 = numpy.exp(-(sigma + alpha) * dt)
p1 = sigma / (sigma + alpha) * (p0 - 1)
return p0[expand_slice], p1[expand_slice]
p0e, p1e = par(xe)
p0h, p1h = par(xh)
region = [slice(None)] * 3
if polarity < 0:
region[direction] = slice(None, thickness)
elif polarity > 0:
region[direction] = slice(-thickness, None)
else:
raise Exception('Bad polarity!')
if direction == 1:
se = 1
else:
se = -1
# TODO check if epsilon is uniform?
shape = list(epsilon[0].shape)
shape[direction] = thickness
psi_e = [numpy.zeros(shape, dtype=dtype), numpy.zeros(shape, dtype=dtype)]
psi_h = [numpy.zeros(shape, dtype=dtype), numpy.zeros(shape, dtype=dtype)]
fields = {
'psi_e_u': psi_e[0],
'psi_e_v': psi_e[1],
'psi_h_u': psi_h[0],
'psi_h_v': psi_h[1],
}
def pml_e(e: field_t, h: field_t, epsilon: field_t) -> Tuple[field_t, field_t]:
psi_e[0] *= p0e
psi_e[0] += p1e * (h[v][region] - numpy.roll(h[v], 1, axis=direction)[region])
psi_e[1] *= p0e
psi_e[1] += p1e * (h[u][region] - numpy.roll(h[u], 1, axis=direction)[region])
e[u][region] += se * dt * psi_e[0] / epsilon[u][region]
e[v][region] -= se * dt * psi_e[1] / epsilon[v][region]
return e, h
def pml_h(e: field_t, h: field_t) -> Tuple[field_t, field_t]:
psi_h[0] *= p0h
psi_h[0] += p1h * (numpy.roll(e[v], -1, axis=direction)[region] - e[v][region])
psi_h[1] *= p0h
psi_h[1] += p1h * (numpy.roll(e[u], -1, axis=direction)[region] - e[u][region])
h[u][region] -= se * dt * psi_h[0]
h[v][region] += se * dt * psi_h[1]
return e, h
return pml_e, pml_h, fields

149
fdfd_tools/functional.py Normal file
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@ -0,0 +1,149 @@
"""
Functional versions of many FDFD operators. These can be useful for performing
FDFD calculations without needing to construct large matrices in memory.
The functions generated here expect inputs in the form E = [E_x, E_y, E_z], where each
component E_* is an ndarray of equal shape.
"""
from typing import List, Callable
import numpy
from . import dx_lists_t, field_t
__author__ = 'Jan Petykiewicz'
functional_matrix = Callable[[field_t], field_t]
def curl_h(dxes: dx_lists_t) -> functional_matrix:
"""
Curl operator for use with the H field.
:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header
:return: Function for taking the discretized curl of the H-field, F(H) -> curlH
"""
dxyz_b = numpy.meshgrid(*dxes[1], indexing='ij')
def dh(f, ax):
return (f - numpy.roll(f, 1, axis=ax)) / dxyz_b[ax]
def ch_fun(h: field_t) -> field_t:
e = [dh(h[2], 1) - dh(h[1], 2),
dh(h[0], 2) - dh(h[2], 0),
dh(h[1], 0) - dh(h[0], 1)]
return e
return ch_fun
def curl_e(dxes: dx_lists_t) -> functional_matrix:
"""
Curl operator for use with the E field.
:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header
:return: Function for taking the discretized curl of the E-field, F(E) -> curlE
"""
dxyz_a = numpy.meshgrid(*dxes[0], indexing='ij')
def de(f, ax):
return (numpy.roll(f, -1, axis=ax) - f) / dxyz_a[ax]
def ce_fun(e: field_t) -> field_t:
h = [de(e[2], 1) - de(e[1], 2),
de(e[0], 2) - de(e[2], 0),
de(e[1], 0) - de(e[0], 1)]
return h
return ce_fun
def e_full(omega: complex,
dxes: dx_lists_t,
epsilon: field_t,
mu: field_t = None
) -> functional_matrix:
"""
Wave operator del x (1/mu * del x) - omega**2 * epsilon, for use with E-field,
with wave equation
(del x (1/mu * del x) - omega**2 * epsilon) E = -i * omega * J
:param omega: Angular frequency of the simulation
:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header
:param epsilon: Dielectric constant
:param mu: Magnetic permeability (default 1 everywhere)
:return: Function implementing the wave operator A(E) -> E
"""
ch = curl_h(dxes)
ce = curl_e(dxes)
def op_1(e):
curls = ch(ce(e))
return [c - omega ** 2 * e * x for c, e, x in zip(curls, epsilon, e)]
def op_mu(e):
curls = ch([m * y for m, y in zip(mu, ce(e))])
return [c - omega ** 2 * p * x for c, p, x in zip(curls, epsilon, e)]
if numpy.any(numpy.equal(mu, None)):
return op_1
else:
return op_mu
def eh_full(omega: complex,
dxes: dx_lists_t,
epsilon: field_t,
mu: field_t = None
) -> functional_matrix:
"""
Wave operator for full (both E and H) field representation.
:param omega: Angular frequency of the simulation
:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header
:param epsilon: Dielectric constant
:param mu: Magnetic permeability (default 1 everywhere)
:return: Function implementing the wave operator A(E, H) -> (E, H)
"""
ch = curl_h(dxes)
ce = curl_e(dxes)
def op_1(e, h):
return ([c - 1j * omega * p * x for c, p, x in zip(ch(h), epsilon, e)],
[c + 1j * omega * y for c, y in zip(ce(e), h)])
def op_mu(e, h):
return ([c - 1j * omega * p * x for c, p, x in zip(ch(h), epsilon, e)],
[c + 1j * omega * m * y for c, m, y in zip(ce(e), mu, h)])
if numpy.any(numpy.equal(mu, None)):
return op_1
else:
return op_mu
def e2h(omega: complex,
dxes: dx_lists_t,
mu: field_t = None,
) -> functional_matrix:
"""
Utility operator for converting the E field into the H field.
For use with e_full -- assumes that there is no magnetic current M.
:param omega: Angular frequency of the simulation
:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header
:param mu: Magnetic permeability (default 1 everywhere)
:return: Function for converting E to H
"""
A2 = curl_e(dxes)
def e2h_1_1(e):
return [y / (-1j * omega) for y in A2(e)]
def e2h_mu(e):
return [y / (-1j * omega * m) for y, m in zip(A2(e), mu)]
if numpy.any(numpy.equal(mu, None)):
return e2h_1_1
else:
return e2h_mu

105
meanas/fdfd/scpml.py → fdfd_tools/grid.py
View File

@ -1,35 +1,29 @@
"""
Functions for creating stretched coordinate perfectly matched layer (PML) absorbers.
Functions for creating stretched coordinate PMLs.
"""
from typing import List, Callable
import numpy
from ..fdmath import dx_lists_t
__author__ = 'Jan Petykiewicz'
s_function_t = Callable[[float], float]
"""Typedef for s-functions, see `prepare_s_function()`"""
dx_lists_t = List[List[numpy.ndarray]]
s_function_type = Callable[[float], float]
def prepare_s_function(ln_R: float = -16,
m: float = 4
) -> s_function_t:
) -> s_function_type:
"""
Create an s_function to pass to the SCPML functions. This is used when you would like to
customize the PML parameters.
Args:
ln_R: Natural logarithm of the desired reflectance
m: Polynomial order for the PML (imaginary part increases as distance ** m)
Returns:
An s_function, which takes an ndarray (distances) and returns an ndarray (complex part
of the cell width; needs to be divided by `sqrt(epilon_effective) * real(omega))`
before use.
:param ln_R: Natural logarithm of the desired reflectance
:param m: Polynomial order for the PML (imaginary part increases as distance ** m)
:return: An s_function, which takes an ndarray (distances) and returns an ndarray (complex part
of the cell width; needs to be divided by sqrt(epilon_effective) * real(omega))
before use.
"""
def s_factor(distance: numpy.ndarray) -> numpy.ndarray:
s_max = (m + 1) * ln_R / 2 # / 2 because we assume periodic boundaries
@ -41,29 +35,26 @@ def uniform_grid_scpml(shape: numpy.ndarray or List[int],
thicknesses: numpy.ndarray or List[int],
omega: float,
epsilon_effective: float = 1.0,
s_function: s_function_t = None,
s_function: s_function_type = None,
) -> dx_lists_t:
"""
Create dx arrays for a uniform grid with a cell width of 1 and a pml.
If you want something more fine-grained, check out `stretch_with_scpml(...)`.
If you want something more fine-grained, check out stretch_with_scpml(...).
Args:
shape: Shape of the grid, including the PMLs (which are 2*thicknesses thick)
thicknesses: `[th_x, th_y, th_z]`
Thickness of the PML in each direction.
Both polarities are added.
Each th_ of pml is applied twice, once on each edge of the grid along the given axis.
`th_*` may be zero, in which case no pml is added.
omega: Angular frequency for the simulation
epsilon_effective: Effective epsilon of the PML. Match this to the material
at the edge of your grid.
Default 1.
s_function: created by `prepare_s_function(...)`, allowing customization of pml parameters.
Default uses `prepare_s_function()` with no parameters.
Returns:
Complex cell widths (dx_lists_t) as discussed in `meanas.fdmath.types`.
:param shape: Shape of the grid, including the PMLs (which are 2*thicknesses thick)
:param thicknesses: [th_x, th_y, th_z] Thickness of the PML in each direction.
Both polarities are added.
Each th_ of pml is applied twice, once on each edge of the grid along the given axis.
th_* may be zero, in which case no pml is added.
:param omega: Angular frequency for the simulation
:param epsilon_effective: Effective epsilon of the PML. Match this to the material
at the edge of your grid.
Default 1.
:param s_function: s_function created by prepare_s_function(...), allowing
customization of pml parameters.
Default uses prepare_s_function() with no parameters.
:return: Complex cell widths (dx_lists)
"""
if s_function is None:
s_function = prepare_s_function()
@ -96,25 +87,21 @@ def stretch_with_scpml(dxes: dx_lists_t,
omega: float,
epsilon_effective: float = 1.0,
thickness: int = 10,
s_function: s_function_t = None,
s_function: s_function_type = None,
) -> dx_lists_t:
"""
Stretch dxes to contain a stretched-coordinate PML (SCPML) in one direction along one axis.
Args:
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types`
axis: axis to stretch (0=x, 1=y, 2=z)
polarity: direction to stretch (-1 for -ve, +1 for +ve)
omega: Angular frequency for the simulation
epsilon_effective: Effective epsilon of the PML. Match this to the material at the
edge of your grid. Default 1.
thickness: number of cells to use for pml (default 10)
s_function: Created by `prepare_s_function(...)`, allowing customization
of pml parameters. Default uses `prepare_s_function()` with no parameters.
Returns:
Complex cell widths (dx_lists_t) as discussed in `meanas.fdmath.types`.
Multiple calls to this function may be necessary if multiple absorpbing boundaries are needed.
:param dxes: dx_tuple with coordinates to stretch
:param axis: axis to stretch (0=x, 1=y, 2=z)
:param polarity: direction to stretch (-1 for -ve, +1 for +ve)
:param omega: Angular frequency for the simulation
:param epsilon_effective: Effective epsilon of the PML. Match this to the material at the
edge of your grid. Default 1.
:param thickness: number of cells to use for pml (default 10)
:param s_function: s_function created by prepare_s_function(...), allowing customization
of pml parameters. Default uses prepare_s_function() with no parameters.
:return: Complex cell widths
"""
if s_function is None:
s_function = prepare_s_function()
@ -159,3 +146,25 @@ def stretch_with_scpml(dxes: dx_lists_t,
dxes[1][axis] = dx_bi
return dxes
def generate_periodic_dx(pos: List[numpy.ndarray]) -> dx_lists_t:
"""
Given a list of 3 ndarrays cell centers, creates the cell width parameters for a periodic grid.
:param pos: List of 3 ndarrays of cell centers
:return: (dx_a, dx_b) cell widths (no pml)
"""
if len(pos) != 3:
raise Exception('Must have len(pos) == 3')
dx_a = [numpy.array(numpy.inf)] * 3
dx_b = [numpy.array(numpy.inf)] * 3
for i, p_orig in enumerate(pos):
p = numpy.array(p_orig, dtype=float)
if p.size != 1:
p_shifted = numpy.hstack((p[1:], p[-1] + (p[1] - p[0])))
dx_a[i] = numpy.diff(p)
dx_b[i] = numpy.diff((p + p_shifted) / 2)
return dx_a, dx_b

507
fdfd_tools/operators.py Normal file
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@ -0,0 +1,507 @@
"""
Sparse matrix operators for use with electromagnetic wave equations.
These functions return sparse-matrix (scipy.sparse.spmatrix) representations of
a variety of operators, intended for use with E and H fields vectorized using the
fdfd_tools.vec() and .unvec() functions (column-major/Fortran ordering).
E- and H-field values are defined on a Yee cell; epsilon values should be calculated for
cells centered at each E component (mu at each H component).
Many of these functions require a 'dxes' parameter, of type fdfd_tools.dx_lists_type,
which contains grid cell width information in the following format:
[[[dx_e_0, dx_e_1, ...], [dy_e_0, ...], [dz_e_0, ...]],
[[dx_h_0, dx_h_1, ...], [dy_h_0, ...], [dz_h_0, ...]]]
where dx_e_0 is the x-width of the x=0 cells, as used when calculating dE/dx,
and dy_h_0 is the y-width of the y=0 cells, as used when calculating dH/dy, etc.
The following operators are included:
- E-only wave operator
- H-only wave operator
- EH wave operator
- Curl for use with E, H fields
- E to H conversion
- M to J conversion
- Poynting cross products
Also available:
- Circular shifts
- Discrete derivatives
- Averaging operators
- Cross product matrices
"""
from typing import List, Tuple
import numpy
import scipy.sparse as sparse
from . import vec, dx_lists_t, vfield_t
__author__ = 'Jan Petykiewicz'
def e_full(omega: complex,
dxes: dx_lists_t,
epsilon: vfield_t,
mu: vfield_t = None,
pec: vfield_t = None,
pmc: vfield_t = None,
) -> sparse.spmatrix:
"""
Wave operator del x (1/mu * del x) - omega**2 * epsilon, for use with E-field,
with wave equation
(del x (1/mu * del x) - omega**2 * epsilon) E = -i * omega * J
To make this matrix symmetric, use the preconditions from e_full_preconditioners().
:param omega: Angular frequency of the simulation
:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header
:param epsilon: Vectorized dielectric constant
:param mu: Vectorized magnetic permeability (default 1 everywhere).
:param pec: Vectorized mask specifying PEC cells. Any cells where pec != 0 are interpreted
as containing a perfect electrical conductor (PEC).
The PEC is applied per-field-component (ie, pec.size == epsilon.size)
:param pmc: Vectorized mask specifying PMC cells. Any cells where pmc != 0 are interpreted
as containing a perfect magnetic conductor (PMC).
The PMC is applied per-field-component (ie, pmc.size == epsilon.size)
:return: Sparse matrix containing the wave operator
"""
ce = curl_e(dxes)
ch = curl_h(dxes)
if numpy.any(numpy.equal(pec, None)):
pe = sparse.eye(epsilon.size)
else:
pe = sparse.diags(numpy.where(pec, 0, 1)) # Set pe to (not PEC)
if numpy.any(numpy.equal(pmc, None)):
pm = sparse.eye(epsilon.size)
else:
pm = sparse.diags(numpy.where(pmc, 0, 1)) # set pm to (not PMC)
e = sparse.diags(epsilon)
if numpy.any(numpy.equal(mu, None)):
m_div = sparse.eye(epsilon.size)
else:
m_div = sparse.diags(1 / mu)
op = pe @ (ch @ pm @ m_div @ ce - omega**2 * e) @ pe
return op
def e_full_preconditioners(dxes: dx_lists_t
) -> Tuple[sparse.spmatrix, sparse.spmatrix]:
"""
Left and right preconditioners (Pl, Pr) for symmetrizing the e_full wave operator.
The preconditioned matrix A_symm = (Pl @ A @ Pr) is complex-symmetric
(non-Hermitian unless there is no loss or PMLs).
The preconditioner matrices are diagonal and complex, with Pr = 1 / Pl
:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header
:return: Preconditioner matrices (Pl, Pr)
"""
p_squared = [dxes[0][0][:, None, None] * dxes[1][1][None, :, None] * dxes[1][2][None, None, :],
dxes[1][0][:, None, None] * dxes[0][1][None, :, None] * dxes[1][2][None, None, :],
dxes[1][0][:, None, None] * dxes[1][1][None, :, None] * dxes[0][2][None, None, :]]
p_vector = numpy.sqrt(vec(p_squared))
P_left = sparse.diags(p_vector)
P_right = sparse.diags(1 / p_vector)
return P_left, P_right
def h_full(omega: complex,
dxes: dx_lists_t,
epsilon: vfield_t,
mu: vfield_t = None,
pec: vfield_t = None,
pmc: vfield_t = None,
) -> sparse.spmatrix:
"""
Wave operator del x (1/epsilon * del x) - omega**2 * mu, for use with H-field,
with wave equation
(del x (1/epsilon * del x) - omega**2 * mu) H = i * omega * M
:param omega: Angular frequency of the simulation
:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header
:param epsilon: Vectorized dielectric constant
:param mu: Vectorized magnetic permeability (default 1 everywhere)
:param pec: Vectorized mask specifying PEC cells. Any cells where pec != 0 are interpreted
as containing a perfect electrical conductor (PEC).
The PEC is applied per-field-component (ie, pec.size == epsilon.size)
:param pmc: Vectorized mask specifying PMC cells. Any cells where pmc != 0 are interpreted
as containing a perfect magnetic conductor (PMC).
The PMC is applied per-field-component (ie, pmc.size == epsilon.size)
:return: Sparse matrix containing the wave operator
"""
ec = curl_e(dxes)
hc = curl_h(dxes)
if numpy.any(numpy.equal(pec, None)):
pe = sparse.eye(epsilon.size)
else:
pe = sparse.diags(numpy.where(pec, 0, 1)) # set pe to (not PEC)
if numpy.any(numpy.equal(pmc, None)):
pm = sparse.eye(epsilon.size)
else:
pm = sparse.diags(numpy.where(pmc, 0, 1)) # Set pe to (not PMC)
e_div = sparse.diags(1 / epsilon)
if mu is None:
m = sparse.eye(epsilon.size)
else:
m = sparse.diags(mu)
A = pm @ (ec @ pe @ e_div @ hc - omega**2 * m) @ pm
return A
def eh_full(omega: complex,
dxes: dx_lists_t,
epsilon: vfield_t,
mu: vfield_t = None,
pec: vfield_t = None,
pmc: vfield_t = None
) -> sparse.spmatrix:
"""
Wave operator for [E, H] field representation. This operator implements Maxwell's
equations without cancelling out either E or H. The operator is
[[-i * omega * epsilon, del x],
[del x, i * omega * mu]]
for use with a field vector of the form hstack(vec(E), vec(H)).
:param omega: Angular frequency of the simulation
:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header
:param epsilon: Vectorized dielectric constant
:param mu: Vectorized magnetic permeability (default 1 everywhere)
:param pec: Vectorized mask specifying PEC cells. Any cells where pec != 0 are interpreted
as containing a perfect electrical conductor (PEC).
The PEC is applied per-field-component (i.e., pec.size == epsilon.size)
:param pmc: Vectorized mask specifying PMC cells. Any cells where pmc != 0 are interpreted
as containing a perfect magnetic conductor (PMC).
The PMC is applied per-field-component (i.e., pmc.size == epsilon.size)
:return: Sparse matrix containing the wave operator
"""
if numpy.any(numpy.equal(pec, None)):
pe = sparse.eye(epsilon.size)
else:
pe = sparse.diags(numpy.where(pec, 0, 1)) # set pe to (not PEC)
if numpy.any(numpy.equal(pmc, None)):
pm = sparse.eye(epsilon.size)
else:
pm = sparse.diags(numpy.where(pmc, 0, 1)) # set pm to (not PMC)
iwe = pe @ (1j * omega * sparse.diags(epsilon)) @ pe
iwm = 1j * omega
if not numpy.any(numpy.equal(mu, None)):
iwm *= sparse.diags(mu)
iwm = pm @ iwm @ pm
A1 = pe @ curl_h(dxes) @ pm
A2 = pm @ curl_e(dxes) @ pe
A = sparse.bmat([[-iwe, A1],
[A2, iwm]])
return A
def curl_h(dxes: dx_lists_t) -> sparse.spmatrix:
"""
Curl operator for use with the H field.
:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header
:return: Sparse matrix for taking the discretized curl of the H-field
"""
return cross(deriv_back(dxes[1]))
def curl_e(dxes: dx_lists_t) -> sparse.spmatrix:
"""
Curl operator for use with the E field.
:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header
:return: Sparse matrix for taking the discretized curl of the E-field
"""
return cross(deriv_forward(dxes[0]))
def e2h(omega: complex,
dxes: dx_lists_t,
mu: vfield_t = None,
pmc: vfield_t = None,
) -> sparse.spmatrix:
"""
Utility operator for converting the E field into the H field.
For use with e_full -- assumes that there is no magnetic current M.
:param omega: Angular frequency of the simulation
:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header
:param mu: Vectorized magnetic permeability (default 1 everywhere)
:param pmc: Vectorized mask specifying PMC cells. Any cells where pmc != 0 are interpreted
as containing a perfect magnetic conductor (PMC).
The PMC is applied per-field-component (ie, pmc.size == epsilon.size)
:return: Sparse matrix for converting E to H
"""
op = curl_e(dxes) / (-1j * omega)
if not numpy.any(numpy.equal(mu, None)):
op = sparse.diags(1 / mu) @ op
if not numpy.any(numpy.equal(pmc, None)):
op = sparse.diags(numpy.where(pmc, 0, 1)) @ op
return op
def m2j(omega: complex,
dxes: dx_lists_t,
mu: vfield_t = None
) -> sparse.spmatrix:
"""
Utility operator for converting M field into J.
Converts a magnetic current M into an electric current J.
For use with eg. e_full.
:param omega: Angular frequency of the simulation
:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header
:param mu: Vectorized magnetic permeability (default 1 everywhere)
:return: Sparse matrix for converting E to H
"""
op = curl_h(dxes) / (1j * omega)
if not numpy.any(numpy.equal(mu, None)):
op = op @ sparse.diags(1 / mu)
return op
def rotation(axis: int, shape: List[int], shift_distance: int=1) -> sparse.spmatrix:
"""
Utility operator for performing a circular shift along a specified axis by a
specified number of elements.
:param axis: Axis to shift along. x=0, y=1, z=2
:param shape: Shape of the grid being shifted
:param shift_distance: Number of cells to shift by. May be negative. Default 1.
:return: Sparse matrix for performing the circular shift
"""
if len(shape) not in (2, 3):
raise Exception('Invalid shape: {}'.format(shape))
if axis not in range(len(shape)):
raise Exception('Invalid direction: {}, shape is {}'.format(axis, shape))
shifts = [abs(shift_distance) if a == axis else 0 for a in range(3)]
shifted_diags = [(numpy.arange(n) + s) % n for n, s in zip(shape, shifts)]
ijk = numpy.meshgrid(*shifted_diags, indexing='ij')
n = numpy.prod(shape)
i_ind = numpy.arange(n)
j_ind = numpy.ravel_multi_index(ijk, shape, order='C')
vij = (numpy.ones(n), (i_ind, j_ind.ravel(order='C')))
d = sparse.csr_matrix(vij, shape=(n, n))
if shift_distance < 0:
d = d.T
return d
def shift_with_mirror(axis: int, shape: List[int], shift_distance: int=1) -> sparse.spmatrix:
"""
Utility operator for performing an n-element shift along a specified axis, with mirror
boundary conditions applied to the cells beyond the receding edge.
:param axis: Axis to shift along. x=0, y=1, z=2
:param shape: Shape of the grid being shifted
:param shift_distance: Number of cells to shift by. May be negative. Default 1.
:return: Sparse matrix for performing the circular shift
"""
if len(shape) not in (2, 3):
raise Exception('Invalid shape: {}'.format(shape))
if axis not in range(len(shape)):
raise Exception('Invalid direction: {}, shape is {}'.format(axis, shape))
if shift_distance >= shape[axis]:
raise Exception('Shift ({}) is too large for axis {} of size {}'.format(
shift_distance, axis, shape[axis]))
def mirrored_range(n, s):
v = numpy.arange(n) + s
v = numpy.where(v >= n, 2 * n - v - 1, v)
v = numpy.where(v < 0, - 1 - v, v)
return v
shifts = [shift_distance if a == axis else 0 for a in range(3)]
shifted_diags = [mirrored_range(n, s) for n, s in zip(shape, shifts)]
ijk = numpy.meshgrid(*shifted_diags, indexing='ij')
n = numpy.prod(shape)
i_ind = numpy.arange(n)
j_ind = ijk[0] + ijk[1] * shape[0]
if len(shape) == 3:
j_ind += ijk[2] * shape[0] * shape[1]
vij = (numpy.ones(n), (i_ind, j_ind.ravel(order='C')))
d = sparse.csr_matrix(vij, shape=(n, n))
return d
def deriv_forward(dx_e: List[numpy.ndarray]) -> List[sparse.spmatrix]:
"""
Utility operators for taking discretized derivatives (forward variant).
:param dx_e: Lists of cell sizes for all axes [[dx_0, dx_1, ...], ...].
:return: List of operators for taking forward derivatives along each axis.
"""
shape = [s.size for s in dx_e]
n = numpy.prod(shape)
dx_e_expanded = numpy.meshgrid(*dx_e, indexing='ij')
def deriv(axis):
return rotation(axis, shape, 1) - sparse.eye(n)
Ds = [sparse.diags(+1 / dx.ravel(order='C')) @ deriv(a)
for a, dx in enumerate(dx_e_expanded)]
return Ds
def deriv_back(dx_h: List[numpy.ndarray]) -> List[sparse.spmatrix]:
"""
Utility operators for taking discretized derivatives (backward variant).
:param dx_h: Lists of cell sizes for all axes [[dx_0, dx_1, ...], ...].
:return: List of operators for taking forward derivatives along each axis.
"""
shape = [s.size for s in dx_h]
n = numpy.prod(shape)
dx_h_expanded = numpy.meshgrid(*dx_h, indexing='ij')
def deriv(axis):
return rotation(axis, shape, -1) - sparse.eye(n)
Ds = [sparse.diags(-1 / dx.ravel(order='C')) @ deriv(a)
for a, dx in enumerate(dx_h_expanded)]
return Ds
def cross(B: List[sparse.spmatrix]) -> sparse.spmatrix:
"""
Cross product operator
:param B: List [Bx, By, Bz] of sparse matrices corresponding to the x, y, z
portions of the operator on the left side of the cross product.
:return: Sparse matrix corresponding to (B x), where x is the cross product
"""
n = B[0].shape[0]
zero = sparse.csr_matrix((n, n))
return sparse.bmat([[zero, -B[2], B[1]],
[B[2], zero, -B[0]],
[-B[1], B[0], zero]])
def vec_cross(b: vfield_t) -> sparse.spmatrix:
"""
Vector cross product operator
:param b: Vector on the left side of the cross product
:return: Sparse matrix corresponding to (b x), where x is the cross product
"""
B = [sparse.diags(c) for c in numpy.split(b, 3)]
return cross(B)
def avgf(axis: int, shape: List[int]) -> sparse.spmatrix:
"""
Forward average operator (x4 = (x4 + x5) / 2)
:param axis: Axis to average along (x=0, y=1, z=2)
:param shape: Shape of the grid to average
:return: Sparse matrix for forward average operation
"""
if len(shape) not in (2, 3):
raise Exception('Invalid shape: {}'.format(shape))
n = numpy.prod(shape)
return 0.5 * (sparse.eye(n) + rotation(axis, shape))
def avgb(axis: int, shape: List[int]) -> sparse.spmatrix:
"""
Backward average operator (x4 = (x4 + x3) / 2)
:param axis: Axis to average along (x=0, y=1, z=2)
:param shape: Shape of the grid to average
:return: Sparse matrix for backward average operation
"""
return avgf(axis, shape).T
def poynting_e_cross(e: vfield_t, dxes: dx_lists_t) -> sparse.spmatrix:
"""
Operator for computing the Poynting vector, contining the (E x) portion of the Poynting vector.
:param e: Vectorized E-field for the ExH cross product
:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header
:return: Sparse matrix containing (E x) portion of Poynting cross product
"""
shape = [len(dx) for dx in dxes[0]]
fx, fy, fz = [avgf(i, shape) for i in range(3)]
bx, by, bz = [avgb(i, shape) for i in range(3)]
dxag = [dx.ravel(order='C') for dx in numpy.meshgrid(*dxes[0], indexing='ij')]
dbgx, dbgy, dbgz = [sparse.diags(dx.ravel(order='C'))
for dx in numpy.meshgrid(*dxes[1], indexing='ij')]
Ex, Ey, Ez = [sparse.diags(ei * da) for ei, da in zip(numpy.split(e, 3), dxag)]
n = numpy.prod(shape)
zero = sparse.csr_matrix((n, n))
P = sparse.bmat(
[[ zero, -fx @ Ez @ bz @ dbgy, fx @ Ey @ by @ dbgz],
[ fy @ Ez @ bz @ dbgx, zero, -fy @ Ex @ bx @ dbgz],
[-fz @ Ey @ by @ dbgx, fz @ Ex @ bx @ dbgy, zero]])
return P
def poynting_h_cross(h: vfield_t, dxes: dx_lists_t) -> sparse.spmatrix:
"""
Operator for computing the Poynting vector, containing the (H x) portion of the Poynting vector.
:param h: Vectorized H-field for the HxE cross product
:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header
:return: Sparse matrix containing (H x) portion of Poynting cross product
"""
shape = [len(dx) for dx in dxes[0]]
fx, fy, fz = [avgf(i, shape) for i in range(3)]
bx, by, bz = [avgb(i, shape) for i in range(3)]
dxbg = [dx.ravel(order='C') for dx in numpy.meshgrid(*dxes[1], indexing='ij')]
dagx, dagy, dagz = [sparse.diags(dx.ravel(order='C'))
for dx in numpy.meshgrid(*dxes[0], indexing='ij')]
Hx, Hy, Hz = [sparse.diags(hi * db) for hi, db in zip(numpy.split(h, 3), dxbg)]
n = numpy.prod(shape)
zero = sparse.csr_matrix((n, n))
P = sparse.bmat(
[[ zero, -by @ Hz @ fx @ dagy, bz @ Hy @ fx @ dagz],
[ bx @ Hz @ fy @ dagx, zero, -bz @ Hx @ fy @ dagz],
[-bx @ Hy @ fz @ dagx, by @ Hx @ fz @ dagy, zero]])
return P

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"""
Solvers for FDFD problems.
"""
from typing import List, Callable, Dict, Any
import logging
import numpy
from numpy.linalg import norm
import scipy.sparse.linalg
from . import operators
logger = logging.getLogger(__name__)
def _scipy_qmr(A: scipy.sparse.csr_matrix,
b: numpy.ndarray,
**kwargs
) -> numpy.ndarray:
"""
Wrapper for scipy.sparse.linalg.qmr
:param A: Sparse matrix
:param b: Right-hand-side vector
:param kwargs: Passed as **kwargs to the wrapped function
:return: Guess for solution (returned even if didn't converge)
"""
'''
Report on our progress
'''
iter = 0
def log_residual(xk):
nonlocal iter
iter += 1
if iter % 100 == 0:
logger.info('Solver residual at iteration {} : {}'.format(iter, norm(A @ xk - b)))
if 'callback' in kwargs:
def augmented_callback(xk):
log_residual(xk)
kwargs['callback'](xk)
kwargs['callback'] = augmented_callback
else:
kwargs['callback'] = log_residual
'''
Run the actual solve
'''
x, _ = scipy.sparse.linalg.qmr(A, b, **kwargs)
return x
def generic(omega: complex,
dxes: List[List[numpy.ndarray]],
J: numpy.ndarray,
epsilon: numpy.ndarray,
mu: numpy.ndarray = None,
pec: numpy.ndarray = None,
pmc: numpy.ndarray = None,
adjoint: bool = False,
matrix_solver: Callable[..., numpy.ndarray] = _scipy_qmr,
matrix_solver_opts: Dict[str, Any] = None,
) -> numpy.ndarray:
"""
Conjugate gradient FDFD solver using CSR sparse matrices.
All ndarray arguments should be 1D array, as returned by fdfd_tools.vec().
:param omega: Complex frequency to solve at.
:param dxes: [[dx_e, dy_e, dz_e], [dx_h, dy_h, dz_h]] (complex cell sizes)
:param J: Electric current distribution (at E-field locations)
:param epsilon: Dielectric constant distribution (at E-field locations)
:param mu: Magnetic permeability distribution (at H-field locations)
:param pec: Perfect electric conductor distribution
(at E-field locations; non-zero value indicates PEC is present)
:param pmc: Perfect magnetic conductor distribution
(at H-field locations; non-zero value indicates PMC is present)
:param adjoint: If true, solves the adjoint problem.
:param matrix_solver: Called as matrix_solver(A, b, **matrix_solver_opts) -> x
Where A: scipy.sparse.csr_matrix
b: numpy.ndarray
x: numpy.ndarray
Default is a wrapped version of scipy.sparse.linalg.qmr()
which doesn't return convergence info and logs the residual
every 100 iterations.
:param matrix_solver_opts: Passed as kwargs to matrix_solver(...)
:return: E-field which solves the system.
"""
if matrix_solver_opts is None:
matrix_solver_opts = dict()
b0 = -1j * omega * J
A0 = operators.e_full(omega, dxes, epsilon=epsilon, mu=mu, pec=pec, pmc=pmc)
Pl, Pr = operators.e_full_preconditioners(dxes)
if adjoint:
A = (Pl @ A0 @ Pr).H
b = Pr.H @ b0
else:
A = Pl @ A0 @ Pr
b = Pl @ b0
x = matrix_solver(A.tocsr(), b, **matrix_solver_opts)
if adjoint:
x0 = Pl.H @ x
else:
x0 = Pr @ x
return x0

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"""
Functions for moving between a vector field (list of 3 ndarrays, [f_x, f_y, f_z])
and a 1D array representation of that field [f_x0, f_x1, f_x2,... f_y0,... f_z0,...].
Vectorized versions of the field use row-major (ie., C-style) ordering.
"""
from typing import List
import numpy
__author__ = 'Jan Petykiewicz'
# Types
field_t = List[numpy.ndarray] # vector field (eg. [E_x, E_y, E_z]
vfield_t = numpy.ndarray # linearized vector field
def vec(f: field_t) -> vfield_t:
"""
Create a 1D ndarray from a 3D vector field which spans a 1-3D region.
Returns None if called with f=None.
:param f: A vector field, [f_x, f_y, f_z] where each f_ component is a 1 to
3D ndarray (f_* should all be the same size). Doesn't fail with f=None.
:return: A 1D ndarray containing the linearized field (or None)
"""
if numpy.any(numpy.equal(f, None)):
return None
return numpy.hstack(tuple((fi.ravel(order='C') for fi in f)))
def unvec(v: vfield_t, shape: numpy.ndarray) -> field_t:
"""
Perform the inverse of vec(): take a 1D ndarray and output a 3D field
of form [f_x, f_y, f_z] where each of f_* is a len(shape)-dimensional
ndarray.
Returns None if called with v=None.
:param v: 1D ndarray representing a 3D vector field of shape shape (or None)
:param shape: shape of the vector field
:return: [f_x, f_y, f_z] where each f_ is a len(shape) dimensional ndarray
(or None)
"""
if numpy.any(numpy.equal(v, None)):
return None
return [vi.reshape(shape, order='C') for vi in numpy.split(v, 3)]

