[eigensolvers] Increase number of lanczos vectors (ncv) based on number of requested eigenvalues

- Properly account for rmin vs r0
- Change return values to match waveguide_2d
- Change operator definition to look more like waveguide_2d

remaining TODO:
- Fix docs
- Further consolidate operators vs waveguide_2d
- Figure out E/H field conversions

examples/fdfd0.py

@@ -0,0 +1,103 @@
+import numpy
+from numpy.linalg import norm
+from matplotlib import pyplot, colors
+import logging
+
+import meanas
+from meanas import fdtd
+from meanas.fdmath import vec, unvec
+from meanas.fdfd import waveguide_3d, functional, scpml, operators
+from meanas.fdfd.solvers import generic as generic_solver
+
+import gridlock
+
+
+logging.basicConfig(level=logging.DEBUG)
+logging.getLogger('matplotlib').setLevel(logging.WARNING)
+
+__author__ = 'Jan Petykiewicz'
+
+
+def pcolor(ax, v) -> None:
+    mappable = ax.pcolor(v, cmap='seismic', norm=colors.CenteredNorm())
+    ax.axis('equal')
+    ax.get_figure().colorbar(mappable)
+
+
+def test0(solver=generic_solver):
+    dx = 50                # discretization (nm/cell)
+    pml_thickness = 10     # (number of cells)
+
+    wl = 1550               # Excitation wavelength
+    omega = 2 * numpy.pi / wl
+
+    # Device design parameters
+    radii = (1, 0.6)
+    th = 220
+    center = [0, 0, 0]
+
+    # refractive indices
+    n_ring = numpy.sqrt(12.6)  # ~Si
+    n_air = 4.0   # air
+
+    # Half-dimensions of the simulation grid
+    xyz_max = numpy.array([1.2, 1.2, 0.3]) * 1000 + pml_thickness * dx
+
+    # Coordinates of the edges of the cells.
+    half_edge_coords = [numpy.arange(dx/2, m + dx, step=dx) for m in xyz_max]
+    edge_coords = [numpy.hstack((-h[::-1], h)) for h in half_edge_coords]
+
+    # #### Create the grid, mask, and draw the device ####
+    grid = gridlock.Grid(edge_coords)
+    epsilon = grid.allocate(n_air**2, dtype=numpy.float32)
+    grid.draw_cylinder(
+        epsilon,
+        h = dict(axis='z', center=center[2], span=th),
+        radius = max(radii),
+        center2d = center[:2],
+        foreground = n_ring ** 2,
+        num_points = 24,
+        )
+    grid.draw_cylinder(
+        epsilon,
+        h = dict(axis='z', center=center[2], span=th * 1.1),
+        radius = min(radii),
+        center2d = center[:2],
+        foreground = n_air ** 2,
+        num_points = 24,
+        )
+
+    dxes = [grid.dxyz, grid.autoshifted_dxyz()]
+    for a in (0, 1, 2):
+        for p in (-1, 1):
+            dxes = meanas.fdfd.scpml.stretch_with_scpml(dxes, axis=a, polarity=p, omega=omega,
+                                                        thickness=pml_thickness)
+
+    J = [numpy.zeros_like(epsilon[0], dtype=complex) for _ in range(3)]
+    J[1][15, grid.shape[1]//2, grid.shape[2]//2] = 1
+
+
+    #
+    # Solve!
+    #
+    sim_args = dict(
+        omega = omega,
+        dxes = dxes,
+        epsilon = vec(epsilon),
+        )
+    x = solver(J=vec(J), **sim_args)
+
+    A = operators.e_full(omega, dxes, vec(epsilon)).tocsr()
+    b = -1j * omega * vec(J)
+    print('Norm of the residual is ', norm(A @ x - b) / norm(b))
+
+    E = unvec(x, grid.shape)
+
+    #
+    # Plot results
+    #
+    grid.visualize_slice(E.real, plane=dict(z=0), which_shifts=1, pcolormesh_args=dict(norm=colors.CenteredNorm(), cmap='bwr'))
+
+
+if __name__ == '__main__':
+    test0()

examples/fdfd1.py

@@ -1,6 +1,8 @@
 import importlib
 import numpy
 from numpy.linalg import norm
+from matplotlib import pyplot, colors
+import logging
 
 import meanas
 from meanas import fdtd
@@ -10,9 +12,6 @@ from meanas.fdfd.solvers import generic as generic_solver
 
 import gridlock
 
-from matplotlib import pyplot
-
-import logging
 
 logging.basicConfig(level=logging.DEBUG)
 logging.getLogger('matplotlib').setLevel(logging.WARNING)
@@ -20,82 +19,6 @@ logging.getLogger('matplotlib').setLevel(logging.WARNING)
 __author__ = 'Jan Petykiewicz'
 
 
-def test0(solver=generic_solver):
-    dx = 50                # discretization (nm/cell)
-    pml_thickness = 10     # (number of cells)
-
-    wl = 1550               # Excitation wavelength
-    omega = 2 * numpy.pi / wl
-
-    # Device design parameters
-    radii = (1, 0.6)
-    th = 220
-    center = [0, 0, 0]
-
-    # refractive indices
-    n_ring = numpy.sqrt(12.6)  # ~Si
-    n_air = 4.0   # air
-
-    # Half-dimensions of the simulation grid
-    xyz_max = numpy.array([1.2, 1.2, 0.3]) * 1000 + pml_thickness * dx
-
-    # Coordinates of the edges of the cells.
-    half_edge_coords = [numpy.arange(dx/2, m + dx, step=dx) for m in xyz_max]
-    edge_coords = [numpy.hstack((-h[::-1], h)) for h in half_edge_coords]
-
-    # #### Create the grid, mask, and draw the device ####
-    grid = gridlock.Grid(edge_coords)
-    epsilon = grid.allocate(n_air**2, dtype=numpy.float32)
-    grid.draw_cylinder(epsilon,
-                       surface_normal=2,
-                       center=center,
-                       radius=max(radii),
-                       thickness=th,
-                       eps=n_ring**2,
-                       num_points=24)
-    grid.draw_cylinder(epsilon,
-                       surface_normal=2,
-                       center=center,
-                       radius=min(radii),
-                       thickness=th*1.1,
-                       eps=n_air ** 2,
-                       num_points=24)
-
-    dxes = [grid.dxyz, grid.autoshifted_dxyz()]
-    for a in (0, 1, 2):
-        for p in (-1, 1):
-            dxes = meanas.fdfd.scpml.stretch_with_scpml(dxes, axis=a, polarity=p, omega=omega,
-                                                        thickness=pml_thickness)
-
-    J = [numpy.zeros_like(epsilon[0], dtype=complex) for _ in range(3)]
-    J[1][15, grid.shape[1]//2, grid.shape[2]//2] = 1
-
-
-    '''
-    Solve!
-    '''
-    sim_args = {
-        'omega': omega,
-        'dxes': dxes,
-        'epsilon': vec(epsilon),
-    }
-    x = solver(J=vec(J), **sim_args)
-
-    A = operators.e_full(omega, dxes, vec(epsilon)).tocsr()
-    b = -1j * omega * vec(J)
-    print('Norm of the residual is ', norm(A @ x - b))
-
-    E = unvec(x, grid.shape)
-
-    '''
-    Plot results
-    '''
-    pyplot.figure()
-    pyplot.pcolor(numpy.real(E[1][:, :, grid.shape[2]//2]), cmap='seismic')
-    pyplot.axis('equal')
-    pyplot.show()
-
-
 def test1(solver=generic_solver):
     dx = 40                 # discretization (nm/cell)
     pml_thickness = 10      # (number of cells)
@@ -122,7 +45,7 @@ def test1(solver=generic_solver):
     # #### Create the grid and draw the device ####
     grid = gridlock.Grid(edge_coords)
     epsilon = grid.allocate(n_air**2, dtype=numpy.float32)
-    grid.draw_cuboid(epsilon, center=center, dimensions=[8e3, w, th], eps=n_wg**2)
+    grid.draw_cuboid(epsilon, x=dict(center=0, span=8e3), y=dict(center=0, span=w), z=dict(center=0, span=th), foreground=n_wg**2)
 
     dxes = [grid.dxyz, grid.autoshifted_dxyz()]
     for a in (0, 1, 2):
@@ -156,22 +79,14 @@ def test1(solver=generic_solver):
     # grid.draw_cuboid(pmcg, center=[700, 0, 0], dimensions=[80, 1e8, 1e8], eps=1)
     # grid.visualize_isosurface(pmcg)
 
-    def pcolor(v) -> None:
+    grid.visualize_slice(J.imag, plane=dict(y=6*dx), which_shifts=1, pcolormesh_args=dict(norm=colors.CenteredNorm(), cmap='bwr'))
-        vmax = numpy.max(numpy.abs(v))
+    fig, ax = pyplot.subplots()
-        pyplot.pcolor(v, cmap='seismic', vmin=-vmax, vmax=vmax)
+    ax.pcolormesh((numpy.abs(J).sum(axis=2).sum(axis=0) > 0).astype(float).T, cmap='hot')
-        pyplot.axis('equal')
-        pyplot.colorbar()
-
-    ss = (1, slice(None), J.shape[2]//2+6, slice(None))
-#    pyplot.figure()
-#    pcolor(J3[ss].T.imag)
-#    pyplot.figure()
-#    pcolor((numpy.abs(J3).sum(axis=2).sum(axis=0) > 0).astype(float).T)
     pyplot.show(block=True)
 
-    '''
+    #
-    Solve!
+    # Solve!
-    '''
+    #
     sim_args = {
         'omega': omega,
         'dxes': dxes,
@@ -188,20 +103,18 @@ def test1(solver=generic_solver):
 
     E = unvec(x, grid.shape)
 
-    '''
+    #
-    Plot results
+    # Plot results
-    '''
+    #
     center = grid.pos2ind([0, 0, 0], None).astype(int)
-    pyplot.figure()
+    fig, axes = pyplot.subplots(2, 2)
-    pyplot.subplot(2, 2, 1)
+    grid.visualize_slice(E.real, plane=dict(x=0), which_shifts=1, ax=axes[0, 0], finalize=False, pcolormesh_args=dict(norm=colors.CenteredNorm(), cmap='bwr'))
-    pcolor(numpy.real(E[1][center[0], :, :]).T)
+    grid.visualize_slice(E.real, plane=dict(z=0), which_shifts=1, ax=axes[0, 1], finalize=False, pcolormesh_args=dict(norm=colors.CenteredNorm(), cmap='bwr'))
-    pyplot.subplot(2, 2, 2)
+#    pcolor(axes[0, 0], numpy.real(E[1][center[0], :, :]).T)
-    pyplot.plot(numpy.log10(numpy.abs(E[1][:, center[1], center[2]]) + 1e-10))
+#    pcolor(axes[0, 1], numpy.real(E[1][:, :, center[2]]).T)
-    pyplot.grid(alpha=0.6)
+    axes[1, 0].plot(numpy.log10(numpy.abs(E[1][:, center[1], center[2]]) + 1e-10))
-    pyplot.ylabel('log10 of field')
+    axes[1, 0].grid(alpha=0.6)
-    pyplot.subplot(2, 2, 3)
+    axes[1, 0].set_ylabel('log10 of field')
-    pcolor(numpy.real(E[1][:, :, center[2]]).T)
-    pyplot.subplot(2, 2, 4)
 
     def poyntings(E):
         H = functional.e2h(omega, dxes)(E)
@@ -215,24 +128,28 @@ def test1(solver=generic_solver):
         return s0, s1, s2
 
     s0x, s1x, s2x = poyntings(E)
-    pyplot.plot(s0x[0].sum(axis=2).sum(axis=1), label='s0', marker='.')
+    ax = axes[1, 1]
-    pyplot.plot(s1x[0].sum(axis=2).sum(axis=1), label='s1', marker='.')
+    ax.plot(s0x[0].sum(axis=2).sum(axis=1), label='s0', marker='.')
-    pyplot.plot(s2x[0].sum(axis=2).sum(axis=1), label='s2', marker='.')
+    ax.plot(s1x[0].sum(axis=2).sum(axis=1), label='s1', marker='.')
-    pyplot.plot(E[1][:, center[1], center[2]].real.T, label='Ey', marker='x')
+    ax.plot(s2x[0].sum(axis=2).sum(axis=1), label='s2', marker='.')
-    pyplot.grid(alpha=0.6)
+    ax.plot(E[1][:, center[1], center[2]].real.T, label='Ey', marker='x')
-    pyplot.legend()
+    ax.grid(alpha=0.6)
-    pyplot.show()
+    ax.legend()
+
+    p_in = (-E * J.conj()).sum() / 2 * (dx * dx * dx)
+    print(f'{p_in=}')
 
     q = []
     for i in range(-5, 30):
         e_ovl_rolled = numpy.roll(e_overlap, i, axis=1)
-        q += [numpy.abs(vec(E) @ vec(e_ovl_rolled).conj())]
+        q += [numpy.abs(vec(E).conj() @ vec(e_ovl_rolled))]
-    pyplot.figure()
+    fig, ax = pyplot.subplots()
-    pyplot.plot(q, marker='.')
+    ax.plot(q, marker='.')
-    pyplot.grid(alpha=0.6)
+    ax.grid(alpha=0.6)
-    pyplot.title('Overlap with mode')
+    ax.set_title('Overlap with mode')
-    pyplot.show()
+    print('Average overlap with mode:', sum(q[8:32])/len(q[8:32]))
-    print('Average overlap with mode:', sum(q)/len(q))
+
+    pyplot.show(block=True)
 
 
 def module_available(name):
@@ -240,9 +157,6 @@ def module_available(name):
 
 
 if __name__ == '__main__':
-    #test0()
-#    test1()
-
     if module_available('opencl_fdfd'):
         from opencl_fdfd import cg_solver as opencl_solver
         test1(opencl_solver)
@@ -253,3 +167,4 @@ if __name__ == '__main__':
     #     test1(magma_solver)
     else:
         test1()
+

meanas/__init__.py

@@ -6,12 +6,13 @@ See the readme or `import meanas; help(meanas)` for more info.
 
