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514 lines
19 KiB
Python
514 lines
19 KiB
Python
'''
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Bloch eigenmode solver/operators
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This module contains functions for generating and solving the
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3D Bloch eigenproblem. The approach is to transform the problem
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into the (spatial) fourier domain, transforming the equation
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1/mu * curl(1/eps * curl(H)) = (w/c)^2 H
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into
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conv(1/mu_k, ik x conv(1/eps_k, ik x H_k)) = (w/c)^2 H_k
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where:
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- the _k subscript denotes a 3D fourier transformed field
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- each component of H_k corresponds to a plane wave with wavevector k
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- x is the cross product
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- conv denotes convolution
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Since k and H are orthogonal for each plane wave, we can use each
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k to create an orthogonal basis (k, m, n), with k x m = n, and
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|m| = |n| = 1. The cross products are then simplified with
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k @ h = kx hx + ky hy + kz hz = 0 = hk
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h = hk + hm + hn = hm + hn
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k = kk + km + kn = kk = |k|
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k x h = (ky hz - kz hy,
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kz hx - kx hz,
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kx hy - ky hx)
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= ((k x h) @ k, (k x h) @ m, (k x h) @ n)_kmn
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= (0, (m x k) @ h, (n x k) @ h)_kmn # triple product ordering
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= (0, kk (-n @ h), kk (m @ h))_kmn # (m x k) = -|k| n, etc.
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= |k| (0, -h @ n, h @ m)_kmn
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k x h = (km hn - kn hm,
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kn hk - kk hn,
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kk hm - km hk)_kmn
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= (0, -kk hn, kk hm)_kmn
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= (-kk hn)(mx, my, mz) + (kk hm)(nx, ny, nz)
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= |k| (hm * (nx, ny, nz) - hn * (mx, my, mz))
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where h is shorthand for H_k, (...)_kmn deontes the (k, m, n) basis,
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and e.g. hm is the component of h in the m direction.
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We can also simplify conv(X_k, Y_k) as fftn(X * ifftn(Y_k)).
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Using these results and storing H_k as h = (hm, hn), we have
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e_xyz = fftn(1/eps * ifftn(|k| (hm * n - hn * m)))
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b_mn = |k| (-e_xyz @ n, e_xyz @ m)
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h_mn = fftn(1/mu * ifftn(b_m * m + b_n * n))
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which forms the operator from the left side of the equation.
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We can then use a preconditioned block Rayleigh iteration algorithm, as in
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SG Johnson and JD Joannopoulos, Block-iterative frequency-domain methods
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for Maxwell's equations in a planewave basis, Optics Express 8, 3, 173-190 (2001)
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(similar to that used in MPB) to find the eigenvectors for this operator.
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===
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Typically you will want to do something like
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recip_lattice = numpy.diag(1/numpy.array(epsilon[0].shape * dx))
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n, v = bloch.eigsolve(5, k0, recip_lattice, epsilon)
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f = numpy.sqrt(-numpy.real(n[0]))
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n_eff = norm(recip_lattice @ k0) / f
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v2e = bloch.hmn_2_exyz(k0, recip_lattice, epsilon)
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e_field = v2e(v[0])
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k, f = find_k(frequency=1/1550,
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tolerance=(1/1550 - 1/1551),
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direction=[1, 0, 0],
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G_matrix=recip_lattice,
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epsilon=epsilon,
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band=0)
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'''
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from typing import List, Tuple, Callable, Dict
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import logging
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import numpy
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from numpy.fft import fftn, ifftn, fftfreq
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import scipy
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import scipy.optimize
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from scipy.linalg import norm
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import scipy.sparse.linalg as spalg
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from .eigensolvers import rayleigh_quotient_iteration
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from . import field_t
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logger = logging.getLogger(__name__)
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def generate_kmn(k0: numpy.ndarray,
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G_matrix: numpy.ndarray,
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shape: numpy.ndarray
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) -> Tuple[numpy.ndarray, numpy.ndarray, numpy.ndarray]:
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"""
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Generate a (k, m, n) orthogonal basis for each k-vector in the simulation grid.
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:param k0: [k0x, k0y, k0z], Bloch wavevector, in G basis.
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:param G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns.
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:param shape: [nx, ny, nz] shape of the simulation grid.
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:return: (|k|, m, n) where |k| has shape tuple(shape) + (1,)
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and m, n have shape tuple(shape) + (3,).
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All are given in the xyz basis (e.g. |k|[0,0,0] = norm(G_matrix @ k0)).
