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493 lines
18 KiB
Python
493 lines
18 KiB
Python
"""
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Various operators and helper functions for solving for waveguide modes.
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Assuming a z-dependence of the from exp(-i * wavenumber * z), we can simplify Maxwell's
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equations in the absence of sources to the form
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A @ [H_x, H_y] = wavenumber**2 * [H_x, H_y]
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with A =
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omega**2 * epsilon * mu +
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epsilon * [[-Dy], [Dx]] / epsilon * [-Dy, Dx] +
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[[Dx], [Dy]] / mu * [Dx, Dy] * mu
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which is the form used in this file.
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As the z-dependence is known, all the functions in this file assume a 2D grid
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(ie. dxes = [[[dx_e_0, dx_e_1, ...], [dy_e_0, ...]], [[dx_h_0, ...], [dy_h_0, ...]]])
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with propagation along the z axis.
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"""
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# TODO update module docs
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from typing import List, Tuple
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import numpy
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from numpy.linalg import norm
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import scipy.sparse as sparse
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from .. import vec, unvec, dx_lists_t, field_t, vfield_t
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from . import operators
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__author__ = 'Jan Petykiewicz'
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def operator_e(omega: complex,
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dxes: dx_lists_t,
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epsilon: vfield_t,
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mu: vfield_t = None,
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) -> sparse.spmatrix:
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if numpy.any(numpy.equal(mu, None)):
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mu = numpy.ones_like(epsilon)
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Dfx, Dfy = operators.deriv_forward(dxes[0])
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Dbx, Dby = operators.deriv_back(dxes[1])
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eps_parts = numpy.split(epsilon, 3)
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eps_xy = sparse.diags(numpy.hstack((eps_parts[0], eps_parts[1])))
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eps_z_inv = sparse.diags(1 / eps_parts[2])
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mu_parts = numpy.split(mu, 3)
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mu_yx = sparse.diags(numpy.hstack((mu_parts[1], mu_parts[0])))
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mu_z_inv = sparse.diags(1 / mu_parts[2])
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op = omega * omega * mu_yx @ eps_xy + \
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mu_yx @ sparse.vstack((-Dby, Dbx)) @ mu_z_inv @ sparse.hstack((-Dfy, Dfx)) + \
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sparse.vstack((Dfx, Dfy)) @ eps_z_inv @ sparse.hstack((Dbx, Dby)) @ eps_xy
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return op
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def operator_h(omega: complex,
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dxes: dx_lists_t,
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epsilon: vfield_t,
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mu: vfield_t = None,
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) -> sparse.spmatrix:
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"""
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Waveguide operator of the form
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omega**2 * epsilon * mu +
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epsilon * [[-Dy], [Dx]] / epsilon * [-Dy, Dx] +
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[[Dx], [Dy]] / mu * [Dx, Dy] * mu
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for use with a field vector of the form [H_x, H_y].
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This operator can be used to form an eigenvalue problem of the form
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A @ [H_x, H_y] = wavenumber**2 * [H_x, H_y]
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which can then be solved for the eigenmodes of the system (an exp(-i * wavenumber * z)
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z-dependence is assumed for the fields).
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:param omega: The angular frequency of the system
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:param dxes: Grid parameters [dx_e, dx_h] as described in meanas.types (2D)
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:param epsilon: Vectorized dielectric constant grid
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:param mu: Vectorized magnetic permeability grid (default 1 everywhere)
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:return: Sparse matrix representation of the operator
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"""
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if numpy.any(numpy.equal(mu, None)):
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mu = numpy.ones_like(epsilon)
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Dfx, Dfy = operators.deriv_forward(dxes[0])
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Dbx, Dby = operators.deriv_back(dxes[1])
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eps_parts = numpy.split(epsilon, 3)
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eps_yx = sparse.diags(numpy.hstack((eps_parts[1], eps_parts[0])))
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eps_z_inv = sparse.diags(1 / eps_parts[2])
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mu_parts = numpy.split(mu, 3)
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mu_xy = sparse.diags(numpy.hstack((mu_parts[0], mu_parts[1])))
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mu_z_inv = sparse.diags(1 / mu_parts[2])
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op = omega * omega * eps_yx @ mu_xy + \
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eps_yx @ sparse.vstack((-Dfy, Dfx)) @ eps_z_inv @ sparse.hstack((-Dby, Dbx)) + \
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sparse.vstack((Dbx, Dby)) @ mu_z_inv @ sparse.hstack((Dfx, Dfy)) @ mu_xy
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return op
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def normalized_fields_e(e_xy: numpy.ndarray,
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wavenumber: complex,
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omega: complex,
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dxes: dx_lists_t,
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epsilon: vfield_t,
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mu: vfield_t = None,
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prop_phase: float = 0,
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) -> Tuple[vfield_t, vfield_t]:
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"""
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Given a vector e_xy containing the vectorized E_x and E_y fields,
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returns normalized, vectorized E and H fields for the system.