 import pathlib
 
-__version__ = '0.9'
+__version__ = '0.10'
 __author__ = 'Jan Petykiewicz'
 
 
 try:
-    with open(pathlib.Path(__file__).parent / 'README.md', 'r') as f:
+    readme_path = pathlib.Path(__file__).parent / 'README.md'
+    with readme_path.open('r') as f:
         __doc__ = f.read()
 except Exception:
     pass

meanas/eigensolvers.py

@@ -1,12 +1,12 @@
 """
 Solvers for eigenvalue / eigenvector problems
 """
-from typing import Callable
+from collections.abc import Callable
 import numpy
 from numpy.typing import NDArray, ArrayLike
 from numpy.linalg import norm
-from scipy import sparse              # type: ignore
+from scipy import sparse
-import scipy.sparse.linalg as spalg   # type: ignore
+import scipy.sparse.linalg as spalg
 
 
 def power_iteration(
@@ -25,8 +25,9 @@ def power_iteration(
     Returns:
         (Largest-magnitude eigenvalue, Corresponding eigenvector estimate)
     """
+    rng = numpy.random.default_rng()
     if guess_vector is None:
-        v = numpy.random.rand(operator.shape[0]) + 1j * numpy.random.rand(operator.shape[0])
+        v = rng.random(operator.shape[0]) + 1j * rng.random(operator.shape[0])
     else:
         v = guess_vector
 
@@ -63,10 +64,10 @@ def rayleigh_quotient_iteration(
         (eigenvalues, eigenvectors)
     """
     try:
-        (operator - sparse.eye(operator.shape[0]))
+        (operator - sparse.eye_array(operator.shape[0]))
 
-        def shift(eigval: float) -> sparse:
+        def shift(eigval: float) -> sparse.sparray:
-            return eigval * sparse.eye(operator.shape[0])
+            return eigval * sparse.eye_array(operator.shape[0])
 
         if solver is None:
             solver = spalg.spsolve
@@ -129,12 +130,12 @@ def signed_eigensolve(
 
     # Try to combine, use general LinearOperator if we fail
     try:
-        shifted_operator = operator + shift * sparse.eye(operator.shape[0])
+        shifted_operator = operator + shift * sparse.eye_array(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)
+    shifted_eigenvalues, eigenvectors = spalg.eigs(shifted_operator, which='LM', k=how_many, ncv=2 * how_many + 50)
     eigenvalues = shifted_eigenvalues - shift
 
     k = eigenvalues.argsort()

meanas/fdfd/__init__.py

@@ -91,5 +91,12 @@ $$
 
 
 """
-from . import solvers, operators, functional, scpml, waveguide_2d, waveguide_3d
+from . import (
+    solvers as solvers,
+    operators as operators,
+    functional as functional,
+    scpml as scpml,
+    waveguide_2d as waveguide_2d,
+    waveguide_3d as waveguide_3d,
+    )
 # from . import farfield, bloch TODO

meanas/fdfd/bloch.py

@@ -94,18 +94,19 @@ This module contains functions for generating and solving the
 
 """
 
-from typing import Callable, Any, cast, Sequence
+from typing import Any, cast
+from collections.abc import Callable, Sequence
 import logging
 import numpy
 from numpy import pi, real, trace
 from numpy.fft import fftfreq
 from numpy.typing import NDArray, ArrayLike
-import scipy                            # type: ignore
+import scipy
-import scipy.optimize                   # type: ignore
+import scipy.optimize
-from scipy.linalg import norm           # type: ignore
+from scipy.linalg import norm
-import scipy.sparse.linalg as spalg     # type: ignore
+import scipy.sparse.linalg as spalg
 
-from ..fdmath import fdfield_t, cfdfield_t
+from ..fdmath import fdfield, cfdfield, cfdfield_t
 
 
 logger = logging.getLogger(__name__)
@@ -114,7 +115,6 @@ logger = logging.getLogger(__name__)
 try:
     import pyfftw.interfaces.numpy_fft  # type: ignore
     import pyfftw.interfaces            # type: ignore
-    import multiprocessing
     logger.info('Using pyfftw')
 
     pyfftw.interfaces.cache.enable()
@@ -155,7 +155,7 @@ def generate_kmn(
             All are given in the xyz basis (e.g. `|k|[0,0,0] = norm(G_matrix @ k0)`).
     """
     k0 = numpy.array(k0)
-    G_matrix = numpy.array(G_matrix, copy=False)
+    G_matrix = numpy.asarray(G_matrix)
 
     Gi_grids = numpy.array(numpy.meshgrid(*(fftfreq(n, 1 / n) for n in shape[:3]), indexing='ij'))
     Gi = numpy.moveaxis(Gi_grids, 0, -1)
@@ -183,8 +183,8 @@ def generate_kmn(
 def maxwell_operator(
         k0: ArrayLike,
         G_matrix: ArrayLike,
-        epsilon: fdfield_t,
+        epsilon: fdfield,
-        mu: fdfield_t | None = None
+        mu: fdfield | None = None
         ) -> Callable[[NDArray[numpy.complex128]], NDArray[numpy.complex128]]:
     """
     Generate the Maxwell operator
@@ -232,13 +232,13 @@ def maxwell_operator(
             Raveled conv(1/mu_k, ik x conv(1/eps_k, ik x h_mn)), returned
             and overwritten in-place of `h`.
         """
-        hin_m, hin_n = [hi.reshape(shape) for hi in numpy.split(h, 2)]
+        hin_m, hin_n = (hi.reshape(shape) for hi in numpy.split(h, 2))
 
         #{d,e,h}_xyz fields are complex 3-fields in (1/x, 1/y, 1/z) basis
 
         # cross product and transform into xyz basis
         d_xyz = (n * hin_m
-               - m * hin_n) * k_mag         # noqa: E128
+               - m * hin_n) * k_mag
 
         # divide by epsilon
         temp = ifftn(d_xyz, axes=range(3))   # reuses d_xyz if using pyfftw
@@ -254,7 +254,7 @@ def maxwell_operator(
         else:
             # transform from mn to xyz
             b_xyz = (m * b_m[:, :, :, None]
-                   + n * b_n[:, :, :, None])    # noqa: E128
+                   + n * b_n[:, :, :, None])
 
             # divide by mu
             temp = ifftn(b_xyz, axes=range(3))
@@ -276,7 +276,7 @@ def maxwell_operator(
 def hmn_2_exyz(
         k0: ArrayLike,
         G_matrix: ArrayLike,
-        epsilon: fdfield_t,
+        epsilon: fdfield,
         ) -> Callable[[NDArray[numpy.complex128]], cfdfield_t]:
     """
     Generate an operator which converts a vectorized spatial-frequency-space
@@ -303,12 +303,13 @@ def hmn_2_exyz(
     k_mag, m, n = generate_kmn(k0, G_matrix, shape)
 
     def operator(h: NDArray[numpy.complex128]) -> cfdfield_t:
-        hin_m, hin_n = [hi.reshape(shape) for hi in numpy.split(h, 2)]
+        hin_m, hin_n = (hi.reshape(shape) for hi in numpy.split(h, 2))
         d_xyz = (n * hin_m
-               - m * hin_n) * k_mag         # noqa: E128
+               - m * hin_n) * k_mag
 
         # divide by epsilon
-        return numpy.array([ei for ei in numpy.moveaxis(ifftn(d_xyz, axes=range(3)) / epsilon, 3, 0)])         # TODO avoid copy
+        exyz = numpy.moveaxis(ifftn(d_xyz, axes=range(3)) / epsilon, 3, 0)
+        return cfdfield_t(exyz)
 
     return operator
 
@@ -316,7 +317,7 @@ def hmn_2_exyz(
 def hmn_2_hxyz(
         k0: ArrayLike,
         G_matrix: ArrayLike,
-        epsilon: fdfield_t
+        epsilon: fdfield,
         ) -> Callable[[NDArray[numpy.complex128]], cfdfield_t]:
     """
     Generate an operator which converts a vectorized spatial-frequency-space
@@ -341,10 +342,10 @@ def hmn_2_hxyz(
     _k_mag, m, n = generate_kmn(k0, G_matrix, shape)
 
     def operator(h: NDArray[numpy.complex128]) -> cfdfield_t:
-        hin_m, hin_n = [hi.reshape(shape) for hi in numpy.split(h, 2)]
+        hin_m, hin_n = (hi.reshape(shape) for hi in numpy.split(h, 2))
         h_xyz = (m * hin_m
-               + n * hin_n)     # noqa: E128
+               + n * hin_n)
-        return numpy.array([ifftn(hi) for hi in numpy.moveaxis(h_xyz, 3, 0)])
+        return cfdfield_t(numpy.array([ifftn(hi) for hi in numpy.moveaxis(h_xyz, 3, 0)]))
 
     return operator
 
@@ -352,8 +353,8 @@ def hmn_2_hxyz(
 def inverse_maxwell_operator_approx(
         k0: ArrayLike,
         G_matrix: ArrayLike,
-        epsilon: fdfield_t,
+        epsilon: fdfield,
-        mu: fdfield_t | None = None,
+        mu: fdfield | None = None,
         ) -> Callable[[NDArray[numpy.complex128]], NDArray[numpy.complex128]]:
     """
     Generate an approximate inverse of the Maxwell operator,
@@ -394,7 +395,7 @@ def inverse_maxwell_operator_approx(
         Returns:
             Raveled ik x conv(eps_k, ik x conv(mu_k, h_mn))
         """
-        hin_m, hin_n = [hi.reshape(shape) for hi in numpy.split(h, 2)]
+        hin_m, hin_n = (hi.reshape(shape) for hi in numpy.split(h, 2))
 
         #{d,e,h}_xyz fields are complex 3-fields in (1/x, 1/y, 1/z) basis
 
@@ -403,7 +404,7 @@ def inverse_maxwell_operator_approx(
         else:
             # transform from mn to xyz
             h_xyz = (m * hin_m[:, :, :, None]
-                   + n * hin_n[:, :, :, None])  # noqa: E128
+                   + n * hin_n[:, :, :, None])
 
             # multiply by mu
             temp = ifftn(h_xyz, axes=range(3))
@@ -416,7 +417,7 @@ def inverse_maxwell_operator_approx(
 
         # cross product and transform into xyz basis
         e_xyz = (n * b_m
-               - m * b_n) / k_mag  # noqa: E128
+               - m * b_n) / k_mag
 
         # multiply by epsilon
         temp = ifftn(e_xyz, axes=range(3))
@@ -440,8 +441,8 @@ def find_k(
         tolerance: float,
         direction: ArrayLike,
         G_matrix: ArrayLike,
-        epsilon: fdfield_t,
+        epsilon: fdfield,
-        mu: fdfield_t | None = None,
+        mu: fdfield | None = None,
         band: int = 0,
         k_bounds: tuple[float, float] = (0, 0.5),
         k_guess: float | None = None,
@@ -508,8 +509,8 @@ def eigsolve(
         num_modes: int,
         k0: ArrayLike,
         G_matrix: ArrayLike,
-        epsilon: fdfield_t,
+        epsilon: fdfield,
-        mu: fdfield_t | None = None,
+        mu: fdfield | None = None,
         tolerance: float = 1e-7,
         max_iters: int = 10000,
         reset_iters: int = 100,
@@ -538,7 +539,7 @@ def eigsolve(
         `(eigenvalues, eigenvectors)` where `eigenvalues[i]` corresponds to the
         vector `eigenvectors[i, :]`
     """
-    k0 = numpy.array(k0, copy=False)
+    k0 = numpy.asarray(k0)
 
     h_size = 2 * epsilon[0].size
 
@@ -561,11 +562,12 @@ def eigsolve(
     prev_theta = 0.5
     D = numpy.zeros(shape=y_shape, dtype=complex)
 
+    rng = numpy.random.default_rng()
     Z: NDArray[numpy.complex128]
     if y0 is None:
-        Z = numpy.random.rand(*y_shape) + 1j * numpy.random.rand(*y_shape)
+        Z = rng.random(y_shape) + 1j * rng.random(y_shape)
     else:
-        Z = numpy.array(y0, copy=False).T
+        Z = numpy.asarray(y0).T
 
     while True:
         Z *= num_modes / norm(Z)
@@ -573,7 +575,7 @@ def eigsolve(
         try:
             U = numpy.linalg.inv(ZtZ)
         except numpy.linalg.LinAlgError:
-            Z = numpy.random.rand(*y_shape) + 1j * numpy.random.rand(*y_shape)
+            Z = rng.random(y_shape) + 1j * rng.random(y_shape)
             continue
 
         trace_U = real(trace(U))
@@ -646,17 +648,16 @@ def eigsolve(
 
         Qi_memo: list[float | None] = [None, None]
 
-        def Qi_func(theta: float) -> float:
+        def Qi_func(theta: float, Qi_memo=Qi_memo, ZtZ=ZtZ, DtD=DtD, symZtD=symZtD) -> float:   # noqa: ANN001
-            nonlocal Qi_memo
             if Qi_memo[0] == theta:
-                return cast(float, Qi_memo[1])
+                return cast('float', Qi_memo[1])
 
             c = numpy.cos(theta)
             s = numpy.sin(theta)
             Q = c * c * ZtZ + s * s * DtD + 2 * s * c * symZtD
             try:
                 Qi = numpy.linalg.inv(Q)
-            except numpy.linalg.LinAlgError:
+            except numpy.linalg.LinAlgError as err:
                 logger.info('taylor Qi')
                 # if c or s small, taylor expand
                 if c < 1e-4 * s and c != 0:
@@ -666,12 +667,12 @@ def eigsolve(
                     ZtZi = numpy.linalg.inv(ZtZ)
                     Qi = ZtZi / (c * c) - 2 * s / (c * c * c) * (ZtZi @ (ZtZi @ symZtD).conj().T)
                 else:
-                    raise Exception('Inexplicable singularity in trace_func')
+                    raise Exception('Inexplicable singularity in trace_func') from err
             Qi_memo[0] = theta
-            Qi_memo[1] = cast(float, Qi)
+            Qi_memo[1] = cast('float', Qi)
-            return cast(float, Qi)
+            return cast('float', Qi)
 