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"""
Various operators and helper functions for solving for waveguide modes.
Assuming a z-dependence of the from exp(-i * wavenumber * z), we can simplify Maxwell's
equations in the absence of sources to the form
A @ [H_x, H_y] = wavenumber**2 * [H_x, H_y]
with A =
omega**2 * epsilon * mu +
epsilon * [[-Dy], [Dx]] / epsilon * [-Dy, Dx] +
[[Dx], [Dy]] / mu * [Dx, Dy] * mu
which is the form used in this file.
As the z-dependence is known, all the functions in this file assume a 2D grid
(ie. dxes = [[[dx_e_0, dx_e_1, ...], [dy_e_0, ...]], [[dx_h_0, ...], [dy_h_0, ...]]])
with propagation along the z axis.
"""
from typing import List, Tuple
import numpy
from numpy.linalg import norm
import scipy.sparse as sparse
from . import vec, unvec, dx_lists_t, field_t, vfield_t
from . import operators
__author__ = 'Jan Petykiewicz'
def operator(omega: complex,
dxes: dx_lists_t,
epsilon: vfield_t,
mu: vfield_t = None,
) -> sparse.spmatrix:
"""
Waveguide operator of the form
omega**2 * epsilon * mu +
epsilon * [[-Dy], [Dx]] / epsilon * [-Dy, Dx] +
[[Dx], [Dy]] / mu * [Dx, Dy] * mu
for use with a field vector of the form [H_x, H_y].
This operator can be used to form an eigenvalue problem of the form
A @ [H_x, H_y] = wavenumber**2 * [H_x, H_y]
which can then be solved for the eigenmodes of the system (an exp(-i * wavenumber * z)
z-dependence is assumed for the fields).
:param omega: The angular frequency of the system
:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header (2D)
:param epsilon: Vectorized dielectric constant grid
:param mu: Vectorized magnetic permeability grid (default 1 everywhere)
:return: Sparse matrix representation of the operator
"""
if numpy.any(numpy.equal(mu, None)):
mu = numpy.ones_like(epsilon)
Dfx, Dfy = operators.deriv_forward(dxes[0])
Dbx, Dby = operators.deriv_back(dxes[1])
eps_parts = numpy.split(epsilon, 3)
eps_yx = sparse.diags(numpy.hstack((eps_parts[1], eps_parts[0])))
eps_z_inv = sparse.diags(1 / eps_parts[2])
mu_parts = numpy.split(mu, 3)
mu_xy = sparse.diags(numpy.hstack((mu_parts[0], mu_parts[1])))
mu_z_inv = sparse.diags(1 / mu_parts[2])
op = omega ** 2 * eps_yx @ mu_xy + \
eps_yx @ sparse.vstack((-Dfy, Dfx)) @ eps_z_inv @ sparse.hstack((-Dby, Dbx)) + \
sparse.vstack((Dbx, Dby)) @ mu_z_inv @ sparse.hstack((Dfx, Dfy)) @ mu_xy
return op
def normalized_fields(v: numpy.ndarray,
wavenumber: complex,
omega: complex,
dxes: dx_lists_t,
epsilon: vfield_t,
mu: vfield_t = None
) -> Tuple[vfield_t, vfield_t]:
"""
Given a vector v containing the vectorized H_x and H_y fields,
returns normalized, vectorized E and H fields for the system.
:param v: Vector containing H_x and H_y fields
:param wavenumber: Wavenumber satisfying A @ v == wavenumber**2 * v
:param omega: The angular frequency of the system
:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header (2D)
:param epsilon: Vectorized dielectric constant grid
:param mu: Vectorized magnetic permeability grid (default 1 everywhere)
:return: Normalized, vectorized (e, h) containing all vector components.
"""
e = v2e(v, wavenumber, omega, dxes, epsilon, mu=mu)
h = v2h(v, wavenumber, dxes, mu=mu)
shape = [s.size for s in dxes[0]]
dxes_real = [[numpy.real(d) for d in numpy.meshgrid(*dxes[v], indexing='ij')] for v in (0, 1)]
E = unvec(e, shape)
H = unvec(h, shape)
S1 = E[0] * numpy.roll(numpy.conj(H[1]), 1, axis=0) * dxes_real[0][1] * dxes_real[1][0]
S2 = E[1] * numpy.roll(numpy.conj(H[0]), 1, axis=1) * dxes_real[0][0] * dxes_real[1][1]
S = 0.25 * ((S1 + numpy.roll(S1, 1, axis=0)) -
(S2 + numpy.roll(S2, 1, axis=1)))
P = 0.5 * numpy.real(S.sum())
assert P > 0, 'Found a mode propagating in the wrong direction! P={}'.format(P)
norm_amplitude = 1 / numpy.sqrt(P)
norm_angle = -numpy.angle(e[e.size//2])
norm_factor = norm_amplitude * numpy.exp(1j * norm_angle)
e *= norm_factor
h *= norm_factor
return e, h
def v2h(v: numpy.ndarray,
wavenumber: complex,
dxes: dx_lists_t,
mu: vfield_t = None
) -> vfield_t:
"""
Given a vector v containing the vectorized H_x and H_y fields,
returns a vectorized H including all three H components.
:param v: Vector containing H_x and H_y fields
:param wavenumber: Wavenumber satisfying A @ v == wavenumber**2 * v
:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header (2D)
:param mu: Vectorized magnetic permeability grid (default 1 everywhere)
:return: Vectorized H field with all vector components
"""
Dfx, Dfy = operators.deriv_forward(dxes[0])
op = sparse.hstack((Dfx, Dfy))
if not numpy.any(numpy.equal(mu, None)):
mu_parts = numpy.split(mu, 3)
mu_xy = sparse.diags(numpy.hstack((mu_parts[0], mu_parts[1])))
mu_z_inv = sparse.diags(1 / mu_parts[2])
op = mu_z_inv @ op @ mu_xy
w = op @ v / (1j * wavenumber)
return numpy.hstack((v, w)).flatten()
def v2e(v: numpy.ndarray,
wavenumber: complex,
omega: complex,
dxes: dx_lists_t,
epsilon: vfield_t,
mu: vfield_t = None
) -> vfield_t:
"""
Given a vector v containing the vectorized H_x and H_y fields,
returns a vectorized E containing all three E components
:param v: Vector containing H_x and H_y fields
:param wavenumber: Wavenumber satisfying A @ v == wavenumber**2 * v
:param omega: The angular frequency of the system
:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header (2D)
:param epsilon: Vectorized dielectric constant grid
:param mu: Vectorized magnetic permeability grid (default 1 everywhere)
:return: Vectorized E field with all vector components.
"""
h2eop = h2e(wavenumber, omega, dxes, epsilon)
return h2eop @ v2h(v, wavenumber, dxes, mu)
def e2h(wavenumber: complex,
omega: complex,
dxes: dx_lists_t,
mu: vfield_t = None
) -> sparse.spmatrix:
"""
Returns an operator which, when applied to a vectorized E eigenfield, produces
the vectorized H eigenfield.
:param wavenumber: Wavenumber satisfying A @ v == wavenumber**2 * v
:param omega: The angular frequency of the system
:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header (2D)
:param mu: Vectorized magnetic permeability grid (default 1 everywhere)
:return: Sparse matrix representation of the operator
"""
op = curl_e(wavenumber, dxes) / (-1j * omega)
if not numpy.any(numpy.equal(mu, None)):
op = sparse.diags(1 / mu) @ op
return op
def h2e(wavenumber: complex,
omega: complex,
dxes: dx_lists_t,
epsilon: vfield_t
) -> sparse.spmatrix:
"""
Returns an operator which, when applied to a vectorized H eigenfield, produces
the vectorized E eigenfield.
:param wavenumber: Wavenumber satisfying A @ v == wavenumber**2 * v
:param omega: The angular frequency of the system
:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header (2D)
:param epsilon: Vectorized dielectric constant grid
:return: Sparse matrix representation of the operator
"""
op = sparse.diags(1 / (1j * omega * epsilon)) @ curl_h(wavenumber, dxes)
return op
def curl_e(wavenumber: complex, dxes: dx_lists_t) -> sparse.spmatrix:
"""
Discretized curl operator for use with the waveguide E field.
:param wavenumber: Wavenumber satisfying A @ v == wavenumber**2 * v
:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header (2D)
:return: Sparse matrix representation of the operator
"""
n = 1
for d in dxes[0]:
n *= len(d)
Bz = -1j * wavenumber * sparse.eye(n)
Dfx, Dfy = operators.deriv_forward(dxes[0])
return operators.cross([Dfx, Dfy, Bz])
def curl_h(wavenumber: complex, dxes: dx_lists_t) -> sparse.spmatrix:
"""
Discretized curl operator for use with the waveguide H field.
:param wavenumber: Wavenumber satisfying A @ v == wavenumber**2 * v
:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header (2D)
:return: Sparse matrix representation of the operator
"""
n = 1
for d in dxes[1]:
n *= len(d)
Bz = -1j * wavenumber * sparse.eye(n)
Dbx, Dby = operators.deriv_back(dxes[1])
return operators.cross([Dbx, Dby, Bz])
def h_err(h: vfield_t,
wavenumber: complex,
omega: complex,
dxes: dx_lists_t,
epsilon: vfield_t,
mu: vfield_t = None
) -> float:
"""
Calculates the relative error in the H field
:param h: Vectorized H field
:param wavenumber: Wavenumber satisfying A @ v == wavenumber**2 * v
:param omega: The angular frequency of the system
:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header (2D)
:param epsilon: Vectorized dielectric constant grid
:param mu: Vectorized magnetic permeability grid (default 1 everywhere)
:return: Relative error norm(OP @ h) / norm(h)
"""
ce = curl_e(wavenumber, dxes)
ch = curl_h(wavenumber, dxes)
eps_inv = sparse.diags(1 / epsilon)
if numpy.any(numpy.equal(mu, None)):
op = ce @ eps_inv @ ch @ h - omega ** 2 * h
else:
op = ce @ eps_inv @ ch @ h - omega ** 2 * (mu * h)
return norm(op) / norm(h)
def e_err(e: vfield_t,
wavenumber: complex,
omega: complex,
dxes: dx_lists_t,
epsilon: vfield_t,
mu: vfield_t = None
) -> float:
"""
Calculates the relative error in the E field
:param e: Vectorized E field
:param wavenumber: Wavenumber satisfying A @ v == wavenumber**2 * v
:param omega: The angular frequency of the system
:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header (2D)
:param epsilon: Vectorized dielectric constant grid
:param mu: Vectorized magnetic permeability grid (default 1 everywhere)
:return: Relative error norm(OP @ e) / norm(e)
"""
ce = curl_e(wavenumber, dxes)
ch = curl_h(wavenumber, dxes)
if numpy.any(numpy.equal(mu, None)):
op = ch @ ce @ e - omega ** 2 * (epsilon * e)
else:
mu_inv = sparse.diags(1 / mu)
op = ch @ mu_inv @ ce @ e - omega ** 2 * (epsilon * e)
return norm(op) / norm(e)
def cylindrical_operator(omega: complex,
dxes: dx_lists_t,
epsilon: vfield_t,
r0: float,
) -> sparse.spmatrix:
"""
Cylindrical coordinate waveguide operator of the form
TODO
for use with a field vector of the form [E_r, E_y].
This operator can be used to form an eigenvalue problem of the form
A @ [E_r, E_y] = wavenumber**2 * [E_r, E_y]
which can then be solved for the eigenmodes of the system (an exp(-i * wavenumber * theta)
theta-dependence is assumed for the fields).
:param omega: The angular frequency of the system
:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header (2D)
:param epsilon: Vectorized dielectric constant grid
:param r0: Radius of curvature for the simulation. This should be the minimum value of
r within the simulation domain.
:return: Sparse matrix representation of the operator
"""
Dfx, Dfy = operators.deriv_forward(dxes[0])
Dbx, Dby = operators.deriv_back(dxes[1])
rx = r0 + numpy.cumsum(dxes[0][0])
ry = r0 + dxes[0][0]/2.0 + numpy.cumsum(dxes[1][0])
tx = rx/r0
ty = ry/r0
Tx = sparse.diags(vec(tx[:, None].repeat(dxes[0][1].size, axis=1)))
Ty = sparse.diags(vec(ty[:, None].repeat(dxes[1][1].size, axis=1)))
eps_parts = numpy.split(epsilon, 3)
eps_x = sparse.diags(eps_parts[0])
eps_y = sparse.diags(eps_parts[1])
eps_z_inv = sparse.diags(1 / eps_parts[2])
pa = sparse.vstack((Dfx, Dfy)) @ Tx @ eps_z_inv @ sparse.hstack((Dbx, Dby))
pb = sparse.vstack((Dfx, Dfy)) @ Tx @ eps_z_inv @ sparse.hstack((Dby, Dbx))
a0 = Ty @ eps_x + omega**-2 * Dby @ Ty @ Dfy
a1 = Tx @ eps_y + omega**-2 * Dbx @ Ty @ Dfx
b0 = Dbx @ Ty @ Dfy
b1 = Dby @ Ty @ Dfx
diag = sparse.block_diag
op = (omega**2 * diag((Tx, Ty)) + pa) @ diag((a0, a1)) + \
- (sparse.bmat(((None, Ty), (Tx, None))) + omega**-2 * pb) @ diag((b0, b1))
return op

340
fdfd_tools/waveguide_mode.py Normal file
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from typing import Dict, List
import numpy
import scipy.sparse as sparse
from . import vec, unvec, dx_lists_t, vfield_t, field_t
from . import operators, waveguide, functional
from .eigensolvers import signed_eigensolve, rayleigh_quotient_iteration
def solve_waveguide_mode_2d(mode_number: int,
omega: complex,
dxes: dx_lists_t,
epsilon: vfield_t,
mu: vfield_t = None,
wavenumber_correction: bool = True,
) -> Dict[str, complex or field_t]:
"""
Given a 2d region, attempts to solve for the eigenmode with the specified mode number.
:param mode_number: Number of the mode, 0-indexed.
:param omega: Angular frequency of the simulation
:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header
:param epsilon: Dielectric constant
:param mu: Magnetic permeability (default 1 everywhere)
:param wavenumber_correction: Whether to correct the wavenumber to
account for numerical dispersion (default True)
:return: {'E': List[numpy.ndarray], 'H': List[numpy.ndarray], 'wavenumber': complex}
"""
'''
Solve for the largest-magnitude eigenvalue of the real operator
'''
dxes_real = [[numpy.real(dx) for dx in dxi] for dxi in dxes]
A_r = waveguide.operator(numpy.real(omega), dxes_real, numpy.real(epsilon), numpy.real(mu))
eigvals, eigvecs = signed_eigensolve(A_r, mode_number+3)
v = eigvecs[:, -(mode_number + 1)]
'''
Now solve for the eigenvector of the full operator, using the real operator's
eigenvector as an initial guess for Rayleigh quotient iteration.
'''
A = waveguide.operator(omega, dxes, epsilon, mu)
eigval, v = rayleigh_quotient_iteration(A, v)
# Calculate the wave-vector (force the real part to be positive)
wavenumber = numpy.sqrt(eigval)
wavenumber *= numpy.sign(numpy.real(wavenumber))
e, h = waveguide.normalized_fields(v, wavenumber, omega, dxes, epsilon, mu)
'''
Perform correction on wavenumber to account for numerical dispersion.
See Numerical Dispersion in Taflove's FDTD book.
This correction term reduces the error in emitted power, but additional
error is introduced into the E_err and H_err terms. This effect becomes
more pronounced as beta increases.
'''
if wavenumber_correction:
wavenumber -= 2 * numpy.sin(numpy.real(wavenumber / 2)) - numpy.real(wavenumber)
shape = [d.size for d in dxes[0]]
fields = {
'wavenumber': wavenumber,
'E': unvec(e, shape),
'H': unvec(h, shape),
}
return fields
def solve_waveguide_mode(mode_number: int,
omega: complex,
dxes: dx_lists_t,
axis: int,
polarity: int,
slices: List[slice],
epsilon: field_t,
mu: field_t = None,
wavenumber_correction: bool = True
) -> Dict[str, complex or numpy.ndarray]:
"""
Given a 3D grid, selects a slice from the grid and attempts to
solve for an eigenmode propagating through that slice.
:param mode_number: Number of the mode, 0-indexed
:param omega: Angular frequency of the simulation
:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header
:param axis: Propagation axis (0=x, 1=y, 2=z)
:param polarity: Propagation direction (+1 for +ve, -1 for -ve)
:param slices: epsilon[tuple(slices)] is used to select the portion of the grid to use
as the waveguide cross-section. slices[axis] should select only one
:param epsilon: Dielectric constant
:param mu: Magnetic permeability (default 1 everywhere)
:param wavenumber_correction: Whether to correct the wavenumber to
account for numerical dispersion (default True)
:return: {'E': List[numpy.ndarray], 'H': List[numpy.ndarray], 'wavenumber': complex}
"""
if mu is None:
mu = [numpy.ones_like(epsilon[0])] * 3
'''
Solve the 2D problem in the specified plane
'''
# Define rotation to set z as propagation direction
order = numpy.roll(range(3), 2 - axis)
reverse_order = numpy.roll(range(3), axis - 2)
# Reduce to 2D and solve the 2D problem
args_2d = {
'dxes': [[dx[i][slices[i]] for i in order[:2]] for dx in dxes],
'epsilon': vec([epsilon[i][slices].transpose(order) for i in order]),
'mu': vec([mu[i][slices].transpose(order) for i in order]),
'wavenumber_correction': wavenumber_correction,
}
fields_2d = solve_waveguide_mode_2d(mode_number, omega=omega, **args_2d)
'''
Apply corrections and expand to 3D
'''
# Scale based on dx in propagation direction
dxab_forward = numpy.array([dx[order[2]][slices[order[2]]] for dx in dxes])
# Adjust for propagation direction
fields_2d['E'][2] *= polarity
fields_2d['H'][2] *= polarity
# Apply phase shift to H-field
d_prop = 0.5 * sum(dxab_forward)
for a in range(3):
fields_2d['H'][a] *= numpy.exp(-polarity * 1j * 0.5 * fields_2d['wavenumber'] * d_prop)
# Expand E, H to full epsilon space we were given
E = [None]*3
H = [None]*3
for a, o in enumerate(reverse_order):
E[a] = numpy.zeros_like(epsilon[0], dtype=complex)
H[a] = numpy.zeros_like(epsilon[0], dtype=complex)
E[a][slices] = fields_2d['E'][o][:, :, None].transpose(reverse_order)
H[a][slices] = fields_2d['H'][o][:, :, None].transpose(reverse_order)
results = {
'wavenumber': fields_2d['wavenumber'],
'H': H,
'E': E,
}
return results
def compute_source(E: field_t,
H: field_t,
wavenumber: complex,
omega: complex,
dxes: dx_lists_t,
axis: int,
polarity: int,
slices: List[slice],
mu: field_t = None,
) -> field_t:
"""
Given an eigenmode obtained by solve_waveguide_mode, returns the current source distribution
necessary to position a unidirectional source at the slice location.
:param E: E-field of the mode
:param H: H-field of the mode (advanced by half of a Yee cell from E)
:param wavenumber: Wavenumber of the mode
:param omega: Angular frequency of the simulation
:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header
:param axis: Propagation axis (0=x, 1=y, 2=z)
:param polarity: Propagation direction (+1 for +ve, -1 for -ve)
:param slices: epsilon[tuple(slices)] is used to select the portion of the grid to use
as the waveguide cross-section. slices[axis] should select only one
:param mu: Magnetic permeability (default 1 everywhere)
:return: J distribution for the unidirectional source
"""
if mu is None:
mu = [1] * 3
J = [None]*3
M = [None]*3
src_order = numpy.roll(range(3), axis)
exp_iphi = numpy.exp(1j * polarity * wavenumber * dxes[1][axis][slices[axis]])
J[src_order[0]] = numpy.zeros_like(E[0])
J[src_order[1]] = +exp_iphi * H[src_order[2]] * polarity
J[src_order[2]] = -exp_iphi * H[src_order[1]] * polarity
M[src_order[0]] = numpy.zeros_like(E[0])
M[src_order[1]] = +numpy.roll(E[src_order[2]], -1, axis=axis)
M[src_order[2]] = -numpy.roll(E[src_order[1]], -1, axis=axis)
A1f = functional.curl_h(dxes)
Jm_iw = A1f([M[k] / mu[k] for k in range(3)])
for k in range(3):
J[k] += Jm_iw[k] / (-1j * omega)
return J
def compute_overlap_e(E: field_t,
H: field_t,
wavenumber: complex,
omega: complex,
dxes: dx_lists_t,
axis: int,
polarity: int,
slices: List[slice],
mu: field_t = None,
) -> field_t:
"""
Given an eigenmode obtained by solve_waveguide_mode, calculates overlap_e for the
mode orthogonality relation Integrate(((E x H_mode) + (E_mode x H)) dot dn)
[assumes reflection symmetry].
overlap_e makes use of the e2h operator to collapse the above expression into
(vec(E) @ vec(overlap_e)), allowing for simple calculation of the mode overlap.
:param E: E-field of the mode
:param H: H-field of the mode (advanced by half of a Yee cell from E)
:param wavenumber: Wavenumber of the mode
:param omega: Angular frequency of the simulation
:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header
:param axis: Propagation axis (0=x, 1=y, 2=z)
:param polarity: Propagation direction (+1 for +ve, -1 for -ve)
:param slices: epsilon[tuple(slices)] is used to select the portion of the grid to use
as the waveguide cross-section. slices[axis] should select only one
:param mu: Magnetic permeability (default 1 everywhere)
:return: overlap_e for calculating the mode overlap
"""
cross_plane = [slice(None)] * 3
cross_plane[axis] = slices[axis]
# Determine phase factors for parallel slices
a_shape = numpy.roll([-1, 1, 1], axis)
a_E = numpy.real(dxes[0][axis]).cumsum()
a_H = numpy.real(dxes[1][axis]).cumsum()
iphi = -polarity * 1j * wavenumber
phase_E = numpy.exp(iphi * (a_E - a_E[slices[axis]])).reshape(a_shape)
phase_H = numpy.exp(iphi * (a_H - a_H[slices[axis]])).reshape(a_shape)
# Expand our slice to the entire grid using the calculated phase factors
Ee = [None]*3
He = [None]*3
for k in range(3):
Ee[k] = phase_E * E[k][tuple(cross_plane)]
He[k] = phase_H * H[k][tuple(cross_plane)]
# Write out the operator product for the mode orthogonality integral
domain = numpy.zeros_like(E[0], dtype=int)
domain[slices] = 1
npts = E[0].size
dn = numpy.zeros(npts * 3, dtype=int)
dn[0:npts] = 1
dn = numpy.roll(dn, npts * axis)
e2h = operators.e2h(omega, dxes, mu)
ds = sparse.diags(vec([domain]*3))
h_cross_ = operators.poynting_h_cross(vec(He), dxes)
e_cross_ = operators.poynting_e_cross(vec(Ee), dxes)
overlap_e = dn @ ds @ (-h_cross_ + e_cross_ @ e2h)
# Normalize
dx_forward = dxes[0][axis][slices[axis]]
norm_factor = numpy.abs(overlap_e @ vec(Ee))
overlap_e /= norm_factor * dx_forward
return unvec(overlap_e, E[0].shape)
def solve_waveguide_mode_cylindrical(mode_number: int,
omega: complex,
dxes: dx_lists_t,
epsilon: vfield_t,
r0: float,
wavenumber_correction: bool = True,
) -> Dict[str, complex or field_t]:
"""
Given a 2d (r, y) slice of epsilon, attempts to solve for the eigenmode
of the bent waveguide with the specified mode number.
:param mode_number: Number of the mode, 0-indexed
:param omega: Angular frequency of the simulation
:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header.
The first coordinate is assumed to be r, the second is y.
:param epsilon: Dielectric constant
:param r0: Radius of curvature for the simulation. This should be the minimum value of
r within the simulation domain.
:param wavenumber_correction: Whether to correct the wavenumber to
account for numerical dispersion (default True)
:return: {'E': List[numpy.ndarray], 'H': List[numpy.ndarray], 'wavenumber': complex}
"""
'''
Solve for the largest-magnitude eigenvalue of the real operator
'''
dxes_real = [[numpy.real(dx) for dx in dxi] for dxi in dxes]
A_r = waveguide.cylindrical_operator(numpy.real(omega), dxes_real, numpy.real(epsilon), r0)
eigvals, eigvecs = signed_eigensolve(A_r, mode_number + 3)
v = eigvecs[:, -(mode_number+1)]
'''
Now solve for the eigenvector of the full operator, using the real operator's
eigenvector as an initial guess for Rayleigh quotient iteration.
'''
A = waveguide.cylindrical_operator(omega, dxes, epsilon, r0)
eigval, v = rayleigh_quotient_iteration(A, v)
# Calculate the wave-vector (force the real part to be positive)
wavenumber = numpy.sqrt(eigval)
wavenumber *= numpy.sign(numpy.real(wavenumber))
'''
Perform correction on wavenumber to account for numerical dispersion.
See Numerical Dispersion in Taflove's FDTD book.
This correction term reduces the error in emitted power, but additional
error is introduced into the E_err and H_err terms. This effect becomes
more pronounced as the wavenumber increases.
'''
if wavenumber_correction:
wavenumber -= 2 * numpy.sin(numpy.real(wavenumber / 2)) - numpy.real(wavenumber)
shape = [d.size for d in dxes[0]]
v = numpy.hstack((v, numpy.zeros(shape[0] * shape[1])))
fields = {
'wavenumber': wavenumber,
'E': unvec(v, shape),
# 'E': unvec(e, shape),
# 'H': unvec(h, shape),
}
return fields

15
make_docs.sh
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@ -1,15 +0,0 @@
#!/bin/bash
cd ~/projects/meanas
# Approach 1: pdf to html?
#pdoc3 --pdf --force --template-dir pdoc_templates -o doc . | \
# pandoc --metadata=title:"meanas" --toc --toc-depth=4 --from=markdown+abbreviations --to=html --output=doc.html --gladtex -s -
# Approach 2: pdf to html with gladtex
pdoc3 --pdf --force --template-dir pdoc_templates -o doc . > doc.md
pandoc --metadata=title:"meanas" --from=markdown+abbreviations --to=html --output=doc.html --gladtex -s --css pdoc_templates/pdoc.css doc.md
gladtex -a -n -d _doc_mathimg -c white doc.html
# Approach 3: html with gladtex
#pdoc3 --html --force --template-dir pdoc_templates -o doc .
#find doc -iname '*.html' -exec gladtex -a -n -d _mathimg -c white {} \;

1
meanas/VERSION
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0.5

16
meanas/__init__.py
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@ -1,16 +0,0 @@
"""
Electromagnetic simulation tools
See the readme or `import meanas; help(meanas)` for more info.
"""
import pathlib
__author__ = 'Jan Petykiewicz'
with open(pathlib.Path(__file__).parent / 'VERSION', 'r') as f:
__version__ = f.read().strip()
with open(pathlib.Path(__file__).parent.parent / 'README.md', 'r') as f:
__doc__ = f.read()

38
meanas/fdfd/__init__.py
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@ -1,38 +0,0 @@
"""
Tools for finite difference frequency-domain (FDFD) simulations and calculations.
These mostly involve picking a single frequency, then setting up and solving a
matrix equation (Ax=b) or eigenvalue problem.
Submodules:
- `operators`, `functional`: General FDFD problem setup.
- `solvers`: Solver interface and reference implementation.
- `scpml`: Stretched-coordinate perfectly matched layer (scpml) boundary conditions
- `waveguide_2d`: Operators and mode-solver for waveguides with constant cross-section.
- `waveguide_3d`: Functions for transforming `waveguide_2d` results into 3D.
================================================================
From the "Frequency domain" section of `meanas.fdmath`, we have
$$
\\begin{aligned}
\\tilde{E}_{l, \\vec{r}} &= \\tilde{E}_{\\vec{r}} e^{-\\imath \\omega l \\Delta_t} \\\\
\\tilde{J}_{l, \\vec{r}} &= \\tilde{J}_{\\vec{r}} e^{-\\imath \\omega (l - \\frac{1}{2}) \\Delta_t} \\\\
\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}})
-\\Omega^2 \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{\\vec{r}} &= \\imath \\Omega \\tilde{J}_{\\vec{r}} \\\\
\\Omega &= 2 \\sin(\\omega \\Delta_t / 2) / \\Delta_t
\\end{aligned}
$$
# TODO FDFD?
# TODO PML
"""
from . import solvers, operators, functional, scpml, waveguide_2d, waveguide_3d
# from . import farfield, bloch TODO