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:param e_xy: Vector containing E_x and E_y fields
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:param wavenumber: Wavenumber satisfying `operator_e(...) @ e_xy == wavenumber**2 * e_xy`
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:param omega: The angular frequency of the system
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:param dxes: Grid parameters [dx_e, dx_h] as described in meanas.types (2D)
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:param epsilon: Vectorized dielectric constant grid
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:param mu: Vectorized magnetic permeability grid (default 1 everywhere)
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:param prop_phase: Phase shift (dz * corrected_wavenumber) over 1 cell in propagation direction.
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Default 0 (continuous propagation direction, i.e. dz->0).
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:return: Normalized, vectorized (e, h) containing all vector components.
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"""
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e = exy2e(wavenumber=wavenumber, dxes=dxes, epsilon=epsilon) @ e_xy
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h = exy2h(wavenumber=wavenumber, omega=omega, dxes=dxes, epsilon=epsilon, mu=mu) @ e_xy
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e_norm, h_norm = _normalized_fields(e=e, h=h, omega=omega, dxes=dxes, epsilon=epsilon,
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mu=mu, prop_phase=prop_phase)
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return e_norm, h_norm
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def normalized_fields_h(h_xy: numpy.ndarray,
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wavenumber: complex,
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omega: complex,
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dxes: dx_lists_t,
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epsilon: vfield_t,
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mu: vfield_t = None,
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prop_phase: float = 0,
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) -> Tuple[vfield_t, vfield_t]:
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"""
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Given a vector e_xy containing the vectorized E_x and E_y fields,
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returns normalized, vectorized E and H fields for the system.
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:param e_xy: Vector containing E_x and E_y fields
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:param wavenumber: Wavenumber satisfying `operator_e(...) @ e_xy == wavenumber**2 * e_xy`
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:param omega: The angular frequency of the system
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:param dxes: Grid parameters [dx_e, dx_h] as described in meanas.types (2D)
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:param epsilon: Vectorized dielectric constant grid
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:param mu: Vectorized magnetic permeability grid (default 1 everywhere)
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:param dxes_prop: Grid cell width in the propagation direction. Default 0 (continuous).
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:return: Normalized, vectorized (e, h) containing all vector components.
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"""
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e = hxy2e(wavenumber=wavenumber, omega=omega, dxes=dxes, epsilon=epsilon, mu=mu) @ h_xy
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h = hxy2h(wavenumber=wavenumber, dxes=dxes, mu=mu) @ h_xy
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e_norm, h_norm = _normalized_fields(e=e, h=h, omega=omega, dxes=dxes, epsilon=epsilon,
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mu=mu, prop_phase=prop_phase)
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return e_norm, h_norm
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def _normalized_fields(e: numpy.ndarray,
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h: numpy.ndarray,
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omega: complex,
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dxes: dx_lists_t,
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epsilon: vfield_t,
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mu: vfield_t = None,
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prop_phase: float = 0,
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) -> Tuple[vfield_t, vfield_t]:
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# TODO documentation
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shape = [s.size for s in dxes[0]]
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dxes_real = [[numpy.real(d) for d in numpy.meshgrid(*dxes[v], indexing='ij')] for v in (0, 1)]
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E = unvec(e, shape)
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H = unvec(h, shape)
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phase = numpy.exp(-1j * prop_phase / 2)
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S1 = E[0] * numpy.conj(H[1] * phase) * dxes_real[0][1] * dxes_real[1][0]
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S2 = E[1] * numpy.conj(H[0] * phase) * dxes_real[0][0] * dxes_real[1][1]
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P = numpy.real(S1.sum() - S2.sum())
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assert P > 0, 'Found a mode propagating in the wrong direction! P={}'.format(P)
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energy = epsilon * e.conj() * e
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norm_amplitude = 1 / numpy.sqrt(P)
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norm_angle = -numpy.angle(e[energy.argmax()]) # Will randomly add a negative sign when mode is symmetric
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# Try to break symmetry to assign a consistent sign [experimental TODO]
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E_weighted = unvec(e * energy * numpy.exp(1j * norm_angle), shape)
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sign = numpy.sign(E_weighted[:, :max(shape[0]//2, 1), :max(shape[1]//2, 1)].real.sum())
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norm_factor = sign * norm_amplitude * numpy.exp(1j * norm_angle)
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e *= norm_factor
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h *= norm_factor
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return e, h
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def exy2h(wavenumber: complex,
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omega: complex,
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dxes: dx_lists_t,
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epsilon: vfield_t,
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mu: vfield_t = None
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) -> sparse.spmatrix:
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"""
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Operator which transforms the vector e_xy containing the vectorized E_x and E_y fields,
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into a vectorized H containing all three H components