-        def trace_func(theta: float) -> float:
+        def trace_func(theta: float, ZtAZ=ZtAZ, DtAD=DtAD, symZtAD=symZtAD) -> float:           # noqa: ANN001
             c = numpy.cos(theta)
             s = numpy.sin(theta)
             Qi = Qi_func(theta)
@@ -680,7 +681,7 @@ def eigsolve(
             return numpy.abs(trace)
 
         if False:
-            def trace_deriv(theta):
+            def trace_deriv(theta, sgn: int = sgn, ZtAZ=ZtAZ, DtAD=DtAD, symZtD=symZtD, symZtAD=symZtAD, ZtZ=ZtZ, DtD=DtD):     # noqa: ANN001
                 Qi = Qi_func(theta)
                 c2 = numpy.cos(2 * theta)
                 s2 = numpy.sin(2 * theta)
@@ -799,3 +800,62 @@ def _rtrace_AtB(
 def _symmetrize(A: NDArray[numpy.complex128]) -> NDArray[numpy.complex128]:
     return (A + A.conj().T) * 0.5
 
+
+
+def inner_product(
+        eL: cfdfield,
+        hL: cfdfield,
+        eR: cfdfield,
+        hR: cfdfield,
+        ) -> complex:
+    # assumes x-axis propagation
+
+    assert numpy.array_equal(eR.shape, hR.shape)
+    assert numpy.array_equal(eL.shape, hL.shape)
+    assert numpy.array_equal(eR.shape, eL.shape)
+
+    # Cross product, times 2 since it's <p | n>, then divide by 4. # TODO might want to abs() this?
+    norm2R = (eR[1] * hR[2] - eR[2] * hR[1]).sum() / 2
+    norm2L = (eL[1] * hL[2] - eL[2] * hL[1]).sum() / 2
+
+    # eRxhR_x = numpy.cross(eR.reshape(3, -1), hR.reshape(3, -1), axis=0).reshape(eR.shape)[0] / normR
+    # logger.info(f'power {eRxhR_x.sum() / 2})
+
+    eR /= numpy.sqrt(norm2R)
+    hR /= numpy.sqrt(norm2R)
+    eL /= numpy.sqrt(norm2L)
+    hL /= numpy.sqrt(norm2L)
+
+    # (eR x hL)[0] and (eL x hR)[0]
+    eRxhL_x = eR[1] * hL[2] - eR[2] - hL[1]
+    eLxhR_x = eL[1] * hR[2] - eL[2] - hR[1]
+
+    #return 1j *  (eRxhL_x - eLxhR_x).sum() / numpy.sqrt(norm2R * norm2L)
+    #return (eRxhL_x.sum() - eLxhR_x.sum()) / numpy.sqrt(norm2R * norm2L)
+    return eRxhL_x.sum() - eLxhR_x.sum()
+
+
+def trq(
+        eI: cfdfield,
+        hI: cfdfield,
+        eO: cfdfield,
+        hO: cfdfield,
+        ) -> tuple[complex, complex]:
+    pp = inner_product(eO,  hO, eI,  hI)
+    pn = inner_product(eO,  hO, eI, -hI)
+    np = inner_product(eO, -hO, eI,  hI)
+    nn = inner_product(eO, -hO, eI, -hI)
+
+    assert pp == -nn
+    assert pn == -np
+
+    logger.info(f'''
+        {pp=:4g} {pn=:4g}
+        {nn=:4g} {np=:4g}
+            {nn * pp / pn=:4g}    {-np=:4g}
+        ''')
+
+    r = -pp / pn    # -<Pp|Bp>/<Pn/Bp> = -(-pp) / (-pn)
+    t = (np - nn * pp / pn) / 4
+
+    return t, r

meanas/fdfd/eme.py

@@ -0,0 +1,84 @@
+from collections.abc import Sequence
+import numpy
+from numpy.typing import NDArray
+from scipy import sparse
+
+from ..fdmath import dx_lists2_t, vcfdfield2
+from .waveguide_2d import inner_product
+
+
+def get_tr(
+        ehLs: Sequence[Sequence[vcfdfield2]],
+        wavenumbers_L: Sequence[complex],
+        ehRs: Sequence[Sequence[vcfdfield2]],
+        wavenumbers_R: Sequence[complex],
+        dxes: dx_lists2_t,
+        ) -> tuple[NDArray[numpy.complex128], NDArray[numpy.complex128]]:
+    nL = len(wavenumbers_L)
+    nR = len(wavenumbers_R)
+    A12 = numpy.zeros((nL, nR), dtype=complex)
+    A21 = numpy.zeros((nL, nR), dtype=complex)
+    B11 = numpy.zeros((nL,), dtype=complex)
+    for ll in range(nL):
+        eL, hL = ehLs[ll]
+        B11[ll] = inner_product(eL, hL, dxes=dxes, conj_h=False)
+        for rr in range(nR):
+            eR, hR = ehRs[rr]
+            A12[ll, rr] = inner_product(eL, hR, dxes=dxes, conj_h=False)    # TODO optimize loop?
+            A21[ll, rr] = inner_product(eR, hL, dxes=dxes, conj_h=False)
+
+    # tt0 = 2 * numpy.linalg.pinv(A21 + numpy.conj(A12))
+    tt0, _resid, _rank, _sing = numpy.linalg.lstsq(A21 + A12, numpy.diag(2 * B11), rcond=None)
+
+    U, st, V = numpy.linalg.svd(tt0)
+    gain = st > 1
+    st[gain] = 1 / st[gain]
+    tt = U @ numpy.diag(st) @ V
+
+    # rr = 0.5 * (A21 - numpy.conj(A12)) @ tt
+    rr = numpy.diag(0.5 / B11) @ (A21 - A12) @ tt
+
+    return tt, rr
+
+
+def get_abcd(
+        ehLs: Sequence[Sequence[vcfdfield2]],
+        wavenumbers_L: Sequence[complex],
+        ehRs: Sequence[Sequence[vcfdfield2]],
+        wavenumbers_R: Sequence[complex],
+        **kwargs,
+        ) -> sparse.sparray:
+    t12, r12 = get_tr(ehLs, wavenumbers_L, ehRs, wavenumbers_R, **kwargs)
+    t21, r21 = get_tr(ehRs, wavenumbers_R, ehLs, wavenumbers_L, **kwargs)
+    t21i = numpy.linalg.pinv(t21)
+    A = t12 - r21 @ t21i @ r12
+    B = r21 @ t21i
+    C = -t21i @ r12
+    D = t21i
+    return sparse.block_array(((A, B), (C, D)))
+
+
+def get_s(
+        ehLs: Sequence[Sequence[vcfdfield2]],
+        wavenumbers_L: Sequence[complex],
+        ehRs: Sequence[Sequence[vcfdfield2]],
+        wavenumbers_R: Sequence[complex],
+        force_nogain: bool = False,
+        force_reciprocal: bool = False,
+        **kwargs,
+        ) -> NDArray[numpy.complex128]:
+    t12, r12 = get_tr(ehLs, wavenumbers_L, ehRs, wavenumbers_R, **kwargs)
+    t21, r21 = get_tr(ehRs, wavenumbers_R, ehLs, wavenumbers_L, **kwargs)
+
+    ss = numpy.block([[r12, t12],
+                      [t21, r21]])
+
+    if force_nogain:
+        # force S @ S.H diagonal
+        U, sing, V = numpy.linalg.svd(ss)
+        ss = numpy.diag(sing) @ U @ V
+
+    if force_reciprocal:
+        ss = 0.5 * (ss + ss.T)
+
+    return ss

meanas/fdfd/farfield.py

@@ -1,13 +1,16 @@
 """
 Functions for performing near-to-farfield transformation (and the reverse).
 """
-from typing import Any, Sequence, cast
+from typing import Any, cast, TYPE_CHECKING
 import numpy
 from numpy.fft import fft2, fftshift, fftfreq, ifft2, ifftshift
 from numpy import pi
 
 from ..fdmath import cfdfield_t
 
+if TYPE_CHECKING:
+    from collections.abc import Sequence
+
 
 def near_to_farfield(
         E_near: cfdfield_t,
@@ -62,7 +65,7 @@ def near_to_farfield(
         padded_size = (2**numpy.ceil(numpy.log2(s))).astype(int)
     if not hasattr(padded_size, '__len__'):
         padded_size = (padded_size, padded_size)            # type: ignore  # checked if sequence
-    padded_shape = cast(Sequence[int], padded_size)
+    padded_shape = cast('Sequence[int]', padded_size)
 
     En_fft = [fftshift(fft2(fftshift(Eni), s=padded_shape)) for Eni in E_near]
     Hn_fft = [fftshift(fft2(fftshift(Hni), s=padded_shape)) for Hni in H_near]
@@ -171,7 +174,7 @@ def far_to_nearfield(
         padded_size = (2 ** numpy.ceil(numpy.log2(s))).astype(int)
     if not hasattr(padded_size, '__len__'):
         padded_size = (padded_size, padded_size)            # type: ignore  # checked if sequence
-    padded_shape = cast(Sequence[int], padded_size)
+    padded_shape = cast('Sequence[int]', padded_size)
 
     k = 2 * pi
     kxs = fftshift(fftfreq(s[0], 1 / (s[0] * dkx)))

meanas/fdfd/functional.py

@@ -5,10 +5,10 @@ Functional versions of many FDFD operators. These can be useful for performing
 The functions generated here expect `cfdfield_t` inputs with shape (3, X, Y, Z),
 e.g. E = [E_x, E_y, E_z] where each (complex) component has shape (X, Y, Z)
 """
-from typing import Callable
+from collections.abc import Callable
 import numpy
 
-from ..fdmath import dx_lists_t, fdfield_t, cfdfield_t, cfdfield_updater_t
+from ..fdmath import dx_lists_t, cfdfield_t, fdfield, cfdfield, cfdfield_updater_t
 from ..fdmath.functional import curl_forward, curl_back
 
 
@@ -18,8 +18,8 @@ __author__ = 'Jan Petykiewicz'
 def e_full(
         omega: complex,
         dxes: dx_lists_t,
-        epsilon: fdfield_t,
+        epsilon: fdfield,
-        mu: fdfield_t | None = None,
+        mu: fdfield | None = None,
         ) -> cfdfield_updater_t:
     """
     Wave operator for use with E-field. See `operators.e_full` for details.
@@ -37,26 +37,25 @@ def e_full(
     ch = curl_back(dxes[1])
     ce = curl_forward(dxes[0])
 
-    def op_1(e: cfdfield_t) -> cfdfield_t:
+    def op_1(e: cfdfield) -> cfdfield_t:
         curls = ch(ce(e))
-        return curls - omega ** 2 * epsilon * e
+        return cfdfield_t(curls - omega ** 2 * epsilon * e)
 
-    def op_mu(e: cfdfield_t) -> cfdfield_t:
+    def op_mu(e: cfdfield) -> cfdfield_t:
         curls = ch(mu * ce(e))          # type: ignore   # mu = None ok because we don't return the function
-        return curls - omega ** 2 * epsilon * e
+        return cfdfield_t(curls - omega ** 2 * epsilon * e)
 
     if mu is None:
         return op_1
-    else:
     return op_mu
 
 
 def eh_full(
         omega: complex,
         dxes: dx_lists_t,
-        epsilon: fdfield_t,
+        epsilon: fdfield,
-        mu: fdfield_t | None = None,
+        mu: fdfield | None = None,
-        ) -> Callable[[cfdfield_t, cfdfield_t], tuple[cfdfield_t, cfdfield_t]]:
+        ) -> Callable[[cfdfield, cfdfield], tuple[cfdfield_t, cfdfield_t]]:
     """
     Wave operator for full (both E and H) field representation.
     See `operators.eh_full`.
@@ -74,24 +73,23 @@ def eh_full(
     ch = curl_back(dxes[1])
     ce = curl_forward(dxes[0])
 
-    def op_1(e: cfdfield_t, h: cfdfield_t) -> tuple[cfdfield_t, cfdfield_t]:
+    def op_1(e: cfdfield, h: cfdfield) -> tuple[cfdfield_t, cfdfield_t]:
-        return (ch(h) - 1j * omega * epsilon * e,
+        return (cfdfield_t(ch(h) - 1j * omega * epsilon * e),
-                ce(e) + 1j * omega * h)
+                cfdfield_t(ce(e) + 1j * omega * h))
 
-    def op_mu(e: cfdfield_t, h: cfdfield_t) -> tuple[cfdfield_t, cfdfield_t]:
+    def op_mu(e: cfdfield, h: cfdfield) -> tuple[cfdfield_t, cfdfield_t]:
-        return (ch(h) - 1j * omega * epsilon * e,
+        return (cfdfield_t(ch(h) - 1j * omega * epsilon * e),
-                ce(e) + 1j * omega * mu * h)            # type: ignore   # mu=None ok
+                cfdfield_t(ce(e) + 1j * omega * mu * h))            # type: ignore   # mu=None ok
 
     if mu is None:
         return op_1
-    else:
     return op_mu
 
 
 def e2h(
         omega: complex,
         dxes: dx_lists_t,
-        mu: fdfield_t | None = None,
+        mu: fdfield | None = None,
         ) -> cfdfield_updater_t:
     """
     Utility operator for converting the `E` field into the `H` field.
@@ -108,22 +106,21 @@ def e2h(
     """
     ce = curl_forward(dxes[0])
 
-    def e2h_1_1(e: cfdfield_t) -> cfdfield_t:
+    def e2h_1_1(e: cfdfield) -> cfdfield_t:
-        return ce(e) / (-1j * omega)
+        return cfdfield_t(ce(e) / (-1j * omega))
 
-    def e2h_mu(e: cfdfield_t) -> cfdfield_t:
+    def e2h_mu(e: cfdfield) -> cfdfield_t:
-        return ce(e) / (-1j * omega * mu)       # type: ignore   # mu=None ok
+        return cfdfield_t(ce(e) / (-1j * omega * mu))       # type: ignore   # mu=None ok
 
     if mu is None:
         return e2h_1_1
-    else:
     return e2h_mu
 
 
 def m2j(
         omega: complex,
         dxes: dx_lists_t,
-        mu: fdfield_t | None = None,
+        mu: fdfield | None = None,
         ) -> cfdfield_updater_t:
     """
     Utility operator for converting magnetic current `M` distribution
@@ -141,26 +138,25 @@ def m2j(
     """
     ch = curl_back(dxes[1])
 
