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@ -1,11 +1,5 @@
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# fdfd_tools
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** DEPRECATED **
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The functionality in this module is now provided by [meanas](https://mpxd.net/code/jan/meanas).
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-----------------------
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**fdfd_tools** is a python package containing utilities for
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creating and analyzing 2D and 3D finite-difference frequency-domain (FDFD)
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electromagnetic simulations.
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@ -1,68 +0,0 @@
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import numpy, scipy, gridlock, fdfd_tools
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from fdfd_tools import bloch
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from numpy.linalg import norm
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import logging
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logging.basicConfig(level=logging.DEBUG)
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logger = logging.getLogger(__name__)
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dx = 40
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x_period = 400
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y_period = z_period = 2000
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g = gridlock.Grid([numpy.arange(-x_period/2, x_period/2, dx),
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numpy.arange(-1000, 1000, dx),
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numpy.arange(-1000, 1000, dx)],
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shifts=numpy.array([[0,0,0]]),
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initial=1.445**2,
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periodic=True)
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g.draw_cuboid([0,0,0], [200e8, 220, 220], eps=3.47**2)
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#x_period = y_period = z_period = 13000
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#g = gridlock.Grid([numpy.arange(3), ]*3,
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# shifts=numpy.array([[0, 0, 0]]),
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# initial=2.0**2,
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# periodic=True)
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g2 = g.copy()
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g2.shifts = numpy.zeros((6,3))
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g2.grids = [numpy.zeros(g.shape) for _ in range(6)]
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epsilon = [g.grids[0],] * 3
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reciprocal_lattice = numpy.diag(1e6/numpy.array([x_period, y_period, z_period])) #cols are vectors
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#print('Finding k at 1550nm')
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#k, f = bloch.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=reciprocal_lattice,
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# epsilon=epsilon,
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# band=0)
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#
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#print("k={}, f={}, 1/f={}, k/f={}".format(k, f, 1/f, norm(reciprocal_lattice @ k) / f ))
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print('Finding f at [0.25, 0, 0]')
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for k0x in [.25]:
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k0 = numpy.array([k0x, 0, 0])
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kmag = norm(reciprocal_lattice @ k0)
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tolerance = (1e6/1550) * 1e-4/1.5 # df = f * dn_eff / n
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logger.info('tolerance {}'.format(tolerance))
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n, v = bloch.eigsolve(4, k0, G_matrix=reciprocal_lattice, epsilon=epsilon, tolerance=tolerance)
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v2e = bloch.hmn_2_exyz(k0, G_matrix=reciprocal_lattice, epsilon=epsilon)
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v2h = bloch.hmn_2_hxyz(k0, G_matrix=reciprocal_lattice, epsilon=epsilon)
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ki = bloch.generate_kmn(k0, reciprocal_lattice, g.shape)
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z = 0
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e = v2e(v[0])
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for i in range(3):
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g2.grids[i] += numpy.real(e[i])
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g2.grids[i+3] += numpy.imag(e[i])
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f = numpy.sqrt(numpy.real(numpy.abs(n))) # TODO
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print('k0x = {:3g}\n eigval = {}\n f = {}\n'.format(k0x, n, f))
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n_eff = norm(reciprocal_lattice @ k0) / f
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print('kmag/f = n_eff = {} \n wl = {}\n'.format(n_eff, 1/f ))
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@ -190,6 +190,7 @@ def test1(solver=generic_solver):
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s1x, s2x = poyntings(E)
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pyplot.plot(s1x[0].sum(axis=2).sum(axis=1))
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pyplot.hold(True)
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pyplot.plot(s2x[0].sum(axis=2).sum(axis=1))
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pyplot.show()
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@ -1,513 +0,0 @@
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'''
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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
|
||||
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
|
||||
|
||||
|
||||
@ -1,120 +0,0 @@
|
||||
"""
|
||||
Solvers for eigenvalue / eigenvector problems
|
||||
"""
|
||||
from typing import Tuple, List
|
||||
import numpy
|
||||
from numpy.linalg import norm
|
||||
from scipy import sparse
|
||||
import scipy.sparse.linalg as spalg
|
||||
|
||||
|
||||
def power_iteration(operator: sparse.spmatrix,
|
||||
guess_vector: numpy.ndarray = None,
|
||||
iterations: int = 20,
|
||||
) -> Tuple[complex, numpy.ndarray]:
|
||||
"""
|
||||
Use power iteration to estimate the dominant eigenvector of a matrix.
