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Add E-field versions of waveguide mode operators, rename v->e_xy or h_xy, and add ability to specify mode margin in solve_waveguide_mode_2d

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fdtd_extras
Jan Petykiewicz 5 years ago
parent 41bec05d4e
commit af8efd00eb
2 changed files with 170 additions and 51 deletions
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205
meanas/fdfd/waveguide.py
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@ -31,11 +31,36 @@ from . import operators
__author__ = 'Jan Petykiewicz'
def operator(omega: complex,
def operator_e(omega: complex,
dxes: dx_lists_t,
epsilon: vfield_t,
mu: vfield_t = None,
) -> sparse.spmatrix:
if numpy.any(numpy.equal(mu, None)):
mu = numpy.ones_like(epsilon)
Dfx, Dfy = operators.deriv_forward(dxes[0])
Dbx, Dby = operators.deriv_back(dxes[1])
eps_parts = numpy.split(epsilon, 3)
eps_xy = sparse.diags(numpy.hstack((eps_parts[0], eps_parts[1])))
eps_z_inv = sparse.diags(1 / eps_parts[2])
mu_parts = numpy.split(mu, 3)
mu_yx = sparse.diags(numpy.hstack((mu_parts[1], mu_parts[0])))
mu_z_inv = sparse.diags(1 / mu_parts[2])
op = omega * omega * mu_yx @ eps_xy + \
mu_yx @ sparse.vstack((-Dby, Dbx)) @ mu_z_inv @ sparse.hstack((-Dfy, Dfx)) + \
sparse.vstack((Dfx, Dfy)) @ eps_z_inv @ sparse.hstack((Dbx, Dby)) @ eps_xy
return op
def operator_h(omega: complex,
dxes: dx_lists_t,
epsilon: vfield_t,
mu: vfield_t = None,
) -> sparse.spmatrix:
"""
Waveguide operator of the form
@ -71,27 +96,27 @@ def operator(omega: complex,
mu_xy = sparse.diags(numpy.hstack((mu_parts[0], mu_parts[1])))
mu_z_inv = sparse.diags(1 / mu_parts[2])
op = omega ** 2 * eps_yx @ mu_xy + \
op = omega * omega * eps_yx @ mu_xy + \
eps_yx @ sparse.vstack((-Dfy, Dfx)) @ eps_z_inv @ sparse.hstack((-Dby, Dbx)) + \
sparse.vstack((Dbx, Dby)) @ mu_z_inv @ sparse.hstack((Dfx, Dfy)) @ mu_xy
return op
def normalized_fields(v: numpy.ndarray,
wavenumber: complex,
omega: complex,
dxes: dx_lists_t,
epsilon: vfield_t,
mu: vfield_t = None,
dx_prop: float = 0,
) -> Tuple[vfield_t, vfield_t]:
def normalized_fields_e(e_xy: numpy.ndarray,
wavenumber: complex,
omega: complex,
dxes: dx_lists_t,
epsilon: vfield_t,
mu: vfield_t = None,
dx_prop: float = 0,
) -> Tuple[vfield_t, vfield_t]:
"""
Given a vector v containing the vectorized H_x and H_y fields,
Given a vector e_xy containing the vectorized E_x and E_y fields,
returns normalized, vectorized E and H fields for the system.
:param v: Vector containing H_x and H_y fields
:param wavenumber: Wavenumber satisfying A @ v == wavenumber**2 * v
:param e_xy: Vector containing E_x and E_y fields
:param wavenumber: Wavenumber satisfying `operator_e(...) @ e_xy == wavenumber**2 * e_xy`
:param omega: The angular frequency of the system
:param dxes: Grid parameters [dx_e, dx_h] as described in meanas.types (2D)
:param epsilon: Vectorized dielectric constant grid
@ -99,9 +124,51 @@ def normalized_fields(v: numpy.ndarray,
:param dxes_prop: Grid cell width in the propagation direction. Default 0 (continuous).
:return: Normalized, vectorized (e, h) containing all vector components.
"""
e = v2e(v, wavenumber, omega, dxes, epsilon, mu=mu)
h = v2h(v, wavenumber, dxes, mu=mu)
e = exy2e(wavenumber=wavenumber, dxes=dxes, epsilon=epsilon) @ e_xy
h = exy2h(wavenumber=wavenumber, omega=omega, dxes=dxes, epsilon=epsilon, mu=mu) @ e_xy
e_norm, h_norm = _normalized_fields(e=e, h=h, wavenumber=wavenumber, omega=omega,
dxes=dxes, epsilon=epsilon, mu=mu, dx_prop=dx_prop)
return e_norm, h_norm
def normalized_fields_h(h_xy: numpy.ndarray,
wavenumber: complex,
omega: complex,
dxes: dx_lists_t,
epsilon: vfield_t,
mu: vfield_t = None,
dx_prop: float = 0,
) -> Tuple[vfield_t, vfield_t]:
"""
Given a vector e_xy containing the vectorized E_x and E_y fields,
returns normalized, vectorized E and H fields for the system.