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"""
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k0 = numpy.array(k0)
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Gi_grids = numpy.meshgrid(*(fftfreq(n, 1/n) for n in shape[:3]), indexing='ij')
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Gi = numpy.stack(Gi_grids, axis=3)
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k_G = k0[None, None, None, :] - Gi
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k_xyz = numpy.rollaxis(G_matrix @ numpy.rollaxis(k_G, 3, 2), 3, 2)
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m = numpy.broadcast_to([0, 1, 0], tuple(shape[:3]) + (3,)).astype(float)
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n = numpy.broadcast_to([0, 0, 1], tuple(shape[:3]) + (3,)).astype(float)
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xy_non0 = numpy.any(k_xyz[:, :, :, 0:1] != 0, axis=3)
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if numpy.any(xy_non0):
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u = numpy.cross(k_xyz[xy_non0], [0, 0, 1])
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m[xy_non0, :] = u / norm(u, axis=1)[:, None]
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z_non0 = numpy.any(k_xyz != 0, axis=3)
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if numpy.any(z_non0):
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v = numpy.cross(k_xyz[z_non0], m[z_non0])
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n[z_non0, :] = v / norm(v, axis=1)[:, None]
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k_mag = norm(k_xyz, axis=3)[:, :, :, None]
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return k_mag, m, n
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def maxwell_operator(k0: numpy.ndarray,
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G_matrix: numpy.ndarray,
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epsilon: field_t,
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mu: field_t = None
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) -> Callable[[numpy.ndarray], numpy.ndarray]:
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"""
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Generate the Maxwell operator
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conv(1/mu_k, ik x conv(1/eps_k, ik x ___))
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which is the spatial-frequency-space representation of
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1/mu * curl(1/eps * curl(___))
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The operator is a function that acts on a vector h_mn of size (2 * epsilon[0].size)
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See the module-level docstring for more information.
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:param k0: Bloch wavevector, [k0x, k0y, k0z].
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:param G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns.
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:param epsilon: Dielectric constant distribution for the simulation.
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All fields are sampled at cell centers (i.e., NOT Yee-gridded)
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:param mu: Magnetic permability distribution for the simulation.
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Default None (1 everywhere).
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:return: Function which applies the maxwell operator to h_mn.
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"""
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shape = epsilon[0].shape + (1,)
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k_mag, m, n = generate_kmn(k0, G_matrix, shape)
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epsilon = numpy.stack(epsilon, 3)
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if mu is not None:
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mu = numpy.stack(mu, 3)
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def operator(h: numpy.ndarray):
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"""
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Maxwell operator for Bloch eigenmode simulation.
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h is complex 2-field in (m, n) basis, vectorized
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:param h: Raveled h_mn; size (2 * epsilon[0].size).
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:return: Raveled conv(1/mu_k, ik x conv(1/eps_k, ik x h_mn)).
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"""
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hin_m, hin_n = [hi.reshape(shape) for hi in numpy.split(h, 2)]
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#{d,e,h}_xyz fields are complex 3-fields in (1/x, 1/y, 1/z) basis
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# cross product and transform into xyz basis
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d_xyz = (n * hin_m -
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m * hin_n) * k_mag
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# divide by epsilon
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e_xyz = fftn(ifftn(d_xyz, axes=range(3)) / epsilon, axes=range(3))
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# cross product and transform into mn basis
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b_m = numpy.sum(e_xyz * n, axis=3)[:, :, :, None] * -k_mag
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b_n = numpy.sum(e_xyz * m, axis=3)[:, :, :, None] * +k_mag
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if mu is None:
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h_m, h_n = b_m, b_n
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else:
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# transform from mn to xyz
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b_xyz = (m * b_m[:, :, :, None] +
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n * b_n[:, :, :, None])
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# divide by mu
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h_xyz = fftn(ifftn(b_xyz, axes=range(3)) / mu, axes=range(3))
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# transform back to mn
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h_m = numpy.sum(h_xyz * m, axis=3)
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h_n = numpy.sum(h_xyz * n, axis=3)
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return numpy.hstack((h_m.ravel(), h_n.ravel()))
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return operator
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def hmn_2_exyz(k0: numpy.ndarray,
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G_matrix: numpy.ndarray,
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epsilon: field_t,
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) -> Callable[[numpy.ndarray], field_t]:
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"""
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Generate an operator which converts a vectorized spatial-frequency-space
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h_mn into an E-field distribution, i.e.
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ifft(conv(1/eps_k, ik x h_mn))
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The operator is a function that acts on a vector h_mn of size (2 * epsilon[0].size)
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See the module-level docstring for more information.