|
||||
|
||||
:param operator: Matrix to analyze.
|
||||
:param guess_vector: Starting point for the eigenvector. Default is a randomly chosen vector.
|
||||
:param iterations: Number of iterations to perform. Default 20.
|
||||
:return: (Largest-magnitude eigenvalue, Corresponding eigenvector estimate)
|
||||
"""
|
||||
if numpy.any(numpy.equal(guess_vector, None)):
|
||||
v = numpy.random.rand(operator.shape[0])
|
||||
else:
|
||||
v = guess_vector
|
||||
|
||||
for _ in range(iterations):
|
||||
v = operator @ v
|
||||
v /= norm(v)
|
||||
|
||||
lm_eigval = v.conj() @ (operator @ v)
|
||||
return lm_eigval, v
|
||||
|
||||
|
||||
def rayleigh_quotient_iteration(operator: sparse.spmatrix or spalg.LinearOperator,
|
||||
guess_vector: numpy.ndarray,
|
||||
iterations: int = 40,
|
||||
tolerance: float = 1e-13,
|
||||
solver=None,
|
||||
) -> Tuple[complex, numpy.ndarray]:
|
||||
"""
|
||||
Use Rayleigh quotient iteration to refine an eigenvector guess.
|
||||
|
||||
:param operator: Matrix to analyze.
|
||||
:param guess_vector: Eigenvector to refine.
|
||||
:param iterations: Maximum number of iterations to perform. Default 40.
|
||||
:param tolerance: Stop iteration if (A - I*eigenvalue) @ v < tolerance.
|
||||
Default 1e-13.
|
||||
:param solver: Solver function of the form x = solver(A, b).
|
||||
By default, use scipy.sparse.spsolve for sparse matrices and
|
||||
scipy.sparse.bicgstab for general LinearOperator instances.
|
||||
:return: (eigenvalue, eigenvector)
|
||||
"""
|
||||
try:
|
||||
_test = operator - sparse.eye(operator.shape[0])
|
||||
shift = lambda eigval: eigval * sparse.eye(operator.shape[0])
|
||||
if solver is None:
|
||||
solver = spalg.spsolve
|
||||
except TypeError:
|
||||
shift = lambda eigval: spalg.LinearOperator(shape=operator.shape,
|
||||
dtype=operator.dtype,
|
||||
matvec=lambda v: eigval * v)
|
||||
if solver is None:
|
||||
solver = lambda A, b: spalg.bicgstab(A, b)[0]
|
||||
|
||||
v = guess_vector
|
||||
v /= norm(v)
|
||||
for _ in range(iterations):
|
||||
eigval = v.conj() @ (operator @ v)
|
||||
if norm(operator @ v - eigval * v) < tolerance:
|
||||
break
|
||||
|
||||
shifted_operator = operator - shift(eigval)
|
||||
v = solver(shifted_operator, v)
|
||||
v /= norm(v)
|
||||
return eigval, v
|
||||
|
||||
|
||||
def signed_eigensolve(operator: sparse.spmatrix or spalg.LinearOperator,
|
||||
how_many: int,
|
||||
negative: bool = False,
|
||||
) -> Tuple[numpy.ndarray, numpy.ndarray]:
|
||||
"""
|
||||
Find the largest-magnitude positive-only (or negative-only) eigenvalues and
|
||||
eigenvectors of the provided matrix.
|
||||
|
||||
:param operator: Matrix to analyze.
|
||||
:param how_many: How many eigenvalues to find.
|
||||
:param negative: Whether to find negative-only eigenvalues.
|
||||
Default False (positive only).
