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310 lines
11 KiB
Python
310 lines
11 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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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 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(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 fdfd_tools.operators header (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 ** 2 * 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(v: 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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) -> Tuple[vfield_t, vfield_t]:
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"""
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Given a vector v containing the vectorized H_x and H_y fields,
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returns normalized, vectorized E and H fields for the system.
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:param v: Vector containing H_x and H_y fields
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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 fdfd_tools.operators header (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: Normalized, vectorized (e, h) containing all vector components.
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"""
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e = v2e(v, wavenumber, omega, dxes, epsilon, mu=mu)
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h = v2h(v, wavenumber, dxes, mu=mu)
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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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S1 = E[0] * numpy.roll(numpy.conj(H[1]), 1, axis=0) * dxes_real[0][1] * dxes_real[1][0]
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S2 = E[1] * numpy.roll(numpy.conj(H[0]), 1, axis=1) * dxes_real[0][0] * dxes_real[1][1]
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S = 0.25 * ((S1 + numpy.roll(S1, 1, axis=0)) -
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(S2 + numpy.roll(S2, 1, axis=1)))
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P = 0.5 * numpy.real(S.sum())
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assert P > 0, 'Found a mode propagating in the wrong direction! P={}'.format(P)
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norm_amplitude = 1 / numpy.sqrt(P)
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norm_angle = -numpy.angle(e[e.size//2])
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norm_factor = 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 v2h(v: numpy.ndarray,
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wavenumber: complex,
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dxes: dx_lists_t,
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mu: vfield_t = None
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) -> vfield_t:
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"""
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Given a vector v containing the vectorized H_x and H_y fields,
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returns a vectorized H including all three H components.
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:param v: Vector containing H_x and H_y fields
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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 fdfd_tools.operators header (2D)
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:param mu: Vectorized magnetic permeability grid (default 1 everywhere)
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:return: Vectorized H field with all vector components
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"""
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Dfx, Dfy = operators.deriv_forward(dxes[0])
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op = sparse.hstack((Dfx, Dfy))
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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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op = mu_z_inv @ op @ mu_xy
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w = op @ v / (1j * wavenumber)
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return numpy.hstack((v, w)).flatten()
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def v2e(v: 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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) -> vfield_t:
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"""
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Given a vector v containing the vectorized H_x and H_y fields,
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returns a vectorized E containing all three E components
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:param v: Vector containing H_x and H_y fields
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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 fdfd_tools.operators header (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: Vectorized E field with all vector components.
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"""
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h2eop = h2e(wavenumber, omega, dxes, epsilon)
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return h2eop @ v2h(v, wavenumber, dxes, mu)
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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 fdfd_tools.operators header (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 fdfd_tools.operators header (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 fdfd_tools.operators header (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 fdfd_tools.operators header (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 fdfd_tools.operators header (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 fdfd_tools.operators header (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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