Jan Petykiewicz
d6e7e3dee1
- Split math into fdmath package - Rename waveguide into _2d _3d and _cyl variants - pdoc-based documentation
237 lines
8.5 KiB
Python
237 lines
8.5 KiB
Python
"""
|
|
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 .. import vec, unvec, dx_lists_t, vfield_t, field_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: field_t,
|
|
mu: field_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.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.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: field_t,
|
|
wavenumber: complex,
|
|
omega: complex,
|
|
dxes: dx_lists_t,
|
|
axis: int,
|
|
polarity: int,
|
|
slices: List[slice],
|
|
epsilon: field_t,
|
|
mu: field_t = None,
|
|
) -> field_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.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: field_t,
|
|
wavenumber: complex,
|
|
dxes: dx_lists_t,
|
|
axis: int,
|
|
polarity: int,
|
|
slices: List[slice],
|
|
) -> field_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.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: field_t,
|
|
wavenumber: complex,
|
|
dxes: dx_lists_t,
|
|
axis: int,
|
|
polarity: int,
|
|
slices: List[slice],
|
|
) -> field_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.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
|