|
||||
:return: (sorted list of eigenvalues, 2D ndarray of corresponding eigenvectors)
|
||||
eigenvectors[:, k] corresponds to the k-th eigenvalue
|
||||
"""
|
||||
# Use power iteration to estimate the dominant eigenvector
|
||||
lm_eigval, _ = power_iteration(operator)
|
||||
|
||||
'''
|
||||
Shift by the absolute value of the largest eigenvalue, then find a few of the
|
||||
largest-magnitude (shifted) eigenvalues. A positive 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)
|
||||
'''
|
||||
shift = numpy.abs(lm_eigval)
|
||||
if negative:
|
||||
shift *= -1
|
||||
|
||||
# Try to combine, use general LinearOperator if we fail
|
||||
try:
|
||||
shifted_operator = operator + shift * sparse.eye(operator.shape[0])
|
||||
except TypeError:
|
||||
shifted_operator = operator + spalg.LinearOperator(shape=operator.shape,
|
||||
matvec=lambda v: shift * v)
|
||||
|
||||
shifted_eigenvalues, eigenvectors = spalg.eigs(shifted_operator, which='LM', k=how_many, ncv=50)
|
||||
eigenvalues = shifted_eigenvalues - shift
|
||||
|
||||
k = eigenvalues.argsort()
|
||||
return eigenvalues[k], eigenvectors[:, k]
|
||||
|
||||
@ -1,220 +0,0 @@
|
||||
"""
|
||||
Functions for performing near-to-farfield transformation (and the reverse).
|
||||
"""
|
||||
from typing import Dict, List
|
||||
import numpy
|
||||
from numpy.fft import fft2, fftshift, fftfreq, ifft2, ifftshift
|
||||
from numpy import pi
|
||||
|
||||
|
||||
def near_to_farfield(E_near: List[numpy.ndarray],
|
||||
H_near: List[numpy.ndarray],
|
||||
dx: float,
|
||||
dy: float,
|
||||
padded_size: List[int] = None
|
||||
) -> Dict[str]:
|
||||
"""
|
||||
Compute the farfield, i.e. the distribution of the fields after propagation
|
||||
through several wavelengths of uniform medium.
|
||||
|
||||
The input fields should be complex phasors.
|
||||
|
||||
:param E_near: List of 2 ndarrays containing the 2D phasor field slices for the transverse
|
||||
E fields (e.g. [Ex, Ey] for calculating the farfield toward the z-direction).
|
||||
:param H_near: List of 2 ndarrays containing the 2D phasor field slices for the transverse
|
||||
H fields (e.g. [Hx, hy] for calculating the farfield towrad the z-direction).
|
||||
:param dx: Cell size along x-dimension, in units of wavelength.
|
||||
:param dy: Cell size along y-dimension, in units of wavelength.
|
||||
:param padded_size: Shape of the output. A single integer `n` will be expanded to `(n, n)`.
|
||||
Powers of 2 are most efficient for FFT computation.
|
||||
Default is the smallest power of 2 larger than the input, for each axis.
|
||||
:returns: Dict with keys
|
||||
'E_far': Normalized E-field farfield; multiply by
|
||||
(i k exp(-i k r) / (4 pi r)) to get the actual field value.
|
||||
'H_far': Normalized H-field farfield; multiply by
|
||||
(i k exp(-i k r) / (4 pi r)) to get the actual field value.
|
||||
'kx', 'ky': Wavevector values corresponding to the x- and y- axes in E_far and H_far,
|
||||
normalized to wavelength (dimensionless).
|
||||
'dkx', 'dky': step size for kx and ky, normalized to wavelength.
|
||||
'theta': arctan2(ky, kx) corresponding to each (kx, ky).
|
||||
This is the angle in the x-y plane, counterclockwise from above, starting from +x.
|
||||
'phi': arccos(kz / k) corresponding to each (kx, ky).
|
||||
This is the angle away from +z.
|
||||
"""
|
||||
|
||||
if not len(E_near) == 2:
|
||||
raise Exception('E_near must be a length-2 list of ndarrays')
|
||||
if not len(H_near) == 2:
|
||||
raise Exception('H_near must be a length-2 list of ndarrays')
|
||||
|
||||
s = E_near[0].shape
|
||||
if not all(s == f.shape for f in E_near + H_near):
|
||||
raise Exception('All fields must be the same shape!')
