302 lines
11 KiB
Python
302 lines
11 KiB
Python
from typing import Dict, List
|
|
import numpy
|
|
import scipy.sparse as sparse
|
|
import scipy.sparse.linalg as spalg
|
|
|
|
from . import vec, unvec, dx_lists_t, vfield_t, field_t
|
|
from . import operators, waveguide, functional
|
|
|
|
|
|
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
|
|
by using power iteration.
|
|
'''
|
|
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))
|
|
|
|
# Use power iteration for 20 steps to estimate the dominant eigenvector
|
|
v = numpy.random.rand(A_r.shape[0])
|
|
for _ in range(20):
|
|
v = A_r @ v
|
|
v /= numpy.linalg.norm(v)
|
|
|
|
lm_eigval = v @ A_r @ v
|
|
|
|
'''
|
|
Shift by the absolute value of the largest eigenvalue, then find a few of the
|
|
largest (shifted) eigenvalues. The 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)
|
|
'''
|
|
shifted_A_r = A_r + abs(lm_eigval) * sparse.eye(A_r.shape[0])
|
|
eigvals, eigvecs = spalg.eigs(shifted_A_r, which='LM', k=mode_number + 3, ncv=50)
|
|
|
|
# Pick the eigenvalue we want from the few we found
|
|
k = eigvals.argsort()[-(mode_number+1)]
|
|
v = eigvecs[:, k]
|
|
|
|
'''
|
|
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 = None
|
|
for _ in range(40):
|
|
eigval = v @ A @ v
|
|
if numpy.linalg.norm(A @ v - eigval * v) < 1e-13:
|
|
break
|
|
w = spalg.spsolve(A - eigval * sparse.eye(A.shape[0]), v)
|
|
v = w / numpy.linalg.norm(w)
|
|
|
|
# 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)
|