-    def m2j_mu(m: cfdfield_t) -> cfdfield_t:
+    def m2j_mu(m: cfdfield) -> cfdfield_t:
         J = ch(m / mu) / (-1j * omega)          # type: ignore  # mu=None ok
-        return J
+        return cfdfield_t(J)
 
-    def m2j_1(m: cfdfield_t) -> cfdfield_t:
+    def m2j_1(m: cfdfield) -> cfdfield_t:
         J = ch(m) / (-1j * omega)
-        return J
+        return cfdfield_t(J)
 
     if mu is None:
         return m2j_1
-    else:
     return m2j_mu
 
 
 def e_tfsf_source(
-        TF_region: fdfield_t,
+        TF_region: fdfield,
         omega: complex,
         dxes: dx_lists_t,
-        epsilon: fdfield_t,
+        epsilon: fdfield,
-        mu: fdfield_t | None = None,
+        mu: fdfield | None = None,
         ) -> cfdfield_updater_t:
     """
     Operator that turns an E-field distribution into a total-field/scattered-field
@@ -182,13 +178,13 @@ def e_tfsf_source(
     # TODO documentation
     A = e_full(omega, dxes, epsilon, mu)
 
-    def op(e: cfdfield_t) -> cfdfield_t:
+    def op(e: cfdfield) -> cfdfield_t:
         neg_iwj = A(TF_region * e) - TF_region * A(e)
-        return neg_iwj / (-1j * omega)
+        return cfdfield_t(neg_iwj / (-1j * omega))
     return op
 
 
-def poynting_e_cross_h(dxes: dx_lists_t) -> Callable[[cfdfield_t, cfdfield_t], cfdfield_t]:
+def poynting_e_cross_h(dxes: dx_lists_t) -> Callable[[cfdfield, cfdfield], cfdfield_t]:
     r"""
     Generates a function that takes the single-frequency `E` and `H` fields
     and calculates the cross product `E` x `H` = $E \times H$ as required
@@ -210,7 +206,7 @@ def poynting_e_cross_h(dxes: dx_lists_t) -> Callable[[cfdfield_t, cfdfield_t], c
     Returns:
         Function `f` that returns E x H as required for the poynting vector.
     """
-    def exh(e: cfdfield_t, h: cfdfield_t) -> cfdfield_t:
+    def exh(e: cfdfield, h: cfdfield) -> cfdfield_t:
         s = numpy.empty_like(e)
         ex = e[0] * dxes[0][0][:, None, None]
         ey = e[1] * dxes[0][1][None, :, None]
@@ -221,5 +217,5 @@ def poynting_e_cross_h(dxes: dx_lists_t) -> Callable[[cfdfield_t, cfdfield_t], c
         s[0] = numpy.roll(ey, -1, axis=0) * hz - numpy.roll(ez, -1, axis=0) * hy
         s[1] = numpy.roll(ez, -1, axis=1) * hx - numpy.roll(ex, -1, axis=1) * hz
         s[2] = numpy.roll(ex, -1, axis=2) * hy - numpy.roll(ey, -1, axis=2) * hx
-        return s
+        return cfdfield_t(s)
     return exh

meanas/fdfd/operators.py

@@ -1,7 +1,7 @@
 """
 Sparse matrix operators for use with electromagnetic wave equations.
 
-These functions return sparse-matrix (`scipy.sparse.spmatrix`) representations of
+These functions return sparse-matrix (`scipy.sparse.sparray`) representations of
  a variety of operators, intended for use with E and H fields vectorized using the
  `meanas.fdmath.vectorization.vec()` and `meanas.fdmath.vectorization.unvec()` functions.
 
@@ -28,9 +28,9 @@ The following operators are included:
 """
 
 import numpy
-import scipy.sparse as sparse       # type: ignore
+from scipy import sparse
 
-from ..fdmath import vec, dx_lists_t, vfdfield_t, vcfdfield_t
+from ..fdmath import vec, dx_lists_t, vfdfield, vcfdfield
 from ..fdmath.operators import shift_with_mirror, shift_circ, curl_forward, curl_back
 
 
@@ -40,11 +40,11 @@ __author__ = 'Jan Petykiewicz'
 def e_full(
         omega: complex,
         dxes: dx_lists_t,
-        epsilon: vfdfield_t,
+        epsilon: vfdfield | vcfdfield,
-        mu: vfdfield_t | None = None,
+        mu: vfdfield | None = None,
-        pec: vfdfield_t | None = None,
+        pec: vfdfield | None = None,
-        pmc: vfdfield_t | None = None,
+        pmc: vfdfield | None = None,
-        ) -> sparse.spmatrix:
+        ) -> sparse.sparray:
     r"""
     Wave operator
      $$ \nabla \times (\frac{1}{\mu} \nabla \times) - \Omega^2 \epsilon $$
@@ -77,20 +77,20 @@ def e_full(
     ce = curl_forward(dxes[0])
 
     if pec is None:
-        pe = sparse.eye(epsilon.size)
+        pe = sparse.eye_array(epsilon.size)
     else:
-        pe = sparse.diags(numpy.where(pec, 0, 1))     # Set pe to (not PEC)
+        pe = sparse.diags_array(numpy.where(pec, 0, 1))     # Set pe to (not PEC)
 
     if pmc is None:
-        pm = sparse.eye(epsilon.size)
+        pm = sparse.eye_array(epsilon.size)
     else:
-        pm = sparse.diags(numpy.where(pmc, 0, 1))     # set pm to (not PMC)
+        pm = sparse.diags_array(numpy.where(pmc, 0, 1))     # set pm to (not PMC)
 
-    e = sparse.diags(epsilon)
+    e = sparse.diags_array(epsilon)
     if mu is None:
-        m_div = sparse.eye(epsilon.size)
+        m_div = sparse.eye_array(epsilon.size)
     else:
-        m_div = sparse.diags(1 / mu)
+        m_div = sparse.diags_array(1 / mu)
 
     op = pe @ (ch @ pm @ m_div @ ce - omega**2 * e) @ pe
     return op
@@ -98,7 +98,7 @@ def e_full(
 
 def e_full_preconditioners(
         dxes: dx_lists_t,
-        ) -> tuple[sparse.spmatrix, sparse.spmatrix]:
+        ) -> tuple[sparse.sparray, sparse.sparray]:
     """
     Left and right preconditioners `(Pl, Pr)` for symmetrizing the `e_full` wave operator.
 
@@ -118,19 +118,19 @@ def e_full_preconditioners(
                  dxes[1][0][:, None, None] * dxes[1][1][None, :, None] * dxes[0][2][None, None, :]]
 
     p_vector = numpy.sqrt(vec(p_squared))
-    P_left = sparse.diags(p_vector)
+    P_left = sparse.diags_array(p_vector)
-    P_right = sparse.diags(1 / p_vector)
+    P_right = sparse.diags_array(1 / p_vector)
     return P_left, P_right
 
 
 def h_full(
         omega: complex,
         dxes: dx_lists_t,
-        epsilon: vfdfield_t,
+        epsilon: vfdfield,
-        mu: vfdfield_t | None = None,
+        mu: vfdfield | None = None,
-        pec: vfdfield_t | None = None,
+        pec: vfdfield | None = None,
-        pmc: vfdfield_t | None = None,
+        pmc: vfdfield | None = None,
-        ) -> sparse.spmatrix:
+        ) -> sparse.sparray:
     r"""
     Wave operator
      $$ \nabla \times (\frac{1}{\epsilon} \nabla \times) - \omega^2 \mu $$
@@ -161,20 +161,20 @@ def h_full(
     ce = curl_forward(dxes[0])
 
     if pec is None:
-        pe = sparse.eye(epsilon.size)
+        pe = sparse.eye_array(epsilon.size)
     else:
-        pe = sparse.diags(numpy.where(pec, 0, 1))    # set pe to (not PEC)
+        pe = sparse.diags_array(numpy.where(pec, 0, 1))    # set pe to (not PEC)
 
     if pmc is None:
-        pm = sparse.eye(epsilon.size)
+        pm = sparse.eye_array(epsilon.size)
     else:
-        pm = sparse.diags(numpy.where(pmc, 0, 1))    # Set pe to (not PMC)
+        pm = sparse.diags_array(numpy.where(pmc, 0, 1))    # Set pe to (not PMC)
 
-    e_div = sparse.diags(1 / epsilon)
+    e_div = sparse.diags_array(1 / epsilon)
     if mu is None:
-        m = sparse.eye(epsilon.size)
+        m = sparse.eye_array(epsilon.size)
     else:
-        m = sparse.diags(mu)
+        m = sparse.diags_array(mu)
 
     A = pm @ (ce @ pe @ e_div @ ch - omega**2 * m) @ pm
     return A
@@ -183,11 +183,11 @@ def h_full(
 def eh_full(
         omega: complex,
         dxes: dx_lists_t,
-        epsilon: vfdfield_t,
+        epsilon: vfdfield,
-        mu: vfdfield_t | None = None,
+        mu: vfdfield | None = None,
-        pec: vfdfield_t | None = None,
+        pec: vfdfield | None = None,
-        pmc: vfdfield_t | None = None,
+        pmc: vfdfield | None = None,
-        ) -> sparse.spmatrix:
+        ) -> sparse.sparray:
     r"""
     Wave operator for `[E, H]` field representation. This operator implements Maxwell's
      equations without cancelling out either E or H. The operator is
@@ -227,25 +227,25 @@ def eh_full(
         Sparse matrix containing the wave operator.
     """
     if pec is None:
-        pe = sparse.eye(epsilon.size)
+        pe = sparse.eye_array(epsilon.size)
     else:
-        pe = sparse.diags(numpy.where(pec, 0, 1))    # set pe to (not PEC)
+        pe = sparse.diags_array(numpy.where(pec, 0, 1))    # set pe to (not PEC)
 
     if pmc is None:
-        pm = sparse.eye(epsilon.size)
+        pm = sparse.eye_array(epsilon.size)
     else:
-        pm = sparse.diags(numpy.where(pmc, 0, 1))    # set pm to (not PMC)
+        pm = sparse.diags_array(numpy.where(pmc, 0, 1))    # set pm to (not PMC)
 
-    iwe = pe @ (1j * omega * sparse.diags(epsilon)) @ pe
+    iwe = pe @ (1j * omega * sparse.diags_array(epsilon)) @ pe
     iwm = 1j * omega
     if mu is not None:
-        iwm *= sparse.diags(mu)
+        iwm *= sparse.diags_array(mu)
     iwm = pm @ iwm @ pm
 
     A1 = pe @ curl_back(dxes[1]) @ pm
     A2 = pm @ curl_forward(dxes[0]) @ pe
 
-    A = sparse.bmat([[-iwe, A1],
+    A = sparse.block_array([[-iwe, A1],
                             [A2,  iwm]])
     return A
 
@@ -253,9 +253,9 @@ def eh_full(
 def e2h(
         omega: complex,
         dxes: dx_lists_t,
-        mu: vfdfield_t | None = None,
+        mu: vfdfield | None = None,
-        pmc: vfdfield_t | None = None,
+        pmc: vfdfield | None = None,
-        ) -> sparse.spmatrix:
+        ) -> sparse.sparray:
     """
     Utility operator for converting the E field into the H field.
     For use with `e_full()` -- assumes that there is no magnetic current M.
@@ -274,10 +274,10 @@ def e2h(
     op = curl_forward(dxes[0]) / (-1j * omega)
 
     if mu is not None:
-        op = sparse.diags(1 / mu) @ op
+        op = sparse.diags_array(1 / mu) @ op
 
     if pmc is not None:
-        op = sparse.diags(numpy.where(pmc, 0, 1)) @ op
+        op = sparse.diags_array(numpy.where(pmc, 0, 1)) @ op
 
     return op
 
@@ -285,8 +285,8 @@ def e2h(
 def m2j(
         omega: complex,
         dxes: dx_lists_t,
-        mu: vfdfield_t | None = None,
+        mu: vfdfield | None = None,
-        ) -> sparse.spmatrix:
+        ) -> sparse.sparray:
     """
     Operator for converting a magnetic current M into an electric current J.
     For use with eg. `e_full()`.
@@ -302,12 +302,12 @@ def m2j(
     op = curl_back(dxes[1]) / (1j * omega)
 
     if mu is not None:
-        op = op @ sparse.diags(1 / mu)
+        op = op @ sparse.diags_array(1 / mu)
 
     return op
 
 
-def poynting_e_cross(e: vcfdfield_t, dxes: dx_lists_t) -> sparse.spmatrix:
+def poynting_e_cross(e: vcfdfield, dxes: dx_lists_t) -> sparse.sparray:
     """
     Operator for computing the Poynting vector, containing the
     (E x) portion of the Poynting vector.
@@ -321,22 +321,22 @@ def poynting_e_cross(e: vcfdfield_t, dxes: dx_lists_t) -> sparse.spmatrix:
     """
     shape = [len(dx) for dx in dxes[0]]
 
-    fx, fy, fz = [shift_circ(i, shape, 1) for i in range(3)]
+    fx, fy, fz = (shift_circ(i, shape, 1) for i in range(3))
 
     dxag = [dx.ravel(order='C') for dx in numpy.meshgrid(*dxes[0], indexing='ij')]
     dxbg = [dx.ravel(order='C') for dx in numpy.meshgrid(*dxes[1], indexing='ij')]
-    Ex, Ey, Ez = [ei * da for ei, da in zip(numpy.split(e, 3), dxag)]
+    Ex, Ey, Ez = (ei * da for ei, da in zip(numpy.split(e, 3), dxag, strict=True))
 
     block_diags = [[ None,     fx @ -Ez, fx @  Ey],
                    [ fy @  Ez, None,     fy @ -Ex],
                    [ fz @ -Ey, fz @  Ex, None]]
-    block_matrix = sparse.bmat([[sparse.diags(x) if x is not None else None for x in row]
+    block_matrix = sparse.block_array([[sparse.diags_array(x) if x is not None else None for x in row]
                                         for row in block_diags])
-    P = block_matrix @ sparse.diags(numpy.concatenate(dxbg))
+    P = block_matrix @ sparse.diags_array(numpy.concatenate(dxbg))
     return P
 
 
-def poynting_h_cross(h: vcfdfield_t, dxes: dx_lists_t) -> sparse.spmatrix:
+def poynting_h_cross(h: vcfdfield, dxes: dx_lists_t) -> sparse.sparray:
     """
     Operator for computing the Poynting vector, containing the (H x) portion of the Poynting vector.
 