|
||||
|
||||
if padded_size is None:
|
||||
padded_size = (2**numpy.ceil(numpy.log2(s))).astype(int)
|
||||
if not hasattr(padded_size, '__len__'):
|
||||
padded_size = (padded_size, padded_size)
|
||||
|
||||
En_fft = [fftshift(fft2(fftshift(Eni), s=padded_size)) for Eni in E_near]
|
||||
Hn_fft = [fftshift(fft2(fftshift(Hni), s=padded_size)) for Hni in H_near]
|
||||
|
||||
# Propagation vectors kx, ky
|
||||
k = 2 * pi
|
||||
kxx = 2 * pi * fftshift(fftfreq(padded_size[0], dx))
|
||||
kyy = 2 * pi * fftshift(fftfreq(padded_size[1], dy))
|
||||
|
||||
kx, ky = numpy.meshgrid(kxx, kyy, indexing='ij')
|
||||
kxy2 = kx * kx + ky * ky
|
||||
kxy = numpy.sqrt(kxy2)
|
||||
kz = numpy.sqrt(k * k - kxy2)
|
||||
|
||||
sin_th = ky / kxy
|
||||
cos_th = kx / kxy
|
||||
cos_phi = kz / k
|
||||
|
||||
sin_th[numpy.logical_and(kx == 0, ky == 0)] = 0
|
||||
cos_th[numpy.logical_and(kx == 0, ky == 0)] = 1
|
||||
|
||||
# Normalized vector potentials N, L
|
||||
N = [-Hn_fft[1] * cos_phi * cos_th + Hn_fft[0] * cos_phi * sin_th,
|
||||
Hn_fft[1] * sin_th + Hn_fft[0] * cos_th]
|
||||
L = [ En_fft[1] * cos_phi * cos_th - En_fft[0] * cos_phi * sin_th,
|
||||
-En_fft[1] * sin_th - En_fft[0] * cos_th]
|
||||
|
||||
E_far = [-L[1] - N[0],
|
||||
L[0] - N[1]]
|
||||
H_far = [-E_far[1],
|
||||
E_far[0]]
|
||||
|
||||
theta = numpy.arctan2(ky, kx)
|
||||
phi = numpy.arccos(cos_phi)
|
||||
theta[numpy.logical_and(kx == 0, ky == 0)] = 0
|
||||
phi[numpy.logical_and(kx == 0, ky == 0)] = 0
|
||||
|
||||
# Zero fields beyond valid (phi, theta)
|
||||
invalid_ind = kxy2 >= k * k
|
||||
theta[invalid_ind] = 0
|
||||
phi[invalid_ind] = 0
|
||||
for i in range(2):
|
||||
E_far[i][invalid_ind] = 0
|
||||
H_far[i][invalid_ind] = 0
|
||||
|
||||
outputs = {
|
||||
'E': E_far,
|
||||
'H': H_far,
|
||||
'dkx': kx[1]-kx[0],
|
||||
'dky': ky[1]-ky[0],
|
||||
'kx': kx,
|
||||
'ky': ky,
|
||||
'theta': theta,
|
||||
'phi': phi,
|
||||
}
|
||||
|
||||
return outputs
|
||||
|
||||
|
||||
|
||||
def far_to_nearfield(E_far: List[numpy.ndarray],
|
||||
H_far: List[numpy.ndarray],
|
||||
dkx: float,
|
||||
dky: float,
|
||||
padded_size: List[int] = None
|
||||
) -> Dict[str]:
|
||||
"""
|
||||
Compute the farfield, i.e. the distribution of the fields after propagation
|
||||
through several wavelengths of uniform medium.
|
||||
|
||||
The input fields should be complex phasors.
|
||||
|
||||
:param E_far: List of 2 ndarrays containing the 2D phasor field slices for the transverse
|
||||
E fields (e.g. [Ex, Ey] for calculating the nearfield toward the z-direction).
|
||||
Fields should be normalized so that
|
||||
E_far = E_far_actual / (i k exp(-i k r) / (4 pi r))
|
||||
:param H_far: List of 2 ndarrays containing the 2D phasor field slices for the transverse
|
||||
H fields (e.g. [Hx, hy] for calculating the nearfield toward the z-direction).
|
||||
Fields should be normalized so that
|
||||
H_far = H_far_actual / (i k exp(-i k r) / (4 pi r))
|
||||
:param dkx: kx discretization, in units of wavelength.
|
||||
:param dky: ky discretization, in units of wavelength.
|
||||
:param padded_size: Shape of the output. A single integer `n` will be expanded to `(n, n)`.