691
meanas/fdfd/bloch.py
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@ -1,691 +0,0 @@
'''
Bloch eigenmode solver/operators
This module contains functions for generating and solving the
3D Bloch eigenproblem. The approach is to transform the problem
into the (spatial) fourier domain, transforming the equation
1/mu * curl(1/eps * curl(H)) = (w/c)^2 H
into
conv(1/mu_k, ik x conv(1/eps_k, ik x H_k)) = (w/c)^2 H_k
where:
- the `_k` subscript denotes a 3D fourier transformed field
- each component of `H_k` corresponds to a plane wave with wavevector `k`
- `x` is the cross product
- `conv()` denotes convolution
Since `k` and `H` are orthogonal for each plane wave, we can use each
`k` to create an orthogonal basis (k, m, n), with `k x m = n`, and
`|m| = |n| = 1`. The cross products are then simplified with
k @ h = kx hx + ky hy + kz hz = 0 = hk
h = hk + hm + hn = hm + hn
k = kk + km + kn = kk = |k|
k x h = (ky hz - kz hy,
kz hx - kx hz,
kx hy - ky hx)
= ((k x h) @ k, (k x h) @ m, (k x h) @ n)_kmn
= (0, (m x k) @ h, (n x k) @ h)_kmn # triple product ordering
= (0, kk (-n @ h), kk (m @ h))_kmn # (m x k) = -|k| n, etc.
= |k| (0, -h @ n, h @ m)_kmn
k x h = (km hn - kn hm,
kn hk - kk hn,
kk hm - km hk)_kmn
= (0, -kk hn, kk hm)_kmn
= (-kk hn)(mx, my, mz) + (kk hm)(nx, ny, nz)
= |k| (hm * (nx, ny, nz) - hn * (mx, my, mz))
where `h` is shorthand for `H_k`, `(...)_kmn` deontes the `(k, m, n)` basis,
and e.g. `hm` is the component of `h` in the `m` direction.
We can also simplify `conv(X_k, Y_k)` as `fftn(X * ifftn(Y_k))`.
Using these results and storing `H_k` as `h = (hm, hn)`, we have
e_xyz = fftn(1/eps * ifftn(|k| (hm * n - hn * m)))
b_mn = |k| (-e_xyz @ n, e_xyz @ m)
h_mn = fftn(1/mu * ifftn(b_m * m + b_n * n))
which forms the operator from the left side of the equation.
We can then use a preconditioned block Rayleigh iteration algorithm, as in
SG Johnson and JD Joannopoulos, Block-iterative frequency-domain methods
for Maxwell's equations in a planewave basis, Optics Express 8, 3, 173-190 (2001)
(similar to that used in MPB) to find the eigenvectors for this operator.
===
Typically you will want to do something like
recip_lattice = numpy.diag(1/numpy.array(epsilon[0].shape * dx))
n, v = bloch.eigsolve(5, k0, recip_lattice, epsilon)
f = numpy.sqrt(-numpy.real(n[0]))
n_eff = norm(recip_lattice @ k0) / f
v2e = bloch.hmn_2_exyz(k0, recip_lattice, epsilon)
e_field = v2e(v[0])
k, f = find_k(frequency=1/1550,
tolerance=(1/1550 - 1/1551),
direction=[1, 0, 0],
G_matrix=recip_lattice,
epsilon=epsilon,
band=0)
'''
from typing import Tuple, Callable
import logging
import numpy
from numpy import pi, real, trace
from numpy.fft import fftfreq
import scipy
import scipy.optimize
from scipy.linalg import norm
import scipy.sparse.linalg as spalg
from ..fdmath import fdfield_t
logger = logging.getLogger(__name__)
try:
import pyfftw.interfaces.numpy_fft
import pyfftw.interfaces
import multiprocessing
logger.info('Using pyfftw')
pyfftw.interfaces.cache.enable()
pyfftw.interfaces.cache.set_keepalive_time(3600)
fftw_args = {
'threads': multiprocessing.cpu_count(),
'overwrite_input': True,
'planner_effort': 'FFTW_EXHAUSTIVE',
}
def fftn(*args, **kwargs):
return pyfftw.interfaces.numpy_fft.fftn(*args, **kwargs, **fftw_args)
def ifftn(*args, **kwargs):
return pyfftw.interfaces.numpy_fft.ifftn(*args, **kwargs, **fftw_args)
except ImportError:
from numpy.fft import fftn, ifftn
logger.info('Using numpy fft')
def generate_kmn(k0: numpy.ndarray,
G_matrix: numpy.ndarray,
shape: numpy.ndarray
) -> Tuple[numpy.ndarray, numpy.ndarray, numpy.ndarray]:
"""
Generate a (k, m, n) orthogonal basis for each k-vector in the simulation grid.
Args:
k0: [k0x, k0y, k0z], Bloch wavevector, in G basis.
G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns.
shape: [nx, ny, nz] shape of the simulation grid.
Returns:
`(|k|, m, n)` where `|k|` has shape `tuple(shape) + (1,)`
and `m`, `n` have shape `tuple(shape) + (3,)`.
All are given in the xyz basis (e.g. `|k|[0,0,0] = norm(G_matrix @ k0)`).
"""
k0 = numpy.array(k0)
Gi_grids = numpy.meshgrid(*(fftfreq(n, 1/n) for n in shape[:3]), indexing='ij')
Gi = numpy.stack(Gi_grids, axis=3)
k_G = k0[None, None, None, :] - Gi
k_xyz = numpy.rollaxis(G_matrix @ numpy.rollaxis(k_G, 3, 2), 3, 2)
m = numpy.broadcast_to([0, 1, 0], tuple(shape[:3]) + (3,)).astype(float)
n = numpy.broadcast_to([0, 0, 1], tuple(shape[:3]) + (3,)).astype(float)
xy_non0 = numpy.any(k_xyz[:, :, :, 0:1] != 0, axis=3)
if numpy.any(xy_non0):
u = numpy.cross(k_xyz[xy_non0], [0, 0, 1])
m[xy_non0, :] = u / norm(u, axis=1)[:, None]
z_non0 = numpy.any(k_xyz != 0, axis=3)
if numpy.any(z_non0):
v = numpy.cross(k_xyz[z_non0], m[z_non0])
n[z_non0, :] = v / norm(v, axis=1)[:, None]
k_mag = norm(k_xyz, axis=3)[:, :, :, None]
return k_mag, m, n
def maxwell_operator(k0: numpy.ndarray,
G_matrix: numpy.ndarray,
epsilon: fdfield_t,
mu: fdfield_t = None
) -> Callable[[numpy.ndarray], numpy.ndarray]:
"""
Generate the Maxwell operator
conv(1/mu_k, ik x conv(1/eps_k, ik x ___))
which is the spatial-frequency-space representation of
1/mu * curl(1/eps * curl(___))
The operator is a function that acts on a vector h_mn of size `2 * epsilon[0].size`
See the `meanas.fdfd.bloch` docstring for more information.
Args:
k0: Bloch wavevector, `[k0x, k0y, k0z]`.
G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns.
epsilon: Dielectric constant distribution for the simulation.
All fields are sampled at cell centers (i.e., NOT Yee-gridded)
mu: Magnetic permability distribution for the simulation.
Default None (1 everywhere).
Returns:
Function which applies the maxwell operator to h_mn.
"""
shape = epsilon[0].shape + (1,)
k_mag, m, n = generate_kmn(k0, G_matrix, shape)
epsilon = numpy.stack(epsilon, 3)
if mu is not None:
mu = numpy.stack(mu, 3)
def operator(h: numpy.ndarray):
"""
Maxwell operator for Bloch eigenmode simulation.
h is complex 2-field in (m, n) basis, vectorized
Args:
h: Raveled h_mn; size `2 * epsilon[0].size`.
Returns:
Raveled conv(1/mu_k, ik x conv(1/eps_k, ik x h_mn)).
"""
hin_m, hin_n = [hi.reshape(shape) for hi in numpy.split(h, 2)]
#{d,e,h}_xyz fields are complex 3-fields in (1/x, 1/y, 1/z) basis
# cross product and transform into xyz basis
d_xyz = (n * hin_m -
m * hin_n) * k_mag
# divide by epsilon
e_xyz = fftn(ifftn(d_xyz, axes=range(3)) / epsilon, axes=range(3))
# cross product and transform into mn basis
b_m = numpy.sum(e_xyz * n, axis=3)[:, :, :, None] * -k_mag
b_n = numpy.sum(e_xyz * m, axis=3)[:, :, :, None] * +k_mag
if mu is None:
h_m, h_n = b_m, b_n
else:
# transform from mn to xyz
b_xyz = (m * b_m[:, :, :, None] +
n * b_n[:, :, :, None])
# divide by mu
h_xyz = fftn(ifftn(b_xyz, axes=range(3)) / mu, axes=range(3))
# transform back to mn
h_m = numpy.sum(h_xyz * m, axis=3)
h_n = numpy.sum(h_xyz * n, axis=3)
return numpy.hstack((h_m.ravel(), h_n.ravel()))
return operator
def hmn_2_exyz(k0: numpy.ndarray,
G_matrix: numpy.ndarray,
epsilon: fdfield_t,
) -> Callable[[numpy.ndarray], fdfield_t]:
"""
Generate an operator which converts a vectorized spatial-frequency-space
`h_mn` into an E-field distribution, i.e.
ifft(conv(1/eps_k, ik x h_mn))
The operator is a function that acts on a vector `h_mn` of size `2 * epsilon[0].size`.
See the `meanas.fdfd.bloch` docstring for more information.
Args:
k0: Bloch wavevector, `[k0x, k0y, k0z]`.
G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns.
epsilon: Dielectric constant distribution for the simulation.
All fields are sampled at cell centers (i.e., NOT Yee-gridded)
Returns:
Function for converting `h_mn` into `E_xyz`
"""
shape = epsilon[0].shape + (1,)
epsilon = numpy.stack(epsilon, 3)
k_mag, m, n = generate_kmn(k0, G_matrix, shape)
def operator(h: numpy.ndarray) -> fdfield_t:
hin_m, hin_n = [hi.reshape(shape) for hi in numpy.split(h, 2)]
d_xyz = (n * hin_m -
m * hin_n) * k_mag
# divide by epsilon
return [ei for ei in numpy.rollaxis(ifftn(d_xyz, axes=range(3)) / epsilon, 3)]
return operator
def hmn_2_hxyz(k0: numpy.ndarray,
G_matrix: numpy.ndarray,
epsilon: fdfield_t
) -> Callable[[numpy.ndarray], fdfield_t]:
"""
Generate an operator which converts a vectorized spatial-frequency-space
`h_mn` into an H-field distribution, i.e.
ifft(h_mn)
The operator is a function that acts on a vector `h_mn` of size `2 * epsilon[0].size`.
See the `meanas.fdfd.bloch` docstring for more information.
Args:
k0: Bloch wavevector, `[k0x, k0y, k0z]`.
G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns.
epsilon: Dielectric constant distribution for the simulation.
Only `epsilon[0].shape` is used.
Returns:
Function for converting `h_mn` into `H_xyz`
"""
shape = epsilon[0].shape + (1,)
_k_mag, m, n = generate_kmn(k0, G_matrix, shape)
def operator(h: numpy.ndarray):
hin_m, hin_n = [hi.reshape(shape) for hi in numpy.split(h, 2)]
h_xyz = (m * hin_m +
n * hin_n)
return [ifftn(hi) for hi in numpy.rollaxis(h_xyz, 3)]
return operator
def inverse_maxwell_operator_approx(k0: numpy.ndarray,
G_matrix: numpy.ndarray,
epsilon: fdfield_t,
mu: fdfield_t = None
) -> Callable[[numpy.ndarray], numpy.ndarray]:
"""
Generate an approximate inverse of the Maxwell operator,
ik x conv(eps_k, ik x conv(mu_k, ___))
which can be used to improve the speed of ARPACK in shift-invert mode.
See the `meanas.fdfd.bloch` docstring for more information.
Args:
k0: Bloch wavevector, `[k0x, k0y, k0z]`.
G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns.
epsilon: Dielectric constant distribution for the simulation.
All fields are sampled at cell centers (i.e., NOT Yee-gridded)
mu: Magnetic permability distribution for the simulation.
Default None (1 everywhere).
Returns:
Function which applies the approximate inverse of the maxwell operator to `h_mn`.
"""
shape = epsilon[0].shape + (1,)
epsilon = numpy.stack(epsilon, 3)
k_mag, m, n = generate_kmn(k0, G_matrix, shape)
if mu is not None:
mu = numpy.stack(mu, 3)
def operator(h: numpy.ndarray):
"""
Approximate inverse Maxwell operator for Bloch eigenmode simulation.
h is complex 2-field in (m, n) basis, vectorized
Args:
h: Raveled h_mn; size `2 * epsilon[0].size`.
Returns:
Raveled ik x conv(eps_k, ik x conv(mu_k, h_mn))
"""
hin_m, hin_n = [hi.reshape(shape) for hi in numpy.split(h, 2)]
#{d,e,h}_xyz fields are complex 3-fields in (1/x, 1/y, 1/z) basis
if mu is None:
b_m, b_n = hin_m, hin_n
else:
# transform from mn to xyz
h_xyz = (m * hin_m[:, :, :, None] +
n * hin_n[:, :, :, None])
# multiply by mu
b_xyz = fftn(ifftn(h_xyz, axes=range(3)) * mu, axes=range(3))
# transform back to mn
b_m = numpy.sum(b_xyz * m, axis=3)
b_n = numpy.sum(b_xyz * n, axis=3)
# cross product and transform into xyz basis
e_xyz = (n * b_m -
m * b_n) / k_mag
# multiply by epsilon
d_xyz = fftn(ifftn(e_xyz, axes=range(3)) * epsilon, axes=range(3))
# cross product and transform into mn basis crossinv_t2c
h_m = numpy.sum(d_xyz * n, axis=3)[:, :, :, None] / +k_mag
h_n = numpy.sum(d_xyz * m, axis=3)[:, :, :, None] / -k_mag
return numpy.hstack((h_m.ravel(), h_n.ravel()))
return operator
def find_k(frequency: float,
tolerance: float,
direction: numpy.ndarray,
G_matrix: numpy.ndarray,
epsilon: fdfield_t,
mu: fdfield_t = None,
band: int = 0,
k_min: float = 0,
k_max: float = 0.5,
solve_callback: Callable = None
) -> Tuple[numpy.ndarray, float]:
"""
Search for a bloch vector that has a given frequency.
Args:
frequency: Target frequency.
tolerance: Target frequency tolerance.
direction: k-vector direction to search along.
G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns.
epsilon: Dielectric constant distribution for the simulation.
All fields are sampled at cell centers (i.e., NOT Yee-gridded)
mu: Magnetic permability distribution for the simulation.
Default None (1 everywhere).
band: Which band to search in. Default 0 (lowest frequency).
Returns:
`(k, actual_frequency)`
The found k-vector and its frequency.
"""
direction = numpy.array(direction) / norm(direction)
def get_f(k0_mag: float, band: int = 0):
k0 = direction * k0_mag
n, v = eigsolve(band + 1, k0, G_matrix=G_matrix, epsilon=epsilon, mu=mu)
f = numpy.sqrt(numpy.abs(numpy.real(n[band])))
if solve_callback:
solve_callback(k0_mag, n, v, f)
return f
res = scipy.optimize.minimize_scalar(lambda x: abs(get_f(x, band) - frequency),
(k_min + k_max) / 2,
method='Bounded',
bounds=(k_min, k_max),
options={'xatol': abs(tolerance)})
return res.x * direction, res.fun + frequency
def eigsolve(num_modes: int,
k0: numpy.ndarray,
G_matrix: numpy.ndarray,
epsilon: fdfield_t,
mu: fdfield_t = None,
tolerance: float = 1e-20,
max_iters: int = 10000,
reset_iters: int = 100,
) -> Tuple[numpy.ndarray, numpy.ndarray]:
"""
Find the first (lowest-frequency) num_modes eigenmodes with Bloch wavevector
k0 of the specified structure.
Args:
k0: Bloch wavevector, `[k0x, k0y, k0z]`.
G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns.
epsilon: Dielectric constant distribution for the simulation.
All fields are sampled at cell centers (i.e., NOT Yee-gridded)
mu: Magnetic permability distribution for the simulation.
Default `None` (1 everywhere).
tolerance: Solver stops when fractional change in the objective
`trace(Z.H @ A @ Z @ inv(Z Z.H))` is smaller than the tolerance
Returns:
`(eigenvalues, eigenvectors)` where `eigenvalues[i]` corresponds to the
vector `eigenvectors[i, :]`
"""
h_size = 2 * epsilon[0].size
kmag = norm(G_matrix @ k0)
'''
Generate the operators
'''
mop = maxwell_operator(k0=k0, G_matrix=G_matrix, epsilon=epsilon, mu=mu)
imop = inverse_maxwell_operator_approx(k0=k0, G_matrix=G_matrix, epsilon=epsilon, mu=mu)
scipy_op = spalg.LinearOperator(dtype=complex, shape=(h_size, h_size), matvec=mop)
scipy_iop = spalg.LinearOperator(dtype=complex, shape=(h_size, h_size), matvec=imop)
y_shape = (h_size, num_modes)
prev_E = 0
d_scale = 1
prev_traceGtKG = 0
#prev_theta = 0.5
D = numpy.zeros(shape=y_shape, dtype=complex)
y0 = None
if y0 is None:
Z = numpy.random.rand(*y_shape) + 1j * numpy.random.rand(*y_shape)
else:
Z = y0
while True:
Z *= num_modes / norm(Z)
ZtZ = Z.conj().T @ Z
try:
U = numpy.linalg.inv(ZtZ)
except numpy.linalg.LinAlgError:
Z = numpy.random.rand(*y_shape) + 1j * numpy.random.rand(*y_shape)
continue
trace_U = real(trace(U))
if trace_U > 1e8 * num_modes:
Z = Z @ scipy.linalg.sqrtm(U).conj().T
prev_traceGtKG = 0
continue
break
for i in range(max_iters):
ZtZ = Z.conj().T @ Z
U = numpy.linalg.inv(ZtZ)
AZ = scipy_op @ Z
AZU = AZ @ U
ZtAZU = Z.conj().T @ AZU
E_signed = real(trace(ZtAZU))
sgn = numpy.sign(E_signed)
E = numpy.abs(E_signed)
G = (AZU - Z @ U @ ZtAZU) * sgn
if i > 0 and abs(E - prev_E) < tolerance * 0.5 * (E + prev_E + 1e-7):
logger.info('Optimization succeded: {} - 5e-8 < {} * {} / 2'.format(abs(E - prev_E), tolerance, E + prev_E))
break
KG = scipy_iop @ G
traceGtKG = _rtrace_AtB(G, KG)
if prev_traceGtKG == 0 or i % reset_iters == 0:
logger.info('CG reset')
gamma = 0
else:
gamma = traceGtKG / prev_traceGtKG
D = gamma / d_scale * D + KG
d_scale = num_modes / norm(D)
D *= d_scale
ZtAZ = Z.conj().T @ AZ
AD = scipy_op @ D
DtD = D.conj().T @ D
DtAD = D.conj().T @ AD
symZtD = _symmetrize(Z.conj().T @ D)
symZtAD = _symmetrize(Z.conj().T @ AD)
Qi_memo = [None, None]
def Qi_func(theta):
nonlocal Qi_memo
if Qi_memo[0] == theta:
return Qi_memo[1]
c = numpy.cos(theta)
s = numpy.sin(theta)
Q = c*c * ZtZ + s*s * DtD + 2*s*c * symZtD
try:
Qi = numpy.linalg.inv(Q)
except numpy.linalg.LinAlgError:
logger.info('taylor Qi')
# if c or s small, taylor expand
if c < 1e-4 * s and c != 0:
DtDi = numpy.linalg.inv(DtD)
Qi = DtDi / (s*s) - 2*c/(s*s*s) * (DtDi @ (DtDi @ symZtD).conj().T)
elif s < 1e-4 * c and s != 0:
ZtZi = numpy.linalg.inv(ZtZ)
Qi = ZtZi / (c*c) - 2*s/(c*c*c) * (ZtZi @ (ZtZi @ symZtD).conj().T)
else:
raise Exception('Inexplicable singularity in trace_func')
Qi_memo[0] = theta
Qi_memo[1] = Qi
return Qi
def trace_func(theta):
c = numpy.cos(theta)
s = numpy.sin(theta)
Qi = Qi_func(theta)
R = c*c * ZtAZ + s*s * DtAD + 2*s*c * symZtAD
trace = _rtrace_AtB(R, Qi)
return numpy.abs(trace)
'''
def trace_deriv(theta):
Qi = Qi_func(theta)
c2 = numpy.cos(2 * theta)
s2 = numpy.sin(2 * theta)
F = -0.5*s2 * (ZtAZ - DtAD) + c2 * symZtAD
trace_deriv = _rtrace_AtB(Qi, F)
G = Qi @ F.conj().T @ Qi.conj().T
H = -0.5*s2 * (ZtZ - DtD) + c2 * symZtD
trace_deriv -= _rtrace_AtB(G, H)
trace_deriv *= 2
return trace_deriv * sgn
U_sZtD = U @ symZtD
dE = 2.0 * (_rtrace_AtB(U, symZtAD) -
_rtrace_AtB(ZtAZU, U_sZtD))
d2E = 2 * (_rtrace_AtB(U, DtAD) -
_rtrace_AtB(ZtAZU, U @ (DtD - 4 * symZtD @ U_sZtD)) -
4 * _rtrace_AtB(U, symZtAD @ U_sZtD))
# Newton-Raphson to find a root of the first derivative:
theta = -dE/d2E
if d2E < 0 or abs(theta) >= pi:
theta = -abs(prev_theta) * numpy.sign(dE)
# theta, new_E, new_dE = linmin(theta, E, dE, 0.1, min(tolerance, 1e-6), 1e-14, 0, -numpy.sign(dE) * K_PI, trace_func)
theta, n, _, new_E, _, _new_dE = scipy.optimize.line_search(trace_func, trace_deriv, xk=theta, pk=numpy.ones((1,1)), gfk=dE, old_fval=E, c1=min(tolerance, 1e-6), c2=0.1, amax=pi)
'''
result = scipy.optimize.minimize_scalar(trace_func, bounds=(0, pi), tol=tolerance)
new_E = result.fun
theta = result.x
improvement = numpy.abs(E - new_E) * 2 / numpy.abs(E + new_E)
logger.info('linmin improvement {}'.format(improvement))
Z *= numpy.cos(theta)
Z += D * numpy.sin(theta)
prev_traceGtKG = traceGtKG
#prev_theta = theta
prev_E = E
'''
Recover eigenvectors from Z
'''
U = numpy.linalg.inv(Z.conj().T @ Z)
Y = Z @ scipy.linalg.sqrtm(U)
W = Y.conj().T @ (scipy_op @ Y)
eigvals, W_eigvecs = numpy.linalg.eig(W)
eigvecs = Y @ W_eigvecs
for i in range(len(eigvals)):
v = eigvecs[:, i]
n = eigvals[i]
v /= norm(v)
eigness = norm(scipy_op @ v - (v.conj() @ (scipy_op @ v)) * v )
f = numpy.sqrt(-numpy.real(n))
df = numpy.sqrt(-numpy.real(n + eigness))
neff_err = kmag * (1/df - 1/f)
logger.info('eigness {}: {}\n neff_err: {}'.format(i, eigness, neff_err))
order = numpy.argsort(numpy.abs(eigvals))
return eigvals[order], eigvecs.T[order]
'''
def linmin(x_guess, f0, df0, x_max, f_tol=0.1, df_tol=min(tolerance, 1e-6), x_tol=1e-14, x_min=0, linmin_func):
if df0 > 0:
x0, f0, df0 = linmin(-x_guess, f0, -df0, -x_max, f_tol, df_tol, x_tol, -x_min, lambda q, dq: -linmin_func(q, dq))
return -x0, f0, -df0
elif df0 == 0:
return 0, f0, df0
else:
x = x_guess
fx = f0
dfx = df0
isave = numpy.zeros((2,), numpy.intc)
dsave = numpy.zeros((13,), float)
x, fx, dfx, task = minpack2.dsrch(x, fx, dfx, f_tol, df_tol, x_tol, task,
x_min, x_max, isave, dsave)
for i in range(int(1e6)):
if task != 'F':
logging.info('search converged in {} iterations'.format(i))
break
fx = f(x, dfx)
x, fx, dfx, task = minpack2.dsrch(x, fx, dfx, f_tol, df_tol, x_tol, task,
x_min, x_max, isave, dsave)
return x, fx, dfx
'''
def _rtrace_AtB(A, B):
return real(numpy.sum(A.conj() * B))
def _symmetrize(A):
return (A + A.conj().T) * 0.5

219
meanas/fdfd/functional.py
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@ -1,219 +0,0 @@
"""
Functional versions of many FDFD operators. These can be useful for performing
FDFD calculations without needing to construct large matrices in memory.
The functions generated here expect `fdfield_t` inputs with shape (3, X, Y, Z),
e.g. E = [E_x, E_y, E_z] where each component has shape (X, Y, Z)
"""
from typing import List, Callable, Tuple
import numpy
from ..fdmath import dx_lists_t, fdfield_t, fdfield_updater_t
from ..fdmath.functional import curl_forward, curl_back
__author__ = 'Jan Petykiewicz'
def e_full(omega: complex,
dxes: dx_lists_t,
epsilon: fdfield_t,
mu: fdfield_t = None
) -> fdfield_updater_t:
"""
Wave operator for use with E-field. See `operators.e_full` for details.
Args:
omega: Angular frequency of the simulation
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types`
epsilon: Dielectric constant
mu: Magnetic permeability (default 1 everywhere)
Return:
Function `f` implementing the wave operator
`f(E)` -> `-i * omega * J`
"""
ch = curl_back(dxes[1])
ce = curl_forward(dxes[0])
def op_1(e):
curls = ch(ce(e))
return curls - omega ** 2 * epsilon * e
def op_mu(e):
curls = ch(mu * ce(e))
return curls - omega ** 2 * epsilon * e
if numpy.any(numpy.equal(mu, None)):
return op_1
else:
return op_mu
def eh_full(omega: complex,
dxes: dx_lists_t,
epsilon: fdfield_t,
mu: fdfield_t = None
) -> Callable[[fdfield_t, fdfield_t], Tuple[fdfield_t, fdfield_t]]:
"""
Wave operator for full (both E and H) field representation.
See `operators.eh_full`.
Args:
omega: Angular frequency of the simulation
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types`
epsilon: Dielectric constant
mu: Magnetic permeability (default 1 everywhere)
Returns:
Function `f` implementing the wave operator
`f(E, H)` -> `(J, -M)`
"""
ch = curl_back(dxes[1])
ce = curl_forward(dxes[0])
def op_1(e, h):
return (ch(h) - 1j * omega * epsilon * e,
ce(e) + 1j * omega * h)
def op_mu(e, h):
return (ch(h) - 1j * omega * epsilon * e,
ce(e) + 1j * omega * mu * h)
if numpy.any(numpy.equal(mu, None)):
return op_1
else:
return op_mu
def e2h(omega: complex,
dxes: dx_lists_t,
mu: fdfield_t = None,
) -> fdfield_updater_t:
"""
Utility operator for converting the `E` field into the `H` field.
For use with `e_full` -- assumes that there is no magnetic current `M`.
Args:
omega: Angular frequency of the simulation
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types`
mu: Magnetic permeability (default 1 everywhere)
Return:
Function `f` for converting `E` to `H`,
`f(E)` -> `H`
"""
ce = curl_forward(dxes[0])
def e2h_1_1(e):
return ce(e) / (-1j * omega)
def e2h_mu(e):
return ce(e) / (-1j * omega * mu)
if numpy.any(numpy.equal(mu, None)):
return e2h_1_1
else:
return e2h_mu
def m2j(omega: complex,
dxes: dx_lists_t,
mu: fdfield_t = None,
) -> fdfield_updater_t:
"""
Utility operator for converting magnetic current `M` distribution
into equivalent electric current distribution `J`.
For use with e.g. `e_full`.
Args:
omega: Angular frequency of the simulation
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types`
mu: Magnetic permeability (default 1 everywhere)
Returns:
Function `f` for converting `M` to `J`,
`f(M)` -> `J`
"""
ch = curl_back(dxes[1])
def m2j_mu(m):
J = ch(m / mu) / (-1j * omega)
return J
def m2j_1(m):
J = ch(m) / (-1j * omega)
return J
if numpy.any(numpy.equal(mu, None)):
return m2j_1
else:
return m2j_mu
def e_tfsf_source(TF_region: fdfield_t,
omega: complex,
dxes: dx_lists_t,
epsilon: fdfield_t,
mu: fdfield_t = None,
) -> fdfield_updater_t:
"""
Operator that turns an E-field distribution into a total-field/scattered-field
(TFSF) source.
Args:
TF_region: mask which is set to 1 in the total-field region, and 0 elsewhere
(i.e. in the scattered-field region).
Should have the same shape as the simulation grid, e.g. `epsilon[0].shape`.
omega: Angular frequency of the simulation
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types`
epsilon: Dielectric constant distribution
mu: Magnetic permeability (default 1 everywhere)
Returns:
Function `f` which takes an E field and returns a current distribution,
`f(E)` -> `J`
"""
# TODO documentation
A = e_full(omega, dxes, epsilon, mu)
def op(e):
neg_iwj = A(TF_region * e) - TF_region * A(e)
return neg_iwj / (-1j * omega)
def poynting_e_cross_h(dxes: dx_lists_t) -> Callable[[fdfield_t, fdfield_t], fdfield_t]:
"""
Generates a function that takes the single-frequency `E` and `H` fields
and calculates the cross product `E` x `H` = $E \\times H$ as required
for the Poynting vector, $S = E \\times H$
Note:
This function also shifts the input `E` field by one cell as required
for computing the Poynting cross product (see `meanas.fdfd` module docs).
Note:
If `E` and `H` are peak amplitudes as assumed elsewhere in this code,
the time-average of the poynting vector is `<S> = Re(S)/2 = Re(E x H) / 2`.
The factor of `1/2` can be omitted if root-mean-square quantities are used
instead.
Args:
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types`
Returns:
Function `f` that returns E x H as required for the poynting vector.
"""
def exh(e: fdfield_t, h: fdfield_t):
s = numpy.empty_like(e)
ex = e[0] * dxes[0][0][:, None, None]
ey = e[1] * dxes[0][1][None, :, None]
ez = e[2] * dxes[0][2][None, None, :]
hx = h[0] * dxes[1][0][:, None, None]
hy = h[1] * dxes[1][1][None, :, None]
hz = h[2] * dxes[1][2][None, None, :]
s[0] = numpy.roll(ey, -1, axis=0) * hz - numpy.roll(ez, -1, axis=0) * hy
s[1] = numpy.roll(ez, -1, axis=1) * hx - numpy.roll(ex, -1, axis=1) * hz
s[2] = numpy.roll(ex, -1, axis=2) * hy - numpy.roll(ey, -1, axis=2) * hx
return s
return exh

441
meanas/fdfd/operators.py
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"""
Sparse matrix operators for use with electromagnetic wave equations.
These functions return sparse-matrix (`scipy.sparse.spmatrix`) representations of
a variety of operators, intended for use with E and H fields vectorized using the
`meanas.vec()` and `meanas.unvec()` functions.
E- and H-field values are defined on a Yee cell; `epsilon` values should be calculated for
cells centered at each E component (`mu` at each H component).
Many of these functions require a `dxes` parameter, of type `dx_lists_t`; see
the `meanas.fdmath.types` submodule for details.
The following operators are included:
- E-only wave operator
- H-only wave operator
- EH wave operator
- Curl for use with E, H fields
- E to H conversion
- M to J conversion
- Poynting cross products
- Circular shifts
- Discrete derivatives
- Averaging operators
- Cross product matrices
"""
from typing import List, Tuple
import numpy
import scipy.sparse as sparse
from ..fdmath import vec, dx_lists_t, vfdfield_t
from ..fdmath.operators import shift_with_mirror, rotation, curl_forward, curl_back
__author__ = 'Jan Petykiewicz'
def e_full(omega: complex,
dxes: dx_lists_t,
epsilon: vfdfield_t,
mu: vfdfield_t = None,
pec: vfdfield_t = None,
pmc: vfdfield_t = None,
) -> sparse.spmatrix:
"""
Wave operator
$$ \\nabla \\times (\\frac{1}{\\mu} \\nabla \\times) - \\Omega^2 \\epsilon $$
del x (1/mu * del x) - omega**2 * epsilon
for use with the E-field, with wave equation
$$ (\\nabla \\times (\\frac{1}{\\mu} \\nabla \\times) - \\Omega^2 \\epsilon) E = -\\imath \\omega J $$
(del x (1/mu * del x) - omega**2 * epsilon) E = -i * omega * J
To make this matrix symmetric, use the preconditioners from `e_full_preconditioners()`.
Args:
omega: Angular frequency of the simulation
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types`
epsilon: Vectorized dielectric constant
mu: Vectorized magnetic permeability (default 1 everywhere).
pec: Vectorized mask specifying PEC cells. Any cells where `pec != 0` are interpreted
as containing a perfect electrical conductor (PEC).
The PEC is applied per-field-component (i.e. `pec.size == epsilon.size`)
pmc: Vectorized mask specifying PMC cells. Any cells where `pmc != 0` are interpreted
as containing a perfect magnetic conductor (PMC).
The PMC is applied per-field-component (i.e. `pmc.size == epsilon.size`)
Returns:
Sparse matrix containing the wave operator.
"""
ch = curl_back(dxes[1])
ce = curl_forward(dxes[0])
if numpy.any(numpy.equal(pec, None)):
pe = sparse.eye(epsilon.size)
else:
pe = sparse.diags(numpy.where(pec, 0, 1)) # Set pe to (not PEC)
if numpy.any(numpy.equal(pmc, None)):
pm = sparse.eye(epsilon.size)
else:
pm = sparse.diags(numpy.where(pmc, 0, 1)) # set pm to (not PMC)
e = sparse.diags(epsilon)
if numpy.any(numpy.equal(mu, None)):
m_div = sparse.eye(epsilon.size)
else:
m_div = sparse.diags(1 / mu)
op = pe @ (ch @ pm @ m_div @ ce - omega**2 * e) @ pe
return op
def e_full_preconditioners(dxes: dx_lists_t
) -> Tuple[sparse.spmatrix, sparse.spmatrix]:
"""
Left and right preconditioners `(Pl, Pr)` for symmetrizing the `e_full` wave operator.
The preconditioned matrix `A_symm = (Pl @ A @ Pr)` is complex-symmetric
(non-Hermitian unless there is no loss or PMLs).
The preconditioner matrices are diagonal and complex, with `Pr = 1 / Pl`
Args:
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types`
Returns:
Preconditioner matrices `(Pl, Pr)`.
"""
p_squared = [dxes[0][0][:, None, None] * dxes[1][1][None, :, None] * dxes[1][2][None, None, :],
dxes[1][0][:, None, None] * dxes[0][1][None, :, None] * dxes[1][2][None, None, :],
dxes[1][0][:, None, None] * dxes[1][1][None, :, None] * dxes[0][2][None, None, :]]
p_vector = numpy.sqrt(vec(p_squared))
P_left = sparse.diags(p_vector)
P_right = sparse.diags(1 / p_vector)
return P_left, P_right
def h_full(omega: complex,
dxes: dx_lists_t,
epsilon: vfdfield_t,
mu: vfdfield_t = None,
pec: vfdfield_t = None,
pmc: vfdfield_t = None,
) -> sparse.spmatrix:
"""
Wave operator
$$ \\nabla \\times (\\frac{1}{\\epsilon} \\nabla \\times) - \\omega^2 \\mu $$
del x (1/epsilon * del x) - omega**2 * mu
for use with the H-field, with wave equation
$$ (\\nabla \\times (\\frac{1}{\\epsilon} \\nabla \\times) - \\omega^2 \\mu) E = \\imath \\omega M $$
(del x (1/epsilon * del x) - omega**2 * mu) E = i * omega * M
Args:
omega: Angular frequency of the simulation
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types`
epsilon: Vectorized dielectric constant
mu: Vectorized magnetic permeability (default 1 everywhere)
pec: Vectorized mask specifying PEC cells. Any cells where `pec != 0` are interpreted
as containing a perfect electrical conductor (PEC).
The PEC is applied per-field-component (i.e. `pec.size == epsilon.size`)
pmc: Vectorized mask specifying PMC cells. Any cells where `pmc != 0` are interpreted
as containing a perfect magnetic conductor (PMC).
The PMC is applied per-field-component (i.e. `pmc.size == epsilon.size`)
Returns:
Sparse matrix containing the wave operator.
"""
ch = curl_back(dxes[1])
ce = curl_forward(dxes[0])
if numpy.any(numpy.equal(pec, None)):
pe = sparse.eye(epsilon.size)
else:
pe = sparse.diags(numpy.where(pec, 0, 1)) # set pe to (not PEC)
if numpy.any(numpy.equal(pmc, None)):
pm = sparse.eye(epsilon.size)
else:
pm = sparse.diags(numpy.where(pmc, 0, 1)) # Set pe to (not PMC)
e_div = sparse.diags(1 / epsilon)
if mu is None:
m = sparse.eye(epsilon.size)
else:
m = sparse.diags(mu)
A = pm @ (ce @ pe @ e_div @ ch - omega**2 * m) @ pm
return A
def eh_full(omega: complex,
dxes: dx_lists_t,
epsilon: vfdfield_t,
mu: vfdfield_t = None,
pec: vfdfield_t = None,
pmc: vfdfield_t = None
) -> sparse.spmatrix:
"""
Wave operator for `[E, H]` field representation. This operator implements Maxwell's
equations without cancelling out either E or H. The operator is
$$ \\begin{bmatrix}
-\\imath \\omega \\epsilon & \\nabla \\times \\\\
\\nabla \\times & \\imath \\omega \\mu
\\end{bmatrix} $$
[[-i * omega * epsilon, del x ],
[del x, i * omega * mu]]
for use with a field vector of the form `cat(vec(E), vec(H))`:
$$ \\begin{bmatrix}
-\\imath \\omega \\epsilon & \\nabla \\times \\\\
\\nabla \\times & \\imath \\omega \\mu
\\end{bmatrix}
\\begin{bmatrix} E \\\\
H
\\end{bmatrix}
= \\begin{bmatrix} J \\\\
-M
\\end{bmatrix} $$
Args:
omega: Angular frequency of the simulation
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types`
epsilon: Vectorized dielectric constant
mu: Vectorized magnetic permeability (default 1 everywhere)
pec: Vectorized mask specifying PEC cells. Any cells where `pec != 0` are interpreted
as containing a perfect electrical conductor (PEC).
The PEC is applied per-field-component (i.e. `pec.size == epsilon.size`)
pmc: Vectorized mask specifying PMC cells. Any cells where `pmc != 0` are interpreted
as containing a perfect magnetic conductor (PMC).
The PMC is applied per-field-component (i.e. `pmc.size == epsilon.size`)
Returns:
Sparse matrix containing the wave operator.
"""
if numpy.any(numpy.equal(pec, None)):
pe = sparse.eye(epsilon.size)
else:
pe = sparse.diags(numpy.where(pec, 0, 1)) # set pe to (not PEC)
if numpy.any(numpy.equal(pmc, None)):
pm = sparse.eye(epsilon.size)
else:
pm = sparse.diags(numpy.where(pmc, 0, 1)) # set pm to (not PMC)
iwe = pe @ (1j * omega * sparse.diags(epsilon)) @ pe
iwm = 1j * omega
if not numpy.any(numpy.equal(mu, None)):
iwm *= sparse.diags(mu)
iwm = pm @ iwm @ pm
A1 = pe @ curl_back(dxes[1]) @ pm
A2 = pm @ curl_forward(dxes[0]) @ pe
A = sparse.bmat([[-iwe, A1],
[A2, iwm]])
return A
def e2h(omega: complex,
dxes: dx_lists_t,
mu: vfdfield_t = None,
pmc: vfdfield_t = None,
) -> sparse.spmatrix:
"""
Utility operator for converting the E field into the H field.
For use with `e_full()` -- assumes that there is no magnetic current M.
Args:
omega: Angular frequency of the simulation
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types`
mu: Vectorized magnetic permeability (default 1 everywhere)
pmc: Vectorized mask specifying PMC cells. Any cells where `pmc != 0` are interpreted
as containing a perfect magnetic conductor (PMC).
The PMC is applied per-field-component (i.e. `pmc.size == epsilon.size`)
Returns:
Sparse matrix for converting E to H.
"""
op = curl_forward(dxes[0]) / (-1j * omega)
if not numpy.any(numpy.equal(mu, None)):
op = sparse.diags(1 / mu) @ op
if not numpy.any(numpy.equal(pmc, None)):
op = sparse.diags(numpy.where(pmc, 0, 1)) @ op
return op
def m2j(omega: complex,
dxes: dx_lists_t,
mu: vfdfield_t = None
) -> sparse.spmatrix:
"""
Operator for converting a magnetic current M into an electric current J.
For use with eg. `e_full()`.
Args:
omega: Angular frequency of the simulation
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types`
mu: Vectorized magnetic permeability (default 1 everywhere)
Returns:
Sparse matrix for converting M to J.
"""
op = curl_back(dxes[1]) / (1j * omega)
if not numpy.any(numpy.equal(mu, None)):
op = op @ sparse.diags(1 / mu)
return op
def poynting_e_cross(e: vfdfield_t, dxes: dx_lists_t) -> sparse.spmatrix:
"""
Operator for computing the Poynting vector, containing the
(E x) portion of the Poynting vector.
Args:
e: Vectorized E-field for the ExH cross product
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types`
Returns:
Sparse matrix containing (E x) portion of Poynting cross product.
"""
shape = [len(dx) for dx in dxes[0]]
fx, fy, fz = [rotation(i, shape, 1) for i in range(3)]
dxag = [dx.ravel(order='C') for dx in numpy.meshgrid(*dxes[0], indexing='ij')]
dxbg = [dx.ravel(order='C') for dx in numpy.meshgrid(*dxes[1], indexing='ij')]
Ex, Ey, Ez = [ei * da for ei, da in zip(numpy.split(e, 3), dxag)]
block_diags = [[ None, fx @ -Ez, fx @ Ey],
[ fy @ Ez, None, fy @ -Ex],
[ fz @ -Ey, fz @ Ex, None]]
block_matrix = sparse.bmat([[sparse.diags(x) if x is not None else None for x in row]
for row in block_diags])
P = block_matrix @ sparse.diags(numpy.concatenate(dxag))
return P
def poynting_h_cross(h: vfdfield_t, dxes: dx_lists_t) -> sparse.spmatrix:
"""
Operator for computing the Poynting vector, containing the (H x) portion of the Poynting vector.
Args:
h: Vectorized H-field for the HxE cross product
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types`
Returns:
Sparse matrix containing (H x) portion of Poynting cross product.
"""
shape = [len(dx) for dx in dxes[0]]
fx, fy, fz = [rotation(i, shape, 1) for i in range(3)]
dxag = [dx.ravel(order='C') for dx in numpy.meshgrid(*dxes[0], indexing='ij')]
dxbg = [dx.ravel(order='C') for dx in numpy.meshgrid(*dxes[1], indexing='ij')]
Hx, Hy, Hz = [sparse.diags(hi * db) for hi, db in zip(numpy.split(h, 3), dxbg)]
P = (sparse.bmat(
[[ None, -Hz @ fx, Hy @ fx],
[ Hz @ fy, None, -Hx @ fy],
[-Hy @ fz, Hx @ fz, None]])
@ sparse.diags(numpy.concatenate(dxag)))
return P
def e_tfsf_source(TF_region: vfdfield_t,
omega: complex,
dxes: dx_lists_t,
epsilon: vfdfield_t,
mu: vfdfield_t = None,
) -> sparse.spmatrix:
"""
Operator that turns a desired E-field distribution into a
total-field/scattered-field (TFSF) source.
TODO: Reference Rumpf paper
Args:
TF_region: Mask, which is set to 1 inside the total-field region and 0 in the
scattered-field region
omega: Angular frequency of the simulation
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types`
epsilon: Vectorized dielectric constant
mu: Vectorized magnetic permeability (default 1 everywhere).
Returns:
Sparse matrix that turns an E-field into a current (J) distribution.
"""
# TODO documentation
A = e_full(omega, dxes, epsilon, mu)
Q = sparse.diags(TF_region)
return (A @ Q - Q @ A) / (-1j * omega)
def e_boundary_source(mask: vfdfield_t,
omega: complex,
dxes: dx_lists_t,
epsilon: vfdfield_t,
mu: vfdfield_t = None,
periodic_mask_edges: bool = False,
) -> sparse.spmatrix:
"""
Operator that turns an E-field distrubtion into a current (J) distribution
along the edges (external and internal) of the provided mask. This is just an
`e_tfsf_source()` with an additional masking step.
Args:
mask: The current distribution is generated at the edges of the mask,
i.e. any points where shifting the mask by one cell in any direction
would change its value.
omega: Angular frequency of the simulation
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types`
epsilon: Vectorized dielectric constant
mu: Vectorized magnetic permeability (default 1 everywhere).
Returns:
Sparse matrix that turns an E-field into a current (J) distribution.
"""
full = e_tfsf_source(TF_region=mask, omega=omega, dxes=dxes, epsilon=epsilon, mu=mu)
shape = [len(dxe) for dxe in dxes[0]]
jmask = numpy.zeros_like(mask, dtype=bool)
if periodic_mask_edges:
shift = lambda axis, polarity: rotation(axis=axis, shape=shape, shift_distance=polarity)
else:
shift = lambda axis, polarity: shift_with_mirror(axis=axis, shape=shape, shift_distance=polarity)
for axis in (0, 1, 2):
if shape[axis] == 1:
continue
for polarity in (-1, +1):
r = shift(axis, polarity) - sparse.eye(numpy.prod(shape)) # shifted minus original
r3 = sparse.block_diag((r, r, r))
jmask = numpy.logical_or(jmask, numpy.abs(r3 @ mask))
# jmask = ((numpy.roll(mask, -1, axis=0) != mask) |
# (numpy.roll(mask, +1, axis=0) != mask) |
# (numpy.roll(mask, -1, axis=1) != mask) |
# (numpy.roll(mask, +1, axis=1) != mask) |
# (numpy.roll(mask, -1, axis=2) != mask) |
# (numpy.roll(mask, +1, axis=2) != mask))
return sparse.diags(jmask.astype(int)) @ full

