139 lines
4.6 KiB
Python
139 lines
4.6 KiB
Python
|
"""
|
||
|
|
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 .. import vec, unvec, dx_lists_t, field_t, vfield_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: 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).
|
||
|
|
|
||
|
|
Args:
|
||
|
|
omega: The angular frequency of the system
|
||
|
|
dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.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: 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.
|
||
|
|
|
||
|
|
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.
|
||
|
|
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
|