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:param wavenumber: Wavenumber satisfying `operator_e(...) @ e_xy == wavenumber**2 * e_xy`
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:param omega: The angular frequency of the system
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:param dxes: Grid parameters [dx_e, dx_h] as described in meanas.types (2D)
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:param epsilon: Vectorized dielectric constant grid
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:param mu: Vectorized magnetic permeability grid (default 1 everywhere)
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:return: Sparse matrix representing the operator
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"""
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e2hop = e2h(wavenumber=wavenumber, omega=omega, dxes=dxes, mu=mu)
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return e2hop @ exy2e(wavenumber=wavenumber, dxes=dxes, epsilon=epsilon)
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def hxy2e(wavenumber: complex,
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omega: complex,
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dxes: dx_lists_t,
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epsilon: vfield_t,
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mu: vfield_t = None
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) -> sparse.spmatrix:
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"""
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Operator which transforms the vector h_xy containing the vectorized H_x and H_y fields,
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into a vectorized E containing all three E components
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:param wavenumber: Wavenumber satisfying `operator_h(...) @ h_xy == wavenumber**2 * h_xy`
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:param omega: The angular frequency of the system
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:param dxes: Grid parameters [dx_e, dx_h] as described in meanas.types (2D)
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:param epsilon: Vectorized dielectric constant grid
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:param mu: Vectorized magnetic permeability grid (default 1 everywhere)
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:return: Sparse matrix representing the operator
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"""
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h2eop = h2e(wavenumber=wavenumber, omega=omega, dxes=dxes, epsilon=epsilon)
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return h2eop @ hxy2h(wavenumber=wavenumber, dxes=dxes, mu=mu)
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def hxy2h(wavenumber: complex,
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dxes: dx_lists_t,
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mu: vfield_t = None
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) -> sparse.spmatrix:
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"""
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Operator which transforms the vector h_xy containing the vectorized H_x and H_y fields,
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into a vectorized H containing all three H components
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:param wavenumber: Wavenumber satisfying `operator_h(...) @ h_xy == wavenumber**2 * h_xy`
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:param dxes: Grid parameters [dx_e, dx_h] as described in meanas.types (2D)
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:param mu: Vectorized magnetic permeability grid (default 1 everywhere)
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:return: Sparse matrix representing the operator
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"""
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Dfx, Dfy = operators.deriv_forward(dxes[0])
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hxy2hz = sparse.hstack((Dfx, Dfy)) / (1j * wavenumber)
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if not numpy.any(numpy.equal(mu, None)):
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mu_parts = numpy.split(mu, 3)
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mu_xy = sparse.diags(numpy.hstack((mu_parts[0], mu_parts[1])))
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mu_z_inv = sparse.diags(1 / mu_parts[2])
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hxy2hz = mu_z_inv @ hxy2hz @ mu_xy
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n_pts = dxes[1][0].size * dxes[1][1].size
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op = sparse.vstack((sparse.eye(2 * n_pts),
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hxy2hz))
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return op
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def exy2e(wavenumber: complex,
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dxes: dx_lists_t,
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epsilon: vfield_t,
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) -> sparse.spmatrix:
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"""
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Operator which transforms the vector e_xy containing the vectorized E_x and E_y fields,
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into a vectorized E containing all three E components
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:param wavenumber: Wavenumber satisfying `operator_e(...) @ e_xy == wavenumber**2 * e_xy`
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:param dxes: Grid parameters [dx_e, dx_h] as described in meanas.types (2D)
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:param epsilon: Vectorized dielectric constant grid
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:return: Sparse matrix representing the operator
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"""
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Dbx, Dby = operators.deriv_back(dxes[1])
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exy2ez = sparse.hstack((Dbx, Dby)) / (1j * wavenumber)
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if not numpy.any(numpy.equal(epsilon, None)):
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epsilon_parts = numpy.split(epsilon, 3)
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epsilon_xy = sparse.diags(numpy.hstack((epsilon_parts[0], epsilon_parts[1])))
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epsilon_z_inv = sparse.diags(1 / epsilon_parts[2])
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exy2ez = epsilon_z_inv @ exy2ez @ epsilon_xy
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n_pts = dxes[0][0].size * dxes[0][1].size
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op = sparse.vstack((sparse.eye(2 * n_pts),
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exy2ez))
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return op
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def e2h(wavenumber: complex,
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omega: complex,
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dxes: dx_lists_t,
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mu: vfield_t = None
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) -> sparse.spmatrix:
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"""
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Returns an operator which, when applied to a vectorized E eigenfield, produces
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the vectorized H eigenfield.