:param e_xy: Vector containing E_x and E_y fields
:param wavenumber: Wavenumber satisfying `operator_e(...) @ e_xy == wavenumber**2 * e_xy`
:param omega: The angular frequency of the system
:param dxes: Grid parameters [dx_e, dx_h] as described in meanas.types (2D)
:param epsilon: Vectorized dielectric constant grid
:param mu: Vectorized magnetic permeability grid (default 1 everywhere)
:param dxes_prop: Grid cell width in the propagation direction. Default 0 (continuous).
:return: Normalized, vectorized (e, h) containing all vector components.
"""
e = hxy2e(wavenumber=wavenumber, omega=omega, dxes=dxes, epsilon=epsilon, mu=mu) @ h_xy
h = hxy2h(wavenumber=wavenumber, dxes=dxes, mu=mu) @ h_xy
e_norm, h_norm = _normalized_fields(e=e, h=h, wavenumber=wavenumber, omega=omega,
dxes=dxes, epsilon=epsilon, mu=mu, dx_prop=dx_prop)
return e_norm, h_norm
def _normalized_fields(e: numpy.ndarray,
h: numpy.ndarray,
wavenumber: complex,
omega: complex,
dxes: dx_lists_t,
epsilon: vfield_t,
mu: vfield_t = None,
dx_prop: float = 0,
) -> Tuple[vfield_t, vfield_t]:
# TODO documentation
shape = [s.size for s in dxes[0]]
dxes_real = [[numpy.real(d) for d in numpy.meshgrid(*dxes[v], indexing='ij')] for v in (0, 1)]
@ -131,56 +198,104 @@ def normalized_fields(v: numpy.ndarray,
return e, h
def v2h(v: numpy.ndarray,
wavenumber: complex,
dxes: dx_lists_t,
mu: vfield_t = None
) -> vfield_t:
def exy2h(wavenumber: complex,
omega: complex,
dxes: dx_lists_t,
epsilon: vfield_t,
mu: vfield_t = None
) -> sparse.spmatrix:
"""
Given a vector v containing the vectorized H_x and H_y fields,
returns a vectorized H including all three H components.
Operator which transforms the vector e_xy containing the vectorized E_x and E_y fields,
into a vectorized H containing all three H components
:param v: Vector containing H_x and H_y fields
:param wavenumber: Wavenumber satisfying A @ v == wavenumber**2 * v
:param wavenumber: Wavenumber satisfying `operator_e(...) @ e_xy == wavenumber**2 * e_xy`
:param omega: The angular frequency of the system
:param dxes: Grid parameters [dx_e, dx_h] as described in meanas.types (2D)
:param epsilon: Vectorized dielectric constant grid
:param mu: Vectorized magnetic permeability grid (default 1 everywhere)
:return: Sparse matrix representing the operator
"""
e2hop = e2h(wavenumber=wavenumber, omega=omega, dxes=dxes, mu=mu)
return e2hop @ exy2e(wavenumber=wavenumber, dxes=dxes, epsilon=epsilon)
def hxy2e(wavenumber: complex,
omega: complex,
dxes: dx_lists_t,
epsilon: vfield_t,
mu: vfield_t = None
) -> sparse.spmatrix:
"""
Operator which transforms the vector h_xy containing the vectorized H_x and H_y fields,
into a vectorized E containing all three E components
:param wavenumber: Wavenumber satisfying `operator_h(...) @ h_xy == wavenumber**2 * h_xy`
:param omega: The angular frequency of the system
:param dxes: Grid parameters [dx_e, dx_h] as described in meanas.types (2D)
:param epsilon: Vectorized dielectric constant grid
:param mu: Vectorized magnetic permeability grid (default 1 everywhere)
:return: Sparse matrix representing the operator
"""
h2eop = h2e(wavenumber=wavenumber, omega=omega, dxes=dxes, epsilon=epsilon)
return h2eop @ hxy2h(wavenumber=wavenumber, dxes=dxes, mu=mu)
def hxy2h(wavenumber: complex,
dxes: dx_lists_t,
mu: vfield_t = None
) -> sparse.spmatrix:
"""
Operator which transforms the vector h_xy containing the vectorized H_x and H_y fields,
into a vectorized H containing all three H components
:param wavenumber: Wavenumber satisfying `operator_h(...) @ h_xy == wavenumber**2 * h_xy`
:param dxes: Grid parameters [dx_e, dx_h] as described in meanas.types (2D)
:param mu: Vectorized magnetic permeability grid (default 1 everywhere)
:return: Vectorized H field with all vector components
:return: Sparse matrix representing the operator
"""
Dfx, Dfy = operators.deriv_forward(dxes[0])
op = sparse.hstack((Dfx, Dfy))
hxy2hz = sparse.hstack((Dfx, Dfy)) / (1j * wavenumber)
