You cannot select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
310 lines
12 KiB
Python
310 lines
12 KiB
Python
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, functional
|
|
from ..eigensolvers import signed_eigensolve, rayleigh_quotient_iteration
|
|
|
|
|
|
def vsolve_waveguide_mode_2d(mode_number: int,
|
|
omega: complex,
|
|
dxes: dx_lists_t,
|
|
epsilon: vfield_t,
|
|
mu: vfield_t = None,
|
|
mode_margin: int = 2,
|
|
) -> Tuple[vfield_t, complex]:
|
|
"""
|
|
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 meanas.types
|
|
:param epsilon: Dielectric constant
|
|
:param mu: Magnetic permeability (default 1 everywhere)
|
|
:param mode_margin: The eigensolver will actually solve for (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.
|
|
:return: (e_xy, wavenumber)
|
|
"""
|
|
|
|
'''
|
|
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_e(numpy.real(omega), dxes_real, numpy.real(epsilon), numpy.real(mu))
|
|
|
|
eigvals, eigvecs = signed_eigensolve(A_r, mode_number + mode_margin)
|
|
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.operator_e(omega, dxes, epsilon, mu)
|
|
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))
|
|
|
|
return e_xy, wavenumber
|
|
|
|
|
|
|
|
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,
|
|
) -> 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 meanas.types
|
|
: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)
|
|
:return: {'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 = vsolve_waveguide_mode_2d(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)
|
|
print(wavenumber_2d / wavenumber)
|
|
|
|
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_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 wavenumber: Wavenumber of the mode
|
|
:param omega: Angular frequency of the simulation
|
|
:param dxes: Grid parameters [dx_e, dx_h] as described in meanas.types
|
|
: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
|
|
"""
|
|
E_expanded = expand_wgmode_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_waveguide_mode, calculates overlap_e for the
|
|
mode orthogonality relation Integrate(((E x H_mode) + (E_mode x H)) dot dn)
|
|
[assumes reflection symmetry].i
|
|
|
|
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 meanas.types
|
|
: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
|
|
"""
|
|
slices = tuple(slices)
|
|
|
|
Ee = expand_wgmode_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 solve_waveguide_mode_cylindrical(mode_number: int,
|
|
omega: complex,
|
|
dxes: dx_lists_t,
|
|
epsilon: vfield_t,
|
|
r0: float,
|
|
) -> Dict[str, complex or field_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.
|
|
|
|
: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 meanas.types.
|
|
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.
|
|
: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)
|
|
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
|
|
|
|
|
|
def expand_wgmode_e(E: field_t,
|
|
wavenumber: complex,
|
|
dxes: dx_lists_t,
|
|
axis: int,
|
|
polarity: int,
|
|
slices: List[slice],
|
|
) -> field_t:
|
|
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
|
|
|
|
|