@@ -349,27 +349,27 @@ def poynting_h_cross(h: vcfdfield_t, dxes: dx_lists_t) -> sparse.spmatrix:
     """
     shape = [len(dx) for dx in dxes[0]]
 
-    fx, fy, fz = [shift_circ(i, shape, 1) for i in range(3)]
+    fx, fy, fz = (shift_circ(i, shape, 1) for i in range(3))
 
     dxag = [dx.ravel(order='C') for dx in numpy.meshgrid(*dxes[0], indexing='ij')]
     dxbg = [dx.ravel(order='C') for dx in numpy.meshgrid(*dxes[1], indexing='ij')]
-    Hx, Hy, Hz = [sparse.diags(hi * db) for hi, db in zip(numpy.split(h, 3), dxbg)]
+    Hx, Hy, Hz = (sparse.diags_array(hi * db) for hi, db in zip(numpy.split(h, 3), dxbg, strict=True))
 
-    P = (sparse.bmat(
+    P = (sparse.block_array(
         [[ None,    -Hz @ fx,   Hy @ fx],
          [ Hz @ fy,  None,     -Hx @ fy],
          [-Hy @ fz,  Hx @ fz,   None]])
-         @ sparse.diags(numpy.concatenate(dxag)))
+         @ sparse.diags_array(numpy.concatenate(dxag)))
     return P
 
 
 def e_tfsf_source(
-        TF_region: vfdfield_t,
+        TF_region: vfdfield,
         omega: complex,
         dxes: dx_lists_t,
-        epsilon: vfdfield_t,
+        epsilon: vfdfield,
-        mu: vfdfield_t | None = None,
+        mu: vfdfield | None = None,
-        ) -> sparse.spmatrix:
+        ) -> sparse.sparray:
     """
     Operator that turns a desired E-field distribution into a
      total-field/scattered-field (TFSF) source.
@@ -390,18 +390,18 @@ def e_tfsf_source(
     """
     # TODO documentation
     A = e_full(omega, dxes, epsilon, mu)
-    Q = sparse.diags(TF_region)
+    Q = sparse.diags_array(TF_region)
     return (A @ Q - Q @ A) / (-1j * omega)
 
 
 def e_boundary_source(
-        mask: vfdfield_t,
+        mask: vfdfield,
         omega: complex,
         dxes: dx_lists_t,
-        epsilon: vfdfield_t,
+        epsilon: vfdfield,
-        mu: vfdfield_t | None = None,
+        mu: vfdfield | None = None,
         periodic_mask_edges: bool = False,
-        ) -> sparse.spmatrix:
+        ) -> sparse.sparray:
     """
     Operator that turns an E-field distrubtion into a current (J) distribution
       along the edges (external and internal) of the provided mask. This is just an
@@ -424,10 +424,10 @@ def e_boundary_source(
     shape = [len(dxe) for dxe in dxes[0]]
     jmask = numpy.zeros_like(mask, dtype=bool)
 
-    def shift_rot(axis: int, polarity: int) -> sparse.spmatrix:
+    def shift_rot(axis: int, polarity: int) -> sparse.sparray:
         return shift_circ(axis=axis, shape=shape, shift_distance=polarity)
 
-    def shift_mir(axis: int, polarity: int) -> sparse.spmatrix:
+    def shift_mir(axis: int, polarity: int) -> sparse.sparray:
         return shift_with_mirror(axis=axis, shape=shape, shift_distance=polarity)
 
     shift = shift_rot if periodic_mask_edges else shift_mir
@@ -436,7 +436,7 @@ def e_boundary_source(
         if shape[axis] == 1:
             continue
         for polarity in (-1, +1):
-            r = shift(axis, polarity) - sparse.eye(numpy.prod(shape))  # shifted minus original
+            r = shift(axis, polarity) - sparse.eye_array(numpy.prod(shape))  # shifted minus original
             r3 = sparse.block_diag((r, r, r))
             jmask = numpy.logical_or(jmask, numpy.abs(r3 @ mask))
 
@@ -447,5 +447,5 @@ def e_boundary_source(
 #             (numpy.roll(mask, -1, axis=2) != mask) |
 #             (numpy.roll(mask, +1, axis=2) != mask))
 
-    return sparse.diags(jmask.astype(int)) @ full
+    return sparse.diags_array(jmask.astype(int)) @ full
 

meanas/fdfd/scpml.py

@@ -2,7 +2,7 @@
 Functions for creating stretched coordinate perfectly matched layer (PML) absorbers.
 """
 
-from typing import Sequence, Callable
+from collections.abc import Sequence, Callable
 
 import numpy
 from numpy.typing import NDArray

meanas/fdfd/solvers.py

@@ -2,15 +2,16 @@
 Solvers and solver interface for FDFD problems.
 """
 
-from typing import Callable, Dict, Any, Optional
+from typing import Any
+from collections.abc import Callable
 import logging
 
 import numpy
 from numpy.typing import ArrayLike, NDArray
 from numpy.linalg import norm
-import scipy.sparse.linalg          # type: ignore
+import scipy.sparse.linalg
 
-from ..fdmath import dx_lists_t, vfdfield_t, vcfdfield_t
+from ..fdmath import dx_lists_t, vfdfield, vcfdfield, vcfdfield_t
 from . import operators
 
 
@@ -18,7 +19,7 @@ logger = logging.getLogger(__name__)
 
 
 def _scipy_qmr(
-        A: scipy.sparse.csr_matrix,
+        A: scipy.sparse.csr_array,
         b: ArrayLike,
         **kwargs: Any,
         ) -> NDArray[numpy.float64]:
@@ -34,16 +35,16 @@ def _scipy_qmr(
         Guess for solution (returned even if didn't converge)
     """
 
-    '''
+    #
-    Report on our progress
+    #Report on our progress
-    '''
+    #
     ii = 0
 
     def log_residual(xk: ArrayLike) -> None:
         nonlocal ii
         ii += 1
         if ii % 100 == 0:
-            cur_norm = norm(A @ xk - b)
+            cur_norm = norm(A @ xk - b) / norm(b)
             logger.info(f'Solver residual at iteration {ii} : {cur_norm}')
 
     if 'callback' in kwargs:
@@ -55,10 +56,9 @@ def _scipy_qmr(
     else:
         kwargs['callback'] = log_residual
 
-    '''
+    #
-    Run the actual solve
+    # Run the actual solve
-    '''
+    #
-
     x, _ = scipy.sparse.linalg.qmr(A, b, **kwargs)
     return x
 
@@ -66,14 +66,16 @@ def _scipy_qmr(
 def generic(
         omega: complex,
         dxes: dx_lists_t,
-        J: vcfdfield_t,
+        J: vcfdfield,
-        epsilon: vfdfield_t,
+        epsilon: vfdfield,
-        mu: Optional[vfdfield_t] = None,
+        mu: vfdfield | None = None,
-        pec: Optional[vfdfield_t] = None,
+        *,
-        pmc: Optional[vfdfield_t] = None,
+        pec: vfdfield | None = None,
+        pmc: vfdfield | None = None,
         adjoint: bool = False,
         matrix_solver: Callable[..., ArrayLike] = _scipy_qmr,
-        matrix_solver_opts: Optional[Dict[str, Any]] = None,
+        matrix_solver_opts: dict[str, Any] | None = None,
+        E_guess: vcfdfield | None = None,
         ) -> vcfdfield_t:
     """
     Conjugate gradient FDFD solver using CSR sparse matrices.
@@ -93,13 +95,15 @@ def generic(
              (at H-field locations; non-zero value indicates PMC is present)
         adjoint: If true, solves the adjoint problem.
         matrix_solver: Called as `matrix_solver(A, b, **matrix_solver_opts) -> x`,
-                where `A`: `scipy.sparse.csr_matrix`;
+                where `A`: `scipy.sparse.csr_array`;
                       `b`: `ArrayLike`;
                       `x`: `ArrayLike`;
                 Default is a wrapped version of `scipy.sparse.linalg.qmr()`
                  which doesn't return convergence info and logs the residual
                  every 100 iterations.
         matrix_solver_opts: Passed as kwargs to `matrix_solver(...)`
+        E_guess: Guess at the solution E-field. `matrix_solver` must accept an
+                `x0` argument with the same purpose.
 
     Returns:
         E-field which solves the system.
@@ -120,6 +124,13 @@ def generic(
         A = Pl @ A0 @ Pr
         b = Pl @ b0
 
+    if E_guess is not None:
+        if adjoint:
+            x0 = Pr.H @ E_guess
+        else:
+            x0 = Pl @ E_guess
+        matrix_solver_opts['x0'] = x0
+
     x = matrix_solver(A.tocsr(), b, **matrix_solver_opts)
 
     if adjoint:
@@ -127,4 +138,4 @@ def generic(
     else:
         x0 = Pr @ x
 
-    return x0
+    return vcfdfield_t(x0)

meanas/fdfd/waveguide_2d.py

@@ -18,8 +18,8 @@ $$
 \begin{aligned}
 \nabla \times \vec{E}(x, y, z) &= -\imath \omega \mu \vec{H} \\
 \nabla \times \vec{H}(x, y, z) &=  \imath \omega \epsilon \vec{E} \\
-\vec{E}(x,y,z) &= (\vec{E}_t(x, y) + E_z(x, y)\vec{z}) e^{-\gamma z} \\
+\vec{E}(x,y,z) &= (\vec{E}_t(x, y) + E_z(x, y)\vec{z}) e^{-\imath \beta z} \\
-\vec{H}(x,y,z) &= (\vec{H}_t(x, y) + H_z(x, y)\vec{z}) e^{-\gamma z} \\
+\vec{H}(x,y,z) &= (\vec{H}_t(x, y) + H_z(x, y)\vec{z}) e^{-\imath \beta z} \\
 \end{aligned}
 $$
 
@@ -40,56 +40,57 @@ Substituting in our expressions for $\vec{E}$, $\vec{H}$ and discretizing:
 
 $$
 \begin{aligned}
--\imath \omega \mu_{xx} H_x &= \tilde{\partial}_y E_z + \gamma E_y \\
+-\imath \omega \mu_{xx} H_x &= \tilde{\partial}_y E_z + \imath \beta E_y \\
--\imath \omega \mu_{yy} H_y &= -\gamma E_x - \tilde{\partial}_x E_z \\
+-\imath \omega \mu_{yy} H_y &= -\imath \beta E_x - \tilde{\partial}_x E_z \\
 -\imath \omega \mu_{zz} H_z &= \tilde{\partial}_x E_y - \tilde{\partial}_y E_x \\
-\imath \omega \epsilon_{xx} E_x &= \hat{\partial}_y H_z + \gamma H_y \\
+\imath \omega \epsilon_{xx} E_x &= \hat{\partial}_y H_z + \imath \beta H_y \\
-\imath \omega \epsilon_{yy} E_y &= -\gamma H_x - \hat{\partial}_x H_z \\
+\imath \omega \epsilon_{yy} E_y &= -\imath \beta H_x - \hat{\partial}_x H_z \\
 \imath \omega \epsilon_{zz} E_z &= \hat{\partial}_x H_y - \hat{\partial}_y H_x \\
 \end{aligned}
 $$
 
 Rewrite the last three equations as
+
 $$
 \begin{aligned}
-\gamma H_y &=  \imath \omega \epsilon_{xx} E_x - \hat{\partial}_y H_z \\
+\imath \beta H_y &=  \imath \omega \epsilon_{xx} E_x - \hat{\partial}_y H_z \\
-\gamma H_x &= -\imath \omega \epsilon_{yy} E_y - \hat{\partial}_x H_z \\
+\imath \beta H_x &= -\imath \omega \epsilon_{yy} E_y - \hat{\partial}_x H_z \\
 \imath \omega E_z &= \frac{1}{\epsilon_{zz}} \hat{\partial}_x H_y - \frac{1}{\epsilon_{zz}} \hat{\partial}_y H_x \\
 \end{aligned}
 $$
 
-Now apply $\gamma \tilde{\partial}_x$ to the last equation,
+Now apply $\imath \beta \tilde{\partial}_x$ to the last equation,
-then substitute in for $\gamma H_x$ and $\gamma H_y$:
+then substitute in for $\imath \beta H_x$ and $\imath \beta H_y$:
 
 $$
 \begin{aligned}
-\gamma \tilde{\partial}_x \imath \omega E_z &= \gamma \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x H_y
+\imath \beta \tilde{\partial}_x \imath \omega E_z &= \imath \beta \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x H_y
-                                             - \gamma \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y H_x \\
+                                                   - \imath \beta \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y H_x \\
         &= \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x ( \imath \omega \epsilon_{xx} E_x - \hat{\partial}_y H_z)
          - \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y (-\imath \omega \epsilon_{yy} E_y - \hat{\partial}_x H_z)  \\
         &= \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x ( \imath \omega \epsilon_{xx} E_x)
          - \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y (-\imath \omega \epsilon_{yy} E_y)  \\
-\gamma \tilde{\partial}_x E_z &= \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
+\imath \beta \tilde{\partial}_x E_z &= \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
                                      + \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y) \\
 \end{aligned}
 $$
 
-With a similar approach (but using $\gamma \tilde{\partial}_y$ instead), we can get
+With a similar approach (but using $\imath \beta \tilde{\partial}_y$ instead), we can get
 
 $$
 \begin{aligned}
-\gamma \tilde{\partial}_y E_z &= \tilde{\partial}_y \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
+\imath \beta \tilde{\partial}_y E_z &= \tilde{\partial}_y \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
                                      + \tilde{\partial}_y \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y) \\
 \end{aligned}
 $$
 