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meanas/fdfd/solvers.py
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@ -1,126 +0,0 @@
"""
Solvers and solver interface for FDFD problems.
"""
from typing import List, Callable, Dict, Any
import logging
import numpy
from numpy.linalg import norm
import scipy.sparse.linalg
from ..fdmath import dx_lists_t, vfdfield_t
from . import operators
logger = logging.getLogger(__name__)
def _scipy_qmr(A: scipy.sparse.csr_matrix,
b: numpy.ndarray,
**kwargs
) -> numpy.ndarray:
"""
Wrapper for scipy.sparse.linalg.qmr
Args:
A: Sparse matrix
b: Right-hand-side vector
kwargs: Passed as **kwargs to the wrapped function
Returns:
Guess for solution (returned even if didn't converge)
"""
'''
Report on our progress
'''
iter = 0
def log_residual(xk):
nonlocal iter
iter += 1
if iter % 100 == 0:
logger.info('Solver residual at iteration {} : {}'.format(iter, norm(A @ xk - b)))
if 'callback' in kwargs:
def augmented_callback(xk):
log_residual(xk)
kwargs['callback'](xk)
kwargs['callback'] = augmented_callback
else:
kwargs['callback'] = log_residual
'''
Run the actual solve
'''
x, _ = scipy.sparse.linalg.qmr(A, b, **kwargs)
return x
def generic(omega: complex,
dxes: dx_lists_t,
J: vfdfield_t,
epsilon: vfdfield_t,
mu: vfdfield_t = None,
pec: vfdfield_t = None,
pmc: vfdfield_t = None,
adjoint: bool = False,
matrix_solver: Callable[..., numpy.ndarray] = _scipy_qmr,
matrix_solver_opts: Dict[str, Any] = None,
) -> vfdfield_t:
"""
Conjugate gradient FDFD solver using CSR sparse matrices.
All ndarray arguments should be 1D arrays, as returned by `meanas.vec()`.
Args:
omega: Complex frequency to solve at.
dxes: `[[dx_e, dy_e, dz_e], [dx_h, dy_h, dz_h]]` (complex cell sizes) as
discussed in `meanas.fdmath.types`
J: Electric current distribution (at E-field locations)
epsilon: Dielectric constant distribution (at E-field locations)
mu: Magnetic permeability distribution (at H-field locations)
pec: Perfect electric conductor distribution
(at E-field locations; non-zero value indicates PEC is present)
pmc: Perfect magnetic conductor distribution
(at H-field locations; non-zero value indicates PMC is present)
adjoint: If true, solves the adjoint problem.
matrix_solver: Called as `matrix_solver(A, b, **matrix_solver_opts) -> x`,
where `A`: `scipy.sparse.csr_matrix`;
`b`: `numpy.ndarray`;
`x`: `numpy.ndarray`;
Default is a wrapped version of `scipy.sparse.linalg.qmr()`
which doesn't return convergence info and logs the residual
every 100 iterations.
matrix_solver_opts: Passed as kwargs to `matrix_solver(...)`
Returns:
E-field which solves the system.
"""
if matrix_solver_opts is None:
matrix_solver_opts = dict()
b0 = -1j * omega * J
A0 = operators.e_full(omega, dxes, epsilon=epsilon, mu=mu, pec=pec, pmc=pmc)
Pl, Pr = operators.e_full_preconditioners(dxes)
if adjoint:
A = (Pl @ A0 @ Pr).H
b = Pr.H @ b0
else:
A = Pl @ A0 @ Pr
b = Pl @ b0
x = matrix_solver(A.tocsr(), b, **matrix_solver_opts)
if adjoint:
x0 = Pl.H @ x
else:
x0 = Pr @ x
return x0

737
meanas/fdfd/waveguide_2d.py
View File

@ -1,737 +0,0 @@
"""
Operators and helper functions for waveguides with unchanging cross-section.
The propagation direction is chosen to be along the z axis, and all fields
are given an implicit z-dependence of the form `exp(-1 * wavenumber * z)`.
As the z-dependence is known, all the functions in this file assume a 2D grid
(i.e. `dxes = [[[dx_e[0], dx_e[1], ...], [dy_e[0], ...]], [[dx_h[0], ...], [dy_h[0], ...]]]`).
===============
Consider Maxwell's equations in continuous space, in the frequency domain. Assuming
a structure with some (x, y) cross-section extending uniformly into the z dimension,
with a diagonal $\\epsilon$ tensor, we have
$$
\\begin{aligned}
\\nabla \\times \\vec{E}(x, y, z) &= -\\imath \\omega \\mu \\vec{H} \\\\
\\nabla \\times \\vec{H}(x, y, z) &= \\imath \\omega \\epsilon \\vec{E} \\\\
\\vec{E}(x,y,z) = (\\vec{E}_t(x, y) + E_z(x, y)\\vec{z}) e^{-\\gamma z} \\\\
\\vec{H}(x,y,z) = (\\vec{H}_t(x, y) + H_z(x, y)\\vec{z}) e^{-\\gamma z} \\\\
\\end{aligned}
$$
Expanding the first two equations into vector components, we get
$$
\\begin{aligned}
-\\imath \\omega \\mu_{xx} H_x &= \\partial_y E_z - \\partial_z E_y \\\\
-\\imath \\omega \\mu_{yy} H_y &= \\partial_z E_x - \\partial_x E_z \\\\
-\\imath \\omega \\mu_{zz} H_z &= \\partial_x E_y - \\partial_y E_x \\\\
\\imath \\omega \\epsilon_{xx} E_x &= \\partial_y H_z - \\partial_z H_y \\\\
\\imath \\omega \\epsilon_{yy} E_y &= \\partial_z H_x - \\partial_x H_z \\\\
\\imath \\omega \\epsilon_{zz} E_z &= \\partial_x H_y - \\partial_y H_x \\\\
\\end{aligned}
$$
Substituting in our expressions for $\\vec{E}$, $\\vec{H}$ and discretizing:
$$
\\begin{aligned}
-\\imath \\omega \\mu_{xx} H_x &= \\tilde{\\partial}_y E_z + \\gamma E_y \\\\
-\\imath \\omega \\mu_{yy} H_y &= -\\gamma E_x - \\tilde{\\partial}_x E_z \\\\
-\\imath \\omega \\mu_{zz} H_z &= \\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x \\\\
\\imath \\omega \\epsilon_{xx} E_x &= \\hat{\\partial}_y H_z + \\gamma H_y \\\\
\\imath \\omega \\epsilon_{yy} E_y &= -\\gamma H_x - \\hat{\\partial}_x H_z \\\\
\\imath \\omega \\epsilon_{zz} E_z &= \\hat{\\partial}_x H_y - \\hat{\\partial}_y H_x \\\\
\\end{aligned}
$$
Rewrite the last three equations as
$$
\\begin{aligned}
\\gamma H_y &= \\imath \\omega \\epsilon_{xx} E_x - \\hat{\\partial}_y H_z \\\\
\\gamma H_x &= -\\imath \\omega \\epsilon_{yy} E_y - \\hat{\\partial}_x H_z \\\\
\\imath \\omega E_z &= \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x H_y - \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y H_x \\\\
\\end{aligned}
$$
Now apply $\\gamma \\tilde{\\partial}_x$ to the last equation,
then substitute in for $\\gamma H_x$ and $\\gamma H_y$:
$$
\\begin{aligned}
\\gamma \\tilde{\\partial}_x \\imath \\omega E_z &= \\gamma \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x H_y
- \\gamma \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y H_x \\\\
&= \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x ( \\imath \\omega \\epsilon_{xx} E_x - \\hat{\\partial}_y H_z)
- \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (-\\imath \\omega \\epsilon_{yy} E_y - \\hat{\\partial}_x H_z) \\\\
&= \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_x ( \\imath \\omega \\epsilon_{xx} E_x)
- \\tilde{\\partial}_x \\frac{1}{\\epsilon_{zz}} \\hat{\\partial}_y (-\\imath \\omega \\epsilon_{yy} E_y) \\\\
\\gamma \\tilde{\\partial}_x E_z &= \\tilde{\\partial}_x \\frac{1}{\\epsilon_zz} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
\\tilde{\\partial}_x \\frac{1}{\\epsilon_zz} \\hat{\\partial}_y (\\epsilon_{yy} E_y) \\\\
\\end{aligned}
$$
With a similar approach (but using $\\gamma \\tilde{\\partial}_y$ instead), we can get
$$
\\begin{aligned}
\\gamma \\tilde{\\partial}_y E_z &= \\tilde{\\partial}_y \\frac{1}{\\epsilon_zz} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
\\tilde{\\partial}_y \\frac{1}{\\epsilon_zz} \\hat{\\partial}_y (\\epsilon_{yy} E_y) \\\\
\\end{aligned}
$$
We can combine this equation for $\\gamma \\tilde{\\partial}_y E_z$ with
the unused $\\imath \\omega \\mu_{xx} H_z$ and $\\imath \\omega \\mu_{yy} H_y$ equations to get
$$
\\begin{aligned}
-\\imath \\omega \\mu_{xx} \\gamma H_x &= \\gamma^2 E_y + \\tilde{\\partial}_y (
\\tilde{\\partial}_x \\frac{1}{\\epsilon_zz} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
+ \\tilde{\\partial}_x \\frac{1}{\\epsilon_zz} \\hat{\\partial}_y (\\epsilon_{yy} E_y)
) \\\\
-\\imath \\omega \\mu_{yy} \\gamma H_y &= -\\gamma^2 E_x - \\tilde{\\partial}_x (
\\tilde{\\partial}_y \\frac{1}{\\epsilon_zz} \\hat{\\partial}_x (\\epsilon_{xx} E_x)
+ \\tilde{\\partial}_y \\frac{1}{\\epsilon_zz} \\hat{\\partial}_y (\\epsilon_{yy} E_y)
)\\\\
\\end{aligned}
$$
However, based on our rewritten equation for $\\gamma H_x$ and the so-far unused
equation for $\\imath \\omega \\mu_{zz} H_z$ we can also write
$$
\\begin{aligned}
-\\imath \\omega \\mu_{xx} (\\gamma H_x) &= -\\imath \\omega \\mu_{xx} (-\\imath \\omega \\epsilon_{yy} E_y - \\hat{\\partial}_x H_z) \\\\
&= -\\omega^2 \\mu_{xx} \\epsilon_{yy} E_y
-\\imath \\omega \\mu_{xx} \\hat{\\partial}_x (
\\frac{1}{-\\imath \\omega \\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x)) \\\\
&= -\\omega^2 \\mu_{xx} \\epsilon_{yy} E_y
+\\mu_{xx} \\hat{\\partial}_x \\frac{1}{\\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x) \\\\
\\end{aligned}
$$
and, similarly,
$$
\\begin{aligned}
-\\imath \\omega \\mu_{yy} (\\gamma H_y) &= -\\omega^2 \\mu_{yy} \\epsilon_{xx} E_x
+\\mu_{yy} \\hat{\\partial}_y \\frac{1}{\\mu_{zz}} (\\tilde{\\partial}_x E_y - \\tilde{\\partial}_y E_x) \\\\
\\end{aligned}
$$
Using these, we can construct the eigenvalue problem
$$ \\beta^2 \\begin{bmatrix} E_x \\\\
E_y \\end{bmatrix} =
(\\omega^2 \\begin{bmatrix} \\mu_{yy} \\epsilon_{xx} & 0 \\\\
0 & \\mu_{xx} \\epsilon_{yy} \\end{bmatrix} +
\\begin{bmatrix} -\\mu_{yy} \\hat{\\partial}_y \\\\
\\mu_{xx} \\hat{\\partial}_x \\end{bmatrix} \\mu_{zz}^{-1}
\\begin{bmatrix} -\\tilde{\\partial}_y & \\tilde{\\partial}_x \\end{bmatrix} +
\\begin{bmatrix} \\tilde{\\partial}_x \\\\
\\tilde{\\partial}_y \\end{bmatrix} \\epsilon_{zz}^{-1}
\\begin{bmatrix} \\hat{\\partial}_x \\epsilon_{xx} & \\hat{\\partial}_y \\epsilon_{yy} \\end{bmatrix})
\\begin{bmatrix} E_x \\\\
E_y \\end{bmatrix}
$$
An equivalent eigenvalue problem can be formed using the $H_x$ and $H_y$ fields, if those are more convenient.
Note that $E_z$ was never discretized, so $\\gamma$ and $\\beta$ will need adjustment
to account for numerical dispersion if the result is introduced into a space with a discretized z-axis.
"""
# TODO update module docs
from typing import List, Tuple
import numpy
from numpy.linalg import norm
import scipy.sparse as sparse
from ..fdmath.operators import deriv_forward, deriv_back, curl_forward, curl_back, cross
from ..fdmath import vec, unvec, dx_lists_t, fdfield_t, vfdfield_t
from ..eigensolvers import signed_eigensolve, rayleigh_quotient_iteration
from . import operators
__author__ = 'Jan Petykiewicz'
def operator_e(omega: complex,
dxes: dx_lists_t,
epsilon: vfdfield_t,
mu: vfdfield_t = None,
) -> sparse.spmatrix:
"""
Waveguide operator of the form
omega**2 * mu * epsilon +
mu * [[-Dy], [Dx]] / mu * [-Dy, Dx] +
[[Dx], [Dy]] / epsilon * [Dx, Dy] * epsilon
for use with a field vector of the form `cat([E_x, E_y])`.
More precisely, the operator is
$$
\\omega^2 \\begin{bmatrix} \\mu_{yy} \\epsilon_{xx} & 0 \\\\
0 & \\mu_{xx} \\epsilon_{yy} \\end{bmatrix} +
\\begin{bmatrix} -\\mu_{yy} \\hat{\\partial}_y \\\\
\\mu_{xx} \\hat{\\partial}_x \\end{bmatrix} \\mu_{zz}^{-1}
\\begin{bmatrix} -\\tilde{\\partial}_y & \\tilde{\\partial}_x \\end{bmatrix} +
\\begin{bmatrix} \\tilde{\\partial}_x \\\\
\\tilde{\\partial}_y \\end{bmatrix} \\epsilon_{zz}^{-1}
\\begin{bmatrix} \\hat{\\partial}_x \\epsilon_{xx} & \\hat{\\partial}_y \\epsilon_{yy} \\end{bmatrix}
$$
$\\tilde{\\partial}_x$ and $\\hat{\\partial}_x$ are the forward and backward derivatives along x,
and each $\\epsilon_{xx}$, $\\mu_{yy}$, etc. is a diagonal matrix containing the vectorized material
property distribution.
This operator can be used to form an eigenvalue problem of the form
`operator_e(...) @ [E_x, E_y] = wavenumber**2 * [E_x, E_y]`
which can then be solved for the eigenmodes of the system (an `exp(-i * wavenumber * z)`
z-dependence is assumed for the fields).
Args:
omega: The angular frequency of the system.
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types` (2D)
epsilon: Vectorized dielectric constant grid
mu: Vectorized magnetic permeability grid (default 1 everywhere)
Returns:
Sparse matrix representation of the operator.
"""
if numpy.any(numpy.equal(mu, None)):
mu = numpy.ones_like(epsilon)
Dfx, Dfy = deriv_forward(dxes[0])
Dbx, Dby = deriv_back(dxes[1])
eps_parts = numpy.split(epsilon, 3)
eps_xy = sparse.diags(numpy.hstack((eps_parts[0], eps_parts[1])))
eps_z_inv = sparse.diags(1 / eps_parts[2])
mu_parts = numpy.split(mu, 3)
mu_yx = sparse.diags(numpy.hstack((mu_parts[1], mu_parts[0])))
mu_z_inv = sparse.diags(1 / mu_parts[2])
op = omega * omega * mu_yx @ eps_xy + \
mu_yx @ sparse.vstack((-Dby, Dbx)) @ mu_z_inv @ sparse.hstack((-Dfy, Dfx)) + \
sparse.vstack((Dfx, Dfy)) @ eps_z_inv @ sparse.hstack((Dbx, Dby)) @ eps_xy
return op
def operator_h(omega: complex,
dxes: dx_lists_t,
epsilon: vfdfield_t,
mu: vfdfield_t = None,
) -> sparse.spmatrix:
"""
Waveguide operator of the form
omega**2 * epsilon * mu +
epsilon * [[-Dy], [Dx]] / epsilon * [-Dy, Dx] +
[[Dx], [Dy]] / mu * [Dx, Dy] * mu
for use with a field vector of the form `cat([H_x, H_y])`.
More precisely, the operator is
$$
\\omega^2 \\begin{bmatrix} \\epsilon_{yy} \\mu_{xx} & 0 \\\\
0 & \\epsilon_{xx} \\mu_{yy} \\end{bmatrix} +
\\begin{bmatrix} -\\epsilon_{yy} \\tilde{\\partial}_y \\\\
\\epsilon_{xx} \\tilde{\\partial}_x \\end{bmatrix} \\epsilon_{zz}^{-1}
\\begin{bmatrix} -\\hat{\\partial}_y & \\hat{\\partial}_x \\end{bmatrix} +
\\begin{bmatrix} \\hat{\\partial}_x \\\\
\\hat{\\partial}_y \\end{bmatrix} \\mu_{zz}^{-1}
\\begin{bmatrix} \\tilde{\\partial}_x \\mu_{xx} & \\tilde{\\partial}_y \\mu_{yy} \\end{bmatrix}
$$
$\\tilde{\\partial}_x$ and $\\hat{\\partial}_x$ are the forward and backward derivatives along x,
and each $\\epsilon_{xx}$, $\\mu_{yy}$, etc. is a diagonal matrix containing the vectorized material
property distribution.
This operator can be used to form an eigenvalue problem of the form
`operator_h(...) @ [H_x, H_y] = wavenumber**2 * [H_x, H_y]`
which can then be solved for the eigenmodes of the system (an `exp(-i * wavenumber * z)`
z-dependence is assumed for the fields).
Args:
omega: The angular frequency of the system.
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types` (2D)
epsilon: Vectorized dielectric constant grid
mu: Vectorized magnetic permeability grid (default 1 everywhere)
Returns:
Sparse matrix representation of the operator.
"""
if numpy.any(numpy.equal(mu, None)):
mu = numpy.ones_like(epsilon)
Dfx, Dfy = operators.deriv_forward(dxes[0])
Dbx, Dby = operators.deriv_back(dxes[1])
eps_parts = numpy.split(epsilon, 3)
eps_yx = sparse.diags(numpy.hstack((eps_parts[1], eps_parts[0])))
eps_z_inv = sparse.diags(1 / eps_parts[2])
mu_parts = numpy.split(mu, 3)
mu_xy = sparse.diags(numpy.hstack((mu_parts[0], mu_parts[1])))
mu_z_inv = sparse.diags(1 / mu_parts[2])
op = omega * omega * eps_yx @ mu_xy + \
eps_yx @ sparse.vstack((-Dfy, Dfx)) @ eps_z_inv @ sparse.hstack((-Dby, Dbx)) + \
sparse.vstack((Dbx, Dby)) @ mu_z_inv @ sparse.hstack((Dfx, Dfy)) @ mu_xy
return op
def normalized_fields_e(e_xy: numpy.ndarray,
wavenumber: complex,
omega: complex,
dxes: dx_lists_t,
epsilon: vfdfield_t,
mu: vfdfield_t = None,
prop_phase: float = 0,
) -> Tuple[vfdfield_t, vfdfield_t]:
"""
Given a vector `e_xy` containing the vectorized E_x and E_y fields,
returns normalized, vectorized E and H fields for the system.
Args:
e_xy: Vector containing E_x and E_y fields
wavenumber: Wavenumber assuming fields have z-dependence of `exp(-i * wavenumber * z)`.
It should satisfy `operator_e() @ e_xy == wavenumber**2 * e_xy`
omega: The angular frequency of the system
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types` (2D)
epsilon: Vectorized dielectric constant grid
mu: Vectorized magnetic permeability grid (default 1 everywhere)
prop_phase: Phase shift `(dz * corrected_wavenumber)` over 1 cell in propagation direction.
Default 0 (continuous propagation direction, i.e. dz->0).
Returns:
`(e, h)`, where each field is vectorized, normalized,
and contains all three vector components.
"""
e = exy2e(wavenumber=wavenumber, dxes=dxes, epsilon=epsilon) @ e_xy
h = exy2h(wavenumber=wavenumber, omega=omega, dxes=dxes, epsilon=epsilon, mu=mu) @ e_xy
e_norm, h_norm = _normalized_fields(e=e, h=h, omega=omega, dxes=dxes, epsilon=epsilon,
mu=mu, prop_phase=prop_phase)
return e_norm, h_norm
def normalized_fields_h(h_xy: numpy.ndarray,
wavenumber: complex,
omega: complex,
dxes: dx_lists_t,
epsilon: vfdfield_t,
mu: vfdfield_t = None,
prop_phase: float = 0,
) -> Tuple[vfdfield_t, vfdfield_t]:
"""
Given a vector `h_xy` containing the vectorized H_x and H_y fields,
returns normalized, vectorized E and H fields for the system.
Args:
h_xy: Vector containing H_x and H_y fields
wavenumber: Wavenumber assuming fields have z-dependence of `exp(-i * wavenumber * z)`.
It should satisfy `operator_h() @ h_xy == wavenumber**2 * h_xy`
omega: The angular frequency of the system
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types` (2D)
epsilon: Vectorized dielectric constant grid
mu: Vectorized magnetic permeability grid (default 1 everywhere)
prop_phase: Phase shift `(dz * corrected_wavenumber)` over 1 cell in propagation direction.
Default 0 (continuous propagation direction, i.e. dz->0).
Returns:
`(e, h)`, where each field is vectorized, normalized,
and contains all three vector components.
"""
e = hxy2e(wavenumber=wavenumber, omega=omega, dxes=dxes, epsilon=epsilon, mu=mu) @ h_xy
h = hxy2h(wavenumber=wavenumber, dxes=dxes, mu=mu) @ h_xy
e_norm, h_norm = _normalized_fields(e=e, h=h, omega=omega, dxes=dxes, epsilon=epsilon,
mu=mu, prop_phase=prop_phase)
return e_norm, h_norm
def _normalized_fields(e: numpy.ndarray,
h: numpy.ndarray,
omega: complex,
dxes: dx_lists_t,
epsilon: vfdfield_t,
mu: vfdfield_t = None,
prop_phase: float = 0,
) -> Tuple[vfdfield_t, vfdfield_t]:
# TODO documentation
shape = [s.size for s in dxes[0]]
dxes_real = [[numpy.real(d) for d in numpy.meshgrid(*dxes[v], indexing='ij')] for v in (0, 1)]
E = unvec(e, shape)
H = unvec(h, shape)
# Find time-averaged Sz and normalize to it
# H phase is adjusted by a half-cell forward shift for Yee cell, and 1-cell reverse shift for Poynting
phase = numpy.exp(-1j * -prop_phase / 2)
Sz_a = E[0] * numpy.conj(H[1] * phase) * dxes_real[0][1] * dxes_real[1][0]
Sz_b = E[1] * numpy.conj(H[0] * phase) * dxes_real[0][0] * dxes_real[1][1]
Sz_tavg = numpy.real(Sz_a.sum() - Sz_b.sum()) * 0.5 # 0.5 since E, H are assumed to be peak (not RMS) amplitudes
assert Sz_tavg > 0, 'Found a mode propagating in the wrong direction! Sz_tavg={}'.format(Sz_tavg)
energy = epsilon * e.conj() * e
norm_amplitude = 1 / numpy.sqrt(Sz_tavg)
norm_angle = -numpy.angle(e[energy.argmax()]) # Will randomly add a negative sign when mode is symmetric
# Try to break symmetry to assign a consistent sign [experimental TODO]
E_weighted = unvec(e * energy * numpy.exp(1j * norm_angle), shape)
sign = numpy.sign(E_weighted[:, :max(shape[0]//2, 1), :max(shape[1]//2, 1)].real.sum())
norm_factor = sign * norm_amplitude * numpy.exp(1j * norm_angle)
e *= norm_factor
h *= norm_factor
return e, h
def exy2h(wavenumber: complex,
omega: complex,
dxes: dx_lists_t,
epsilon: vfdfield_t,
mu: vfdfield_t = None
) -> sparse.spmatrix:
"""
Operator which transforms the vector `e_xy` containing the vectorized E_x and E_y fields,
into a vectorized H containing all three H components
Args:
wavenumber: Wavenumber assuming fields have z-dependence of `exp(-i * wavenumber * z)`.
It should satisfy `operator_e() @ e_xy == wavenumber**2 * e_xy`
omega: The angular frequency of the system
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types` (2D)
epsilon: Vectorized dielectric constant grid
mu: Vectorized magnetic permeability grid (default 1 everywhere)
Returns:
Sparse matrix representing the operator.
"""
e2hop = e2h(wavenumber=wavenumber, omega=omega, dxes=dxes, mu=mu)
return e2hop @ exy2e(wavenumber=wavenumber, dxes=dxes, epsilon=epsilon)
def hxy2e(wavenumber: complex,
omega: complex,
dxes: dx_lists_t,
epsilon: vfdfield_t,
mu: vfdfield_t = None
) -> sparse.spmatrix:
"""
Operator which transforms the vector `h_xy` containing the vectorized H_x and H_y fields,
into a vectorized E containing all three E components
Args:
wavenumber: Wavenumber assuming fields have z-dependence of `exp(-i * wavenumber * z)`.
It should satisfy `operator_h() @ h_xy == wavenumber**2 * h_xy`
omega: The angular frequency of the system
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types` (2D)
epsilon: Vectorized dielectric constant grid
mu: Vectorized magnetic permeability grid (default 1 everywhere)
Returns:
Sparse matrix representing the operator.
"""
h2eop = h2e(wavenumber=wavenumber, omega=omega, dxes=dxes, epsilon=epsilon)
return h2eop @ hxy2h(wavenumber=wavenumber, dxes=dxes, mu=mu)
def hxy2h(wavenumber: complex,
dxes: dx_lists_t,
mu: vfdfield_t = None
) -> sparse.spmatrix:
"""
Operator which transforms the vector `h_xy` containing the vectorized H_x and H_y fields,
into a vectorized H containing all three H components
Args:
wavenumber: Wavenumber assuming fields have z-dependence of `exp(-i * wavenumber * z)`.
It should satisfy `operator_h() @ h_xy == wavenumber**2 * h_xy`
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types` (2D)
mu: Vectorized magnetic permeability grid (default 1 everywhere)
Returns:
Sparse matrix representing the operator.
"""
Dfx, Dfy = deriv_forward(dxes[0])
hxy2hz = sparse.hstack((Dfx, Dfy)) / (1j * wavenumber)
if not numpy.any(numpy.equal(mu, None)):
mu_parts = numpy.split(mu, 3)
mu_xy = sparse.diags(numpy.hstack((mu_parts[0], mu_parts[1])))
mu_z_inv = sparse.diags(1 / mu_parts[2])
hxy2hz = mu_z_inv @ hxy2hz @ mu_xy
n_pts = dxes[1][0].size * dxes[1][1].size
op = sparse.vstack((sparse.eye(2 * n_pts),
hxy2hz))
return op
def exy2e(wavenumber: complex,
dxes: dx_lists_t,
epsilon: vfdfield_t,
) -> sparse.spmatrix:
"""
Operator which transforms the vector `e_xy` containing the vectorized E_x and E_y fields,
into a vectorized E containing all three E components
Args:
wavenumber: Wavenumber assuming fields have z-dependence of `exp(-i * wavenumber * z)`
It should satisfy `operator_e() @ e_xy == wavenumber**2 * e_xy`
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types` (2D)
epsilon: Vectorized dielectric constant grid
Returns:
Sparse matrix representing the operator.
"""
Dbx, Dby = deriv_back(dxes[1])
exy2ez = sparse.hstack((Dbx, Dby)) / (1j * wavenumber)
if not numpy.any(numpy.equal(epsilon, None)):
epsilon_parts = numpy.split(epsilon, 3)
epsilon_xy = sparse.diags(numpy.hstack((epsilon_parts[0], epsilon_parts[1])))
epsilon_z_inv = sparse.diags(1 / epsilon_parts[2])
exy2ez = epsilon_z_inv @ exy2ez @ epsilon_xy
n_pts = dxes[0][0].size * dxes[0][1].size
op = sparse.vstack((sparse.eye(2 * n_pts),
exy2ez))
return op
def e2h(wavenumber: complex,
omega: complex,
dxes: dx_lists_t,
mu: vfdfield_t = None
) -> sparse.spmatrix:
"""
Returns an operator which, when applied to a vectorized E eigenfield, produces
the vectorized H eigenfield.
Args:
wavenumber: Wavenumber assuming fields have z-dependence of `exp(-i * wavenumber * z)`
omega: The angular frequency of the system
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types` (2D)
mu: Vectorized magnetic permeability grid (default 1 everywhere)
Returns:
Sparse matrix representation of the operator.
"""
op = curl_e(wavenumber, dxes) / (-1j * omega)
if not numpy.any(numpy.equal(mu, None)):
op = sparse.diags(1 / mu) @ op
return op
def h2e(wavenumber: complex,
omega: complex,
dxes: dx_lists_t,
epsilon: vfdfield_t
) -> sparse.spmatrix:
"""
Returns an operator which, when applied to a vectorized H eigenfield, produces
the vectorized E eigenfield.
Args:
wavenumber: Wavenumber assuming fields have z-dependence of `exp(-i * wavenumber * z)`
omega: The angular frequency of the system
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types` (2D)
epsilon: Vectorized dielectric constant grid
Returns:
Sparse matrix representation of the operator.
"""
op = sparse.diags(1 / (1j * omega * epsilon)) @ curl_h(wavenumber, dxes)
return op
def curl_e(wavenumber: complex, dxes: dx_lists_t) -> sparse.spmatrix:
"""
Discretized curl operator for use with the waveguide E field.
Args:
wavenumber: Wavenumber assuming fields have z-dependence of `exp(-i * wavenumber * z)`
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types` (2D)
Return:
Sparse matrix representation of the operator.
"""
n = 1
for d in dxes[0]:
n *= len(d)
Bz = -1j * wavenumber * sparse.eye(n)
Dfx, Dfy = deriv_forward(dxes[0])
return cross([Dfx, Dfy, Bz])
def curl_h(wavenumber: complex, dxes: dx_lists_t) -> sparse.spmatrix:
"""
Discretized curl operator for use with the waveguide H field.
Args:
wavenumber: Wavenumber assuming fields have z-dependence of `exp(-i * wavenumber * z)`
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types` (2D)
Return:
Sparse matrix representation of the operator.
"""
n = 1
for d in dxes[1]:
n *= len(d)
Bz = -1j * wavenumber * sparse.eye(n)
Dbx, Dby = deriv_back(dxes[1])
return cross([Dbx, Dby, Bz])
def h_err(h: vfdfield_t,
wavenumber: complex,
omega: complex,
dxes: dx_lists_t,
epsilon: vfdfield_t,
mu: vfdfield_t = None
) -> float:
"""
Calculates the relative error in the H field
Args:
h: Vectorized H field
wavenumber: Wavenumber assuming fields have z-dependence of `exp(-i * wavenumber * z)`
omega: The angular frequency of the system
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types` (2D)
epsilon: Vectorized dielectric constant grid
mu: Vectorized magnetic permeability grid (default 1 everywhere)
Returns:
Relative error `norm(A_h @ h) / norm(h)`.
"""
ce = curl_e(wavenumber, dxes)
ch = curl_h(wavenumber, dxes)
eps_inv = sparse.diags(1 / epsilon)
if numpy.any(numpy.equal(mu, None)):
op = ce @ eps_inv @ ch @ h - omega ** 2 * h
else:
op = ce @ eps_inv @ ch @ h - omega ** 2 * (mu * h)
return norm(op) / norm(h)
def e_err(e: vfdfield_t,
wavenumber: complex,
omega: complex,
dxes: dx_lists_t,
epsilon: vfdfield_t,
mu: vfdfield_t = None
) -> float:
"""
Calculates the relative error in the E field
Args:
e: Vectorized E field
wavenumber: Wavenumber assuming fields have z-dependence of `exp(-i * wavenumber * z)`
omega: The angular frequency of the system
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types` (2D)
epsilon: Vectorized dielectric constant grid
mu: Vectorized magnetic permeability grid (default 1 everywhere)
Returns:
Relative error `norm(A_e @ e) / norm(e)`.
"""
ce = curl_e(wavenumber, dxes)
ch = curl_h(wavenumber, dxes)
if numpy.any(numpy.equal(mu, None)):
op = ch @ ce @ e - omega ** 2 * (epsilon * e)
else:
mu_inv = sparse.diags(1 / mu)
op = ch @ mu_inv @ ce @ e - omega ** 2 * (epsilon * e)
return norm(op) / norm(e)
def solve_modes(mode_numbers: List[int],
omega: complex,
dxes: dx_lists_t,
epsilon: vfdfield_t,
mu: vfdfield_t = None,
mode_margin: int = 2,
) -> Tuple[List[vfdfield_t], List[complex]]:
"""
Given a 2D region, attempts to solve for the eigenmode with the specified mode numbers.
Args:
mode_numbers: List of 0-indexed mode numbers to solve for
omega: Angular frequency of the simulation
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types`
epsilon: Dielectric constant
mu: Magnetic permeability (default 1 everywhere)
mode_margin: The eigensolver will actually solve for `(max(mode_number) + mode_margin)`
modes, but only return the target mode. Increasing this value can improve the solver's
ability to find the correct mode. Default 2.
Returns:
(e_xys, wavenumbers)
"""
'''
Solve for the largest-magnitude eigenvalue of the real operator
'''
dxes_real = [[numpy.real(dx) for dx in dxi] for dxi in dxes]
A_r = operator_e(numpy.real(omega), dxes_real, numpy.real(epsilon), numpy.real(mu))
eigvals, eigvecs = signed_eigensolve(A_r, max(mode_numbers) + mode_margin)
e_xys = eigvecs[:, -(numpy.array(mode_numbers) + 1)]
'''
Now solve for the eigenvector of the full operator, using the real operator's
eigenvector as an initial guess for Rayleigh quotient iteration.
'''
A = operator_e(omega, dxes, epsilon, mu)
for nn in range(len(mode_numbers)):
eigvals[nn], e_xys[:, nn] = rayleigh_quotient_iteration(A, e_xys[:, nn])
# Calculate the wave-vector (force the real part to be positive)
wavenumbers = numpy.sqrt(eigvals)
wavenumbers *= numpy.sign(numpy.real(wavenumbers))
return e_xys, wavenumbers
def solve_mode(mode_number: int,
*args,
**kwargs
) -> Tuple[vfdfield_t, complex]:
"""
Wrapper around `solve_modes()` that solves for a single mode.
Args:
mode_number: 0-indexed mode number to solve for
*args: passed to `solve_modes()`
**kwargs: passed to `solve_modes()`
Returns:
(e_xy, wavenumber)
"""
e_xys, wavenumbers = solve_modes(mode_numbers=[mode_number], *args, **kwargs)
return e_xys[:, 0], wavenumbers[0]