|
||||
Powers of 2 are most efficient for FFT computation.
|
||||
Default is the smallest power of 2 larger than the input, for each axis.
|
||||
:returns: Dict with keys
|
||||
'E': E-field nearfield
|
||||
'H': H-field nearfield
|
||||
'dx', 'dy': spatial discretization, normalized to wavelength (dimensionless)
|
||||
"""
|
||||
|
||||
if not len(E_far) == 2:
|
||||
raise Exception('E_far must be a length-2 list of ndarrays')
|
||||
if not len(H_far) == 2:
|
||||
raise Exception('H_far must be a length-2 list of ndarrays')
|
||||
|
||||
s = E_far[0].shape
|
||||
if not all(s == f.shape for f in E_far + H_far):
|
||||
raise Exception('All fields must be the same shape!')
|
||||
|
||||
if padded_size is None:
|
||||
padded_size = (2**numpy.ceil(numpy.log2(s))).astype(int)
|
||||
if not hasattr(padded_size, '__len__'):
|
||||
padded_size = (padded_size, padded_size)
|
||||
|
||||
|
||||
k = 2 * pi
|
||||
kxs = fftshift(fftfreq(s[0], 1/(s[0] * dkx)))
|
||||
kys = fftshift(fftfreq(s[0], 1/(s[1] * dky)))
|
||||
|
||||
kx, ky = numpy.meshgrid(kxs, kys, indexing='ij')
|
||||
kxy2 = kx * kx + ky * ky
|
||||
kxy = numpy.sqrt(kxy2)
|
||||
|
||||
kz = numpy.sqrt(k * k - kxy2)
|
||||
|
||||
sin_th = ky / kxy
|
||||
cos_th = kx / kxy
|
||||
cos_phi = kz / k
|
||||
|
||||
sin_th[numpy.logical_and(kx == 0, ky == 0)] = 0
|
||||
cos_th[numpy.logical_and(kx == 0, ky == 0)] = 1
|
||||
|
||||
# Zero fields beyond valid (phi, theta)
|
||||
invalid_ind = kxy2 >= k * k
|
||||
theta[invalid_ind] = 0
|
||||
phi[invalid_ind] = 0
|
||||
for i in range(2):
|
||||
E_far[i][invalid_ind] = 0
|
||||
H_far[i][invalid_ind] = 0
|
||||
|
||||
|
||||
# Normalized vector potentials N, L
|
||||
L = [0.5 * E_far[1],
|
||||
-0.5 * E_far[0]]
|
||||
N = [L[1],
|
||||
-L[0]]
|
||||
|
||||
En_fft = [-( L[0] * sin_th + L[1] * cos_phi * cos_th)/cos_phi,
|
||||
-(-L[0] * cos_th + L[1] * cos_phi * sin_th)/cos_phi]
|
||||
|
||||
Hn_fft = [( N[0] * sin_th + N[1] * cos_phi * cos_th)/cos_phi,
|
||||
(-N[0] * cos_th + N[1] * cos_phi * sin_th)/cos_phi]
|
||||
|
||||
for i in range(2):
|
||||
En_fft[i][cos_phi == 0] = 0
|
||||
Hn_fft[i][cos_phi == 0] = 0
|
||||
|
||||
E_near = [ifftshift(ifft2(ifftshift(Ei), s=padded_size)) for Ei in En_fft]
|
||||
H_near = [ifftshift(ifft2(ifftshift(Hi), s=padded_size)) for Hi in Hn_fft]
|
||||
|
||||
dx = 2 * pi / (s[0] * dkx)
|
||||
dy = 2 * pi / (s[0] * dky)
|
||||
|
||||
outputs = {
|
||||
'E': E_near,
|
||||
'H': H_near,
|
||||
'dx': dx,
|
||||
'dy': dy,
|
||||
}
|
||||
|
||||
return outputs
|
||||
|
||||
@ -182,10 +182,10 @@ def eh_full(omega: complex,
|
||||
:param mu: Vectorized magnetic permeability (default 1 everywhere)
|
||||
:param pec: Vectorized mask specifying PEC cells. Any cells where pec != 0 are interpreted
|
||||
as containing a perfect electrical conductor (PEC).