-We can combine this equation for $\gamma \tilde{\partial}_y E_z$ with
+We can combine this equation for $\imath \beta \tilde{\partial}_y E_z$ with
 the unused $\imath \omega \mu_{xx} H_x$ and $\imath \omega \mu_{yy} H_y$ equations to get
 
 $$
 \begin{aligned}
--\imath \omega \mu_{xx} \gamma H_x &=  \gamma^2 E_y + \gamma \tilde{\partial}_y E_z \\
+-\imath \omega \mu_{xx} \imath \beta H_x &=  -\beta^2 E_y + \imath \beta \tilde{\partial}_y E_z \\
--\imath \omega \mu_{xx} \gamma H_x &=  \gamma^2 E_y + \tilde{\partial}_y (
+-\imath \omega \mu_{xx} \imath \beta H_x &=  -\beta^2 E_y + \tilde{\partial}_y (
                                       \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
                                     + \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y)
                                     )\\
@@ -100,22 +101,21 @@ and
 
 $$
 \begin{aligned}
--\imath \omega \mu_{yy} \gamma H_y &= -\gamma^2 E_x - \gamma \tilde{\partial}_x E_z \\
+-\imath \omega \mu_{yy} \imath \beta H_y &= \beta^2 E_x - \imath \beta \tilde{\partial}_x E_z \\
--\imath \omega \mu_{yy} \gamma H_y &= -\gamma^2 E_x - \tilde{\partial}_x (
+-\imath \omega \mu_{yy} \imath \beta H_y &= \beta^2 E_x - \tilde{\partial}_x (
                                       \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
                                     + \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y)
                                     )\\
 \end{aligned}
 $$
 
-However, based on our rewritten equation for $\gamma H_x$ and the so-far unused
+However, based on our rewritten equation for $\imath \beta H_x$ and the so-far unused
 equation for $\imath \omega \mu_{zz} H_z$ we can also write
 
 $$
 \begin{aligned}
--\imath \omega \mu_{xx} (\gamma H_x) &= -\imath \omega \mu_{xx} (-\imath \omega \epsilon_{yy} E_y - \hat{\partial}_x H_z) \\
+-\imath \omega \mu_{xx} (\imath \beta H_x) &= -\imath \omega \mu_{xx} (-\imath \omega \epsilon_{yy} E_y - \hat{\partial}_x H_z) \\
-                                     &= -\omega^2 \mu_{xx} \epsilon_{yy} E_y
+                 &= -\omega^2 \mu_{xx} \epsilon_{yy} E_y + \imath \omega \mu_{xx} \hat{\partial}_x (
-                                        +\imath \omega \mu_{xx} \hat{\partial}_x (
                          \frac{1}{-\imath \omega \mu_{zz}} (\tilde{\partial}_x E_y - \tilde{\partial}_y E_x)) \\
                  &= -\omega^2 \mu_{xx} \epsilon_{yy} E_y
                          -\mu_{xx} \hat{\partial}_x \frac{1}{\mu_{zz}} (\tilde{\partial}_x E_y - \tilde{\partial}_y E_x) \\
@@ -126,7 +126,7 @@ and, similarly,
 
 $$
 \begin{aligned}
--\imath \omega \mu_{yy} (\gamma H_y) &= \omega^2 \mu_{yy} \epsilon_{xx} E_x
+-\imath \omega \mu_{yy} (\imath \beta H_y) &= \omega^2 \mu_{yy} \epsilon_{xx} E_x
                                            +\mu_{yy} \hat{\partial}_y \frac{1}{\mu_{zz}} (\tilde{\partial}_x E_y - \tilde{\partial}_y E_x) \\
 \end{aligned}
 $$
@@ -135,12 +135,12 @@ By combining both pairs of expressions, we get
 
 $$
 \begin{aligned}
--\gamma^2 E_x - \tilde{\partial}_x (
+\beta^2 E_x - \tilde{\partial}_x (
     \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
   + \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y)
     ) &= \omega^2 \mu_{yy} \epsilon_{xx} E_x
         +\mu_{yy} \hat{\partial}_y \frac{1}{\mu_{zz}} (\tilde{\partial}_x E_y - \tilde{\partial}_y E_x) \\
-\gamma^2 E_y + \tilde{\partial}_y (
+-\beta^2 E_y + \tilde{\partial}_y (
     \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
   + \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y)
     ) &= -\omega^2 \mu_{xx} \epsilon_{yy} E_y
@@ -165,27 +165,27 @@ $$
                     E_y \end{bmatrix}
 $$
 
-where $\gamma = \imath\beta$. In the literature, $\beta$ is usually used to denote
+In the literature, $\beta$ is usually used to denote the lossless/real part of the propagation constant,
-the lossless/real part of the propagation constant, but in `meanas` it is allowed to
+but in `meanas` it is allowed to be complex.
-be complex.
 
 An equivalent eigenvalue problem can be formed using the $H_x$ and $H_y$ fields, if those are more convenient.
 
-Note that $E_z$ was never discretized, so $\gamma$ and $\beta$ will need adjustment
+Note that $E_z$ was never discretized, so $\beta$ will need adjustment to account for numerical dispersion
-to account for numerical dispersion if the result is introduced into a space with a discretized z-axis.
+if the result is introduced into a space with a discretized z-axis.
 
 
 """
 # TODO update module docs
 
 from typing import Any
+from collections.abc import Sequence
 import numpy
-from numpy.typing import NDArray, ArrayLike
+from numpy.typing import NDArray
 from numpy.linalg import norm
-import scipy.sparse as sparse       # type: ignore
+from scipy import sparse
 
 from ..fdmath.operators import deriv_forward, deriv_back, cross
-from ..fdmath import vec, unvec, dx_lists_t, vfdfield_t, vcfdfield_t
+from ..fdmath import vec, unvec, dx_lists2_t, vcfdfield2_t, vcfdslice_t, vcfdfield2, vfdslice, vcfdslice
 from ..eigensolvers import signed_eigensolve, rayleigh_quotient_iteration
 
 
@@ -194,10 +194,10 @@ __author__ = 'Jan Petykiewicz'
 
 def operator_e(
         omega: complex,
-        dxes: dx_lists_t,
+        dxes: dx_lists2_t,
-        epsilon: vfdfield_t,
+        epsilon: vfdslice,
-        mu: vfdfield_t | None = None,
+        mu: vfdslice | None = None,
-        ) -> sparse.spmatrix:
+        ) -> sparse.sparray:
     r"""
     Waveguide operator of the form
 
@@ -246,12 +246,12 @@ def operator_e(
     Dbx, Dby = deriv_back(dxes[1])
 
     eps_parts = numpy.split(epsilon, 3)
-    eps_xy = sparse.diags(numpy.hstack((eps_parts[0], eps_parts[1])))
+    eps_xy = sparse.diags_array(numpy.hstack((eps_parts[0], eps_parts[1])))
-    eps_z_inv = sparse.diags(1 / eps_parts[2])
+    eps_z_inv = sparse.diags_array(1 / eps_parts[2])
 
     mu_parts = numpy.split(mu, 3)
-    mu_yx = sparse.diags(numpy.hstack((mu_parts[1], mu_parts[0])))
+    mu_yx = sparse.diags_array(numpy.hstack((mu_parts[1], mu_parts[0])))
-    mu_z_inv = sparse.diags(1 / mu_parts[2])
+    mu_z_inv = sparse.diags_array(1 / mu_parts[2])
 
     op = (
         omega * omega * mu_yx @ eps_xy
@@ -263,10 +263,10 @@ def operator_e(
 
 def operator_h(
         omega: complex,
-        dxes: dx_lists_t,
+        dxes: dx_lists2_t,
-        epsilon: vfdfield_t,
+        epsilon: vfdslice,
-        mu: vfdfield_t | None = None,
+        mu: vfdslice | None = None,
-        ) -> sparse.spmatrix:
+        ) -> sparse.sparray:
     r"""
     Waveguide operator of the form
 
@@ -315,12 +315,12 @@ def operator_h(
     Dbx, Dby = deriv_back(dxes[1])
 
     eps_parts = numpy.split(epsilon, 3)
-    eps_yx = sparse.diags(numpy.hstack((eps_parts[1], eps_parts[0])))
+    eps_yx = sparse.diags_array(numpy.hstack((eps_parts[1], eps_parts[0])))
-    eps_z_inv = sparse.diags(1 / eps_parts[2])
+    eps_z_inv = sparse.diags_array(1 / eps_parts[2])
 
     mu_parts = numpy.split(mu, 3)
-    mu_xy = sparse.diags(numpy.hstack((mu_parts[0], mu_parts[1])))
+    mu_xy = sparse.diags_array(numpy.hstack((mu_parts[0], mu_parts[1])))
-    mu_z_inv = sparse.diags(1 / mu_parts[2])
+    mu_z_inv = sparse.diags_array(1 / mu_parts[2])
 
     op = (
         omega * omega * eps_yx @ mu_xy
@@ -331,14 +331,14 @@ def operator_h(
 
 
 def normalized_fields_e(
-        e_xy: ArrayLike,
+        e_xy: vcfdfield2,
         wavenumber: complex,
         omega: complex,
-        dxes: dx_lists_t,
+        dxes: dx_lists2_t,
-        epsilon: vfdfield_t,
+        epsilon: vfdslice,
-        mu: vfdfield_t | None = None,
+        mu: vfdslice | None = None,
         prop_phase: float = 0,
-        ) -> tuple[vcfdfield_t, vcfdfield_t]:
+        ) -> tuple[vcfdslice_t, vcfdslice_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.
@@ -366,14 +366,14 @@ def normalized_fields_e(
 
 
 def normalized_fields_h(
-        h_xy: ArrayLike,
+        h_xy: vcfdfield2,
         wavenumber: complex,
         omega: complex,
-        dxes: dx_lists_t,
+        dxes: dx_lists2_t,
-        epsilon: vfdfield_t,
+        epsilon: vfdslice,
-        mu: vfdfield_t | None = None,
+        mu: vfdslice | None = None,
         prop_phase: float = 0,
-        ) -> tuple[vcfdfield_t, vcfdfield_t]:
+        ) -> tuple[vcfdslice_t, vcfdslice_t]:
     """
     Given a vector `h_xy` containing the vectorized H_x and H_y fields,
      returns normalized, vectorized E and H fields for the system.
@@ -401,30 +401,22 @@ def normalized_fields_h(
 
 
 def _normalized_fields(
-        e: vcfdfield_t,
+        e: vcfdslice,
-        h: vcfdfield_t,
+        h: vcfdslice,
         omega: complex,
-        dxes: dx_lists_t,
+        dxes: dx_lists2_t,
-        epsilon: vfdfield_t,
+        epsilon: vfdslice,
-        mu: vfdfield_t | None = None,
+        mu: vfdslice | None = None,
         prop_phase: float = 0,
-        ) -> tuple[vcfdfield_t, vcfdfield_t]:
+        ) -> tuple[vcfdslice_t, vcfdslice_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)]
-
-    E = unvec(e, shape)
-    H = unvec(h, shape)
 
     # Find time-averaged Sz and normalize to it
-    # H phase is adjusted by a half-cell forward shift for Yee cell, and 1-cell reverse shift for Poynting
+    Sz_tavg = inner_product(e, h, dxes=dxes, prop_phase=prop_phase, conj_h=True).real
-    phase = numpy.exp(-1j * -prop_phase / 2)
-    Sz_a = E[0] * numpy.conj(H[1] * phase) * dxes_real[0][1] * dxes_real[1][0]
-    Sz_b = E[1] * numpy.conj(H[0] * phase) * dxes_real[0][0] * dxes_real[1][1]
-    Sz_tavg = numpy.real(Sz_a.sum() - Sz_b.sum()) * 0.5       # 0.5 since E, H are assumed to be peak (not RMS) amplitudes
     assert Sz_tavg > 0, f'Found a mode propagating in the wrong direction! {Sz_tavg=}'
 
-    energy = epsilon * e.conj() * e
+    energy = numpy.real(epsilon * e.conj() * e)
 
     norm_amplitude = 1 / numpy.sqrt(Sz_tavg)
     norm_angle = -numpy.angle(e[energy.argmax()])       # Will randomly add a negative sign when mode is symmetric
@@ -434,22 +426,23 @@ def _normalized_fields(
     sign = numpy.sign(E_weighted[:,
                                  :max(shape[0] // 2, 1),
                                  :max(shape[1] // 2, 1)].real.sum())
+    assert sign != 0
 
     norm_factor = sign * norm_amplitude * numpy.exp(1j * norm_angle)
 
     e *= norm_factor
     h *= norm_factor
 
-    return e, h
+    return vcfdslice_t(e), vcfdslice_t(h)
 
 
 def exy2h(
         wavenumber: complex,
         omega: complex,
-        dxes: dx_lists_t,
+        dxes: dx_lists2_t,
-        epsilon: vfdfield_t,
+        epsilon: vfdslice,
-        mu: vfdfield_t | None = None
+        mu: vfdslice | None = None
-        ) -> sparse.spmatrix:
+        ) -> sparse.sparray:
     """
     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
@@ -472,10 +465,10 @@ def exy2h(
 def hxy2e(
         wavenumber: complex,
         omega: complex,
-        dxes: dx_lists_t,
+        dxes: dx_lists2_t,
-        epsilon: vfdfield_t,
+        epsilon: vfdslice,
-        mu: vfdfield_t | None = None
+        mu: vfdslice | None = None
-        ) -> sparse.spmatrix:
+        ) -> sparse.sparray:
     """
     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
@@ -497,9 +490,9 @@ def hxy2e(
 
 def hxy2h(
         wavenumber: complex,
-        dxes: dx_lists_t,
+        dxes: dx_lists2_t,
-        mu: vfdfield_t | None = None
+        mu: vfdslice | None = None
-        ) -> sparse.spmatrix:
+        ) -> sparse.sparray:
     """
     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
@@ -518,26 +511,53 @@ def hxy2h(
 
     if mu is not None:
         mu_parts = numpy.split(mu, 3)
-        mu_xy = sparse.diags(numpy.hstack((mu_parts[0], mu_parts[1])))
+        mu_xy = sparse.diags_array(numpy.hstack((mu_parts[0], mu_parts[1])))
-        mu_z_inv = sparse.diags(1 / mu_parts[2])
+        mu_z_inv = sparse.diags_array(1 / mu_parts[2])
 
         hxy2hz = mu_z_inv @ hxy2hz @ mu_xy
 
     n_pts = dxes[1][0].size * dxes[1][1].size
-    op = sparse.vstack((sparse.eye(2 * n_pts),
+    op = sparse.vstack((sparse.eye_array(2 * n_pts),
                         hxy2hz))
     return op
 
 
 def exy2e(
         wavenumber: complex,
-        dxes: dx_lists_t,
+        dxes: dx_lists2_t,
-        epsilon: vfdfield_t,
+        epsilon: vfdslice,
-        ) -> sparse.spmatrix:
+        ) -> sparse.sparray:
-    """
+    r"""
     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
 