236
meanas/fdfd/waveguide_3d.py
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"""
Tools for working with waveguide modes in 3D domains.
This module relies heavily on `waveguide_2d` and mostly just transforms
its parameters into 2D equivalents and expands the results back into 3D.
"""
from typing import Dict, List, Tuple
import numpy
import scipy.sparse as sparse
from ..fdmath import vec, unvec, dx_lists_t, vfdfield_t, fdfield_t
from . import operators, waveguide_2d, functional
def solve_mode(mode_number: int,
omega: complex,
dxes: dx_lists_t,
axis: int,
polarity: int,
slices: List[slice],
epsilon: fdfield_t,
mu: fdfield_t = None,
) -> Dict[str, complex or numpy.ndarray]:
"""
Given a 3D grid, selects a slice from the grid and attempts to
solve for an eigenmode propagating through that slice.
Args:
mode_number: Number of the mode, 0-indexed
omega: Angular frequency of the simulation
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types`
axis: Propagation axis (0=x, 1=y, 2=z)
polarity: Propagation direction (+1 for +ve, -1 for -ve)
slices: `epsilon[tuple(slices)]` is used to select the portion of the grid to use
as the waveguide cross-section. `slices[axis]` should select only one item.
epsilon: Dielectric constant
mu: Magnetic permeability (default 1 everywhere)
Returns:
`{'E': List[numpy.ndarray], 'H': List[numpy.ndarray], 'wavenumber': complex}`
"""
if mu is None:
mu = numpy.ones_like(epsilon)
slices = tuple(slices)
'''
Solve the 2D problem in the specified plane
'''
# Define rotation to set z as propagation direction
order = numpy.roll(range(3), 2 - axis)
reverse_order = numpy.roll(range(3), axis - 2)
# Find dx in propagation direction
dxab_forward = numpy.array([dx[order[2]][slices[order[2]]] for dx in dxes])
dx_prop = 0.5 * sum(dxab_forward)[0]
# Reduce to 2D and solve the 2D problem
args_2d = {
'omega': omega,
'dxes': [[dx[i][slices[i]] for i in order[:2]] for dx in dxes],
'epsilon': vec([epsilon[i][slices].transpose(order) for i in order]),
'mu': vec([mu[i][slices].transpose(order) for i in order]),
}
e_xy, wavenumber_2d = waveguide_2d.solve_mode(mode_number, **args_2d)
'''
Apply corrections and expand to 3D
'''
# Correct wavenumber to account for numerical dispersion.
wavenumber = 2/dx_prop * numpy.arcsin(wavenumber_2d * dx_prop/2)
shape = [d.size for d in args_2d['dxes'][0]]
ve, vh = waveguide_2d.normalized_fields_e(e_xy, wavenumber=wavenumber_2d, **args_2d, prop_phase=dx_prop * wavenumber)
e = unvec(ve, shape)
h = unvec(vh, shape)
# Adjust for propagation direction
h *= polarity
# Apply phase shift to H-field
h[:2] *= numpy.exp(-1j * polarity * 0.5 * wavenumber * dx_prop)
e[2] *= numpy.exp(-1j * polarity * 0.5 * wavenumber * dx_prop)
# Expand E, H to full epsilon space we were given
E = numpy.zeros_like(epsilon, dtype=complex)
H = numpy.zeros_like(epsilon, dtype=complex)
for a, o in enumerate(reverse_order):
E[(a, *slices)] = e[o][:, :, None].transpose(reverse_order)
H[(a, *slices)] = h[o][:, :, None].transpose(reverse_order)
results = {
'wavenumber': wavenumber,
'wavenumber_2d': wavenumber_2d,
'H': H,
'E': E,
}
return results
def compute_source(E: fdfield_t,
wavenumber: complex,
omega: complex,
dxes: dx_lists_t,
axis: int,
polarity: int,
slices: List[slice],
epsilon: fdfield_t,
mu: fdfield_t = None,
) -> fdfield_t:
"""
Given an eigenmode obtained by `solve_mode`, returns the current source distribution
necessary to position a unidirectional source at the slice location.
Args:
E: E-field of the mode
wavenumber: Wavenumber of the mode
omega: Angular frequency of the simulation
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types`
axis: Propagation axis (0=x, 1=y, 2=z)
polarity: Propagation direction (+1 for +ve, -1 for -ve)
slices: `epsilon[tuple(slices)]` is used to select the portion of the grid to use
as the waveguide cross-section. `slices[axis]` should select only one item.
mu: Magnetic permeability (default 1 everywhere)
Returns:
J distribution for the unidirectional source
"""
E_expanded = expand_e(E=E, dxes=dxes, wavenumber=wavenumber, axis=axis,
polarity=polarity, slices=slices)
smask = [slice(None)] * 4
if polarity > 0:
smask[axis + 1] = slice(slices[axis].start, None)
else:
smask[axis + 1] = slice(None, slices[axis].stop)
mask = numpy.zeros_like(E_expanded, dtype=int)
mask[tuple(smask)] = 1
masked_e2j = operators.e_boundary_source(mask=vec(mask), omega=omega, dxes=dxes, epsilon=vec(epsilon), mu=vec(mu))
J = unvec(masked_e2j @ vec(E_expanded), E.shape[1:])
return J
def compute_overlap_e(E: fdfield_t,
wavenumber: complex,
dxes: dx_lists_t,
axis: int,
polarity: int,
slices: List[slice],
) -> fdfield_t: # TODO DOCS
"""
Given an eigenmode obtained by `solve_mode`, calculates an overlap_e for the
mode orthogonality relation Integrate(((E x H_mode) + (E_mode x H)) dot dn)
[assumes reflection symmetry].
Args:
E: E-field of the mode
H: H-field of the mode (advanced by half of a Yee cell from E)
wavenumber: Wavenumber of the mode
omega: Angular frequency of the simulation
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types`
axis: Propagation axis (0=x, 1=y, 2=z)
polarity: Propagation direction (+1 for +ve, -1 for -ve)
slices: `epsilon[tuple(slices)]` is used to select the portion of the grid to use
as the waveguide cross-section. slices[axis] should select only one item.
mu: Magnetic permeability (default 1 everywhere)
Returns:
overlap_e such that `numpy.sum(overlap_e * other_e)` computes the overlap integral
"""
slices = tuple(slices)
Ee = expand_e(E=E, wavenumber=wavenumber, dxes=dxes,
axis=axis, polarity=polarity, slices=slices)
start, stop = sorted((slices[axis].start, slices[axis].start - 2 * polarity))
slices2 = list(slices)
slices2[axis] = slice(start, stop)
slices2 = (slice(None), *slices2)
Etgt = numpy.zeros_like(Ee)
Etgt[slices2] = Ee[slices2]
Etgt /= (Etgt.conj() * Etgt).sum()
return Etgt
def expand_e(E: fdfield_t,
wavenumber: complex,
dxes: dx_lists_t,
axis: int,
polarity: int,
slices: List[slice],
) -> fdfield_t:
"""
Given an eigenmode obtained by `solve_mode`, expands the E-field from the 2D
slice where the mode was calculated to the entire domain (along the propagation
axis). This assumes the epsilon cross-section remains constant throughout the
entire domain; it is up to the caller to truncate the expansion to any regions
where it is valid.
Args:
E: E-field of the mode
wavenumber: Wavenumber of the mode
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types`
axis: Propagation axis (0=x, 1=y, 2=z)
polarity: Propagation direction (+1 for +ve, -1 for -ve)
slices: `epsilon[tuple(slices)]` is used to select the portion of the grid to use
as the waveguide cross-section. slices[axis] should select only one item.
Returns:
`E`, with the original field expanded along the specified `axis`.
"""
slices = tuple(slices)
# Determine phase factors for parallel slices
a_shape = numpy.roll([1, -1, 1, 1], axis)
a_E = numpy.real(dxes[0][axis]).cumsum()
r_E = a_E - a_E[slices[axis]]
iphi = polarity * -1j * wavenumber
phase_E = numpy.exp(iphi * r_E).reshape(a_shape)
# Expand our slice to the entire grid using the phase factors
E_expanded = numpy.zeros_like(E)
slices_exp = list(slices)
slices_exp[axis] = slice(E.shape[axis + 1])
slices_exp = (slice(None), *slices_exp)
slices_in = (slice(None), *slices)
E_expanded[slices_exp] = phase_E * numpy.array(E)[slices_in]
return E_expanded

138
meanas/fdfd/waveguide_cyl.py
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"""
Operators and helper functions for cylindrical waveguides with unchanging cross-section.
WORK IN PROGRESS, CURRENTLY BROKEN
As the z-dependence is known, all the functions in this file assume a 2D grid
(i.e. `dxes = [[[dr_e_0, dx_e_1, ...], [dy_e_0, ...]], [[dr_h_0, ...], [dy_h_0, ...]]]`).
"""
# TODO update module docs
from typing import List, Tuple, Dict
import numpy
from numpy.linalg import norm
import scipy.sparse as sparse
from ..fdmath import vec, unvec, dx_lists_t, fdfield_t, vfdfield_t
from ..eigensolvers import signed_eigensolve, rayleigh_quotient_iteration
from . import operators
__author__ = 'Jan Petykiewicz'
def cylindrical_operator(omega: complex,
dxes: dx_lists_t,
epsilon: vfdfield_t,
r0: float,
) -> sparse.spmatrix:
"""
Cylindrical coordinate waveguide operator of the form
TODO
for use with a field vector of the form `[E_r, E_y]`.
This operator can be used to form an eigenvalue problem of the form
A @ [E_r, E_y] = wavenumber**2 * [E_r, E_y]
which can then be solved for the eigenmodes of the system
(an `exp(-i * wavenumber * theta)` theta-dependence is assumed for the fields).
Args:
omega: The angular frequency of the system
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types` (2D)
epsilon: Vectorized dielectric constant grid
r0: Radius of curvature for the simulation. This should be the minimum value of
r within the simulation domain.
Returns:
Sparse matrix representation of the operator
"""
Dfx, Dfy = operators.deriv_forward(dxes[0])
Dbx, Dby = operators.deriv_back(dxes[1])
rx = r0 + numpy.cumsum(dxes[0][0])
ry = r0 + dxes[0][0]/2.0 + numpy.cumsum(dxes[1][0])
tx = rx/r0
ty = ry/r0
Tx = sparse.diags(vec(tx[:, None].repeat(dxes[0][1].size, axis=1)))
Ty = sparse.diags(vec(ty[:, None].repeat(dxes[1][1].size, axis=1)))
eps_parts = numpy.split(epsilon, 3)
eps_x = sparse.diags(eps_parts[0])
eps_y = sparse.diags(eps_parts[1])
eps_z_inv = sparse.diags(1 / eps_parts[2])
pa = sparse.vstack((Dfx, Dfy)) @ Tx @ eps_z_inv @ sparse.hstack((Dbx, Dby))
pb = sparse.vstack((Dfx, Dfy)) @ Tx @ eps_z_inv @ sparse.hstack((Dby, Dbx))
a0 = Ty @ eps_x + omega**-2 * Dby @ Ty @ Dfy
a1 = Tx @ eps_y + omega**-2 * Dbx @ Ty @ Dfx
b0 = Dbx @ Ty @ Dfy
b1 = Dby @ Ty @ Dfx
diag = sparse.block_diag
op = (omega**2 * diag((Tx, Ty)) + pa) @ diag((a0, a1)) + \
- (sparse.bmat(((None, Ty), (Tx, None))) + omega**-2 * pb) @ diag((b0, b1))
return op
def solve_mode(mode_number: int,
omega: complex,
dxes: dx_lists_t,
epsilon: vfdfield_t,
r0: float,
) -> Dict[str, complex or fdfield_t]:
"""
TODO: fixup
Given a 2d (r, y) slice of epsilon, attempts to solve for the eigenmode
of the bent waveguide with the specified mode number.
Args:
mode_number: Number of the mode, 0-indexed
omega: Angular frequency of the simulation
dxes: Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types.
The first coordinate is assumed to be r, the second is y.
epsilon: Dielectric constant
r0: Radius of curvature for the simulation. This should be the minimum value of
r within the simulation domain.
Returns:
`{'E': List[numpy.ndarray], 'H': List[numpy.ndarray], 'wavenumber': complex}`
"""
'''
Solve for the largest-magnitude eigenvalue of the real operator
'''
dxes_real = [[numpy.real(dx) for dx in dxi] for dxi in dxes]
A_r = waveguide.cylindrical_operator(numpy.real(omega), dxes_real, numpy.real(epsilon), r0)
eigvals, eigvecs = signed_eigensolve(A_r, mode_number + 3)
e_xy = eigvecs[:, -(mode_number+1)]
'''
Now solve for the eigenvector of the full operator, using the real operator's
eigenvector as an initial guess for Rayleigh quotient iteration.
'''
A = waveguide.cylindrical_operator(omega, dxes, epsilon, r0)
eigval, e_xy = rayleigh_quotient_iteration(A, e_xy)
# Calculate the wave-vector (force the real part to be positive)
wavenumber = numpy.sqrt(eigval)
wavenumber *= numpy.sign(numpy.real(wavenumber))
# TODO: Perform correction on wavenumber to account for numerical dispersion.
shape = [d.size for d in dxes[0]]
e_xy = numpy.hstack((e_xy, numpy.zeros(shape[0] * shape[1])))
fields = {
'wavenumber': wavenumber,
'E': unvec(e_xy, shape),
# 'E': unvec(e, shape),
# 'H': unvec(h, shape),
}
return fields

747
meanas/fdmath/__init__.py
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"""
Basic discrete calculus for finite difference (fd) simulations.
Fields, Functions, and Operators
================================
Discrete fields are stored in one of two forms:
- The `fdfield_t` form is a multidimensional `numpy.ndarray`
+ For a scalar field, this is just `U[m, n, p]`, where `m`, `n`, and `p` are
discrete indices referring to positions on the x, y, and z axes respectively.
+ For a vector field, the first index specifies which vector component is accessed:
`E[:, m, n, p] = [Ex[m, n, p], Ey[m, n, p], Ez[m, n, p]]`.
- The `vfdfield_t` form is simply a vectorzied (i.e. 1D) version of the `field_t`,
as obtained by `meanas.fdmath.vectorization.vec` (effectively just `numpy.ravel`)
Operators which act on fields also come in two forms:
+ Python functions, created by the functions in `meanas.fdmath.functional`.
The generated functions act on fields in the `fdfield_t` form.
+ Linear operators, usually 2D sparse matrices using `scipy.sparse`, created
by `meanas.fdmath.operators`. These operators act on vectorized fields in the
`vfdfield_t` form.
The operations performed should be equivalent: `functional.op(*args)(E)` should be
equivalent to `unvec(operators.op(*args) @ vec(E), E.shape[1:])`.
Generally speaking the `field_t` form is easier to work with, but can be harder or less
efficient to compose (e.g. it is easy to generate a single matrix by multiplying a
series of other matrices).
Discrete calculus
=================
This documentation and approach is roughly based on W.C. Chew's excellent
"Electromagnetic Theory on a Lattice" (doi:10.1063/1.355770),
which covers a superset of this material with similar notation and more detail.
Scalar derivatives and cell shifts
----------------------------------
Define the discrete forward derivative as
$$ [\\tilde{\\partial}_x f]_{m + \\frac{1}{2}} = \\frac{1}{\\Delta_{x, m}} (f_{m + 1} - f_m) $$
where $f$ is a function defined at discrete locations on the x-axis (labeled using $m$).
The value at $m$ occupies a length $\\Delta_{x, m}$ along the x-axis. Note that $m$
is an index along the x-axis, _not_ necessarily an x-coordinate, since each length
$\\Delta_{x, m}, \\Delta_{x, m+1}, ...$ is independently chosen.
If we treat `f` as a 1D array of values, with the `i`-th value `f[i]` taking up a length `dx[i]`
along the x-axis, the forward derivative is
deriv_forward(f)[i] = (f[i + 1] - f[i]) / dx[i]
Likewise, discrete reverse derivative is
$$ [\\hat{\\partial}_x f ]_{m - \\frac{1}{2}} = \\frac{1}{\\Delta_{x, m}} (f_{m} - f_{m - 1}) $$
or
deriv_back(f)[i] = (f[i] - f[i - 1]) / dx[i]
The derivatives' values are shifted by a half-cell relative to the original function, and
will have different cell widths if all the `dx[i]` ( $\\Delta_{x, m}$ ) are not
identical:
[figure: derivatives and cell sizes]
dx0 dx1 dx2 dx3 cell sizes for function
----- ----- ----------- -----
______________________________
| | | |
f0 | f1 | f2 | f3 | function
_____|_____|___________|_____|
| | | |
| Df0 | Df1 | Df2 | Df3 forward derivative (periodic boundary)
__|_____|________|________|___
dx'3] dx'0 dx'1 dx'2 [dx'3 cell sizes for forward derivative
-- ----- -------- -------- ---
dx'0] dx'1 dx'2 dx'3 [dx'0 cell sizes for reverse derivative
______________________________
| | | |
| df1 | df2 | df3 | df0 reverse derivative (periodic boundary)
__|_____|________|________|___
Periodic boundaries are used here and elsewhere unless otherwise noted.
In the above figure,
`f0 =` $f_0$, `f1 =` $f_1$
`Df0 =` $[\\tilde{\\partial}f]_{0 + \\frac{1}{2}}$
`Df1 =` $[\\tilde{\\partial}f]_{1 + \\frac{1}{2}}$
`df0 =` $[\\hat{\\partial}f]_{0 - \\frac{1}{2}}$
etc.
The fractional subscript $m + \\frac{1}{2}$ is used to indicate values defined
at shifted locations relative to the original $m$, with corresponding lengths
$$ \\Delta_{x, m + \\frac{1}{2}} = \\frac{1}{2} * (\\Delta_{x, m} + \\Delta_{x, m + 1}) $$
Just as $m$ is not itself an x-coordinate, neither is $m + \\frac{1}{2}$;
carefully note the positions of the various cells in the above figure vs their labels.
If the positions labeled with $m$ are considered the "base" or "original" grid,
the positions labeled with $m + \\frac{1}{2}$ are said to lie on a "dual" or
"derived" grid.
For the remainder of the `Discrete calculus` section, all figures will show
constant-length cells in order to focus on the vector derivatives themselves.
See the `Grid description` section below for additional information on this topic
and generalization to three dimensions.
Gradients and fore-vectors
--------------------------
Expanding to three dimensions, we can define two gradients
$$ [\\tilde{\\nabla} f]_{m,n,p} = \\vec{x} [\\tilde{\\partial}_x f]_{m + \\frac{1}{2},n,p} +
\\vec{y} [\\tilde{\\partial}_y f]_{m,n + \\frac{1}{2},p} +
\\vec{z} [\\tilde{\\partial}_z f]_{m,n,p + \\frac{1}{2}} $$
$$ [\\hat{\\nabla} f]_{m,n,p} = \\vec{x} [\\hat{\\partial}_x f]_{m + \\frac{1}{2},n,p} +
\\vec{y} [\\hat{\\partial}_y f]_{m,n + \\frac{1}{2},p} +
\\vec{z} [\\hat{\\partial}_z f]_{m,n,p + \\frac{1}{2}} $$
or
[code: gradients]
grad_forward(f)[i,j,k] = [Dx_forward(f)[i, j, k],
Dy_forward(f)[i, j, k],
Dz_forward(f)[i, j, k]]
= [(f[i + 1, j, k] - f[i, j, k]) / dx[i],
(f[i, j + 1, k] - f[i, j, k]) / dy[i],
(f[i, j, k + 1] - f[i, j, k]) / dz[i]]
grad_back(f)[i,j,k] = [Dx_back(f)[i, j, k],
Dy_back(f)[i, j, k],
Dz_back(f)[i, j, k]]
= [(f[i, j, k] - f[i - 1, j, k]) / dx[i],
(f[i, j, k] - f[i, j - 1, k]) / dy[i],
(f[i, j, k] - f[i, j, k - 1]) / dz[i]]
The three derivatives in the gradient cause shifts in different
directions, so the x/y/z components of the resulting "vector" are defined
at different points: the x-component is shifted in the x-direction,
y in y, and z in z.
We call the resulting object a "fore-vector" or "back-vector", depending
on the direction of the shift. We write it as
$$ \\tilde{g}_{m,n,p} = \\vec{x} g^x_{m + \\frac{1}{2},n,p} +
\\vec{y} g^y_{m,n + \\frac{1}{2},p} +
\\vec{z} g^z_{m,n,p + \\frac{1}{2}} $$
$$ \\hat{g}_{m,n,p} = \\vec{x} g^x_{m - \\frac{1}{2},n,p} +
\\vec{y} g^y_{m,n - \\frac{1}{2},p} +
\\vec{z} g^z_{m,n,p - \\frac{1}{2}} $$
[figure: gradient / fore-vector]
(m, n+1, p+1) ______________ (m+1, n+1, p+1)
/: /|
/ : / |
/ : / |
(m, n, p+1)/_____________/ | The forward derivatives are defined
| : | | at the Dx, Dy, Dz points,
| :.........|...| but the forward-gradient fore-vector
z y Dz / | / is the set of all three
|/_x | Dy | / and is said to be "located" at (m,n,p)
|/ |/
(m, n, p)|_____Dx______| (m+1, n, p)
Divergences
-----------
There are also two divergences,
$$ d_{n,m,p} = [\\tilde{\\nabla} \\cdot \\hat{g}]_{n,m,p}
= [\\tilde{\\partial}_x g^x]_{m,n,p} +
[\\tilde{\\partial}_y g^y]_{m,n,p} +
[\\tilde{\\partial}_z g^z]_{m,n,p} $$
$$ d_{n,m,p} = [\\hat{\\nabla} \\cdot \\tilde{g}]_{n,m,p}
= [\\hat{\\partial}_x g^x]_{m,n,p} +
[\\hat{\\partial}_y g^y]_{m,n,p} +
[\\hat{\\partial}_z g^z]_{m,n,p} $$
or
[code: divergences]
div_forward(g)[i,j,k] = Dx_forward(gx)[i, j, k] +
Dy_forward(gy)[i, j, k] +
Dz_forward(gz)[i, j, k]
= (gx[i + 1, j, k] - gx[i, j, k]) / dx[i] +
(gy[i, j + 1, k] - gy[i, j, k]) / dy[i] +
(gz[i, j, k + 1] - gz[i, j, k]) / dz[i]
div_back(g)[i,j,k] = Dx_back(gx)[i, j, k] +
Dy_back(gy)[i, j, k] +
Dz_back(gz)[i, j, k]
= (gx[i, j, k] - gx[i - 1, j, k]) / dx[i] +
(gy[i, j, k] - gy[i, j - 1, k]) / dy[i] +
(gz[i, j, k] - gz[i, j, k - 1]) / dz[i]
where `g = [gx, gy, gz]` is a fore- or back-vector field.
Since we applied the forward divergence to the back-vector (and vice-versa), the resulting scalar value
is defined at the back-vector's (fore-vector's) location $(m,n,p)$ and not at the locations of its components
$(m \\pm \\frac{1}{2},n,p)$ etc.
[figure: divergence]
^^
(m-1/2, n+1/2, p+1/2) _____||_______ (m+1/2, n+1/2, p+1/2)
/: || ,, /|
/ : || // / | The divergence at (m, n, p) (the center
/ : // / | of this cube) of a fore-vector field
(m-1/2, n-1/2, p+1/2)/_____________/ | is the sum of the outward-pointing
| : | | fore-vector components, which are
z y <==|== :.........|.====> located at the face centers.
|/_x | / | /
| / // | / Note that in a nonuniform grid, each
|/ // || |/ dimension is normalized by the cell width.
(m-1/2, n-1/2, p-1/2)|____//_______| (m+1/2, n-1/2, p-1/2)
'' ||
VV
Curls
-----
The two curls are then
$$ \\begin{aligned}
\\hat{h}_{m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2}} &= \\\\
[\\tilde{\\nabla} \\times \\tilde{g}]_{m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2}} &=
\\vec{x} (\\tilde{\\partial}_y g^z_{m,n,p + \\frac{1}{2}} - \\tilde{\\partial}_z g^y_{m,n + \\frac{1}{2},p}) \\\\
&+ \\vec{y} (\\tilde{\\partial}_z g^x_{m + \\frac{1}{2},n,p} - \\tilde{\\partial}_x g^z_{m,n,p + \\frac{1}{2}}) \\\\
&+ \\vec{z} (\\tilde{\\partial}_x g^y_{m,n + \\frac{1}{2},p} - \\tilde{\\partial}_y g^z_{m + \\frac{1}{2},n,p})
\\end{aligned} $$
and
$$ \\tilde{h}_{m - \\frac{1}{2}, n - \\frac{1}{2}, p - \\frac{1}{2}} =
[\\hat{\\nabla} \\times \\hat{g}]_{m - \\frac{1}{2}, n - \\frac{1}{2}, p - \\frac{1}{2}} $$
where $\\hat{g}$ and $\\tilde{g}$ are located at $(m,n,p)$
with components at $(m \\pm \\frac{1}{2},n,p)$ etc.,
while $\\hat{h}$ and $\\tilde{h}$ are located at $(m \\pm \\frac{1}{2}, n \\pm \\frac{1}{2}, p \\pm \\frac{1}{2})$
with components at $(m, n \\pm \\frac{1}{2}, p \\pm \\frac{1}{2})$ etc.
[code: curls]
curl_forward(g)[i,j,k] = [Dy_forward(gz)[i, j, k] - Dz_forward(gy)[i, j, k],
Dz_forward(gx)[i, j, k] - Dx_forward(gz)[i, j, k],
Dx_forward(gy)[i, j, k] - Dy_forward(gx)[i, j, k]]
curl_back(g)[i,j,k] = [Dy_back(gz)[i, j, k] - Dz_back(gy)[i, j, k],
Dz_back(gx)[i, j, k] - Dx_back(gz)[i, j, k],
Dx_back(gy)[i, j, k] - Dy_back(gx)[i, j, k]]
For example, consider the forward curl, at (m, n, p), of a back-vector field `g`, defined
on a grid containing (m + 1/2, n + 1/2, p + 1/2).
The curl will be a fore-vector, so its z-component will be defined at (m, n, p + 1/2).
Take the nearest x- and y-components of `g` in the xy plane where the curl's z-component
is located; these are
[curl components]
(m, n + 1/2, p + 1/2) : x-component of back-vector at (m + 1/2, n + 1/2, p + 1/2)
(m + 1, n + 1/2, p + 1/2) : x-component of back-vector at (m + 3/2, n + 1/2, p + 1/2)
(m + 1/2, n , p + 1/2) : y-component of back-vector at (m + 1/2, n + 1/2, p + 1/2)
(m + 1/2, n + 1 , p + 1/2) : y-component of back-vector at (m + 1/2, n + 3/2, p + 1/2)
These four xy-components can be used to form a loop around the curl's z-component; its magnitude and sign
is set by their loop-oriented sum (i.e. two have their signs flipped to complete the loop).
[figure: z-component of curl]
: |
z y : ^^ |
|/_x :....||.<.....| (m+1, n+1, p+1/2)
/ || /
| v || | ^
|/ |/
(m, n, p+1/2) |_____>______| (m+1, n, p+1/2)
Maxwell's Equations
===================
If we discretize both space (m,n,p) and time (l), Maxwell's equations become
$$ \\begin{aligned}
\\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}} &= -\\tilde{\\partial}_t \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}}
- \\hat{M}_{l, \\vec{r} + \\frac{1}{2}} \\\\
\\hat{\\nabla} \\times \\hat{H}_{l-\\frac{1}{2},\\vec{r} + \\frac{1}{2}} &= \\hat{\\partial}_t \\tilde{D}_{l, \\vec{r}}
+ \\tilde{J}_{l-\\frac{1}{2},\\vec{r}} \\\\
\\tilde{\\nabla} \\cdot \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}} &= 0 \\\\
\\hat{\\nabla} \\cdot \\tilde{D}_{l,\\vec{r}} &= \\rho_{l,\\vec{r}}
\\end{aligned} $$
with
$$ \\begin{aligned}
\\hat{B}_{\\vec{r}} &= \\mu_{\\vec{r} + \\frac{1}{2}} \\cdot \\hat{H}_{\\vec{r} + \\frac{1}{2}} \\\\
\\tilde{D}_{\\vec{r}} &= \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{\\vec{r}}
\\end{aligned} $$
where the spatial subscripts are abbreviated as $\\vec{r} = (m, n, p)$ and
$\\vec{r} + \\frac{1}{2} = (m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2})$,
$\\tilde{E}$ and $\\hat{H}$ are the electric and magnetic fields,
$\\tilde{J}$ and $\\hat{M}$ are the electric and magnetic current distributions,
and $\\epsilon$ and $\\mu$ are the dielectric permittivity and magnetic permeability.
The above is Yee's algorithm, written in a form analogous to Maxwell's equations.
The time derivatives can be expanded to form the update equations:
[code: Maxwell's equations updates]
H[i, j, k] -= dt * (curl_forward(E)[i, j, k] + M[t, i, j, k]) / mu[i, j, k]
E[i, j, k] += dt * (curl_back( H)[i, j, k] + J[t, i, j, k]) / epsilon[i, j, k]
Note that the E-field fore-vector and H-field back-vector are offset by a half-cell, resulting
in distinct locations for all six E- and H-field components:
[figure: Field components]
(m - 1/2,=> ____________Hx__________[H] <= r + 1/2 = (m + 1/2,
n + 1/2, /: /: /| n + 1/2,
z y p + 1/2) / : / : / | p + 1/2)
|/_x / : / : / |
/ : Ez__________Hy | Locations of the E- and
/ : : : /| | H-field components for the
(m - 1/2, / : : Ey...../.|..Hz [E] fore-vector at r = (m,n,p)
n - 1/2, =>/________________________/ | /| (the large cube's center)
p + 1/2) | : : / | | / | and [H] back-vector at r + 1/2
| : :/ | |/ | (the top right corner)
| : [E].......|.Ex |
| :.................|......| <= (m + 1/2, n + 1/2, p + 1/2)
| / | /
| / | /
| / | / This is the Yee discretization
| / | / scheme ("Yee cell").
r - 1/2 = | / | /
(m - 1/2, |/ |/
n - 1/2,=> |________________________| <= (m + 1/2, n - 1/2, p - 1/2)
p - 1/2)
Each component forms its own grid, offset from the others:
[figure: E-fields for adjacent cells]
H1__________Hx0_________H0
z y /: /|
|/_x / : / | This figure shows H back-vector locations
/ : / | H0, H1, etc. and their associated components
Hy1 : Hy0 | H0 = (Hx0, Hy0, Hz0) etc.
/ : / |
/ Hz1 / Hz0
H2___________Hx3_________H3 | The equivalent drawing for E would have
| : | | fore-vectors located at the cube's
| : | | center (and the centers of adjacent cubes),
| : | | with components on the cube's faces.
| H5..........Hx4...|......H4
| / | /
Hz2 / Hz2 /
| / | /
| Hy6 | Hy4
| / | /
|/ |/
H6__________Hx7__________H7
The divergence equations can be derived by taking the divergence of the curl equations
and combining them with charge continuity,
$$ \\hat{\\nabla} \\cdot \\tilde{J} + \\hat{\\partial}_t \\rho = 0 $$
implying that the discrete Maxwell's equations do not produce spurious charges.
Wave equation
-------------
Taking the backward curl of the $\\tilde{\\nabla} \\times \\tilde{E}$ equation and
replacing the resulting $\\hat{\\nabla} \\times \\hat{H}$ term using its respective equation,
and setting $\\hat{M}$ to zero, we can form the discrete wave equation:
$$
\\begin{aligned}
\\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}} &=
-\\tilde{\\partial}_t \\hat{B}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}}
- \\hat{M}_{l-1, \\vec{r} + \\frac{1}{2}} \\\\
\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}} &=
-\\tilde{\\partial}_t \\hat{H}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}} \\\\
\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) &=
\\hat{\\nabla} \\times (-\\tilde{\\partial}_t \\hat{H}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}}) \\\\
\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) &=
-\\tilde{\\partial}_t \\hat{\\nabla} \\times \\hat{H}_{l-\\frac{1}{2}, \\vec{r} + \\frac{1}{2}} \\\\
\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}}) &=
-\\tilde{\\partial}_t \\hat{\\partial}_t \\epsilon_{\\vec{r}} \\tilde{E}_{l, \\vec{r}} + \\hat{\\partial}_t \\tilde{J}_{l-\\frac{1}{2},\\vec{r}} \\\\
\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l,\\vec{r}})
+ \\tilde{\\partial}_t \\hat{\\partial}_t \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{l, \\vec{r}}
&= \\tilde{\\partial}_t \\tilde{J}_{l - \\frac{1}{2}, \\vec{r}}
\\end{aligned}
$$
Frequency domain
----------------
We can substitute in a time-harmonic fields
$$
\\begin{aligned}
\\tilde{E}_{l, \\vec{r}} &= \\tilde{E}_{\\vec{r}} e^{-\\imath \\omega l \\Delta_t} \\\\
\\tilde{J}_{l, \\vec{r}} &= \\tilde{J}_{\\vec{r}} e^{-\\imath \\omega (l - \\frac{1}{2}) \\Delta_t}
\\end{aligned}
$$
resulting in
$$
\\begin{aligned}
\\tilde{\\partial}_t &\\Rightarrow (e^{ \\imath \\omega \\Delta_t} - 1) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_t} \\sin(\\omega \\Delta_t / 2) e^{-\\imath \\omega \\Delta_t / 2} = -\\imath \\Omega e^{-\\imath \\omega \\Delta_t / 2}\\\\
\\hat{\\partial}_t &\\Rightarrow (1 - e^{-\\imath \\omega \\Delta_t}) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_t} \\sin(\\omega \\Delta_t / 2) e^{ \\imath \\omega \\Delta_t / 2} = -\\imath \\Omega e^{ \\imath \\omega \\Delta_t / 2}\\\\
\\Omega &= 2 \\sin(\\omega \\Delta_t / 2) / \\Delta_t
\\end{aligned}
$$
This gives the frequency-domain wave equation,
$$
\\hat{\\nabla} \\times (\\mu^{-1}_{\\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}})
-\\Omega^2 \\epsilon_{\\vec{r}} \\cdot \\tilde{E}_{\\vec{r}} = \\imath \\Omega \\tilde{J}_{\\vec{r}}
$$
Plane waves and Dispersion relation
------------------------------------
With uniform material distribution and no sources
$$
\\begin{aligned}
\\mu_{\\vec{r} + \\frac{1}{2}} &= \\mu \\\\
\\epsilon_{\\vec{r}} &= \\epsilon \\\\
\\tilde{J}_{\\vec{r}} &= 0 \\\\
\\end{aligned}
$$
the frequency domain wave equation simplifies to
$$ \\hat{\\nabla} \\times \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}} - \\Omega^2 \\epsilon \\mu \\tilde{E}_{\\vec{r}} = 0 $$
Since $\\hat{\\nabla} \\cdot \\tilde{E}_{\\vec{r}} = 0$, we can simplify
$$
\\begin{aligned}
\\hat{\\nabla} \\times \\tilde{\\nabla} \\times \\tilde{E}_{\\vec{r}}
&= \\tilde{\\nabla}(\\hat{\\nabla} \\cdot \\tilde{E}_{\\vec{r}}) - \\hat{\\nabla} \\cdot \\tilde{\\nabla} \\tilde{E}_{\\vec{r}} \\\\
&= - \\hat{\\nabla} \\cdot \\tilde{\\nabla} \\tilde{E}_{\\vec{r}} \\\\
&= - \\tilde{\\nabla}^2 \\tilde{E}_{\\vec{r}}
\\end{aligned}
$$
and we get
$$ \\tilde{\\nabla}^2 \\tilde{E}_{\\vec{r}} + \\Omega^2 \\epsilon \\mu \\tilde{E}_{\\vec{r}} = 0 $$
We can convert this to three scalar-wave equations of the form
$$ (\\tilde{\\nabla}^2 + K^2) \\phi_{\\vec{r}} = 0 $$
with $K^2 = \\Omega^2 \\mu \\epsilon$. Now we let
$$ \\phi_{\\vec{r}} = A e^{\\imath (k_x m \\Delta_x + k_y n \\Delta_y + k_z p \\Delta_z)} $$
resulting in
$$
\\begin{aligned}
\\tilde{\\partial}_x &\\Rightarrow (e^{ \\imath k_x \\Delta_x} - 1) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_x} \\sin(k_x \\Delta_x / 2) e^{ \\imath k_x \\Delta_x / 2} = \\imath K_x e^{ \\imath k_x \\Delta_x / 2}\\\\
\\hat{\\partial}_x &\\Rightarrow (1 - e^{-\\imath k_x \\Delta_x}) / \\Delta_t = \\frac{-2 \\imath}{\\Delta_x} \\sin(k_x \\Delta_x / 2) e^{-\\imath k_x \\Delta_x / 2} = \\imath K_x e^{-\\imath k_x \\Delta_x / 2}\\\\
K_x &= 2 \\sin(k_x \\Delta_x / 2) / \\Delta_x \\\\
\\end{aligned}
$$
with similar expressions for the y and z dimnsions (and $K_y, K_z$).
This implies
$$
\\tilde{\\nabla}^2 = -(K_x^2 + K_y^2 + K_z^2) \\phi_{\\vec{r}} \\\\
K_x^2 + K_y^2 + K_z^2 = \\Omega^2 \\mu \\epsilon = \\Omega^2 / c^2
$$
where $c = \\sqrt{\\mu \\epsilon}$.
Assuming real $(k_x, k_y, k_z), \\omega$ will be real only if
$$ c^2 \\Delta_t^2 = \\frac{\\Delta_t^2}{\\mu \\epsilon} < 1/(\\frac{1}{\\Delta_x^2} + \\frac{1}{\\Delta_y^2} + \\frac{1}{\\Delta_z^2}) $$
If $\\Delta_x = \\Delta_y = \\Delta_z$, this simplifies to $c \\Delta_t < \\Delta_x / \\sqrt{3}$.
This last form can be interpreted as enforcing causality; the distance that light
travels in one timestep (i.e., $c \\Delta_t$) must be less than the diagonal
of the smallest cell ( $\\Delta_x / \\sqrt{3}$ when on a uniform cubic grid).
Grid description
================
As described in the section on scalar discrete derivatives above, cell widths
(`dx[i]`, `dy[j]`, `dz[k]`) along each axis can be arbitrary and independently
defined. Moreover, all field components are actually defined at "derived" or "dual"
positions, in-between the "base" grid points on one or more axes.
To get a better sense of how this works, let's start by drawing a grid with uniform
`dy` and `dz` and nonuniform `dx`. We will only draw one cell in the y and z dimensions
to make the illustration simpler; we need at least two cells in the x dimension to
demonstrate how nonuniform `dx` affects the various components.
Place the E fore-vectors at integer indices $r = (m, n, p)$ and the H back-vectors
at fractional indices $r + \\frac{1}{2} = (m + \\frac{1}{2}, n + \\frac{1}{2},
p + \\frac{1}{2})$. Remember that these are indices and not coordinates; they can
correspond to arbitrary (monotonically increasing) coordinates depending on the cell widths.
Draw lines to denote the planes on which the H components and back-vectors are defined.
For simplicity, don't draw the equivalent planes for the E components and fore-vectors,
except as necessary to show their locations -- it's easiest to just connect them to their
associated H-equivalents.
The result looks something like this:
[figure: Component centers]
p=
[H]__________Hx___________[H]_____Hx______[H] __ +1/2
z y /: /: /: /: /| | |
|/_x / : / : / : / : / | | |
/ : / : / : / : / | | |
Hy : Ez...........Hy : Ez......Hy | | |
/: : : : /: : : : /| | | |
/ : Hz : Ey....../.:..Hz : Ey./.|..Hz __ 0 | dz[0]
/ : /: : / / : /: : / / | /| | |
/_________________________/_______________/ | / | | |
| :/ : :/ | :/ : :/ | |/ | | |
| Ex : [E].......|..Ex : [E]..|..Ex | | |
| : | : | | | |
| [H]..........Hx....|......[H].....H|x.....[H] __ --------- (n=+1/2, p=-1/2)
| / | / | / / /
Hz / Hz / Hz / / /
| / | / | / / /
| Hy | Hy | Hy __ 0 / dy[0]
| / | / | / / /
| / | / | / / /
|/ |/ |/ / /
[H]__________Hx___________[H]_____Hx______[H] __ -1/2 /
=n
|------------|------------|-------|-------|
-1/2 0 +1/2 +1 +3/2 = m
------------------------- ----------------
dx[0] dx[1]
Part of a nonuniform "base grid", with labels specifying
positions of the various field components. [E] fore-vectors
are at the cell centers, and [H] back-vectors are at the
vertices. H components along the near (-y) top (+z) edge
have been omitted to make the insides of the cubes easier
to visualize.
The above figure shows where all the components are located; however, it is also useful to show
what volumes those components correspond to. Consider the Ex component at `m = +1/2`: it is
shifted in the x-direction by a half-cell from the E fore-vector at `m = 0` (labeled `[E]`
in the figure). It corresponds to a volume between `m = 0` and `m = +1` (the other
dimensions are not shifted, i.e. they are still bounded by `n, p = +-1/2`). (See figure
below). Since `m` is an index and not an x-coordinate, the Ex component is not necessarily
at the center of the volume it represents, and the x-length of its volume is the derived
quantity `dx'[0] = (dx[0] + dx[1]) / 2` rather than the base `dx`.
(See also `Scalar derivatives and cell shifts`).
[figure: Ex volumes]
p=
<_________________________________________> __ +1/2
z y << /: / /: >> | |
|/_x < < / : / / : > > | |
< < / : / / : > > | |
< < / : / / : > > | |
<: < / : : / : >: > | |
< : < / : : / : > : > __ 0 | dz[0]
< : < / : : / :> : > | |
<____________/____________________/_______> : > | |
< : < | : : | > : > | |
< Ex < | : Ex | > Ex > | |
< : < | : : | > : > | |
< : <....|.......:........:...|.......>...:...> __ --------- (n=+1/2, p=-1/2)
< : < | / : /| /> : > / /
< : < | / : / | / > : > / /
< :< | / :/ | / > :> / /
< < | / : | / > > _ 0 / dy[0]
< < | / | / > > / /
< < | / | / > > / /
<< |/ |/ >> / /
<____________|____________________|_______> __ -1/2 /
=n
|------------|------------|-------|-------|
-1/2 0 +1/2 +1 +3/2 = m
~------------ -------------------- -------~
dx'[-1] dx'[0] dx'[1]
The Ex values are positioned on the x-faces of the base
grid. They represent the Ex field in volumes shifted by
a half-cell in the x-dimension, as shown here. Only the
center cell (with width dx'[0]) is fully shown; the
other two are truncated (shown using >< markers).
Note that the Ex positions are the in the same positions
as the previous figure; only the cell boundaries have moved.
Also note that the points at which Ex is defined are not
necessarily centered in the volumes they represent; non-
uniform cell sizes result in off-center volumes like the
center cell here.
The next figure shows the volumes corresponding to the Hy components, which
are shifted in two dimensions (x and z) compared to the base grid.
[figure: Hy volumes]
p=
z y mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm __ +1/2 s
|/_x << m: m: >> | |
< < m : m : > > | | dz'[1]
< < m : m : > > | |
Hy........... m........Hy...........m......Hy > | |
< < m : m : > > | |
< ______ m_____:_______________m_____:_>______ __ 0
< < m /: m / > > | |
mmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmmm > | |
< < | / : | / > > | | dz'[0]
< < | / : | / > > | |
< < | / : | / > > | |
< wwwww|w/wwwwwwwwwwwwwwwwwww|w/wwwww>wwwwwwww __ s
< < |/ w |/ w> > / /
_____________|_____________________|________ > / /
< < | w | w > > / /
< Hy........|...w........Hy.......|...w...>..Hy _ 0 / dy[0]
< < | w | w > > / /
<< | w | w > > / /
< |w |w >> / /
wwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwww __ -1/2 /
|------------|------------|--------|-------|
-1/2 0 +1/2 +1 +3/2 = m
~------------ --------------------- -------~
dx'[-1] dx'[0] dx'[1]
The Hy values are positioned on the y-edges of the base
grid. Again here, the 'Hy' labels represent the same points
as in the basic grid figure above; the edges have shifted
by a half-cell along the x- and z-axes.
The grid lines _|:/ are edges of the area represented by
each Hy value, and the lines drawn using <m>.w represent
edges where a cell's faces extend beyond the drawn area
(i.e. where the drawing is truncated in the x- or z-
directions).
Datastructure: dx_lists_t
-------------------
In this documentation, the E fore-vectors are placed on the base grid. An
equivalent formulation could place the H back-vectors on the base grid instead.
However, in the case of a non-uniform grid, the operation to get from the "base"
cell widths to "derived" ones is not its own inverse.
The base grid's cell sizes could be fully described by a list of three 1D arrays,
specifying the cell widths along all three axes:
[dx, dy, dz] = [[dx[0], dx[1], ...], [dy[0], ...], [dz[0], ...]]
Note that this is a list-of-arrays rather than a 2D array, as the simulation domain
may have a different number of cells along each axis.
Knowing the base grid's cell widths and the boundary conditions (periodic unless
otherwise noted) is enough information to calculate the cell widths `dx'`, `dy'`,
and `dz'` for the derived grids.
However, since most operations are trivially generalized to allow either E or H
to be defined on the base grid, they are written to take the a full set of base
and derived cell widths, distinguished by which field they apply to rather than
their "base" or "derived" status. This removes the need for each function to
generate the derived widths, and makes the "base" vs "derived" distinction
unnecessary in the code.
The resulting data structure containing all the cell widths takes the form of a
list-of-lists-of-arrays. The first list-of-arrays provides the cell widths for
the E-field fore-vectors, while the second list-of-arrays does the same for the
H-field back-vectors:
[[[dx_e[0], dx_e[1], ...], [dy_e[0], ...], [dz_e[0], ...]],
[[dx_h[0], dx_h[1], ...], [dy_h[0], ...], [dz_h[0], ...]]]
where `dx_e[0]` is the x-width of the `m=0` cells, as used when calculating dE/dx,
and `dy_h[0]` is the y-width of the `n=0` cells, as used when calculating dH/dy, etc.
Permittivity and Permeability
=============================
Since each vector component of E and H is defined in a different location and represents
a different volume, the value of the spatially-discrete `epsilon` and `mu` can also be
different for all three field components, even when representing a simple planar interface
between two isotropic materials.
As a result, `epsilon` and `mu` are taken to have the same dimensions as the field, and
composed of the three diagonal tensor components:
[equations: epsilon_and_mu]
epsilon = [epsilon_xx, epsilon_yy, epsilon_zz]
mu = [mu_xx, mu_yy, mu_zz]
or
$$
\\epsilon = \\begin{bmatrix} \\epsilon_{xx} & 0 & 0 \\\\
0 & \\epsilon_{yy} & 0 \\\\
0 & 0 & \\epsilon_{zz} \\end{bmatrix}
$$
$$
\\mu = \\begin{bmatrix} \\mu_{xx} & 0 & 0 \\\\
0 & \\mu_{yy} & 0 \\\\
0 & 0 & \\mu_{zz} \\end{bmatrix}
$$
where the off-diagonal terms (e.g. `epsilon_xy`) are assumed to be zero.
High-accuracy volumetric integration of shapes on multiple grids can be performed
by the [gridlock](https://mpxd.net/code/jan/gridlock) module.
The values of the vacuum permittivity and permability effectively become scaling
factors that appear in several locations (e.g. between the E and H fields). In
order to limit floating-point inaccuracy and simplify calculations, they are often
set to 1 and relative permittivities and permeabilities are used in their places;
the true values can be multiplied back in after the simulation is complete if non-
normalized results are needed.
"""
from .types import fdfield_t, vfdfield_t, dx_lists_t, fdfield_updater_t
from .vectorization import vec, unvec
from . import operators, functional, types, vectorization