|
||||
The PEC is applied per-field-component (i.e., pec.size == epsilon.size)
|
||||
The PEC is applied per-field-component (ie, pec.size == epsilon.size)
|
||||
:param pmc: Vectorized mask specifying PMC cells. Any cells where pmc != 0 are interpreted
|
||||
as containing a perfect magnetic conductor (PMC).
|
||||
The PMC is applied per-field-component (i.e., pmc.size == epsilon.size)
|
||||
The PMC is applied per-field-component (ie, pmc.size == epsilon.size)
|
||||
:return: Sparse matrix containing the wave operator
|
||||
"""
|
||||
if numpy.any(numpy.equal(pec, None)):
|
||||
@ -284,8 +284,7 @@ def m2j(omega: complex,
|
||||
|
||||
def rotation(axis: int, shape: List[int], shift_distance: int=1) -> sparse.spmatrix:
|
||||
"""
|
||||
Utility operator for performing a circular shift along a specified axis by a
|
||||
specified number of elements.
|
||||
Utility operator for performing a circular shift along a specified axis by 1 element.
|
||||
|
||||
:param axis: Axis to shift along. x=0, y=1, z=2
|
||||
:param shape: Shape of the grid being shifted
|
||||
@ -305,7 +304,7 @@ def rotation(axis: int, shape: List[int], shift_distance: int=1) -> sparse.spmat
|
||||
i_ind = numpy.arange(n)
|
||||
j_ind = numpy.ravel_multi_index(ijk, shape, order='C')
|
||||
|
||||
vij = (numpy.ones(n), (i_ind, j_ind.ravel(order='C')))
|
||||
vij = (numpy.ones(n), (i_ind, j_ind.flatten(order='C')))
|
||||
|
||||
d = sparse.csr_matrix(vij, shape=(n, n))
|
||||
|
||||
@ -349,7 +348,7 @@ def shift_with_mirror(axis: int, shape: List[int], shift_distance: int=1) -> spa
|
||||
if len(shape) == 3:
|
||||
j_ind += ijk[2] * shape[0] * shape[1]
|
||||
|
||||
vij = (numpy.ones(n), (i_ind, j_ind.ravel(order='C')))
|
||||
vij = (numpy.ones(n), (i_ind, j_ind.flatten(order='C')))
|
||||
|
||||
d = sparse.csr_matrix(vij, shape=(n, n))
|
||||
return d
|
||||
@ -370,7 +369,7 @@ def deriv_forward(dx_e: List[numpy.ndarray]) -> List[sparse.spmatrix]:
|
||||
def deriv(axis):
|
||||
return rotation(axis, shape, 1) - sparse.eye(n)
|
||||
|
||||
Ds = [sparse.diags(+1 / dx.ravel(order='C')) @ deriv(a)
|
||||
Ds = [sparse.diags(+1 / dx.flatten(order='C')) @ deriv(a)
|
||||
for a, dx in enumerate(dx_e_expanded)]
|
||||
|
||||
return Ds
|
||||
@ -391,7 +390,7 @@ def deriv_back(dx_h: List[numpy.ndarray]) -> List[sparse.spmatrix]:
|
||||
def deriv(axis):
|
||||
return rotation(axis, shape, -1) - sparse.eye(n)
|
||||
|
||||
Ds = [sparse.diags(-1 / dx.ravel(order='C')) @ deriv(a)
|
||||
Ds = [sparse.diags(-1 / dx.flatten(order='C')) @ deriv(a)
|
||||
for a, dx in enumerate(dx_h_expanded)]
|
||||
|
||||
return Ds
|
||||
@ -462,8 +461,8 @@ def poynting_e_cross(e: vfield_t, dxes: dx_lists_t) -> sparse.spmatrix:
|
||||
fx, fy, fz = [avgf(i, shape) for i in range(3)]
|
||||
bx, by, bz = [avgb(i, shape) for i in range(3)]
|
||||
|
||||
dxag = [dx.ravel(order='C') for dx in numpy.meshgrid(*dxes[0], indexing='ij')]
|
||||
dbgx, dbgy, dbgz = [sparse.diags(dx.ravel(order='C'))
|
||||