+    From the operator derivation (see module docs), we have
+
+    $$
+    \imath \omega \epsilon_{zz} E_z = \hat{\partial}_x H_y - \hat{\partial}_y H_x \\
+    $$
+
+    as well as the intermediate equations
+
+    $$
+    \begin{aligned}
+    \imath \beta H_y &=  \imath \omega \epsilon_{xx} E_x - \hat{\partial}_y H_z \\
+    \imath \beta H_x &= -\imath \omega \epsilon_{yy} E_y - \hat{\partial}_x H_z \\
+    \end{aligned}
+    $$
+
+    Combining these, we get
+
+    $$
+    \begin{aligned}
+    E_z &= \frac{1}{- \omega \beta \epsilon_{zz}} ((
+             \hat{\partial}_y \hat{\partial}_x H_z
+            -\hat{\partial}_x \hat{\partial}_y H_z)
+          + \imath \omega (\hat{\partial}_x \epsilon_{xx} E_x + \hat{\partial}_y \epsilon{yy} E_y))
+        &= \frac{1}{\imath \beta \epsilon_{zz}} (\hat{\partial}_x \epsilon_{xx} E_x + \hat{\partial}_y \epsilon{yy} E_y)
+    \end{aligned}
+    $$
+
     Args:
         wavenumber: Wavenumber assuming fields have z-dependence of `exp(-i * wavenumber * z)`
                     It should satisfy `operator_e() @ e_xy == wavenumber**2 * e_xy`
@@ -552,13 +572,13 @@ def exy2e(
 
     if epsilon is not None:
         epsilon_parts = numpy.split(epsilon, 3)
-        epsilon_xy = sparse.diags(numpy.hstack((epsilon_parts[0], epsilon_parts[1])))
+        epsilon_xy = sparse.diags_array(numpy.hstack((epsilon_parts[0], epsilon_parts[1])))
-        epsilon_z_inv = sparse.diags(1 / epsilon_parts[2])
+        epsilon_z_inv = sparse.diags_array(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),
+    op = sparse.vstack((sparse.eye_array(2 * n_pts),
                         exy2ez))
     return op
 
@@ -566,12 +586,12 @@ def exy2e(
 def e2h(
         wavenumber: complex,
         omega: complex,
-        dxes: dx_lists_t,
+        dxes: dx_lists2_t,
-        mu: vfdfield_t | None = None
+        mu: vfdslice | None = None
-        ) -> sparse.spmatrix:
+        ) -> sparse.sparray:
     """
     Returns an operator which, when applied to a vectorized E eigenfield, produces
-     the vectorized H eigenfield.
+     the vectorized H eigenfield slice.
 
     Args:
         wavenumber: Wavenumber assuming fields have z-dependence of `exp(-i * wavenumber * z)`
@@ -584,19 +604,19 @@ def e2h(
     """
     op = curl_e(wavenumber, dxes) / (-1j * omega)
     if mu is not None:
-        op = sparse.diags(1 / mu) @ op
+        op = sparse.diags_array(1 / mu) @ op
     return op
 
 
 def h2e(
         wavenumber: complex,
         omega: complex,
-        dxes: dx_lists_t,
+        dxes: dx_lists2_t,
-        epsilon: vfdfield_t
+        epsilon: vfdslice,
-        ) -> sparse.spmatrix:
+        ) -> sparse.sparray:
     """
     Returns an operator which, when applied to a vectorized H eigenfield, produces
-     the vectorized E eigenfield.
+     the vectorized E eigenfield slice.
 
     Args:
         wavenumber: Wavenumber assuming fields have z-dependence of `exp(-i * wavenumber * z)`
@@ -607,13 +627,13 @@ def h2e(
     Returns:
         Sparse matrix representation of the operator.
     """
-    op = sparse.diags(1 / (1j * omega * epsilon)) @ curl_h(wavenumber, dxes)
+    op = sparse.diags_array(1 / (1j * omega * epsilon)) @ curl_h(wavenumber, dxes)
     return op
 
 
-def curl_e(wavenumber: complex, dxes: dx_lists_t) -> sparse.spmatrix:
+def curl_e(wavenumber: complex, dxes: dx_lists2_t) -> sparse.sparray:
     """
-    Discretized curl operator for use with the waveguide E field.
+    Discretized curl operator for use with the waveguide E field slice.
 
     Args:
         wavenumber: Wavenumber assuming fields have z-dependence of `exp(-i * wavenumber * z)`
@@ -622,18 +642,18 @@ def curl_e(wavenumber: complex, dxes: dx_lists_t) -> sparse.spmatrix:
     Returns:
         Sparse matrix representation of the operator.
     """
-    n = 1
+    nn = 1
-    for d in dxes[0]:
+    for dd in dxes[0]:
-        n *= len(d)
+        nn *= len(dd)
 
-    Bz = -1j * wavenumber * sparse.eye(n)
+    Bz = -1j * wavenumber * sparse.eye_array(nn)
     Dfx, Dfy = deriv_forward(dxes[0])
     return cross([Dfx, Dfy, Bz])
 
 
-def curl_h(wavenumber: complex, dxes: dx_lists_t) -> sparse.spmatrix:
+def curl_h(wavenumber: complex, dxes: dx_lists2_t) -> sparse.sparray:
     """
-    Discretized curl operator for use with the waveguide H field.
+    Discretized curl operator for use with the waveguide H field slice.
 
     Args:
         wavenumber: Wavenumber assuming fields have z-dependence of `exp(-i * wavenumber * z)`
@@ -642,22 +662,22 @@ def curl_h(wavenumber: complex, dxes: dx_lists_t) -> sparse.spmatrix:
     Returns:
         Sparse matrix representation of the operator.
     """
-    n = 1
+    nn = 1
-    for d in dxes[1]:
+    for dd in dxes[1]:
-        n *= len(d)
+        nn *= len(dd)
 
-    Bz = -1j * wavenumber * sparse.eye(n)
+    Bz = -1j * wavenumber * sparse.eye_array(nn)
     Dbx, Dby = deriv_back(dxes[1])
     return cross([Dbx, Dby, Bz])
 
 
 def h_err(
-        h: vcfdfield_t,
+        h: vcfdslice,
         wavenumber: complex,
         omega: complex,
-        dxes: dx_lists_t,
+        dxes: dx_lists2_t,
-        epsilon: vfdfield_t,
+        epsilon: vfdslice,
-        mu: vfdfield_t | None = None
+        mu: vfdslice | None = None
         ) -> float:
     """
     Calculates the relative error in the H field
@@ -676,7 +696,7 @@ def h_err(
     ce = curl_e(wavenumber, dxes)
     ch = curl_h(wavenumber, dxes)
 
-    eps_inv = sparse.diags(1 / epsilon)
+    eps_inv = sparse.diags_array(1 / epsilon)
 
     if mu is None:
         op = ce @ eps_inv @ ch @ h - omega ** 2 * h
@@ -687,12 +707,12 @@ def h_err(
 
 
 def e_err(
-        e: vcfdfield_t,
+        e: vcfdslice,
         wavenumber: complex,
         omega: complex,
-        dxes: dx_lists_t,
+        dxes: dx_lists2_t,
-        epsilon: vfdfield_t,
+        epsilon: vfdslice,
-        mu: vfdfield_t | None = None,
+        mu: vfdslice | None = None,
         ) -> float:
     """
     Calculates the relative error in the E field
@@ -714,21 +734,21 @@ def e_err(
     if mu is None:
         op = ch @ ce @ e - omega ** 2 * (epsilon * e)
     else:
-        mu_inv = sparse.diags(1 / mu)
+        mu_inv = sparse.diags_array(1 / mu)
         op = ch @ mu_inv @ ce @ e - omega ** 2 * (epsilon * e)
 
     return float(norm(op) / norm(e))
 
 
 def sensitivity(
-        e_norm: vcfdfield_t,
+        e_norm: vcfdslice,
-        h_norm: vcfdfield_t,
+        h_norm: vcfdslice,
         wavenumber: complex,
         omega: complex,
-        dxes: dx_lists_t,
+        dxes: dx_lists2_t,
-        epsilon: vfdfield_t,
+        epsilon: vfdslice,
-        mu: vfdfield_t | None = None,
+        mu: vfdslice | None = None,
-        ) -> vcfdfield_t:
+        ) -> vcfdslice_t:
     r"""
       Given a waveguide structure (`dxes`, `epsilon`, `mu`) and mode fields
     (`e_norm`, `h_norm`, `wavenumber`, `omega`), calculates the sensitivity of the wavenumber
@@ -802,11 +822,11 @@ def sensitivity(
     Dbx, Dby = deriv_back(dxes[1])
 
     eps_x, eps_y, eps_z = numpy.split(epsilon, 3)
-    eps_xy = sparse.diags(numpy.hstack((eps_x, eps_y)))
+    eps_xy = sparse.diags_array(numpy.hstack((eps_x, eps_y)))
-    eps_z_inv = sparse.diags(1 / eps_z)
+    eps_z_inv = sparse.diags_array(1 / eps_z)
 
     mu_x, mu_y, _mu_z = numpy.split(mu, 3)
-    mu_yx = sparse.diags(numpy.hstack((mu_y, mu_x)))
+    mu_yx = sparse.diags_array(numpy.hstack((mu_y, mu_x)))
 
     da_exxhyy = vec(dxes[1][0][:, None] * dxes[0][1][None, :])
     da_eyyhxx = vec(dxes[1][1][None, :] * dxes[0][0][:, None])
@@ -820,15 +840,15 @@ def sensitivity(
     norm = hv_yx_conj @ ev_xy
 
     sens_tot = numpy.concatenate([sens_xy1 + sens_xy2, sens_z]) / (2 * wavenumber * norm)
-    return sens_tot
+    return vcfdslice_t(sens_tot)
 
 
 def solve_modes(
-        mode_numbers: list[int],
+        mode_numbers: Sequence[int],
         omega: complex,
-        dxes: dx_lists_t,
+        dxes: dx_lists2_t,
-        epsilon: vfdfield_t,
+        epsilon: vfdslice,
-        mu: vfdfield_t | None = None,
+        mu: vfdslice | None = None,
         mode_margin: int = 2,
         ) -> tuple[NDArray[numpy.complex128], NDArray[numpy.complex128]]:
     """
@@ -845,32 +865,38 @@ def solve_modes(
             ability to find the correct mode. Default 2.
 
     Returns:
-        e_xys: list of vfdfield_t specifying fields
+        e_xys: NDArray of vfdfield_t specifying fields. First dimension is mode number.
         wavenumbers: list of wavenumbers
     """
 
-    '''
+    #
-    Solve for the largest-magnitude eigenvalue of the real operator
+    # Solve for the largest-magnitude eigenvalue of the real operator
-    '''
+    #
     dxes_real = [[numpy.real(dx) for dx in dxi] for dxi in dxes]
     mu_real = None if mu is None else numpy.real(mu)
     A_r = operator_e(numpy.real(omega), dxes_real, numpy.real(epsilon), mu_real)
 
     eigvals, eigvecs = signed_eigensolve(A_r, max(mode_numbers) + mode_margin)
-    e_xys = eigvecs[:, -(numpy.array(mode_numbers) + 1)]
+    keep_inds = -(numpy.array(mode_numbers) + 1)
+    e_xys = eigvecs[:, keep_inds].T
+    eigvals = eigvals[keep_inds]
 
-    '''
+    #
-    Now solve for the eigenvector of the full operator, using the real operator's
+    # Now solve for the eigenvector of the full operator, using the real operator's
-     eigenvector as an initial guess for Rayleigh quotient iteration.
+    #  eigenvector as an initial guess for Rayleigh quotient iteration.
-    '''
+    #
     A = operator_e(omega, dxes, epsilon, mu)
     for nn in range(len(mode_numbers)):
-        eigvals[nn], e_xys[:, nn] = rayleigh_quotient_iteration(A, e_xys[:, nn])
+        eigvals[nn], e_xys[nn, :] = rayleigh_quotient_iteration(A, e_xys[nn, :])
 
     # Calculate the wave-vector (force the real part to be positive)
     wavenumbers = numpy.sqrt(eigvals)
     wavenumbers *= numpy.sign(numpy.real(wavenumbers))
 
+    order = wavenumbers.argsort()[::-1]
+    e_xys = e_xys[order]
+    wavenumbers = wavenumbers[order]
+
     return e_xys, wavenumbers
 
 
@@ -878,7 +904,7 @@ def solve_mode(
         mode_number: int,
         *args: Any,
         **kwargs: Any,
-        ) -> tuple[vcfdfield_t, complex]:
+        ) -> tuple[vcfdfield2_t, complex]:
     """
     Wrapper around `solve_modes()` that solves for a single mode.
 