109
meanas/fdmath/functional.py
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"""
Math functions for finite difference simulations
Basic discrete calculus etc.
"""
from typing import List, Callable, Tuple, Dict
import numpy
from .types import fdfield_t, fdfield_updater_t
def deriv_forward(dx_e: List[numpy.ndarray] = None) -> fdfield_updater_t:
"""
Utility operators for taking discretized derivatives (backward variant).
Args:
dx_e: Lists of cell sizes for all axes
`[[dx_0, dx_1, ...], [dy_0, dy_1, ...], ...]`.
Returns:
List of functions for taking forward derivatives along each axis.
"""
if dx_e:
derivs = [lambda f: (numpy.roll(f, -1, axis=0) - f) / dx_e[0][:, None, None],
lambda f: (numpy.roll(f, -1, axis=1) - f) / dx_e[1][None, :, None],
lambda f: (numpy.roll(f, -1, axis=2) - f) / dx_e[2][None, None, :]]
else:
derivs = [lambda f: numpy.roll(f, -1, axis=0) - f,
lambda f: numpy.roll(f, -1, axis=1) - f,
lambda f: numpy.roll(f, -1, axis=2) - f]
return derivs
def deriv_back(dx_h: List[numpy.ndarray] = None) -> fdfield_updater_t:
"""
Utility operators for taking discretized derivatives (forward variant).
Args:
dx_h: Lists of cell sizes for all axes
`[[dx_0, dx_1, ...], [dy_0, dy_1, ...], ...]`.
Returns:
List of functions for taking forward derivatives along each axis.
"""
if dx_h:
derivs = [lambda f: (f - numpy.roll(f, 1, axis=0)) / dx_h[0][:, None, None],
lambda f: (f - numpy.roll(f, 1, axis=1)) / dx_h[1][None, :, None],
lambda f: (f - numpy.roll(f, 1, axis=2)) / dx_h[2][None, None, :]]
else:
derivs = [lambda f: f - numpy.roll(f, 1, axis=0),
lambda f: f - numpy.roll(f, 1, axis=1),
lambda f: f - numpy.roll(f, 1, axis=2)]
return derivs
def curl_forward(dx_e: List[numpy.ndarray] = None) -> fdfield_updater_t:
"""
Curl operator for use with the E field.
Args:
dx_e: Lists of cell sizes for all axes
`[[dx_0, dx_1, ...], [dy_0, dy_1, ...], ...]`.
Returns:
Function `f` for taking the discrete forward curl of a field,
`f(E)` -> curlE $= \\nabla_f \\times E$
"""
Dx, Dy, Dz = deriv_forward(dx_e)
def ce_fun(e: fdfield_t) -> fdfield_t:
output = numpy.empty_like(e)
output[0] = Dy(e[2])
output[1] = Dz(e[0])
output[2] = Dx(e[1])
output[0] -= Dz(e[1])
output[1] -= Dx(e[2])
output[2] -= Dy(e[0])
return output
return ce_fun
def curl_back(dx_h: List[numpy.ndarray] = None) -> fdfield_updater_t:
"""
Create a function which takes the backward curl of a field.
Args:
dx_h: Lists of cell sizes for all axes
`[[dx_0, dx_1, ...], [dy_0, dy_1, ...], ...]`.
Returns:
Function `f` for taking the discrete backward curl of a field,
`f(H)` -> curlH $= \\nabla_b \\times H$
"""
Dx, Dy, Dz = deriv_back(dx_h)
def ch_fun(h: fdfield_t) -> fdfield_t:
output = numpy.empty_like(h)
output[0] = Dy(h[2])
output[1] = Dz(h[0])
output[2] = Dx(h[1])
output[0] -= Dz(h[1])
output[1] -= Dx(h[2])
output[2] -= Dy(h[0])
return output
return ch_fun

231
meanas/fdmath/operators.py
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"""
Matrix operators for finite difference simulations
Basic discrete calculus etc.
"""
from typing import List, Callable, Tuple, Dict
import numpy
import scipy.sparse as sparse
from .types import fdfield_t, vfdfield_t
def rotation(axis: int, shape: List[int], shift_distance: int=1) -> sparse.spmatrix:
"""
Utility operator for performing a circular shift along a specified axis by a
specified number of elements.
Args:
axis: Axis to shift along. x=0, y=1, z=2
shape: Shape of the grid being shifted
shift_distance: Number of cells to shift by. May be negative. Default 1.
Returns:
Sparse matrix for performing the circular shift.
"""
if len(shape) not in (2, 3):
raise Exception('Invalid shape: {}'.format(shape))
if axis not in range(len(shape)):
raise Exception('Invalid direction: {}, shape is {}'.format(axis, shape))
shifts = [abs(shift_distance) if a == axis else 0 for a in range(3)]
shifted_diags = [(numpy.arange(n) + s) % n for n, s in zip(shape, shifts)]
ijk = numpy.meshgrid(*shifted_diags, indexing='ij')
n = numpy.prod(shape)
i_ind = numpy.arange(n)
j_ind = numpy.ravel_multi_index(ijk, shape, order='C')
vij = (numpy.ones(n), (i_ind, j_ind.ravel(order='C')))
d = sparse.csr_matrix(vij, shape=(n, n))
if shift_distance < 0:
d = d.T
return d
def shift_with_mirror(axis: int, shape: List[int], shift_distance: int=1) -> sparse.spmatrix:
"""
Utility operator for performing an n-element shift along a specified axis, with mirror
boundary conditions applied to the cells beyond the receding edge.
Args:
axis: Axis to shift along. x=0, y=1, z=2
shape: Shape of the grid being shifted
shift_distance: Number of cells to shift by. May be negative. Default 1.
Returns:
Sparse matrix for performing the shift-with-mirror.
"""
if len(shape) not in (2, 3):
raise Exception('Invalid shape: {}'.format(shape))
if axis not in range(len(shape)):
raise Exception('Invalid direction: {}, shape is {}'.format(axis, shape))
if shift_distance >= shape[axis]:
raise Exception('Shift ({}) is too large for axis {} of size {}'.format(
shift_distance, axis, shape[axis]))
def mirrored_range(n, s):
v = numpy.arange(n) + s
v = numpy.where(v >= n, 2 * n - v - 1, v)
v = numpy.where(v < 0, - 1 - v, v)
return v
shifts = [shift_distance if a == axis else 0 for a in range(3)]
shifted_diags = [mirrored_range(n, s) for n, s in zip(shape, shifts)]
ijk = numpy.meshgrid(*shifted_diags, indexing='ij')
n = numpy.prod(shape)
i_ind = numpy.arange(n)
j_ind = numpy.ravel_multi_index(ijk, shape, order='C')
vij = (numpy.ones(n), (i_ind, j_ind.ravel(order='C')))
d = sparse.csr_matrix(vij, shape=(n, n))
return d
def deriv_forward(dx_e: List[numpy.ndarray]) -> List[sparse.spmatrix]:
"""
Utility operators for taking discretized derivatives (forward variant).
Args:
dx_e: Lists of cell sizes for all axes
`[[dx_0, dx_1, ...], [dy_0, dy_1, ...], ...]`.
Returns:
List of operators for taking forward derivatives along each axis.
"""
shape = [s.size for s in dx_e]
n = numpy.prod(shape)
dx_e_expanded = numpy.meshgrid(*dx_e, indexing='ij')
def deriv(axis):
return rotation(axis, shape, 1) - sparse.eye(n)
Ds = [sparse.diags(+1 / dx.ravel(order='C')) @ deriv(a)
for a, dx in enumerate(dx_e_expanded)]
return Ds
def deriv_back(dx_h: List[numpy.ndarray]) -> List[sparse.spmatrix]:
"""
Utility operators for taking discretized derivatives (backward variant).
Args:
dx_h: Lists of cell sizes for all axes
`[[dx_0, dx_1, ...], [dy_0, dy_1, ...], ...]`.
Returns:
List of operators for taking forward derivatives along each axis.
"""
shape = [s.size for s in dx_h]
n = numpy.prod(shape)
dx_h_expanded = numpy.meshgrid(*dx_h, indexing='ij')
def deriv(axis):
return rotation(axis, shape, -1) - sparse.eye(n)
Ds = [sparse.diags(-1 / dx.ravel(order='C')) @ deriv(a)
for a, dx in enumerate(dx_h_expanded)]
return Ds
def cross(B: List[sparse.spmatrix]) -> sparse.spmatrix:
"""
Cross product operator
Args:
B: List `[Bx, By, Bz]` of sparse matrices corresponding to the x, y, z
portions of the operator on the left side of the cross product.
Returns:
Sparse matrix corresponding to (B x), where x is the cross product.
"""
n = B[0].shape[0]
zero = sparse.csr_matrix((n, n))
return sparse.bmat([[zero, -B[2], B[1]],
[B[2], zero, -B[0]],
[-B[1], B[0], zero]])
def vec_cross(b: vfdfield_t) -> sparse.spmatrix:
"""
Vector cross product operator
Args:
b: Vector on the left side of the cross product.
Returns:
Sparse matrix corresponding to (b x), where x is the cross product.
"""
B = [sparse.diags(c) for c in numpy.split(b, 3)]
return cross(B)
def avg_forward(axis: int, shape: List[int]) -> sparse.spmatrix:
"""
Forward average operator `(x4 = (x4 + x5) / 2)`
Args:
axis: Axis to average along (x=0, y=1, z=2)
shape: Shape of the grid to average
Returns:
Sparse matrix for forward average operation.
"""
if len(shape) not in (2, 3):
raise Exception('Invalid shape: {}'.format(shape))
n = numpy.prod(shape)
return 0.5 * (sparse.eye(n) + rotation(axis, shape))
def avg_back(axis: int, shape: List[int]) -> sparse.spmatrix:
"""
Backward average operator `(x4 = (x4 + x3) / 2)`
Args:
axis: Axis to average along (x=0, y=1, z=2)
shape: Shape of the grid to average
Returns:
Sparse matrix for backward average operation.
"""
return avg_forward(axis, shape).T
def curl_forward(dx_e: List[numpy.ndarray]) -> sparse.spmatrix:
"""
Curl operator for use with the E field.
Args:
dx_e: Lists of cell sizes for all axes
`[[dx_0, dx_1, ...], [dy_0, dy_1, ...], ...]`.
Returns:
Sparse matrix for taking the discretized curl of the E-field
"""
return cross(deriv_forward(dx_e))
def curl_back(dx_h: List[numpy.ndarray]) -> sparse.spmatrix:
"""
Curl operator for use with the H field.
Args:
dx_h: Lists of cell sizes for all axes
`[[dx_0, dx_1, ...], [dy_0, dy_1, ...], ...]`.
Returns:
Sparse matrix for taking the discretized curl of the H-field
"""
return cross(deriv_back(dx_h))

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meanas/fdmath/types.py
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"""
Types shared across multiple submodules
"""
import numpy
from typing import List, Callable
# Field types
# TODO: figure out a better way to set the docstrings without creating actual subclasses?
# Probably not a big issue since they're only used for type hinting
class fdfield_t(numpy.ndarray):
"""
Vector field with shape (3, X, Y, Z) (e.g. `[E_x, E_y, E_z]`)
This is actually is just an unaltered `numpy.ndarray`
"""
pass
class vfdfield_t(numpy.ndarray):
"""
Linearized vector field (single vector of length 3*X*Y*Z)
This is actually just an unaltered `numpy.ndarray`
"""
pass
dx_lists_t = List[List[numpy.ndarray]]
'''
'dxes' datastructure which contains grid cell width information in the following format:
[[[dx_e[0], dx_e[1], ...], [dy_e[0], ...], [dz_e[0], ...]],
[[dx_h[0], dx_h[1], ...], [dy_h[0], ...], [dz_h[0], ...]]]
where `dx_e[0]` is the x-width of the `x=0` cells, as used when calculating dE/dx,
and `dy_h[0]` is the y-width of the `y=0` cells, as used when calculating dH/dy, etc.
'''
fdfield_updater_t = Callable[..., fdfield_t]
'''
Convenience type for functions which take and return an fdfield_t
'''

52
meanas/fdmath/vectorization.py
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"""
Functions for moving between a vector field (list of 3 ndarrays, `[f_x, f_y, f_z]`)
and a 1D array representation of that field `[f_x0, f_x1, f_x2,... f_y0,... f_z0,...]`.
Vectorized versions of the field use row-major (ie., C-style) ordering.
"""
from typing import List
import numpy
from .types import fdfield_t, vfdfield_t
__author__ = 'Jan Petykiewicz'
def vec(f: fdfield_t) -> vfdfield_t:
"""
Create a 1D ndarray from a 3D vector field which spans a 1-3D region.
Returns `None` if called with `f=None`.
Args:
f: A vector field, `[f_x, f_y, f_z]` where each `f_` component is a 1- to
3-D ndarray (`f_*` should all be the same size). Doesn't fail with `f=None`.
Returns:
1D ndarray containing the linearized field (or `None`)
"""
if numpy.any(numpy.equal(f, None)):
return None
return numpy.ravel(f, order='C')
def unvec(v: vfdfield_t, shape: numpy.ndarray) -> fdfield_t:
"""
Perform the inverse of vec(): take a 1D ndarray and output a 3D field
of form `[f_x, f_y, f_z]` where each of `f_*` is a len(shape)-dimensional
ndarray.
Returns `None` if called with `v=None`.
Args:
v: 1D ndarray representing a 3D vector field of shape shape (or None)
shape: shape of the vector field
Returns:
`[f_x, f_y, f_z]` where each `f_` is a `len(shape)` dimensional ndarray (or `None`)
"""
if numpy.any(numpy.equal(v, None)):
return None
return v.reshape((3, *shape), order='C')

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meanas/fdtd/__init__.py
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"""
Utilities for running finite-difference time-domain (FDTD) simulations
See the discussion of `Maxwell's Equations` in `meanas.fdmath` for basic
mathematical background.
Timestep
========
From the discussion of "Plane waves and the Dispersion relation" in `meanas.fdmath`,
we have
$$ c^2 \\Delta_t^2 = \\frac{\\Delta_t^2}{\\mu \\epsilon} < 1/(\\frac{1}{\\Delta_x^2} + \\frac{1}{\\Delta_y^2} + \\frac{1}{\\Delta_z^2}) $$
or, if $\\Delta_x = \\Delta_y = \\Delta_z$, then $c \\Delta_t < \\frac{\\Delta_x}{\\sqrt{3}}$.
Based on this, we can set
dt = sqrt(mu.min() * epsilon.min()) / sqrt(1/dx_min**2 + 1/dy_min**2 + 1/dz_min**2)
The `dx_min`, `dy_min`, `dz_min` should be the minimum value across both the base and derived grids.
Poynting Vector and Energy Conservation
=======================================
Let
$$ \\begin{aligned}
\\tilde{S}_{l, l', \\vec{r}} &=& &\\tilde{E}_{l, \\vec{r}} \\otimes \\hat{H}_{l', \\vec{r} + \\frac{1}{2}} \\\\
&=& &\\vec{x} (\\tilde{E}^y_{l,m+1,n,p} \\hat{H}^z_{l',\\vec{r} + \\frac{1}{2}} - \\tilde{E}^z_{l,m+1,n,p} \\hat{H}^y_{l', \\vec{r} + \\frac{1}{2}}) \\\\
& &+ &\\vec{y} (\\tilde{E}^z_{l,m,n+1,p} \\hat{H}^x_{l',\\vec{r} + \\frac{1}{2}} - \\tilde{E}^x_{l,m,n+1,p} \\hat{H}^z_{l', \\vec{r} + \\frac{1}{2}}) \\\\
& &+ &\\vec{z} (\\tilde{E}^x_{l,m,n,p+1} \\hat{H}^y_{l',\\vec{r} + \\frac{1}{2}} - \\tilde{E}^y_{l,m,n,p+1} \\hat{H}^z_{l', \\vec{r} + \\frac{1}{2}})
\\end{aligned}
$$
where $\\vec{r} = (m, n, p)$ and $\\otimes$ is a modified cross product
in which the $\\tilde{E}$ terms are shifted as indicated.
By taking the divergence and rearranging terms, we can show that
$$
\\begin{aligned}
\\hat{\\nabla} \\cdot \\tilde{S}_{l, l', \\vec{r}}
&= \\hat{\\nabla} \\cdot (\\tilde{E}_{l, \\vec{r}} \\otimes \\hat{H}_{l', \\vec{r} + \\frac{1}{2}}) \\\\
&= \\hat{H}_{l', \\vec{r} + \\frac{1}{2}} \\cdot \\tilde{\\nabla} \\times \\tilde{E}_{l, \\vec{r}} -
\\tilde{E}_{l, \\vec{r}} \\cdot \\hat{\\nabla} \\times \\hat{H}_{l', \\vec{r} + \\frac{1}{2}} \\\\
&= \\hat{H}_{l', \\vec{r} + \\frac{1}{2}} \\cdot
(-\\tilde{\\partial}_t \\mu_{\\vec{r} + \\frac{1}{2}} \\hat{H}_{l - \\frac{1}{2}, \\vec{r} + \\frac{1}{2}} -
\\hat{M}_{l-1, \\vec{r} + \\frac{1}{2}}) -
\\tilde{E}_{l, \\vec{r}} \\cdot (\\hat{\\partial}_t \\tilde{\\epsilon}_{\\vec{r}} \\tilde{E}_{l'+\\frac{1}{2}, \\vec{r}} +
\\tilde{J}_{l', \\vec{r}}) \\\\
&= \\hat{H}_{l'} \\cdot (-\\mu / \\Delta_t)(\\hat{H}_{l + \\frac{1}{2}} - \\hat{H}_{l - \\frac{1}{2}}) -
\\tilde{E}_l \\cdot (\\epsilon / \\Delta_t )(\\tilde{E}_{l'+\\frac{1}{2}} - \\tilde{E}_{l'-\\frac{1}{2}})
- \\hat{H}_{l'} \\cdot \\hat{M}_{l-1} - \\tilde{E}_l \\cdot \\tilde{J}_{l'} \\\\
\\end{aligned}
$$
where in the last line the spatial subscripts have been dropped to emphasize
the time subscripts $l, l'$, i.e.
$$
\\begin{aligned}
\\tilde{E}_l &= \\tilde{E}_{l, \\vec{r}} \\\\
\\hat{H}_l &= \\tilde{H}_{l, \\vec{r} + \\frac{1}{2}} \\\\
\\tilde{\\epsilon} &= \\tilde{\\epsilon}_{\\vec{r}} \\\\
\\end{aligned}
$$
etc.
For $l' = l + \\frac{1}{2}$ we get
$$
\\begin{aligned}
\\hat{\\nabla} \\cdot \\tilde{S}_{l, l + \\frac{1}{2}}
&= \\hat{H}_{l + \\frac{1}{2}} \\cdot
(-\\mu / \\Delta_t)(\\hat{H}_{l + \\frac{1}{2}} - \\hat{H}_{l - \\frac{1}{2}}) -
\\tilde{E}_l \\cdot (\\epsilon / \\Delta_t)(\\tilde{E}_{l+1} - \\tilde{E}_l)
- \\hat{H}_{l'} \\cdot \\hat{M}_l - \\tilde{E}_l \\cdot \\tilde{J}_{l + \\frac{1}{2}} \\\\
&= (-\\mu / \\Delta_t)(\\hat{H}^2_{l + \\frac{1}{2}} - \\hat{H}_{l + \\frac{1}{2}} \\cdot \\hat{H}_{l - \\frac{1}{2}}) -
(\\epsilon / \\Delta_t)(\\tilde{E}_{l+1} \\cdot \\tilde{E}_l - \\tilde{E}^2_l)
- \\hat{H}_{l'} \\cdot \\hat{M}_l - \\tilde{E}_l \\cdot \\tilde{J}_{l + \\frac{1}{2}} \\\\
&= -(\\mu \\hat{H}^2_{l + \\frac{1}{2}}
+\\epsilon \\tilde{E}_{l+1} \\cdot \\tilde{E}_l) / \\Delta_t \\ \\
+(\\mu \\hat{H}_{l + \\frac{1}{2}} \\cdot \\hat{H}_{l - \\frac{1}{2}}
+\\epsilon \\tilde{E}^2_l) / \\Delta_t \\ \\
- \\hat{H}_{l+\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\
- \\tilde{E}_l \\cdot \\tilde{J}_{l+\\frac{1}{2}} \\\\
\\end{aligned}
$$
and for $l' = l - \\frac{1}{2}$,
$$
\\begin{aligned}
\\hat{\\nabla} \\cdot \\tilde{S}_{l, l - \\frac{1}{2}}
&= (\\mu \\hat{H}^2_{l - \\frac{1}{2}}
+\\epsilon \\tilde{E}_{l-1} \\cdot \\tilde{E}_l) / \\Delta_t \\ \\
-(\\mu \\hat{H}_{l + \\frac{1}{2}} \\cdot \\hat{H}_{l - \\frac{1}{2}}
+\\epsilon \\tilde{E}^2_l) / \\Delta_t \\ \\
- \\hat{H}_{l-\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\
- \\tilde{E}_l \\cdot \\tilde{J}_{l-\\frac{1}{2}} \\\\
\\end{aligned}
$$
These two results form the discrete time-domain analogue to Poynting's theorem.
They hint at the expressions for the energy, which can be calculated at the same
time-index as either the E or H field:
$$
\\begin{aligned}
U_l &= \\epsilon \\tilde{E}^2_l + \\mu \\hat{H}_{l + \\frac{1}{2}} \\cdot \\hat{H}_{l - \\frac{1}{2}} \\\\
U_{l + \\frac{1}{2}} &= \\epsilon \\tilde{E}_l \\cdot \\tilde{E}_{l + 1} + \\mu \\hat{H}^2_{l + \\frac{1}{2}} \\\\
\\end{aligned}
$$
Rewriting the Poynting theorem in terms of the energy expressions,
$$
\\begin{aligned}
(U_{l+\\frac{1}{2}} - U_l) / \\Delta_t
&= -\\hat{\\nabla} \\cdot \\tilde{S}_{l, l + \\frac{1}{2}} \\ \\
- \\hat{H}_{l+\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\
- \\tilde{E}_l \\cdot \\tilde{J}_{l+\\frac{1}{2}} \\\\
(U_l - U_{l-\\frac{1}{2}}) / \\Delta_t
&= -\\hat{\\nabla} \\cdot \\tilde{S}_{l, l - \\frac{1}{2}} \\ \\
- \\hat{H}_{l-\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\
- \\tilde{E}_l \\cdot \\tilde{J}_{l-\\frac{1}{2}} \\\\
\\end{aligned}
$$
This result is exact an should practically hold to within numerical precision. No time-
or spatial-averaging is necessary.
Note that each value of $J$ contributes to the energy twice (i.e. once per field update)
despite only causing the value of $E$ to change once (same for $M$ and $H$).
Sources
=============
It is often useful to excite the simulation with an arbitrary broadband pulse and then
extract the frequency-domain response by performing an on-the-fly Fourier transform
of the time-domain fields.
The Ricker wavelet (normalized second derivative of a Gaussian) is commonly used for the pulse
shape. It can be written
$$ f_r(t) = (1 - \\frac{1}{2} (\\omega (t - \\tau))^2) e^{-(\\frac{\\omega (t - \\tau)}{2})^2} $$
with $\\tau > \\frac{2 * \\pi}{\\omega}$ as a minimum delay to avoid a discontinuity at
t=0 (assuming the source is off for t<0 this gives $\\sim 10^{-3}$ error at t=0).
Boundary conditions
===================
# TODO notes about boundaries / PMLs
"""
from .base import maxwell_e, maxwell_h
from .pml import cpml
from .energy import (poynting, poynting_divergence, energy_hstep, energy_estep,
delta_energy_h2e, delta_energy_h2e, delta_energy_j)
from .boundaries import conducting_boundary