dxag = [dx.flatten(order='C') for dx in numpy.meshgrid(*dxes[0], indexing='ij')]
|
||||
dbgx, dbgy, dbgz = [sparse.diags(dx.flatten(order='C'))
|
||||
for dx in numpy.meshgrid(*dxes[1], indexing='ij')]
|
||||
|
||||
Ex, Ey, Ez = [sparse.diags(ei * da) for ei, da in zip(numpy.split(e, 3), dxag)]
|
||||
@ -491,8 +490,8 @@ def poynting_h_cross(h: vfield_t, dxes: dx_lists_t) -> sparse.spmatrix:
|
||||
fx, fy, fz = [avgf(i, shape) for i in range(3)]
|
||||
bx, by, bz = [avgb(i, shape) for i in range(3)]
|
||||
|
||||
dxbg = [dx.ravel(order='C') for dx in numpy.meshgrid(*dxes[1], indexing='ij')]
|
||||
dagx, dagy, dagz = [sparse.diags(dx.ravel(order='C'))
|
||||
dxbg = [dx.flatten(order='C') for dx in numpy.meshgrid(*dxes[1], indexing='ij')]
|
||||
dagx, dagy, dagz = [sparse.diags(dx.flatten(order='C'))
|
||||
for dx in numpy.meshgrid(*dxes[0], indexing='ij')]
|
||||
|
||||
Hx, Hy, Hz = [sparse.diags(hi * db) for hi, db in zip(numpy.split(h, 3), dxbg)]
|
||||
|
||||
@ -3,7 +3,6 @@ Solvers for FDFD problems.
|
||||
"""
|
||||
|
||||
from typing import List, Callable, Dict, Any
|
||||
import logging
|
||||
|
||||
import numpy
|
||||
from numpy.linalg import norm
|
||||
@ -12,9 +11,6 @@ import scipy.sparse.linalg
|
||||
from . import operators
|
||||
|
||||
|
||||
logger = logging.getLogger(__name__)
|
||||
|
||||
|
||||
def _scipy_qmr(A: scipy.sparse.csr_matrix,
|
||||
b: numpy.ndarray,
|
||||
**kwargs
|
||||
@ -33,20 +29,20 @@ def _scipy_qmr(A: scipy.sparse.csr_matrix,
|
||||
'''
|
||||
iter = 0
|
||||
|
||||
def log_residual(xk):
|
||||
def print_residual(xk):
|
||||
nonlocal iter
|
||||
iter += 1
|
||||
if iter % 100 == 0:
|
||||
logger.info('Solver residual at iteration {} : {}'.format(iter, norm(A @ xk - b)))
|
||||
print('Solver residual at iteration', iter, ':', norm(A @ xk - b))
|
||||
|
||||
if 'callback' in kwargs:
|
||||
def augmented_callback(xk):
|
||||
log_residual(xk)
|
||||
print_residual(xk)
|
||||
kwargs['callback'](xk)
|
||||
|
||||
kwargs['callback'] = augmented_callback
|
||||
else:
|
||||
kwargs['callback'] = log_residual
|
||||
kwargs['callback'] = print_residual
|
||||
|
||||
'''
|
||||
Run the actual solve
|
||||
@ -87,7 +83,7 @@ def generic(omega: complex,
|
||||
b: numpy.ndarray
|
||||
x: numpy.ndarray
|
||||
Default is a wrapped version of scipy.sparse.linalg.qmr()
|
||||
which doesn't return convergence info and logs the residual
|
||||
which doesn't return convergence info and prints the residual
|
||||
every 100 iterations.
|
||||
:param matrix_solver_opts: Passed as kwargs to matrix_solver(...)
|
||||
:return: E-field which solves the system.
|
||||
|
||||
@ -1,7 +1,7 @@
|
||||
"""
|
||||
Functions for moving between a vector field (list of 3 ndarrays, [f_x, f_y, f_z])
|
||||
and a 1D array representation of that field [f_x0, f_x1, f_x2,... f_y0,... f_z0,...].
|
||||
Vectorized versions of the field use row-major (ie., C-style) ordering.
|
||||
Vectorized versions of the field use column-major (ie., Fortran, Matlab) ordering.