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:param wavenumber: Wavenumber satisfying A @ v == wavenumber**2 * v
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:param omega: The angular frequency of the system
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:param dxes: Grid parameters [dx_e, dx_h] as described in meanas.types (2D)
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:param mu: Vectorized magnetic permeability grid (default 1 everywhere)
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:return: Sparse matrix representation of the operator
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"""
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op = curl_e(wavenumber, dxes) / (-1j * omega)
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if not numpy.any(numpy.equal(mu, None)):
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op = sparse.diags(1 / mu) @ op
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return op
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def h2e(wavenumber: complex,
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omega: complex,
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dxes: dx_lists_t,
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epsilon: vfield_t
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) -> sparse.spmatrix:
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"""
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Returns an operator which, when applied to a vectorized H eigenfield, produces
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the vectorized E eigenfield.
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:param wavenumber: Wavenumber satisfying A @ v == wavenumber**2 * v
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:param omega: The angular frequency of the system
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:param dxes: Grid parameters [dx_e, dx_h] as described in meanas.types (2D)
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:param epsilon: Vectorized dielectric constant grid
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:return: Sparse matrix representation of the operator
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"""
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op = sparse.diags(1 / (1j * omega * epsilon)) @ curl_h(wavenumber, dxes)
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return op
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def curl_e(wavenumber: complex, dxes: dx_lists_t) -> sparse.spmatrix:
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"""
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Discretized curl operator for use with the waveguide E field.
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:param wavenumber: Wavenumber satisfying A @ v == wavenumber**2 * v
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:param dxes: Grid parameters [dx_e, dx_h] as described in meanas.types (2D)
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:return: Sparse matrix representation of the operator
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"""
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n = 1
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for d in dxes[0]:
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n *= len(d)
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Bz = -1j * wavenumber * sparse.eye(n)
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Dfx, Dfy = operators.deriv_forward(dxes[0])
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return operators.cross([Dfx, Dfy, Bz])
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def curl_h(wavenumber: complex, dxes: dx_lists_t) -> sparse.spmatrix:
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"""
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Discretized curl operator for use with the waveguide H field.