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meanas/fdtd/base.py
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"""
Basic FDTD field updates
"""
from typing import List, Callable, Tuple, Dict
import numpy
from ..fdmath import dx_lists_t, fdfield_t, fdfield_updater_t
from ..fdmath.functional import curl_forward, curl_back
__author__ = 'Jan Petykiewicz'
def maxwell_e(dt: float, dxes: dx_lists_t = None) -> fdfield_updater_t:
"""
Build a function which performs a portion the time-domain E-field update,
E += curl_back(H[t]) / epsilon
The full update should be
E += (curl_back(H[t]) + J) / epsilon
which requires an additional step of `E += J / epsilon` which is not performed
by the generated function.
See `meanas.fdmath` for descriptions of
- This update step: "Maxwell's equations" section
- `dxes`: "Datastructure: dx_lists_t" section
- `epsilon`: "Permittivity and Permeability" section
Also see the "Timestep" section of `meanas.fdtd` for a discussion of
the `dt` parameter.
Args:
dt: Timestep. See `meanas.fdtd` for details.
dxes: Grid description; see `meanas.fdmath`.
Returns:
Function `f(E_old, H_old, epsilon) -> E_new`.
"""
if dxes is not None:
curl_h_fun = curl_back(dxes[1])
else:
curl_h_fun = curl_back()
def me_fun(e: fdfield_t, h: fdfield_t, epsilon: fdfield_t):
"""
Update the E-field.
Args:
e: E-field at time t=0
h: H-field at time t=0.5
epsilon: Dielectric constant distribution.
Returns:
E-field at time t=1
"""
e += dt * curl_h_fun(h) / epsilon
return e
return me_fun
def maxwell_h(dt: float, dxes: dx_lists_t = None) -> fdfield_updater_t:
"""
Build a function which performs part of the time-domain H-field update,
H -= curl_forward(E[t]) / mu
The full update should be
H -= (curl_forward(E[t]) + M) / mu
which requires an additional step of `H -= M / mu` which is not performed
by the generated function; this step can be omitted if there is no magnetic
current `M`.
See `meanas.fdmath` for descriptions of
- This update step: "Maxwell's equations" section
- `dxes`: "Datastructure: dx_lists_t" section
- `mu`: "Permittivity and Permeability" section
Also see the "Timestep" section of `meanas.fdtd` for a discussion of
the `dt` parameter.
Args:
dt: Timestep. See `meanas.fdtd` for details.
dxes: Grid description; see `meanas.fdmath`.
Returns:
Function `f(E_old, H_old, epsilon) -> E_new`.
"""
if dxes is not None:
curl_e_fun = curl_forward(dxes[0])
else:
curl_e_fun = curl_forward()
def mh_fun(e: fdfield_t, h: fdfield_t, mu: fdfield_t = None):
"""
Update the H-field.
Args:
e: E-field at time t=1
h: H-field at time t=0.5
mu: Magnetic permeability. Default is 1 everywhere.
Returns:
H-field at time t=1.5
"""
if mu is not None:
h -= dt * curl_e_fun(e) / mu
else:
h -= dt * curl_e_fun(e)
return h
return mh_fun

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meanas/fdtd/boundaries.py
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"""
Boundary conditions
#TODO conducting boundary documentation
"""
from typing import List, Callable, Tuple, Dict
import numpy
from ..fdmath import dx_lists_t, fdfield_t, fdfield_updater_t
def conducting_boundary(direction: int,
polarity: int
) -> Tuple[fdfield_updater_t, fdfield_updater_t]:
dirs = [0, 1, 2]
if direction not in dirs:
raise Exception('Invalid direction: {}'.format(direction))
dirs.remove(direction)
u, v = dirs
if polarity < 0:
boundary_slice = [slice(None)] * 3
shifted1_slice = [slice(None)] * 3
boundary_slice[direction] = 0
shifted1_slice[direction] = 1
def en(e: fdfield_t):
e[direction][boundary_slice] = 0
e[u][boundary_slice] = e[u][shifted1_slice]
e[v][boundary_slice] = e[v][shifted1_slice]
return e
def hn(h: fdfield_t):
h[direction][boundary_slice] = h[direction][shifted1_slice]
h[u][boundary_slice] = 0
h[v][boundary_slice] = 0
return h
return en, hn
if polarity > 0:
boundary_slice = [slice(None)] * 3
shifted1_slice = [slice(None)] * 3
shifted2_slice = [slice(None)] * 3
boundary_slice[direction] = -1
shifted1_slice[direction] = -2
shifted2_slice[direction] = -3
def ep(e: fdfield_t):
e[direction][boundary_slice] = -e[direction][shifted2_slice]
e[direction][shifted1_slice] = 0
e[u][boundary_slice] = e[u][shifted1_slice]
e[v][boundary_slice] = e[v][shifted1_slice]
return e
def hp(h: fdfield_t):
h[direction][boundary_slice] = h[direction][shifted1_slice]
h[u][boundary_slice] = -h[u][shifted2_slice]
h[u][shifted1_slice] = 0
h[v][boundary_slice] = -h[v][shifted2_slice]
h[v][shifted1_slice] = 0
return h
return ep, hp
raise Exception('Bad polarity: {}'.format(polarity))

140
meanas/fdtd/energy.py
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# pylint: disable=unsupported-assignment-operation
from typing import List, Callable, Tuple, Dict
import numpy
from ..fdmath import dx_lists_t, fdfield_t, fdfield_updater_t
from ..fdmath.functional import deriv_back, deriv_forward
def poynting(e: fdfield_t,
h: fdfield_t,
dxes: dx_lists_t = None,
) -> fdfield_t:
"""
Calculate the poynting vector
"""
if dxes is None:
dxes = tuple(tuple(numpy.ones(1) for _ in range(3)) for _ in range(2))
ex = e[0] * dxes[0][0][:, None, None]
ey = e[1] * dxes[0][1][None, :, None]
ez = e[2] * dxes[0][2][None, None, :]
hx = h[0] * dxes[1][0][:, None, None]
hy = h[1] * dxes[1][1][None, :, None]
hz = h[2] * dxes[1][2][None, None, :]
s = numpy.empty_like(e)
s[0] = numpy.roll(ey, -1, axis=0) * hz - numpy.roll(ez, -1, axis=0) * hy
s[1] = numpy.roll(ez, -1, axis=1) * hx - numpy.roll(ex, -1, axis=1) * hz
s[2] = numpy.roll(ex, -1, axis=2) * hy - numpy.roll(ey, -1, axis=2) * hx
return s
def poynting_divergence(s: fdfield_t = None,
*,
e: fdfield_t = None,
h: fdfield_t = None,
dxes: dx_lists_t = None,
) -> fdfield_t:
"""
Calculate the divergence of the poynting vector
"""
if s is None:
s = poynting(e, h, dxes=dxes)
Dx, Dy, Dz = deriv_back()
ds = Dx(s[0]) + Dy(s[1]) + Dz(s[2])
return ds
def energy_hstep(e0: fdfield_t,
h1: fdfield_t,
e2: fdfield_t,
epsilon: fdfield_t = None,
mu: fdfield_t = None,
dxes: dx_lists_t = None,
) -> fdfield_t:
u = dxmul(e0 * e2, h1 * h1, epsilon, mu, dxes)
return u
def energy_estep(h0: fdfield_t,
e1: fdfield_t,
h2: fdfield_t,
epsilon: fdfield_t = None,
mu: fdfield_t = None,
dxes: dx_lists_t = None,
) -> fdfield_t:
u = dxmul(e1 * e1, h0 * h2, epsilon, mu, dxes)
return u
def delta_energy_h2e(dt: float,
e0: fdfield_t,
h1: fdfield_t,
e2: fdfield_t,
h3: fdfield_t,
epsilon: fdfield_t = None,
mu: fdfield_t = None,
dxes: dx_lists_t = None,
) -> fdfield_t:
"""
This is just from (e2 * e2 + h3 * h1) - (h1 * h1 + e0 * e2)
"""
de = e2 * (e2 - e0) / dt
dh = h1 * (h3 - h1) / dt
du = dxmul(de, dh, epsilon, mu, dxes)
return du
def delta_energy_e2h(dt: float,
h0: fdfield_t,
e1: fdfield_t,
h2: fdfield_t,
e3: fdfield_t,
epsilon: fdfield_t = None,
mu: fdfield_t = None,
dxes: dx_lists_t = None,
) -> fdfield_t:
"""
This is just from (h2 * h2 + e3 * e1) - (e1 * e1 + h0 * h2)
"""
de = e1 * (e3 - e1) / dt
dh = h2 * (h2 - h0) / dt
du = dxmul(de, dh, epsilon, mu, dxes)
return du
def delta_energy_j(j0: fdfield_t, e1: fdfield_t, dxes: dx_lists_t = None) -> fdfield_t:
if dxes is None:
dxes = tuple(tuple(numpy.ones(1) for _ in range(3)) for _ in range(2))
du = ((j0 * e1).sum(axis=0) *
dxes[0][0][:, None, None] *
dxes[0][1][None, :, None] *
dxes[0][2][None, None, :])
return du
def dxmul(ee: fdfield_t,
hh: fdfield_t,
epsilon: fdfield_t = None,
mu: fdfield_t = None,
dxes: dx_lists_t = None
) -> fdfield_t:
if epsilon is None:
epsilon = 1
if mu is None:
mu = 1
if dxes is None:
dxes = tuple(tuple(numpy.ones(1) for _ in range(3)) for _ in range(2))
result = ((ee * epsilon).sum(axis=0) *
dxes[0][0][:, None, None] *
dxes[0][1][None, :, None] *
dxes[0][2][None, None, :] +
(hh * mu).sum(axis=0) *
dxes[1][0][:, None, None] *
dxes[1][1][None, :, None] *
dxes[1][2][None, None, :])
return result

127
meanas/fdtd/pml.py
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"""
PML implementations
#TODO discussion of PMLs
#TODO cpml documentation
"""
# TODO retest pmls!
from typing import List, Callable, Tuple, Dict
import numpy
from ..fdmath import dx_lists_t, fdfield_t, fdfield_updater_t
__author__ = 'Jan Petykiewicz'
def cpml(direction: int,
polarity: int,
dt: float,
epsilon: fdfield_t,
thickness: int = 8,
ln_R_per_layer: float = -1.6,
epsilon_eff: float = 1,
mu_eff: float = 1,
m: float = 3.5,
ma: float = 1,
cfs_alpha: float = 0,
dtype: numpy.dtype = numpy.float32,
) -> Tuple[Callable, Callable, Dict[str, fdfield_t]]:
if direction not in range(3):
raise Exception('Invalid direction: {}'.format(direction))
if polarity not in (-1, 1):
raise Exception('Invalid polarity: {}'.format(polarity))
if thickness <= 2:
raise Exception('It would be wise to have a pml with 4+ cells of thickness')
if epsilon_eff <= 0:
raise Exception('epsilon_eff must be positive')
sigma_max = -ln_R_per_layer / 2 * (m + 1)
kappa_max = numpy.sqrt(epsilon_eff * mu_eff)
alpha_max = cfs_alpha
transverse = numpy.delete(range(3), direction)
u, v = transverse
xe = numpy.arange(1, thickness+1, dtype=float)
xh = numpy.arange(1, thickness+1, dtype=float)
if polarity > 0:
xe -= 0.5
elif polarity < 0:
xh -= 0.5
xe = xe[::-1]
xh = xh[::-1]
else:
raise Exception('Bad polarity!')
expand_slice = [None] * 3
expand_slice[direction] = slice(None)
expand_slice = tuple(expand_slice)
def par(x):
scaling = (x / thickness) ** m
sigma = scaling * sigma_max
kappa = 1 + scaling * (kappa_max - 1)
alpha = ((1 - x / thickness) ** ma) * alpha_max
p0 = numpy.exp(-(sigma / kappa + alpha) * dt)
p1 = sigma / (sigma + kappa * alpha) * (p0 - 1)
p2 = 1 / kappa
return p0[expand_slice], p1[expand_slice], p2[expand_slice]
p0e, p1e, p2e = par(xe)
p0h, p1h, p2h = par(xh)
region = [slice(None)] * 3
if polarity < 0:
region[direction] = slice(None, thickness)
elif polarity > 0:
region[direction] = slice(-thickness, None)
else:
raise Exception('Bad polarity!')
region = tuple(region)
se = 1 if direction == 1 else -1
# TODO check if epsilon is uniform in pml region?
shape = list(epsilon[0].shape)
shape[direction] = thickness
psi_e = [numpy.zeros(shape, dtype=dtype), numpy.zeros(shape, dtype=dtype)]
psi_h = [numpy.zeros(shape, dtype=dtype), numpy.zeros(shape, dtype=dtype)]
fields = {
'psi_e_u': psi_e[0],
'psi_e_v': psi_e[1],
'psi_h_u': psi_h[0],
'psi_h_v': psi_h[1],
}
# Note that this is kinda slow -- would be faster to reuse dHv*p2h for the original
# H update, but then you have multiple arrays and a monolithic (field + pml) update operation
def pml_e(e: fdfield_t, h: fdfield_t, epsilon: fdfield_t) -> Tuple[fdfield_t, fdfield_t]:
dHv = h[v][region] - numpy.roll(h[v], 1, axis=direction)[region]
dHu = h[u][region] - numpy.roll(h[u], 1, axis=direction)[region]
psi_e[0] *= p0e
psi_e[0] += p1e * dHv * p2e
psi_e[1] *= p0e
psi_e[1] += p1e * dHu * p2e
e[u][region] += se * dt / epsilon[u][region] * (psi_e[0] + (p2e - 1) * dHv)
e[v][region] -= se * dt / epsilon[v][region] * (psi_e[1] + (p2e - 1) * dHu)
return e, h
def pml_h(e: fdfield_t, h: fdfield_t) -> Tuple[fdfield_t, fdfield_t]:
dEv = (numpy.roll(e[v], -1, axis=direction)[region] - e[v][region])
dEu = (numpy.roll(e[u], -1, axis=direction)[region] - e[u][region])
psi_h[0] *= p0h
psi_h[0] += p1h * dEv * p2h
psi_h[1] *= p0h
psi_h[1] += p1h * dEu * p2h
h[u][region] -= se * dt * (psi_h[0] + (p2h - 1) * dEv)
h[v][region] += se * dt * (psi_h[1] + (p2h - 1) * dEu)
return e, h
return pml_e, pml_h, fields

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meanas/test/__init__.py
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"""
Tests (run with `python3 -m pytest -rxPXs | tee results.txt`)
"""

79
meanas/test/conftest.py
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from typing import List, Tuple
import numpy
import pytest
from .utils import PRNG
#####################################
# Test fixtures
#####################################
@pytest.fixture(scope='module',
params=[(5, 5, 1),
(5, 1, 5),
(5, 5, 5),
#(7, 7, 7),
])
def shape(request):
yield (3, *request.param)
@pytest.fixture(scope='module', params=[1.0, 1.5])
def epsilon_bg(request):
yield request.param
@pytest.fixture(scope='module', params=[1.0, 2.5])
def epsilon_fg(request):
yield request.param
@pytest.fixture(scope='module', params=['center', '000', 'random'])
def epsilon(request, shape, epsilon_bg, epsilon_fg):
is3d = (numpy.array(shape) == 1).sum() == 0
if is3d:
if request.param == '000':
pytest.skip('Skipping 000 epsilon because test is 3D (for speed)')
if epsilon_bg != 1:
pytest.skip('Skipping epsilon_bg != 1 because test is 3D (for speed)')
if epsilon_fg not in (1.0, 2.0):
pytest.skip('Skipping epsilon_fg not in (1, 2) because test is 3D (for speed)')
epsilon = numpy.full(shape, epsilon_bg, dtype=float)
if request.param == 'center':
epsilon[:, shape[1]//2, shape[2]//2, shape[3]//2] = epsilon_fg
elif request.param == '000':
epsilon[:, 0, 0, 0] = epsilon_fg
elif request.param == 'random':
epsilon[:] = PRNG.uniform(low=min(epsilon_bg, epsilon_fg),
high=max(epsilon_bg, epsilon_fg),
size=shape)
yield epsilon
@pytest.fixture(scope='module', params=[1.0])#, 1.5])
def j_mag(request):
yield request.param
@pytest.fixture(scope='module', params=[1.0, 1.5])
def dx(request):
yield request.param
@pytest.fixture(scope='module', params=['uniform', 'centerbig'])
def dxes(request, shape, dx):
if request.param == 'uniform':
dxes = [[numpy.full(s, dx) for s in shape[1:]] for _ in range(2)]
elif request.param == 'centerbig':
dxes = [[numpy.full(s, dx) for s in shape[1:]] for _ in range(2)]
for eh in (0, 1):
for ax in (0, 1, 2):
dxes[eh][ax][dxes[eh][ax].size // 2] *= 1.1
elif request.param == 'random':
dxe = [PRNG.uniform(low=1.0 * dx, high=1.1 * dx, size=s) for s in shape[1:]]
dxh = [(d + numpy.roll(d, -1)) / 2 for d in dxe]
dxes = [dxe, dxh]
yield dxes

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meanas/test/test_fdfd.py
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# pylint: disable=redefined-outer-name
from typing import List, Tuple
import dataclasses
import pytest
import numpy
#from numpy.testing import assert_allclose, assert_array_equal
from .. import fdfd
from ..fdmath import vec, unvec
from .utils import assert_close, assert_fields_close
def test_residual(sim):
A = fdfd.operators.e_full(sim.omega, sim.dxes, vec(sim.epsilon)).tocsr()
b = -1j * sim.omega * vec(sim.j)
residual = A @ vec(sim.e) - b
assert numpy.linalg.norm(residual) < 1e-10
def test_poynting_planes(sim):
mask = (sim.j != 0).any(axis=0)
if mask.sum() != 2:
pytest.skip(f'test_poynting_planes will only test 2-point sources, got {mask.sum()}')
# for dxg in sim.dxes:
# for dxa in dxg:
# if not (dxa == sim.dxes[0][0][0]).all():
# pytest.skip('test_poynting_planes skips nonuniform dxes')
points = numpy.where(mask)
mask[points[0][0], points[1][0], points[2][0]] = 0
mx = numpy.roll(mask, -1, axis=0)
my = numpy.roll(mask, -1, axis=1)
mz = numpy.roll(mask, -1, axis=2)
e2h = fdfd.operators.e2h(omega=sim.omega, dxes=sim.dxes, pmc=sim.pmc)
ev = vec(sim.e)
hv = e2h @ ev
exh = fdfd.operators.poynting_e_cross(e=ev, dxes=sim.dxes) @ hv.conj()
s = unvec(exh.real / 2, sim.shape[1:])
planes = [s[0, mask].sum(), -s[0, mx].sum(),
s[1, mask].sum(), -s[1, my].sum(),
s[2, mask].sum(), -s[2, mz].sum()]
e_dot_j = sim.e * sim.j * sim.dxes[0][0][:, None, None] * sim.dxes[0][1][None, :, None] * sim.dxes[0][2][None, None, :]
src_energy = -e_dot_j[:, mask].real / 2
assert_close(sum(planes), src_energy.sum())
#####################################
# Test fixtures
#####################################
# Also see conftest.py
@pytest.fixture(params=[1/1500])
def omega(request):
yield request.param
@pytest.fixture(params=[None])
def pec(request):
yield request.param
@pytest.fixture(params=[None])
def pmc(request):
yield request.param
#@pytest.fixture(scope='module',
# params=[(25, 5, 5)])
#def shape(request):
# yield (3, *request.param)
@pytest.fixture(params=['diag']) #'center'
def j_distribution(request, shape, j_mag):
j = numpy.zeros(shape, dtype=complex)
center_mask = numpy.zeros(shape, dtype=bool)
center_mask[:, shape[1]//2, shape[2]//2, shape[3]//2] = True
if request.param == 'center':
j[center_mask] = j_mag
elif request.param == 'diag':
j[numpy.roll(center_mask, [1, 1, 1], axis=(1, 2, 3))] = j_mag
j[numpy.roll(center_mask, [-1, -1, -1], axis=(1, 2, 3))] = -1j * j_mag
yield j
@dataclasses.dataclass()
class FDResult:
shape: Tuple[int]
dxes: List[List[numpy.ndarray]]
epsilon: numpy.ndarray
omega: complex
j: numpy.ndarray
e: numpy.ndarray
pmc: numpy.ndarray
pec: numpy.ndarray
@pytest.fixture()
def sim(request, shape, epsilon, dxes, j_distribution, omega, pec, pmc):
# is3d = (numpy.array(shape) == 1).sum() == 0
# if is3d:
# pytest.skip('Skipping dt != 0.3 because test is 3D (for speed)')
# # If no edge currents, add pmls
# src_mask = j_distribution.any(axis=0)
# th = 10
# #if src_mask.sum() - src_mask[th:-th, th:-th, th:-th].sum() == 0:
# if src_mask.sum() - src_mask[th:-th, :, :].sum() == 0:
# for axis in (0,):
# for polarity in (-1, 1):
# dxes = fdfd.scpml.stretch_with_scpml(dxes, axis=axis, polarity=polarity,
j_vec = vec(j_distribution)
eps_vec = vec(epsilon)
e_vec = fdfd.solvers.generic(J=j_vec, omega=omega, dxes=dxes, epsilon=eps_vec,
matrix_solver_opts={'atol': 1e-15, 'tol': 1e-11})
e = unvec(e_vec, shape[1:])
sim = FDResult(
shape=shape,
dxes=dxes,
epsilon=epsilon,
j=j_distribution,
e=e,
pec=pec,
pmc=pmc,
omega=omega,
)
return sim

148
meanas/test/test_fdfd_pml.py
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@ -1,148 +0,0 @@
#####################################
# pylint: disable=redefined-outer-name
from typing import List, Tuple
import dataclasses
import pytest
import numpy
from numpy.testing import assert_allclose, assert_array_equal
from .. import fdfd
from ..fdmath import vec, unvec
from .utils import assert_close, assert_fields_close
from .test_fdfd import FDResult
def test_pml(sim, src_polarity):
dim = numpy.where(numpy.array(sim.shape[1:]) > 1)[0][0] # Propagation axis
e_sqr = numpy.squeeze((sim.e.conj() * sim.e).sum(axis=0))
# from matplotlib import pyplot
# pyplot.figure()
# pyplot.plot(numpy.squeeze(e_sqr))
# pyplot.show(block=True)
e_sqr_tgt = e_sqr[16:19]
e_sqr_rev = e_sqr[10:13]
if src_polarity < 0:
e_sqr_tgt, e_sqr_rev = e_sqr_rev, e_sqr_tgt
assert_allclose(e_sqr_rev, 0, atol=1e-12)
assert_allclose(e_sqr_tgt, 1, rtol=3e-6)
# pyplot.figure()
# pyplot.plot(numpy.squeeze(sim.e[0].real), label='Ex_real')
# pyplot.plot(numpy.squeeze(sim.e[0].imag), label='Ex_imag')
# pyplot.plot(numpy.squeeze(sim.e[1].real), label='Ey_real')
# pyplot.plot(numpy.squeeze(sim.e[1].imag), label='Ey_imag')
# pyplot.plot(numpy.squeeze(sim.e[2].real), label='Ez_real')
# pyplot.plot(numpy.squeeze(sim.e[2].imag), label='Ez_imag')
# pyplot.legend()
# pyplot.show(block=True)
# Test fixtures
#####################################
# Also see conftest.py
@pytest.fixture(params=[1/1500])
def omega(request):
yield request.param
@pytest.fixture(params=[None])
def pec(request):
yield request.param
@pytest.fixture(params=[None])
def pmc(request):
yield request.param
@pytest.fixture(params=[(30, 1, 1),
(1, 30, 1),
(1, 1, 30)])
def shape(request):
yield (3, *request.param)
@pytest.fixture(params=[+1, -1])
def src_polarity(request):
yield request.param
@pytest.fixture()
def j_distribution(request, shape, epsilon, dxes, omega, src_polarity):
j = numpy.zeros(shape, dtype=complex)
dim = numpy.where(numpy.array(shape[1:]) > 1)[0][0] # Propagation axis
other_dims = [0, 1, 2]
other_dims.remove(dim)
dx_prop = (dxes[0][dim][shape[dim + 1] // 2] +
dxes[1][dim][shape[dim + 1] // 2]) / 2 #TODO is this right for nonuniform dxes?
# Mask only contains components orthogonal to propagation direction
center_mask = numpy.zeros(shape, dtype=bool)
center_mask[other_dims, shape[1]//2, shape[2]//2, shape[3]//2] = True
if (epsilon[center_mask] != epsilon[center_mask][0]).any():
center_mask[other_dims[1]] = False # If epsilon is not isotropic, pick only one dimension
wavenumber = omega * numpy.sqrt(epsilon[center_mask].mean())
wavenumber_corrected = 2 / dx_prop * numpy.arcsin(wavenumber * dx_prop / 2)
e = numpy.zeros_like(epsilon, dtype=complex)
e[center_mask] = 1 / numpy.linalg.norm(center_mask[:])
slices = [slice(None), slice(None), slice(None)]
slices[dim] = slice(shape[dim + 1] // 2,
shape[dim + 1] // 2 + 1)
j = fdfd.waveguide_3d.compute_source(E=e, wavenumber=wavenumber_corrected, omega=omega, dxes=dxes,
axis=dim, polarity=src_polarity, slices=slices, epsilon=epsilon)
yield j
@pytest.fixture()
def epsilon(request, shape, epsilon_bg, epsilon_fg):
epsilon = numpy.full(shape, epsilon_fg, dtype=float)
yield epsilon
@pytest.fixture(params=['uniform'])
def dxes(request, shape, dx, omega, epsilon_fg):
if request.param == 'uniform':
dxes = [[numpy.full(s, dx) for s in shape[1:]] for _ in range(2)]
dim = numpy.where(numpy.array(shape[1:]) > 1)[0][0] # Propagation axis
for axis in (dim,):
for polarity in (-1, 1):
dxes = fdfd.scpml.stretch_with_scpml(dxes, axis=axis, polarity=polarity,
omega=omega, epsilon_effective=epsilon_fg,
thickness=10)
yield dxes
@pytest.fixture()
def sim(request, shape, epsilon, dxes, j_distribution, omega, pec, pmc):
j_vec = vec(j_distribution)
eps_vec = vec(epsilon)
e_vec = fdfd.solvers.generic(J=j_vec, omega=omega, dxes=dxes, epsilon=eps_vec,
matrix_solver_opts={'atol': 1e-15, 'tol': 1e-11})
e = unvec(e_vec, shape[1:])
sim = FDResult(
shape=shape,
dxes=dxes,
epsilon=epsilon,
j=j_distribution,
e=e,
pec=pec,
pmc=pmc,
omega=omega,
)
return sim

214
meanas/test/test_fdtd.py
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@ -1,214 +0,0 @@
# pylint: disable=redefined-outer-name, no-member
from typing import List, Tuple
import dataclasses
import pytest
import numpy
from numpy.testing import assert_allclose, assert_array_equal
from .. import fdtd
from .utils import assert_close, assert_fields_close, PRNG
def test_initial_fields(sim):
# Make sure initial fields didn't change
e0 = sim.es[0]
h0 = sim.hs[0]
j0 = sim.js[0]
mask = (j0 != 0)
assert_fields_close(e0[mask], j0[mask] / sim.epsilon[mask])
assert not e0[~mask].any()
assert not h0.any()
def test_initial_energy(sim):
"""
Assumes fields start at 0 before J0 is added
"""
j0 = sim.js[0]
e0 = sim.es[0]
h0 = sim.hs[0]
h1 = sim.hs[1]
mask = (j0 != 0)
dV = numpy.prod(numpy.meshgrid(*sim.dxes[0], indexing='ij'), axis=0)
u0 = (j0 * j0.conj() / sim.epsilon * dV).sum(axis=0)
args = {'dxes': sim.dxes,
'epsilon': sim.epsilon}
# Make sure initial energy and E dot J are correct
energy0 = fdtd.energy_estep(h0=h0, e1=e0, h2=h1, **args)
e0_dot_j0 = fdtd.delta_energy_j(j0=j0, e1=e0, dxes=sim.dxes)
assert_fields_close(energy0, u0)
assert_fields_close(e0_dot_j0, u0)
def test_energy_conservation(sim):
"""
Assumes fields start at 0 before J0 is added
"""
e0 = sim.es[0]
j0 = sim.js[0]
u = fdtd.delta_energy_j(j0=j0, e1=e0, dxes=sim.dxes).sum()
args = {'dxes': sim.dxes,
'epsilon': sim.epsilon}
for ii in range(1, 8):
u_hstep = fdtd.energy_hstep(e0=sim.es[ii-1], h1=sim.hs[ii], e2=sim.es[ii], **args) # pylint: disable=bad-whitespace
u_estep = fdtd.energy_estep(h0=sim.hs[ii], e1=sim.es[ii], h2=sim.hs[ii + 1], **args) # pylint: disable=bad-whitespace
delta_j_A = fdtd.delta_energy_j(j0=sim.js[ii], e1=sim.es[ii-1], dxes=sim.dxes)
delta_j_B = fdtd.delta_energy_j(j0=sim.js[ii], e1=sim.es[ii], dxes=sim.dxes) # pylint: disable=bad-whitespace
u += delta_j_A.sum()
assert_close(u_hstep.sum(), u)
u += delta_j_B.sum()
assert_close(u_estep.sum(), u)
def test_poynting_divergence(sim):
args = {'dxes': sim.dxes,
'epsilon': sim.epsilon}
u_eprev = None
for ii in range(1, 8):
u_hstep = fdtd.energy_hstep(e0=sim.es[ii-1], h1=sim.hs[ii], e2=sim.es[ii], **args) # pylint: disable=bad-whitespace
u_estep = fdtd.energy_estep(h0=sim.hs[ii], e1=sim.es[ii], h2=sim.hs[ii + 1], **args) # pylint: disable=bad-whitespace
delta_j_B = fdtd.delta_energy_j(j0=sim.js[ii], e1=sim.es[ii], dxes=sim.dxes)
du_half_h2e = u_estep - u_hstep - delta_j_B
div_s_h2e = sim.dt * fdtd.poynting_divergence(e=sim.es[ii], h=sim.hs[ii], dxes=sim.dxes)
assert_fields_close(du_half_h2e, -div_s_h2e)
if u_eprev is None:
u_eprev = u_estep
continue
# previous half-step
delta_j_A = fdtd.delta_energy_j(j0=sim.js[ii], e1=sim.es[ii-1], dxes=sim.dxes)
du_half_e2h = u_hstep - u_eprev - delta_j_A
div_s_e2h = sim.dt * fdtd.poynting_divergence(e=sim.es[ii-1], h=sim.hs[ii], dxes=sim.dxes)
assert_fields_close(du_half_e2h, -div_s_e2h)
u_eprev = u_estep
def test_poynting_planes(sim):
mask = (sim.js[0] != 0).any(axis=0)
if mask.sum() > 1:
pytest.skip('test_poynting_planes can only test single point sources, got {}'.format(mask.sum()))
args = {'dxes': sim.dxes,
'epsilon': sim.epsilon}
mx = numpy.roll(mask, -1, axis=0)
my = numpy.roll(mask, -1, axis=1)
mz = numpy.roll(mask, -1, axis=2)
u_eprev = None
for ii in range(1, 8):
u_hstep = fdtd.energy_hstep(e0=sim.es[ii-1], h1=sim.hs[ii], e2=sim.es[ii], **args) # pylint: disable=bad-whitespace
u_estep = fdtd.energy_estep(h0=sim.hs[ii], e1=sim.es[ii], h2=sim.hs[ii + 1], **args) # pylint: disable=bad-whitespace
delta_j_B = fdtd.delta_energy_j(j0=sim.js[ii], e1=sim.es[ii], dxes=sim.dxes)
du_half_h2e = u_estep - u_hstep - delta_j_B
s_h2e = -fdtd.poynting(e=sim.es[ii], h=sim.hs[ii], dxes=sim.dxes) * sim.dt
planes = [s_h2e[0, mask].sum(), -s_h2e[0, mx].sum(),
s_h2e[1, mask].sum(), -s_h2e[1, my].sum(),
s_h2e[2, mask].sum(), -s_h2e[2, mz].sum()]
assert_close(sum(planes), du_half_h2e[mask])
if u_eprev is None:
u_eprev = u_estep
continue
delta_j_A = fdtd.delta_energy_j(j0=sim.js[ii], e1=sim.es[ii-1], dxes=sim.dxes)
du_half_e2h = u_hstep - u_eprev - delta_j_A
s_e2h = -fdtd.poynting(e=sim.es[ii - 1], h=sim.hs[ii], dxes=sim.dxes) * sim.dt
planes = [s_e2h[0, mask].sum(), -s_e2h[0, mx].sum(),
s_e2h[1, mask].sum(), -s_e2h[1, my].sum(),
s_e2h[2, mask].sum(), -s_e2h[2, mz].sum()]
assert_close(sum(planes), du_half_e2h[mask])
# previous half-step
u_eprev = u_estep
#####################################
# Test fixtures
#####################################
# Also see conftest.py
@pytest.fixture(params=[0.3])
def dt(request):
yield request.param
@dataclasses.dataclass()
class TDResult:
shape: Tuple[int]
dt: float
dxes: List[List[numpy.ndarray]]
epsilon: numpy.ndarray
j_distribution: numpy.ndarray
j_steps: Tuple[int]
es: List[numpy.ndarray] = dataclasses.field(default_factory=list)
hs: List[numpy.ndarray] = dataclasses.field(default_factory=list)
js: List[numpy.ndarray] = dataclasses.field(default_factory=list)
@pytest.fixture(params=[(0, 4, 8),]) #(0,)])
def j_steps(request):
yield request.param
@pytest.fixture(params=['center', 'random'])
def j_distribution(request, shape, j_mag):
j = numpy.zeros(shape)
if request.param == 'center':
j[:, shape[1]//2, shape[2]//2, shape[3]//2] = j_mag
elif request.param == '000':
j[:, 0, 0, 0] = j_mag
elif request.param == 'random':
j[:] = PRNG.uniform(low=-j_mag, high=j_mag, size=shape)
yield j
@pytest.fixture()
def sim(request, shape, epsilon, dxes, dt, j_distribution, j_steps):
is3d = (numpy.array(shape) == 1).sum() == 0
if is3d:
if dt != 0.3:
pytest.skip('Skipping dt != 0.3 because test is 3D (for speed)')
sim = TDResult(
shape=shape,
dt=dt,
dxes=dxes,
epsilon=epsilon,
j_distribution=j_distribution,
j_steps=j_steps,
)
e = numpy.zeros_like(epsilon)
h = numpy.zeros_like(epsilon)
assert 0 in j_steps
j_zeros = numpy.zeros_like(j_distribution)
eh2h = fdtd.maxwell_h(dt=dt, dxes=dxes)
eh2e = fdtd.maxwell_e(dt=dt, dxes=dxes)
for tt in range(10):
e = e.copy()
h = h.copy()
eh2h(e, h)
eh2e(e, h, epsilon)
if tt in j_steps:
e += j_distribution / epsilon
sim.js.append(j_distribution)
else:
sim.js.append(j_zeros)
sim.es.append(e)
sim.hs.append(h)
return sim