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:param k0: Bloch wavevector, [k0x, k0y, k0z].
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:param G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns.
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:param epsilon: Dielectric constant distribution for the simulation.
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All fields are sampled at cell centers (i.e., NOT Yee-gridded)
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:return: Function for converting h_mn into E_xyz
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"""
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shape = epsilon[0].shape + (1,)
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epsilon = numpy.stack(epsilon, 3)
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k_mag, m, n = generate_kmn(k0, G_matrix, shape)
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def operator(h: numpy.ndarray) -> field_t:
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hin_m, hin_n = [hi.reshape(shape) for hi in numpy.split(h, 2)]
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d_xyz = (n * hin_m -
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m * hin_n) * k_mag
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# divide by epsilon
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return [ei for ei in numpy.rollaxis(ifftn(d_xyz, axes=range(3)) / epsilon, 3)]
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return operator
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def hmn_2_hxyz(k0: numpy.ndarray,
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G_matrix: numpy.ndarray,
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epsilon: field_t
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) -> Callable[[numpy.ndarray], field_t]:
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"""
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Generate an operator which converts a vectorized spatial-frequency-space
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h_mn into an H-field distribution, i.e.
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ifft(h_mn)
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The operator is a function that acts on a vector h_mn of size (2 * epsilon[0].size)
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See the module-level docstring for more information.
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:param k0: Bloch wavevector, [k0x, k0y, k0z].
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:param G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns.
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:param epsilon: Dielectric constant distribution for the simulation.
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Only epsilon[0].shape is used.
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:return: Function for converting h_mn into H_xyz
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"""
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shape = epsilon[0].shape + (1,)
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k_mag, m, n = generate_kmn(k0, G_matrix, shape)
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def operator(h: numpy.ndarray):
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hin_m, hin_n = [hi.reshape(shape) for hi in numpy.split(h, 2)]
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h_xyz = (m * hin_m +
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n * hin_n)
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return [ifftn(hi) for hi in numpy.rollaxis(h_xyz, 3)]
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return operator
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def inverse_maxwell_operator_approx(k0: numpy.ndarray,
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G_matrix: numpy.ndarray,
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epsilon: field_t,
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mu: field_t = None
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) -> Callable[[numpy.ndarray], numpy.ndarray]:
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"""
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Generate an approximate inverse of the Maxwell operator,
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ik x conv(eps_k, ik x conv(mu_k, ___))
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which can be used to improve the speed of ARPACK in shift-invert mode.
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See the module-level docstring for more information.
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:param k0: Bloch wavevector, [k0x, k0y, k0z].
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:param G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns.
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:param epsilon: Dielectric constant distribution for the simulation.
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All fields are sampled at cell centers (i.e., NOT Yee-gridded)
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:param mu: Magnetic permability distribution for the simulation.
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Default None (1 everywhere).
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:return: Function which applies the approximate inverse of the maxwell operator to h_mn.
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"""
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shape = epsilon[0].shape + (1,)
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epsilon = numpy.stack(epsilon, 3)
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k_mag, m, n = generate_kmn(k0, G_matrix, shape)
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if mu is not None:
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mu = numpy.stack(mu, 3)
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def operator(h: numpy.ndarray):
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"""
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Approximate inverse Maxwell operator for Bloch eigenmode simulation.
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h is complex 2-field in (m, n) basis, vectorized
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:param h: Raveled h_mn; size (2 * epsilon[0].size).
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:return: Raveled ik x conv(eps_k, ik x conv(mu_k, h_mn))
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"""
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hin_m, hin_n = [hi.reshape(shape) for hi in numpy.split(h, 2)]
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#{d,e,h}_xyz fields are complex 3-fields in (1/x, 1/y, 1/z) basis
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if mu is None:
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b_m, b_n = hin_m, hin_n
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else:
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# transform from mn to xyz
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h_xyz = (m * hin_m[:, :, :, None] +
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n * hin_n[:, :, :, None])
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# multiply by mu
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b_xyz = fftn(ifftn(h_xyz, axes=range(3)) * mu, axes=range(3))
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# transform back to mn
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b_m = numpy.sum(b_xyz * m, axis=3)
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b_n = numpy.sum(b_xyz * n, axis=3)
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# cross product and transform into xyz basis
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e_xyz = (n * b_m -
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m * b_n) / k_mag
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# multiply by epsilon
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d_xyz = fftn(ifftn(e_xyz, axes=range(3)) * epsilon, axes=range(3))
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# cross product and transform into mn basis crossinv_t2c
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h_m = numpy.sum(e_xyz * n, axis=3)[:, :, :, None] / +k_mag
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h_n = numpy.sum(e_xyz * m, axis=3)[:, :, :, None] / -k_mag
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return numpy.hstack((h_m.ravel(), h_n.ravel()))
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return operator
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def eigsolve(num_modes: int,
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k0: numpy.ndarray,
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G_matrix: numpy.ndarray,
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epsilon: field_t,
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mu: field_t = None,
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tolerance = 1e-8,
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) -> Tuple[numpy.ndarray, numpy.ndarray]:
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"""
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Find the first (lowest-frequency) num_modes eigenmodes with Bloch wavevector
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k0 of the specified structure.
