514 lines
19 KiB
Python
514 lines
19 KiB
Python
'''
|
|
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
|
|
|
|
|