12
meanas/test/utils.py
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@ -1,12 +0,0 @@
import numpy
PRNG = numpy.random.RandomState(12345)
def assert_fields_close(x, y, *args, **kwargs):
numpy.testing.assert_allclose(x, y, verbose=False,
err_msg='Fields did not match:\n{}\n{}'.format(numpy.rollaxis(x, -1),
numpy.rollaxis(y, -1)), *args, **kwargs)
def assert_close(x, y, *args, **kwargs):
numpy.testing.assert_allclose(x, y, *args, **kwargs)

47
pdoc_templates/config.mako
View File

@ -1,47 +0,0 @@
<%!
# Template configuration. Copy over in your template directory
# (used with --template-dir) and adapt as required.
html_lang = 'en'
show_inherited_members = False
extract_module_toc_into_sidebar = True
list_class_variables_in_index = True
sort_identifiers = True
show_type_annotations = True
# Show collapsed source code block next to each item.
# Disabling this can improve rendering speed of large modules.
show_source_code = True
# If set, format links to objects in online source code repository
# according to this template. Supported keywords for interpolation
# are: commit, path, start_line, end_line.
#git_link_template = 'https://github.com/USER/PROJECT/blob/{commit}/{path}#L{start_line}-L{end_line}'
#git_link_template = 'https://gitlab.com/USER/PROJECT/blob/{commit}/{path}#L{start_line}-L{end_line}'
#git_link_template = 'https://bitbucket.org/USER/PROJECT/src/{commit}/{path}#lines-{start_line}:{end_line}'
#git_link_template = 'https://CGIT_HOSTNAME/PROJECT/tree/{path}?id={commit}#n{start_line}'
#git_link_template = None
git_link_template = 'https://mpxd.net/code/jan/fdfd_tools/src/commit/{commit}/{path}#L{start_line}-L{end_line}'
# A prefix to use for every HTML hyperlink in the generated documentation.
# No prefix results in all links being relative.
link_prefix = ''
# Enable syntax highlighting for code/source blocks by including Highlight.js
syntax_highlighting = True
# Set the style keyword such as 'atom-one-light' or 'github-gist'
# Options: https://github.com/highlightjs/highlight.js/tree/master/src/styles
# Demo: https://highlightjs.org/static/demo/
hljs_style = 'github'
# If set, insert Google Analytics tracking code. Value is GA
# tracking id (UA-XXXXXX-Y).
google_analytics = ''
# If set, render LaTeX math syntax within \(...\) (inline equations),
# or within \[...\] or $$...$$ or `.. math::` (block equations)
# as nicely-formatted math formulas using MathJax.
# Note: in Python docstrings, either all backslashes need to be escaped (\\)
# or you need to use raw r-strings.
latex_math = True
%>

389
pdoc_templates/css.mako
View File

@ -1,389 +0,0 @@
<%!
from pdoc.html_helpers import minify_css
%>
<%def name="mobile()" filter="minify_css">
.flex {
display: flex !important;
}
body {
line-height: 1.5em;
background: black;
color: #DDD;
}
#content {
padding: 20px;
}
#sidebar {
padding: 30px;
overflow: hidden;
}
.http-server-breadcrumbs {
font-size: 130%;
margin: 0 0 15px 0;
}
#footer {
font-size: .75em;
padding: 5px 30px;
border-top: 1px solid #ddd;
text-align: right;
}
#footer p {
margin: 0 0 0 1em;
display: inline-block;
}
#footer p:last-child {
margin-right: 30px;
}
h1, h2, h3, h4, h5 {
font-weight: 300;
}
h1 {
font-size: 2.5em;
line-height: 1.1em;
}
h2 {
font-size: 1.75em;
margin: 1em 0 .50em 0;
}
h3 {
font-size: 1.4em;
margin: 25px 0 10px 0;
}
h4 {
margin: 0;
font-size: 105%;
}
a {
color: #999;
text-decoration: none;
transition: color .3s ease-in-out;
}
a:hover {
color: #18d;
}
.title code {
font-weight: bold;
}
h2[id^="header-"] {
margin-top: 2em;
}
.ident {
color: #7ff;
}
pre code {
background: transparent;
font-size: .8em;
line-height: 1.4em;
}
code {
background: #0d0d0e;
padding: 1px 4px;
overflow-wrap: break-word;
}
h1 code { background: transparent }
pre {
background: #111;
border: 0;
border-top: 1px solid #ccc;
border-bottom: 1px solid #ccc;
margin: 1em 0;
padding: 1ex;
}
#http-server-module-list {
display: flex;
flex-flow: column;
}
#http-server-module-list div {
display: flex;
}
#http-server-module-list dt {
min-width: 10%;
}
#http-server-module-list p {
margin-top: 0;
}
.toc ul,
#index {
list-style-type: none;
margin: 0;
padding: 0;
}
#index code {
background: transparent;
}
#index h3 {
border-bottom: 1px solid #ddd;
}
#index ul {
padding: 0;
}
#index h4 {
font-weight: bold;
}
#index h4 + ul {
margin-bottom:.6em;
}
/* Make TOC lists have 2+ columns when viewport is wide enough.
Assuming ~20-character identifiers and ~30% wide sidebar. */
@media (min-width: 200ex) { #index .two-column { column-count: 2 } }
@media (min-width: 300ex) { #index .two-column { column-count: 3 } }
dl {
margin-bottom: 2em;
}
dl dl:last-child {
margin-bottom: 4em;
}
dd {
margin: 0 0 1em 3em;
}
#header-classes + dl > dd {
margin-bottom: 3em;
}
dd dd {
margin-left: 2em;
}
dd p {
margin: 10px 0;
}
.name {
background: #111;
font-weight: bold;
font-size: .85em;
padding: 5px 10px;
display: inline-block;
min-width: 40%;
}
.name:hover {
background: #101010;
}
.name > span:first-child {
white-space: nowrap;
}
.name.class > span:nth-child(2) {
margin-left: .4em;
}
.inherited {
color: #777;
border-left: 5px solid #eee;
padding-left: 1em;
}
.inheritance em {
font-style: normal;
font-weight: bold;
}
/* Docstrings titles, e.g. in numpydoc format */
.desc h2 {
font-weight: 400;
font-size: 1.25em;
}
.desc h3 {
font-size: 1em;
}
.desc dt code {
background: inherit; /* Don't grey-back parameters */
}
.source summary,
.git-link-div {
color: #aaa;
text-align: right;
font-weight: 400;
font-size: .8em;
text-transform: uppercase;
}
.source summary > * {
white-space: nowrap;
cursor: pointer;
}
.git-link {
color: inherit;
margin-left: 1em;
}
.source pre {
max-height: 500px;
overflow: auto;
margin: 0;
}
.source pre code {
font-size: 12px;
overflow: visible;
}
.hlist {
list-style: none;
}
.hlist li {
display: inline;
}
.hlist li:after {
content: ',\2002';
}
.hlist li:last-child:after {
content: none;
}
.hlist .hlist {
display: inline;
padding-left: 1em;
}
img {
max-width: 100%;
}
.admonition {
padding: .1em .5em;
margin-bottom: 1em;
}
.admonition-title {
font-weight: bold;
}
.admonition.note,
.admonition.info,
.admonition.important {
background: #610;
}
.admonition.todo,
.admonition.versionadded,
.admonition.tip,
.admonition.hint {
background: #202;
}
.admonition.warning,
.admonition.versionchanged,
.admonition.deprecated {
background: #02b;
}
.admonition.error,
.admonition.danger,
.admonition.caution {
background: darkpink;
}
</%def>
<%def name="desktop()" filter="minify_css">
@media screen and (min-width: 700px) {
#sidebar {
width: 30%;
}
#content {
width: 70%;
max-width: 100ch;
padding: 3em 4em;
border-left: 1px solid #ddd;
}
pre code {
font-size: 1em;
}
.item .name {
font-size: 1em;
}
main {
display: flex;
flex-direction: row-reverse;
justify-content: flex-end;
}
.toc ul ul,
#index ul {
padding-left: 1.5em;
}
.toc > ul > li {
margin-top: .5em;
}
}
</%def>
<%def name="print()" filter="minify_css">
@media print {
#sidebar h1 {
page-break-before: always;
}
.source {
display: none;
}
}
@media print {
* {
background: transparent !important;
color: #000 !important; /* Black prints faster: h5bp.com/s */
box-shadow: none !important;
text-shadow: none !important;
}
a[href]:after {
content: " (" attr(href) ")";
font-size: 90%;
}
/* Internal, documentation links, recognized by having a title,
don't need the URL explicity stated. */
a[href][title]:after {
content: none;
}
abbr[title]:after {
content: " (" attr(title) ")";
}
/*
* Don't show links for images, or javascript/internal links
*/
.ir a:after,
a[href^="javascript:"]:after,
a[href^="#"]:after {
content: "";
}
pre,
blockquote {
border: 1px solid #999;
page-break-inside: avoid;
}
thead {
display: table-header-group; /* h5bp.com/t */
}
tr,
img {
page-break-inside: avoid;
}
img {
max-width: 100% !important;
}
@page {
margin: 0.5cm;
}
p,
h2,
h3 {
orphans: 3;
widows: 3;
}
h1,
h2,
h3,
h4,
h5,
h6 {
page-break-after: avoid;
}
}
</%def>

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@ -1,445 +0,0 @@
<%
import os
import pdoc
from pdoc.html_helpers import extract_toc, glimpse, to_html as _to_html, format_git_link, _md, to_markdown
from markdown.inlinepatterns import InlineProcessor
from markdown.util import AtomicString, etree
def link(d, name=None, fmt='{}'):
name = fmt.format(name or d.qualname + ('()' if isinstance(d, pdoc.Function) else ''))
if not isinstance(d, pdoc.Doc) or isinstance(d, pdoc.External) and not external_links:
return name
url = d.url(relative_to=module, link_prefix=link_prefix,
top_ancestor=not show_inherited_members)
return '<a title="{}" href="{}">{}</a>'.format(d.refname, url, name)
# Altered latex delimeters (allow inline $...$, wrap in <eq></eq>)
class _MathPattern(InlineProcessor):
NAME = 'pdoc-math'
PATTERN = r'(?<!\S|\\)(?:\\\((.+?)\\\)|\\\[(.+?)\\\]|\$\$(.+?)\$\$|\$(\S.*?)\$)'
PRIORITY = 181 # Larger than that of 'escape' pattern
def handleMatch(self, m, data):
for value, is_block in zip(m.groups(), (False, True, True, False)):
if value:
break
wrapper = etree.Element('eq')
wrapper.text = AtomicString(value)
return wrapper, m.start(0), m.end(0)
def to_html(text: str):
if not latex_math and _MathPattern.NAME in _md.inlinePatterns:
_md.inlinePatterns.deregister(_MathPattern.NAME)
elif latex_math and _MathPattern.NAME not in _md.inlinePatterns:
_md.inlinePatterns.register(_MathPattern(_MathPattern.PATTERN),
_MathPattern.NAME,
_MathPattern.PRIORITY)
md = to_markdown(text, docformat='numpy,google', module=module, link=link)
return _md.reset().convert(md)
# def to_html(text):
# return _to_html(text, module=module, link=link, latex_math=latex_math)
%>
<%def name="ident(name)"><span class="ident">${name}</span></%def>
<%def name="show_source(d)">
% if (show_source_code or git_link_template) and d.source and d.obj is not getattr(d.inherits, 'obj', None):
<% git_link = format_git_link(git_link_template, d) %>
% if show_source_code:
<details class="source">
<summary>
<span>Expand source code</span>
% if git_link:
<a href="${git_link}" class="git-link">Browse git</a>
%endif
</summary>
<pre><code class="python">${d.source | h}</code></pre>
</details>
% elif git_link:
<div class="git-link-div"><a href="${git_link}" class="git-link">Browse git</a></div>
%endif
%endif
</%def>
<%def name="show_desc(d, short=False)">
<%
inherits = ' inherited' if d.inherits else ''
docstring = glimpse(d.docstring) if short or inherits else d.docstring
%>
% if d.inherits:
<p class="inheritance">
<em>Inherited from:</em>
% if hasattr(d.inherits, 'cls'):
<code>${link(d.inherits.cls)}</code>.<code>${link(d.inherits, d.name)}</code>
% else:
<code>${link(d.inherits)}</code>
% endif
</p>
% endif
<section class="desc${inherits}">${docstring | to_html}</section>
% if not isinstance(d, pdoc.Module):
${show_source(d)}
% endif
</%def>
<%def name="show_module_list(modules)">
<h1>Python module list</h1>
% if not modules:
<p>No modules found.</p>
% else:
<dl id="http-server-module-list">
% for name, desc in modules:
<div class="flex">
<dt><a href="${link_prefix}${name}">${name}</a></dt>
<dd>${desc | glimpse, to_html}</dd>
</div>
% endfor
</dl>
% endif
</%def>
<%def name="show_column_list(items)">
<%
two_column = len(items) >= 6 and all(len(i.name) < 20 for i in items)
%>
<ul class="${'two-column' if two_column else ''}">
% for item in items:
<li><code>${link(item, item.name)}</code></li>
% endfor
</ul>
</%def>
<%def name="show_module(module)">
<%
variables = module.variables(sort=sort_identifiers)
classes = module.classes(sort=sort_identifiers)
functions = module.functions(sort=sort_identifiers)
submodules = module.submodules()
%>
<%def name="show_func(f)">
<dt id="${f.refname}"><code class="name flex">
<%
params = ', '.join(f.params(annotate=show_type_annotations, link=link))
returns = show_type_annotations and f.return_annotation(link=link) or ''
if returns:
returns = ' ->\N{NBSP}' + returns
%>
<span>${f.funcdef()} ${ident(f.name)}</span>(<span>${params})${returns}</span>
</code></dt>
<dd>${show_desc(f)}</dd>
</%def>
<header>
% if http_server:
<nav class="http-server-breadcrumbs">
<a href="/">All packages</a>
<% parts = module.name.split('.')[:-1] %>
% for i, m in enumerate(parts):
<% parent = '.'.join(parts[:i+1]) %>
:: <a href="/${parent.replace('.', '/')}/">${parent}</a>
% endfor
</nav>
% endif
<h1 class="title">${'Namespace' if module.is_namespace else 'Module'} <code>${module.name}</code></h1>
</header>
<section id="section-intro">
${module.docstring | to_html}
${show_source(module)}
</section>
<section>
% if submodules:
<h2 class="section-title" id="header-submodules">Sub-modules</h2>
<dl>
% for m in submodules:
<dt><code class="name">${link(m)}</code></dt>
<dd>${show_desc(m, short=True)}</dd>
% endfor
</dl>
% endif
</section>
<section>
% if variables:
<h2 class="section-title" id="header-variables">Global variables</h2>
<dl>
% for v in variables:
<dt id="${v.refname}"><code class="name">var ${ident(v.name)}</code></dt>
<dd>${show_desc(v)}</dd>
% endfor
</dl>
% endif
</section>
<section>
% if functions:
<h2 class="section-title" id="header-functions">Functions</h2>
<dl>
% for f in functions:
${show_func(f)}
% endfor
</dl>
% endif
</section>
<section>
% if classes:
<h2 class="section-title" id="header-classes">Classes</h2>
<dl>
% for c in classes:
<%
class_vars = c.class_variables(show_inherited_members, sort=sort_identifiers)
smethods = c.functions(show_inherited_members, sort=sort_identifiers)
inst_vars = c.instance_variables(show_inherited_members, sort=sort_identifiers)
methods = c.methods(show_inherited_members, sort=sort_identifiers)
mro = c.mro()
subclasses = c.subclasses()
params = ', '.join(c.params(annotate=show_type_annotations, link=link))
%>
<dt id="${c.refname}"><code class="flex name class">
<span>class ${ident(c.name)}</span>
% if params:
<span>(</span><span>${params})</span>
% endif
</code></dt>
<dd>${show_desc(c)}
% if mro:
<h3>Ancestors</h3>
<ul class="hlist">
% for cls in mro:
<li>${link(cls)}</li>
% endfor
</ul>
%endif
% if subclasses:
<h3>Subclasses</h3>
<ul class="hlist">
% for sub in subclasses:
<li>${link(sub)}</li>
% endfor
</ul>
% endif
% if class_vars:
<h3>Class variables</h3>
<dl>
% for v in class_vars:
<dt id="${v.refname}"><code class="name">var ${ident(v.name)}</code></dt>
<dd>${show_desc(v)}</dd>
% endfor
</dl>
% endif
% if smethods:
<h3>Static methods</h3>
<dl>
% for f in smethods:
${show_func(f)}
% endfor
</dl>
% endif
% if inst_vars:
<h3>Instance variables</h3>
<dl>
% for v in inst_vars:
<dt id="${v.refname}"><code class="name">var ${ident(v.name)}</code></dt>
<dd>${show_desc(v)}</dd>
% endfor
</dl>
% endif
% if methods:
<h3>Methods</h3>
<dl>
% for f in methods:
${show_func(f)}
% endfor
</dl>
% endif
% if not show_inherited_members:
<%
members = c.inherited_members()
%>
% if members:
<h3>Inherited members</h3>
<ul class="hlist">
% for cls, mems in members:
<li><code><b>${link(cls)}</b></code>:
<ul class="hlist">
% for m in mems:
<li><code>${link(m, name=m.name)}</code></li>
% endfor
</ul>
</li>
% endfor
</ul>
% endif
% endif
</dd>
% endfor
</dl>
% endif
</section>
</%def>
<%def name="module_index(module)">
<%
variables = module.variables(sort=sort_identifiers)
classes = module.classes(sort=sort_identifiers)
functions = module.functions(sort=sort_identifiers)
submodules = module.submodules()
supermodule = module.supermodule
%>
<nav id="sidebar">
<%include file="logo.mako"/>
<h1>Index</h1>
${extract_toc(module.docstring) if extract_module_toc_into_sidebar else ''}
<ul id="index">
% if supermodule:
<li><h3>Super-module</h3>
<ul>
<li><code>${link(supermodule)}</code></li>
</ul>
</li>
% endif
% if submodules:
<li><h3><a href="#header-submodules">Sub-modules</a></h3>
<ul>
% for m in submodules:
<li><code>${link(m)}</code></li>
% endfor
</ul>
</li>
% endif
% if variables:
<li><h3><a href="#header-variables">Global variables</a></h3>
${show_column_list(variables)}
</li>
% endif
% if functions:
<li><h3><a href="#header-functions">Functions</a></h3>
${show_column_list(functions)}
</li>
% endif
% if classes:
<li><h3><a href="#header-classes">Classes</a></h3>
<ul>
% for c in classes:
<li>
<h4><code>${link(c)}</code></h4>
<%
members = c.functions(sort=sort_identifiers) + c.methods(sort=sort_identifiers)
if list_class_variables_in_index:
members += (c.instance_variables(sort=sort_identifiers) +
c.class_variables(sort=sort_identifiers))
if not show_inherited_members:
members = [i for i in members if not i.inherits]
if sort_identifiers:
members = sorted(members)
%>
% if members:
${show_column_list(members)}
% endif
</li>
% endfor
</ul>
</li>
% endif
</ul>
</nav>
</%def>
<!doctype html>
<html lang="${html_lang}">
<head>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, initial-scale=1, minimum-scale=1" />
<meta name="generator" content="pdoc ${pdoc.__version__}" />
<%
module_list = 'modules' in context.keys() # Whether we're showing module list in server mode
%>
% if module_list:
<title>Python module list</title>
<meta name="description" content="A list of documented Python modules." />
% else:
<title>${module.name} API documentation</title>
<meta name="description" content="${module.docstring | glimpse, trim, h}" />
% endif
<link href='https://mpxd.net/scripts/normalize.css/normalize.css' rel='stylesheet'>
<link href='https://mpxd.net/scripts/sanitize.css/sanitize.css' rel='stylesheet'>
% if syntax_highlighting:
<link href="https://mpxd.net/scripts/highlightjs/styles/${hljs_style}.min.css" rel="stylesheet">
%endif
<%namespace name="css" file="css.mako" />
<style>${css.mobile()}</style>
<style media="screen and (min-width: 700px)">${css.desktop()}</style>
<style media="print">${css.print()}</style>
% if google_analytics:
<script>
window.ga=window.ga||function(){(ga.q=ga.q||[]).push(arguments)};ga.l=+new Date;
ga('create', '${google_analytics}', 'auto'); ga('send', 'pageview');
</script><script async src='https://www.google-analytics.com/analytics.js'></script>
% endif
<%include file="head.mako"/>
</head>
<body>
<main>
% if module_list:
<article id="content">
${show_module_list(modules)}
</article>
% else:
<article id="content">
${show_module(module)}
</article>
${module_index(module)}
% endif
</main>
<footer id="footer">
<%include file="credits.mako"/>
<p>Generated by <a href="https://pdoc3.github.io/pdoc"><cite>pdoc</cite> ${pdoc.__version__}</a>.</p>
</footer>
% if syntax_highlighting:
<script src="https://mpxd.net/scripts/highlightjs/highlight.pack.js"></script>
<script>hljs.initHighlightingOnLoad()</script>
% endif
% if http_server and module: ## Auto-reload on file change in dev mode
<script>
setInterval(() =>
fetch(window.location.href, {
method: "HEAD",
cache: "no-store",
headers: {"If-None-Match": "${os.stat(module.obj.__file__).st_mtime}"},
}).then(response => response.ok && window.location.reload()), 700);
</script>
% endif
</body>
</html>

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pdoc_templates/pdf.mako
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@ -1,185 +0,0 @@
<%!
import re
import pdoc
from pdoc.html_helpers import to_markdown, format_git_link
def link(d, fmt='{}'):
name = fmt.format(d.qualname + ('()' if isinstance(d, pdoc.Function) else ''))
if isinstance(d, pdoc.External):
return name
return '[{}](#{})'.format(name, d.refname)
def _to_md(text, module):
text = to_markdown(text, module=module, link=link)
# Setext H2 headings to atx H2 headings
text = re.sub(r'\n(.+)\n-{3,}\n', r'\n## \1\n\n', text)
# Convert admonitions into simpler paragraphs, dedent contents
text = re.sub(r'^(?P<indent>( *))!!! \w+ \"([^\"]*)\"(.*(?:\n(?P=indent) +.*)*)',
lambda m: '{}**{}:** {}'.format(m.group(2), m.group(3),
re.sub('\n {,4}', '\n', m.group(4))),
text, flags=re.MULTILINE)
return text
def subh(text, level=2):
# Deepen heading levels so H2 becomes H4 etc.
return re.sub(r'\n(#+) +(.+)\n', r'\n%s\1 \2\n' % ('#' * level), text)
%>
<%def name="title(level, string, id=None)">
<% id = ' {#%s}' % id if id is not None else '' %>
${('#' * level) + ' ' + string + id}
</%def>
<%def name="funcdef(f)">
<%
returns = show_type_annotations and f.return_annotation() or ''
if returns:
returns = ' -> ' + returns
%>
> `${f.funcdef()} ${f.name}(${', '.join(f.params(annotate=show_type_annotations))})${returns}`
</%def>
<%def name="classdef(c)">
> `class ${c.name}(${', '.join(c.params(annotate=show_type_annotations))})`
</%def>
<%def name="show_source(d)">
% if (show_source_code or git_link_template) and d.source and d.obj is not getattr(d.inherits, 'obj', None):
<% git_link = format_git_link(git_link_template, d) %>
[[view code]](${git_link})
%endif
</%def>
---
description: |
API documentation for modules: ${', '.join(m.name for m in modules)}.
lang: en
classoption: oneside
geometry: margin=1in
papersize: a4
linkcolor: blue
links-as-notes: true
...
% for module in modules:
<%
submodules = module.submodules()
variables = module.variables()
functions = module.functions()
classes = module.classes()
def to_md(text):
return _to_md(text, module)
%>
-------------------------------------------
${title(1, ('Namespace' if module.is_namespace else 'Module') + ' `%s`' % module.name, module.refname)}
${module.docstring | to_md}
% if submodules:
${title(2, 'Sub-modules')}
% for m in submodules:
* [${m.name}](#${m.refname})
% endfor
% endif
% if variables:
${title(2, 'Variables')}
% for v in variables:
${title(3, 'Variable `%s`' % v.name, v.refname)}
${show_source(v)}
${v.docstring | to_md, subh, subh}
% endfor
% endif
% if functions:
${title(2, 'Functions')}
% for f in functions:
${title(3, 'Function `%s`' % f.name, f.refname)}
${show_source(f)}
${funcdef(f)}
${f.docstring | to_md, subh, subh}
% endfor
% endif
% if classes:
${title(2, 'Classes')}
% for cls in classes:
${title(3, 'Class `%s`' % cls.name, cls.refname)}
${show_source(cls)}
${classdef(cls)}
${cls.docstring | to_md, subh}
<%
class_vars = cls.class_variables(show_inherited_members, sort=sort_identifiers)
static_methods = cls.functions(show_inherited_members, sort=sort_identifiers)
inst_vars = cls.instance_variables(show_inherited_members, sort=sort_identifiers)
methods = cls.methods(show_inherited_members, sort=sort_identifiers)
mro = cls.mro()
subclasses = cls.subclasses()
%>
% if mro:
${title(4, 'Ancestors (in MRO)')}
% for c in mro:
* [${c.refname}](#${c.refname})
% endfor
% endif
% if subclasses:
${title(4, 'Descendants')}
% for c in subclasses:
* [${c.refname}](#${c.refname})
% endfor
% endif
% if class_vars:
${title(4, 'Class variables')}
% for v in class_vars:
${title(5, 'Variable `%s`' % v.name, v.refname)}
${v.docstring | to_md, subh, subh}
% endfor
% endif
% if inst_vars:
${title(4, 'Instance variables')}
% for v in inst_vars:
${title(5, 'Variable `%s`' % v.name, v.refname)}
${v.docstring | to_md, subh, subh}
% endfor
% endif
% if static_methods:
${title(4, 'Static methods')}
% for f in static_methods:
${title(5, '`Method %s`' % f.name, f.refname)}
${funcdef(f)}
${f.docstring | to_md, subh, subh}
% endfor
% endif
% if methods:
${title(4, 'Methods')}
% for f in methods:
${title(5, 'Method `%s`' % f.name, f.refname)}
${funcdef(f)}
${f.docstring | to_md, subh, subh}
% endfor
% endif
% endfor
% endif
##\## for module in modules:
% endfor
-----
Generated by *pdoc* ${pdoc.__version__} (<https://pdoc3.github.io>).

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@ -1,381 +0,0 @@
.flex {
display: flex !important;
}
body {
line-height: 1.5em;
background: black;
color: #DDD;
max-width: 140ch;
}
#content {
padding: 20px;
}
#sidebar {
padding: 30px;
overflow: hidden;
}
.http-server-breadcrumbs {
font-size: 130%;
margin: 0 0 15px 0;
}
#footer {
font-size: .75em;
padding: 5px 30px;
border-top: 1px solid #ddd;
text-align: right;
}
#footer p {
margin: 0 0 0 1em;
display: inline-block;
}
#footer p:last-child {
margin-right: 30px;
}
h1, h2, h3, h4, h5 {
font-weight: 300;
}
h1 {
font-size: 2.5em;
line-height: 1.1em;
border-top: 20px white;
}
h2 {
font-size: 1.75em;
margin: 1em 0 .50em 0;
}
h3 {
font-size: 1.4em;
margin: 25px 0 10px 0;
}
h4 {
margin: 0;
font-size: 105%;
}
a {
color: #999;
text-decoration: none;
transition: color .3s ease-in-out;
}
a:hover {
color: #18d;
}
.title code {
font-weight: bold;
}
h2[id^="header-"] {
margin-top: 2em;
}
.ident {
color: #7ff;
}
pre code {
background: transparent;
font-size: .8em;
line-height: 1.4em;
}
code {
background: #0d0d0e;
padding: 1px 4px;
overflow-wrap: break-word;
}
h1 code { background: transparent }
pre {
background: #111;
border: 0;
border-top: 1px solid #ccc;
border-bottom: 1px solid #ccc;
margin: 1em 0;
padding: 1ex;
}
#http-server-module-list {
display: flex;
flex-flow: column;
}
#http-server-module-list div {
display: flex;
}
#http-server-module-list dt {
min-width: 10%;
}
#http-server-module-list p {
margin-top: 0;
}
.toc ul,
#index {
list-style-type: none;
margin: 0;
padding: 0;
}
#index code {
background: transparent;
}
#index h3 {
border-bottom: 1px solid #ddd;
}
#index ul {
padding: 0;
}
#index h4 {
font-weight: bold;
}
#index h4 + ul {
margin-bottom:.6em;
}
/* Make TOC lists have 2+ columns when viewport is wide enough.
Assuming ~20-character identifiers and ~30% wide sidebar. */
@media (min-width: 200ex) { #index .two-column { column-count: 2 } }
@media (min-width: 300ex) { #index .two-column { column-count: 3 } }
dl {
margin-bottom: 2em;
}
dl dl:last-child {
margin-bottom: 4em;
}
dd {
margin: 0 0 1em 3em;
}
#header-classes + dl > dd {
margin-bottom: 3em;
}
dd dd {
margin-left: 2em;
}
dd p {
margin: 10px 0;
}
blockquote code {
background: #111;
font-weight: bold;
font-size: .85em;
padding: 5px 10px;
display: inline-block;
min-width: 40%;
}
blockquote code:hover {
background: #101010;
}
.name > span:first-child {
white-space: nowrap;
}
.name.class > span:nth-child(2) {
margin-left: .4em;
}
.inherited {
color: #777;
border-left: 5px solid #eee;
padding-left: 1em;
}
.inheritance em {
font-style: normal;
font-weight: bold;
}
/* Docstrings titles, e.g. in numpydoc format */
.desc h2 {
font-weight: 400;
font-size: 1.25em;
}
.desc h3 {
font-size: 1em;
}
.desc dt code {
background: inherit; /* Don't grey-back parameters */
}
.source summary,
.git-link-div {
color: #aaa;
text-align: right;
font-weight: 400;
font-size: .8em;
text-transform: uppercase;
}
.source summary > * {
white-space: nowrap;
cursor: pointer;
}
.git-link {
color: inherit;
margin-left: 1em;
}
.source pre {
max-height: 500px;
overflow: auto;
margin: 0;
}
.source pre code {
font-size: 12px;
overflow: visible;
}
.hlist {
list-style: none;
}
.hlist li {
display: inline;
}
.hlist li:after {
content: ',\2002';
}
.hlist li:last-child:after {
content: none;
}
.hlist .hlist {
display: inline;
padding-left: 1em;
}
img {
max-width: 100%;
}
.admonition {
padding: .1em .5em;
margin-bottom: 1em;
}
.admonition-title {
font-weight: bold;
}
.admonition.note,
.admonition.info,
.admonition.important {
background: #610;
}
.admonition.todo,
.admonition.versionadded,
.admonition.tip,
.admonition.hint {
background: #202;
}
.admonition.warning,
.admonition.versionchanged,
.admonition.deprecated {
background: #02b;
}
.admonition.error,
.admonition.danger,
.admonition.caution {
background: darkpink;
}
@media screen and (min-width: 700px) {
#sidebar {
width: 30%;
}
#content {
width: 70%;
max-width: 100ch;
padding: 3em 4em;
border-left: 1px solid #ddd;
}
pre code {
font-size: 1em;
}
.item .name {
font-size: 1em;
}
main {
display: flex;
flex-direction: row-reverse;
justify-content: flex-end;
}
.toc ul ul,
#index ul {
padding-left: 1.5em;
}
.toc > ul > li {
margin-top: .5em;
}
}
@media print {
#sidebar h1 {
page-break-before: always;
}
.source {
display: none;
}
}
@media print {
* {
background: transparent !important;
color: #000 !important; /* Black prints faster: h5bp.com/s */
box-shadow: none !important;
text-shadow: none !important;
}
a[href]:after {
content: " (" attr(href) ")";
font-size: 90%;
}
/* Internal, documentation links, recognized by having a title,
don't need the URL explicity stated. */
a[href][title]:after {
content: none;
}
abbr[title]:after {
content: " (" attr(title) ")";
}
/*
* Don't show links for images, or javascript/internal links
*/
.ir a:after,
a[href^="javascript:"]:after,
a[href^="#"]:after {
content: "";
}
pre,
blockquote {
border: 1px solid #999;
page-break-inside: avoid;
}
thead {
display: table-header-group; /* h5bp.com/t */
}
tr,
img {
page-break-inside: avoid;
}
img {
max-width: 100% !important;
}
@page {
margin: 0.5cm;
}
p,
h2,
h3 {
orphans: 3;
widows: 3;
}
h1,
h2,
h3,
h4,
h5,
h6 {
page-break-after: avoid;
}
}

36
setup.py
View File

@ -1,44 +1,18 @@
#!/usr/bin/env python3
#!/usr/bin/env python
from setuptools import setup, find_packages
with open('README.md', 'r') as f:
long_description = f.read()
with open('meanas/VERSION', 'r') as f:
version = f.read().strip()
setup(name='meanas',
version=version,
description='Electromagnetic simulation tools',
long_description=long_description,
long_description_content_type='text/markdown',
setup(name='fdfd_tools',
version='0.4',
description='FDFD Electromagnetic simulation tools',
author='Jan Petykiewicz',
author_email='anewusername@gmail.com',
url='https://mpxd.net/code/jan/meanas',
url='https://mpxd.net/gogs/jan/fdfd_tools',
packages=find_packages(),
package_data={
'meanas': ['VERSION']
},
install_requires=[
'numpy',
'scipy',
],
extras_require={
'test': [
'pytest',
'dataclasses',
],
'examples': [
'gridlock',
],
},
classifiers=[
'Programming Language :: Python :: 3',
'Development Status :: 4 - Beta',
'Intended Audience :: Developers',
'Intended Audience :: Science/Research',
'License :: OSI Approved :: GNU Affero General Public License v3',
'Topic :: Scientific/Engineering :: Physics',
],
)