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:param wavenumber: Wavenumber satisfying A @ v == wavenumber**2 * v
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:param dxes: Grid parameters [dx_e, dx_h] as described in meanas.types (2D)
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:return: Sparse matrix representation of the operator
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"""
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n = 1
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for d in dxes[1]:
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n *= len(d)
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Bz = -1j * wavenumber * sparse.eye(n)
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Dbx, Dby = operators.deriv_back(dxes[1])
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return operators.cross([Dbx, Dby, Bz])
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def h_err(h: vfield_t,
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wavenumber: complex,
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omega: complex,
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dxes: dx_lists_t,
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epsilon: vfield_t,
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mu: vfield_t = None
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) -> float:
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"""
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Calculates the relative error in the H field
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:param h: Vectorized H field
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:param wavenumber: Wavenumber satisfying A @ v == wavenumber**2 * v
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:param omega: The angular frequency of the system
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:param dxes: Grid parameters [dx_e, dx_h] as described in meanas.types (2D)
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:param epsilon: Vectorized dielectric constant grid
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:param mu: Vectorized magnetic permeability grid (default 1 everywhere)
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:return: Relative error norm(OP @ h) / norm(h)
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"""
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ce = curl_e(wavenumber, dxes)
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ch = curl_h(wavenumber, dxes)
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eps_inv = sparse.diags(1 / epsilon)
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if numpy.any(numpy.equal(mu, None)):
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op = ce @ eps_inv @ ch @ h - omega ** 2 * h
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else:
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op = ce @ eps_inv @ ch @ h - omega ** 2 * (mu * h)
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return norm(op) / norm(h)
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def e_err(e: vfield_t,
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wavenumber: complex,
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omega: complex,
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dxes: dx_lists_t,
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epsilon: vfield_t,
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mu: vfield_t = None
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) -> float:
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"""
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Calculates the relative error in the E field
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:param e: Vectorized E field
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:param wavenumber: Wavenumber satisfying A @ v == wavenumber**2 * v
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:param omega: The angular frequency of the system
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:param dxes: Grid parameters [dx_e, dx_h] as described in meanas.types (2D)
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:param epsilon: Vectorized dielectric constant grid
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:param mu: Vectorized magnetic permeability grid (default 1 everywhere)
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:return: Relative error norm(OP @ e) / norm(e)
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"""
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ce = curl_e(wavenumber, dxes)
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ch = curl_h(wavenumber, dxes)
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if numpy.any(numpy.equal(mu, None)):
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op = ch @ ce @ e - omega ** 2 * (epsilon * e)
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else:
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mu_inv = sparse.diags(1 / mu)
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op = ch @ mu_inv @ ce @ e - omega ** 2 * (epsilon * e)
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return norm(op) / norm(e)
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def cylindrical_operator(omega: complex,
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dxes: dx_lists_t,
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epsilon: vfield_t,
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r0: float,
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) -> sparse.spmatrix:
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"""
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Cylindrical coordinate waveguide operator of the form
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TODO
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for use with a field vector of the form [E_r, E_y].
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This operator can be used to form an eigenvalue problem of the form
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A @ [E_r, E_y] = wavenumber**2 * [E_r, E_y]
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which can then be solved for the eigenmodes of the system (an exp(-i * wavenumber * theta)
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theta-dependence is assumed for the fields).
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:param omega: The angular frequency of the system
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:param dxes: Grid parameters [dx_e, dx_h] as described in meanas.types (2D)
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:param epsilon: Vectorized dielectric constant grid
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:param r0: Radius of curvature for the simulation. This should be the minimum value of
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r within the simulation domain.
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:return: Sparse matrix representation of the operator
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"""
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Dfx, Dfy = operators.deriv_forward(dxes[0])
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Dbx, Dby = operators.deriv_back(dxes[1])
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rx = r0 + numpy.cumsum(dxes[0][0])
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ry = r0 + dxes[0][0]/2.0 + numpy.cumsum(dxes[1][0])
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tx = rx/r0
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ty = ry/r0
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Tx = sparse.diags(vec(tx[:, None].repeat(dxes[0][1].size, axis=1)))
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Ty = sparse.diags(vec(ty[:, None].repeat(dxes[1][1].size, axis=1)))
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eps_parts = numpy.split(epsilon, 3)
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eps_x = sparse.diags(eps_parts[0])
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eps_y = sparse.diags(eps_parts[1])
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eps_z_inv = sparse.diags(1 / eps_parts[2])
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pa = sparse.vstack((Dfx, Dfy)) @ Tx @ eps_z_inv @ sparse.hstack((Dbx, Dby))
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pb = sparse.vstack((Dfx, Dfy)) @ Tx @ eps_z_inv @ sparse.hstack((Dby, Dbx))
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a0 = Ty @ eps_x + omega**-2 * Dby @ Ty @ Dfy
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a1 = Tx @ eps_y + omega**-2 * Dbx @ Ty @ Dfx
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b0 = Dbx @ Ty @ Dfy
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b1 = Dby @ Ty @ Dfx
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diag = sparse.block_diag
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op = (omega**2 * diag((Tx, Ty)) + pa) @ diag((a0, a1)) + \
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- (sparse.bmat(((None, Ty), (Tx, None))) + omega**-2 * pb) @ diag((b0, b1))
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return op
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