if not numpy.any(numpy.equal(mu, None)):
mu_parts = numpy.split(mu, 3)
mu_xy = sparse.diags(numpy.hstack((mu_parts[0], mu_parts[1])))
mu_z_inv = sparse.diags(1 / mu_parts[2])
op = mu_z_inv @ op @ mu_xy
hxy2hz = mu_z_inv @ hxy2hz @ mu_xy
w = op @ v / (1j * wavenumber)
return numpy.hstack((v, w)).flatten()
n_pts = dxes[1][0].size * dxes[1][1].size
op = sparse.vstack((sparse.eye(2 * n_pts),
hxy2hz))
return op
def v2e(v: numpy.ndarray,
wavenumber: complex,
omega: complex,
dxes: dx_lists_t,
epsilon: vfield_t,
mu: vfield_t = None
) -> vfield_t:
def exy2e(wavenumber: complex,
dxes: dx_lists_t,
epsilon: vfield_t,
) -> sparse.spmatrix:
"""
Given a vector v containing the vectorized H_x and H_y fields,
returns a vectorized E containing all three E components
Operator which transforms the vector e_xy containing the vectorized E_x and E_y fields,
into a vectorized E containing all three E components
:param v: Vector containing H_x and H_y fields
:param wavenumber: Wavenumber satisfying A @ v == wavenumber**2 * v
:param omega: The angular frequency of the system
:param wavenumber: Wavenumber satisfying `operator_e(...) @ e_xy == wavenumber**2 * e_xy`
:param dxes: Grid parameters [dx_e, dx_h] as described in meanas.types (2D)
:param epsilon: Vectorized dielectric constant grid
:param mu: Vectorized magnetic permeability grid (default 1 everywhere)
:return: Vectorized E field with all vector components.
:return: Sparse matrix representing the operator
"""
h2eop = h2e(wavenumber, omega, dxes, epsilon)
return h2eop @ v2h(v, wavenumber, dxes, mu)
Dbx, Dby = operators.deriv_back(dxes[1])
exy2ez = sparse.hstack((Dbx, Dby)) / (1j * wavenumber)
if not numpy.any(numpy.equal(epsilon, None)):
epsilon_parts = numpy.split(epsilon, 3)
epsilon_xy = sparse.diags(numpy.hstack((epsilon_parts[0], epsilon_parts[1])))
epsilon_z_inv = sparse.diags(1 / epsilon_parts[2])
exy2ez = epsilon_z_inv @ exy2ez @ epsilon_xy
n_pts = dxes[0][0].size * dxes[0][1].size
op = sparse.vstack((sparse.eye(2 * n_pts),
exy2ez))
return op
def e2h(wavenumber: complex,

16
meanas/fdfd/waveguide_mode.py
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@ -12,6 +12,7 @@ def solve_waveguide_mode_2d(mode_number: int,
dxes: dx_lists_t,
epsilon: vfield_t,
mu: vfield_t = None,
mode_margin: int = 2,
) -> Dict[str, complex or field_t]:
"""
Given a 2d region, attempts to solve for the eigenmode with the specified mode number.
@ -21,6 +22,9 @@ def solve_waveguide_mode_2d(mode_number: int,
:param dxes: Grid parameters [dx_e, dx_h] as described in meanas.types
:param epsilon: Dielectric constant
:param mu: Magnetic permeability (default 1 everywhere)
:param mode_margin: The eigensolver will actually solve for (mode_number + mode_margin)
modes, but only return the target mode. Increasing this value can improve the solver's
ability to find the correct mode. Default 2.
:return: {'E': List[numpy.ndarray], 'H': List[numpy.ndarray], 'wavenumber': complex}
"""
@ -28,23 +32,23 @@ def solve_waveguide_mode_2d(mode_number: int,
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.operator(numpy.real(omega), dxes_real, numpy.real(epsilon), numpy.real(mu))
A_r = waveguide.operator_e(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)]
eigvals, eigvecs = signed_eigensolve(A_r, mode_number + mode_margin)
exy = 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.operator(omega, dxes, epsilon, mu)
eigval, v = rayleigh_quotient_iteration(A, v)
A = waveguide.operator_e(omega, dxes, epsilon, mu)
eigval, exy = rayleigh_quotient_iteration(A, exy)
# Calculate the wave-vector (force the real part to be positive)
wavenumber = numpy.sqrt(eigval)
wavenumber *= numpy.sign(numpy.real(wavenumber))
e, h = waveguide.normalized_fields(v, wavenumber, omega, dxes, epsilon, mu)
e, h = waveguide.normalized_fields_e(exy, wavenumber, omega, dxes, epsilon, mu)
shape = [d.size for d in dxes[0]]
fields = {

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