|
||||
"""
|
||||
|
||||
|
||||
@ -27,7 +27,7 @@ def vec(f: field_t) -> vfield_t:
|
||||
"""
|
||||
if numpy.any(numpy.equal(f, None)):
|
||||
return None
|
||||
return numpy.hstack(tuple((fi.ravel(order='C') for fi in f)))
|
||||
return numpy.hstack(tuple((fi.flatten(order='C') for fi in f)))
|
||||
|
||||
|
||||
def unvec(v: vfield_t, shape: numpy.ndarray) -> field_t:
|
||||
|
||||
@ -23,7 +23,7 @@ import numpy
|
||||
from numpy.linalg import norm
|
||||
import scipy.sparse as sparse
|
||||
|
||||
from . import vec, unvec, dx_lists_t, field_t, vfield_t
|
||||
from . import unvec, dx_lists_t, field_t, vfield_t
|
||||
from . import operators
|
||||
|
||||
|
||||
@ -307,62 +307,3 @@ def e_err(e: vfield_t,
|
||||
op = ch @ mu_inv @ ce @ e - omega ** 2 * (epsilon * e)
|
||||
|
||||
return norm(op) / norm(e)
|
||||
|
||||
|
||||
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).
|
||||
|
||||
:param omega: The angular frequency of the system
|
||||
:param dxes: Grid parameters [dx_e, dx_h] as described in fdfd_tools.operators header (2D)
|
||||
:param epsilon: Vectorized dielectric constant grid
|
||||
:param r0: Radius of curvature for the simulation. This should be the minimum value of
|
||||
r within the simulation domain.
|
||||
:return: 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
|
||||
|
||||
|
||||
|
||||
|
||||
@ -1,10 +1,10 @@
|
||||
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
|
||||
from .eigensolvers import signed_eigensolve, rayleigh_quotient_iteration
|
||||
|
||||
|
||||
def solve_waveguide_mode_2d(mode_number: int,
|
||||
@ -12,12 +12,12 @@ def solve_waveguide_mode_2d(mode_number: int,
|
||||
dxes: dx_lists_t,
|
||||
epsilon: vfield_t,
|
||||
mu: vfield_t = None,
|
||||
wavenumber_correction: bool = True,
|
||||
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 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
|
||||
@ -29,19 +29,46 @@ def solve_waveguide_mode_2d(mode_number: int,
|
||||
|
||||
'''
|
||||
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))
|
||||
|
||||
eigvals, eigvecs = signed_eigensolve(A_r, mode_number+3)
|
||||
v = eigvecs[:, -(mode_number + 1)]
|
||||
# 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, v = rayleigh_quotient_iteration(A, v)
|
||||
|
||||
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)
|
||||
@ -272,69 +299,3 @@ def compute_overlap_e(E: field_t,
|
||||
overlap_e /= norm_factor * dx_forward
|
||||
|
||||
return unvec(overlap_e, E[0].shape)
|
||||
|
||||
|
||||
def solve_waveguide_mode_cylindrical(mode_number: int,
|
||||
omega: complex,
|
||||
dxes: dx_lists_t,
|
||||
epsilon: vfield_t,
|
||||
r0: float,
|
||||
wavenumber_correction: bool = True,
|
||||
) -> Dict[str, complex or field_t]:
|
||||
"""
|
||||
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 fdfd_tools.operators header.
|
||||
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.
|
||||
: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
|
||||
'''
|
||||
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)
|
||||
v = 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, v = rayleigh_quotient_iteration(A, v)
|
||||
|
||||
# Calculate the wave-vector (force the real part to be positive)
|
||||
wavenumber = numpy.sqrt(eigval)
|
||||
wavenumber *= numpy.sign(numpy.real(wavenumber))
|
||||
|
||||
'''
|
||||
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 the wavenumber increases.
|
||||
'''
|
||||
if wavenumber_correction:
|
||||
wavenumber -= 2 * numpy.sin(numpy.real(wavenumber / 2)) - numpy.real(wavenumber)
|
||||
|
||||
shape = [d.size for d in dxes[0]]
|
||||
v = numpy.hstack((v, numpy.zeros(shape[0] * shape[1])))
|
||||
fields = {
|
||||
'wavenumber': wavenumber,
|
||||
'E': unvec(v, shape),
|
||||
# 'E': unvec(e, shape),
|
||||
# 'H': unvec(h, shape),
|
||||
}
|
||||
|
||||
return fields
|
||||
|
||||
Loading…
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Reference in New Issue
Block a user