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:param k0: Bloch wavevector, [k0x, k0y, k0z].
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:param G_matrix: 3x3 matrix, with reciprocal lattice vectors as columns.
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:param epsilon: Dielectric constant distribution for the simulation.
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All fields are sampled at cell centers (i.e., NOT Yee-gridded)
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:param mu: Magnetic permability distribution for the simulation.
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Default None (1 everywhere).
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:return: (eigenvalues, eigenvectors) where eigenvalues[i] corresponds to the
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vector eigenvectors[i, :]
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"""
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h_size = 2 * epsilon[0].size
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kmag = norm(G_matrix @ k0)
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'''
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Generate the operators
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'''
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mop = maxwell_operator(k0=k0, G_matrix=G_matrix, epsilon=epsilon, mu=mu)
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imop = inverse_maxwell_operator_approx(k0=k0, G_matrix=G_matrix, epsilon=epsilon, mu=mu)
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scipy_op = spalg.LinearOperator(dtype=complex, shape=(h_size, h_size), matvec=mop)
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scipy_iop = spalg.LinearOperator(dtype=complex, shape=(h_size, h_size), matvec=imop)
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y_shape = (h_size, num_modes)
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def rayleigh_quotient(Z: numpy.ndarray, approx_grad: bool = True):
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"""
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Absolute value of the block Rayleigh quotient, and the associated gradient.
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See Johnson and Joannopoulos, Opt. Expr. 8, 3 (2001) for details (full
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citation in module docstring).
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===
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Notes on my understanding of the procedure:
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Minimize f(Y) = |trace((Y.H @ A @ Y)|, making use of Y = Z @ inv(Z.H @ Z)^(1/2)
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(a polar orthogonalization of Y). This gives f(Z) = |trace(Z.H @ A @ Z @ U)|,
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where U = inv(Z.H @ Z). We minimize the absolute value to find the eigenvalues
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with smallest magnitude.
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The gradient is P @ (A @ Z @ U), where P = (1 - Z @ U @ Z.H) is a projection
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onto the space orthonormal to Z. If approx_grad is True, the approximate
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inverse of the maxwell operator is used to precondition the gradient.
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"""
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z = Z.view(dtype=complex).reshape(y_shape)
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U = numpy.linalg.inv(z.conj().T @ z)
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zU = z @ U
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AzU = scipy_op @ zU
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zTAzU = z.conj().T @ AzU
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f = numpy.real(numpy.trace(zTAzU))
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if approx_grad:
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df_dy = scipy_iop @ (AzU - zU @ zTAzU)
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else:
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df_dy = (AzU - zU @ zTAzU)
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df_dy_flat = df_dy.view(dtype=float).ravel()
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return numpy.abs(f), numpy.sign(f) * df_dy_flat
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'''
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Use the conjugate gradient method and the approximate gradient calculation to
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quickly find approximate eigenvectors.
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'''
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result = scipy.optimize.minimize(rayleigh_quotient,
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numpy.random.rand(*y_shape, 2),
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jac=True,
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method='L-BFGS-B',
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tol=1e-20,
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options={'maxiter': 2000, 'gtol':0, 'ftol':1e-20 , 'disp':True})#, 'maxls':80, 'm':30})
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result = scipy.optimize.minimize(lambda y: rayleigh_quotient(y, True),
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result.x,
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jac=True,
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method='L-BFGS-B',
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tol=1e-20,
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options={'maxiter': 2000, 'gtol':0, 'disp':True})
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result = scipy.optimize.minimize(lambda y: rayleigh_quotient(y, False),
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result.x,
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jac=True,
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method='L-BFGS-B',
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tol=1e-20,
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options={'maxiter': 2000, 'gtol':0, 'disp':True})
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for i in range(20):
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result = scipy.optimize.minimize(lambda y: rayleigh_quotient(y, False),
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result.x,
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jac=True,
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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
|
|
|
|
|