@@ -892,4 +918,38 @@ def solve_mode(
     """
     kwargs['mode_numbers'] = [mode_number]
     e_xys, wavenumbers = solve_modes(*args, **kwargs)
-    return e_xys[:, 0], wavenumbers[0]
+    return vcfdfield2_t(e_xys[0]), wavenumbers[0]
+
+
+def inner_product(    # TODO documentation
+        e1: vcfdfield2,
+        h2: vcfdfield2,
+        dxes: dx_lists2_t,
+        prop_phase: float = 0,
+        conj_h: bool = False,
+        trapezoid: bool = False,
+        ) -> complex:
+
+    shape = [s.size for s in dxes[0]]
+
+    # H phase is adjusted by a half-cell forward shift for Yee cell, and 1-cell reverse shift for Poynting
+    phase = numpy.exp(-1j * -prop_phase / 2)
+
+    E1 = unvec(e1, shape)
+    H2 = unvec(h2, shape) * phase
+
+    if conj_h:
+        H2 = numpy.conj(H2)
+
+    # Find time-averaged Sz and normalize to it
+    dxes_real = [[numpy.real(dxyz) for dxyz in dxeh] for dxeh in dxes]
+    if trapezoid:
+        Sz_a = numpy.trapezoid(numpy.trapezoid(E1[0] * H2[1], numpy.cumsum(dxes_real[0][1])), numpy.cumsum(dxes_real[1][0]))
+        Sz_b = numpy.trapezoid(numpy.trapezoid(E1[1] * H2[0], numpy.cumsum(dxes_real[0][0])), numpy.cumsum(dxes_real[1][1]))
+    else:
+        Sz_a = E1[0] * H2[1] * dxes_real[1][0][:, None] * dxes_real[0][1][None, :]
+        Sz_b = E1[1] * H2[0] * dxes_real[0][0][:, None] * dxes_real[1][1][None, :]
+    Sz = 0.5 * (Sz_a.sum() - Sz_b.sum())
+    return Sz
+
+

meanas/fdfd/waveguide_3d.py

@@ -4,11 +4,13 @@ Tools for working with waveguide modes in 3D domains.
 This module relies heavily on `waveguide_2d` and mostly just transforms
 its parameters into 2D equivalents and expands the results back into 3D.
 """
-from typing import Sequence, Any
+from typing import Any, cast
+from collections.abc import Sequence
 import numpy
 from numpy.typing import NDArray
+from numpy import complexfloating
 
-from ..fdmath import vec, unvec, dx_lists_t, fdfield_t, cfdfield_t
+from ..fdmath import vec, unvec, dx_lists_t, cfdfield_t, fdfield, cfdfield
 from . import operators, waveguide_2d
 
 
@@ -19,9 +21,9 @@ def solve_mode(
         axis: int,
         polarity: int,
         slices: Sequence[slice],
-        epsilon: fdfield_t,
+        epsilon: fdfield,
-        mu: fdfield_t | None = None,
+        mu: fdfield | None = None,
-        ) -> dict[str, complex | NDArray[numpy.float_]]:
+        ) -> dict[str, complex | NDArray[complexfloating]]:
     """
     Given a 3D grid, selects a slice from the grid and attempts to
      solve for an eigenmode propagating through that slice.
@@ -40,9 +42,10 @@ def solve_mode(
     Returns:
         ```
         {
-            'E': list[NDArray[numpy.float_]],
+            'E': NDArray[complexfloating],
-            'H': list[NDArray[numpy.float_]],
+            'H': NDArray[complexfloating],
             'wavenumber': complex,
+            'wavenumber_2d': complex,
         }
         ```
     """
@@ -51,9 +54,9 @@ def solve_mode(
 
     slices = tuple(slices)
 
-    '''
+    #
-    Solve the 2D problem in the specified plane
+    # Solve the 2D problem in the specified plane
-    '''
+    #
     # Define rotation to set z as propagation direction
     order = numpy.roll(range(3), 2 - axis)
     reverse_order = numpy.roll(range(3), axis - 2)
@@ -71,9 +74,10 @@ def solve_mode(
     }
     e_xy, wavenumber_2d = waveguide_2d.solve_mode(mode_number, **args_2d)
 
-    '''
+    #
-    Apply corrections and expand to 3D
+    # Apply corrections and expand to 3D
-    '''
+    #
+
     # Correct wavenumber to account for numerical dispersion.
     wavenumber = 2 / dx_prop * numpy.arcsin(wavenumber_2d * dx_prop / 2)
 
@@ -92,9 +96,10 @@ def solve_mode(
     # Expand E, H to full epsilon space we were given
     E = numpy.zeros_like(epsilon, dtype=complex)
     H = numpy.zeros_like(epsilon, dtype=complex)
-    for a, o in enumerate(reverse_order):
+    for aa, oo in enumerate(reverse_order):
-        E[(a, *slices)] = e[o][:, :, None].transpose(reverse_order)
+        iii = cast('tuple[slice | int]', (aa, *slices))
-        H[(a, *slices)] = h[o][:, :, None].transpose(reverse_order)
+        E[iii] = e[oo][:, :, None].transpose(reverse_order)
+        H[iii] = h[oo][:, :, None].transpose(reverse_order)
 
     results = {
         'wavenumber': wavenumber,
@@ -106,15 +111,15 @@ def solve_mode(
 
 
 def compute_source(
-        E: cfdfield_t,
+        E: cfdfield,
         wavenumber: complex,
         omega: complex,
         dxes: dx_lists_t,
         axis: int,
         polarity: int,
         slices: Sequence[slice],
-        epsilon: fdfield_t,
+        epsilon: fdfield,
-        mu: fdfield_t | None = None,
+        mu: fdfield | None = None,
         ) -> cfdfield_t:
     """
     Given an eigenmode obtained by `solve_mode`, returns the current source distribution
@@ -148,35 +153,32 @@ def compute_source(
 
     masked_e2j = operators.e_boundary_source(mask=vec(mask), omega=omega, dxes=dxes, epsilon=vec(epsilon), mu=vec(mu))
     J = unvec(masked_e2j @ vec(E_expanded), E.shape[1:])
-    return J
+    return cfdfield_t(J)
 
 
 def compute_overlap_e(
-        E: cfdfield_t,
+        E: cfdfield,
         wavenumber: complex,
         dxes: dx_lists_t,
         axis: int,
         polarity: int,
         slices: Sequence[slice],
-        ) -> cfdfield_t:                 # TODO DOCS
+        ) -> cfdfield_t:
     """
     Given an eigenmode obtained by `solve_mode`, calculates an overlap_e for the
     mode orthogonality relation Integrate(((E x H_mode) + (E_mode x H)) dot dn)
     [assumes reflection symmetry].
 
-    TODO: add reference
+    TODO: add reference or derivation for compute_overlap_e
 
     Args:
         E: E-field of the mode
-        H: H-field of the mode (advanced by half of a Yee cell from E)
         wavenumber: Wavenumber of the mode
-        omega: Angular frequency of the simulation
         dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types`
         axis: Propagation axis (0=x, 1=y, 2=z)
         polarity: Propagation direction (+1 for +ve, -1 for -ve)
         slices: `epsilon[tuple(slices)]` is used to select the portion of the grid to use
                 as the waveguide cross-section. slices[axis] should select only one item.
-        mu: Magnetic permeability (default 1 everywhere)
 
     Returns:
         overlap_e such that `numpy.sum(overlap_e * other_e.conj())` computes the overlap integral
@@ -195,12 +197,14 @@ def compute_overlap_e(
     Etgt = numpy.zeros_like(Ee)
     Etgt[slices2] = Ee[slices2]
 
-    Etgt /= (Etgt.conj() * Etgt).sum()
+    # note no sqrt() when normalizing below since we want to get 1.0 after overlapping with the
-    return Etgt
+    # original field, not the normalized one
+    Etgt /= (Etgt.conj() * Etgt).sum()    # type: ignore
+    return cfdfield_t(Etgt)
 
 
 def expand_e(
-        E: cfdfield_t,
+        E: cfdfield,
         wavenumber: complex,
         dxes: dx_lists_t,
         axis: int,
@@ -245,4 +249,4 @@ def expand_e(
     slices_in = (slice(None), *slices)
 
     E_expanded[slices_exp] = phase_E * numpy.array(E)[slices_in]
-    return E_expanded
+    return cfdfield_t(E_expanded)

meanas/fdfd/waveguide_cyl.py

@@ -1,34 +1,102 @@
-"""
+r"""
 Operators and helper functions for cylindrical waveguides with unchanging cross-section.
 
-WORK IN PROGRESS, CURRENTLY BROKEN
+Waveguide operator is derived according to 10.1364/OL.33.001848.
+The curl equations in the complex coordinate system become
 
-As the z-dependence is known, all the functions in this file assume a 2D grid
+$$
+\begin{aligned}
+-\imath \omega \mu_{xx} H_x &= \tilde{\partial}_y E_z + \imath \beta frac{E_y}{\tilde{t}_x} \\
+-\imath \omega \mu_{yy} H_y &= -\imath \beta E_x - \frac{1}{\hat{t}_x} \tilde{\partial}_x \tilde{t}_x E_z \\
+-\imath \omega \mu_{zz} H_z &= \tilde{\partial}_x E_y - \tilde{\partial}_y E_x \\
+\imath \omega \epsilon_{xx} E_x &= \hat{\partial}_y H_z + \imath \beta \frac{H_y}{\hat{T}} \\
+\imath \omega \epsilon_{yy} E_y &= -\imath \beta H_x - \{1}{\tilde{t}_x} \hat{\partial}_x \hat{t}_x} H_z \\
+\imath \omega \epsilon_{zz} E_z &= \hat{\partial}_x H_y - \hat{\partial}_y H_x \\
+\end{aligned}
+$$
+
+where $t_x = 1 + \frac{\Delta_{x, m}}{R_0}$ is the grid spacing adjusted by the nominal radius $R0$.
+
+Rewrite the last three equations as
+
+$$
+\begin{aligned}
+\imath \beta H_y &=  \imath \omega \hat{t}_x \epsilon_{xx} E_x - \hat{t}_x \hat{\partial}_y H_z \\
+\imath \beta H_x &= -\imath \omega \hat{t}_x \epsilon_{yy} E_y - \hat{t}_x \hat{\partial}_x H_z \\
+\imath \omega E_z &= \frac{1}{\epsilon_{zz}} \hat{\partial}_x H_y - \frac{1}{\epsilon_{zz}} \hat{\partial}_y H_x \\
+\end{aligned}
+$$
+
+The derivation then follows the same steps as the straight waveguide, leading to the eigenvalue problem
+
+$$
+\beta^2 \begin{bmatrix} E_x \\
+                        E_y \end{bmatrix} =
+    (\omega^2 \begin{bmatrix} T_b T_b \mu_{yy} \epsilon_{xx} & 0 \\
+                                                            0 & T_a T_a \mu_{xx} \epsilon_{yy} \end{bmatrix} +
+              \begin{bmatrix} -T_b \mu_{yy} \hat{\partial}_y \\
+                               T_a \mu_{xx} \hat{\partial}_x \end{bmatrix} T_b \mu_{zz}^{-1}
+              \begin{bmatrix} -\tilde{\partial}_y & \tilde{\partial}_x \end{bmatrix} +
+      \begin{bmatrix} \tilde{\partial}_x \\
+                      \tilde{\partial}_y \end{bmatrix} T_a \epsilon_{zz}^{-1}
+                 \begin{bmatrix} \hat{\partial}_x T_b \epsilon_{xx} & \hat{\partial}_y T_a \epsilon_{yy} \end{bmatrix})
+    \begin{bmatrix} E_x \\
+                    E_y \end{bmatrix}
+$$
+
+which resembles the straight waveguide eigenproblem with additonal $T_a$ and $T_b$ terms. These
+are diagonal matrices containing the $t_x$ values:
+
+$$
+\begin{aligned}
+T_a &=  1 + \frac{\Delta_{x, m               }}{R_0}
+T_b &=  1 + \frac{\Delta_{x, m + \frac{1}{2} }}{R_0}
+\end{aligned}
+
+
+TODO: consider 10.1364/OE.20.021583 for an alternate approach
+$$
+
+As in the straight waveguide case, all the functions in this file assume a 2D grid
  (i.e. `dxes = [[[dr_e_0, dx_e_1, ...], [dy_e_0, ...]], [[dr_h_0, ...], [dy_h_0, ...]]]`).
 """
-# TODO update module docs
+from typing import Any, cast
+from collections.abc import Sequence
+import logging
 
 import numpy
-import scipy.sparse as sparse       # type: ignore
+from numpy.typing import NDArray, ArrayLike
+from scipy import sparse
 
-from ..fdmath import vec, unvec, dx_lists_t, fdfield_t, vfdfield_t, cfdfield_t
+from ..fdmath import vec, unvec, dx_lists2_t, vcfdslice_t, vcfdfield2_t, vfdslice, vcfdslice, vcfdfield2
 from ..fdmath.operators import deriv_forward, deriv_back
 from ..eigensolvers import signed_eigensolve, rayleigh_quotient_iteration
+from . import waveguide_2d
+
+logger = logging.getLogger(__name__)
 
 
 def cylindrical_operator(
-        omega: complex,
+        omega: float,
-        dxes: dx_lists_t,
+        dxes: dx_lists2_t,
-        epsilon: vfdfield_t,
+        epsilon: vfdslice,
-        r0: float,
+        rmin: float,
-        ) -> sparse.spmatrix:
+        ) -> sparse.sparray:
-    """
+    r"""
     Cylindrical coordinate waveguide operator of the form
 
-    (NOTE: See 10.1364/OL.33.001848)
+    $$
-    TODO: consider 10.1364/OE.20.021583
+        (\omega^2 \begin{bmatrix} T_b T_b \mu_{yy} \epsilon_{xx} & 0 \\
-
+                                                                0 & T_a T_a \mu_{xx} \epsilon_{yy} \end{bmatrix} +
-    TODO
+                  \begin{bmatrix} -T_b \mu_{yy} \hat{\partial}_y \\
+                                   T_a \mu_{xx} \hat{\partial}_x \end{bmatrix} T_b \mu_{zz}^{-1}
+                  \begin{bmatrix} -\tilde{\partial}_y & \tilde{\partial}_x \end{bmatrix} +
+          \begin{bmatrix} \tilde{\partial}_x \\
+                          \tilde{\partial}_y \end{bmatrix} T_a \epsilon_{zz}^{-1}
+                     \begin{bmatrix} \hat{\partial}_x T_b \epsilon_{xx} & \hat{\partial}_y T_a \epsilon_{yy} \end{bmatrix})
+        \begin{bmatrix} E_x \\
+                        E_y \end{bmatrix}
+    $$
 
     for use with a field vector of the form `[E_r, E_y]`.
 
@@ -38,12 +106,13 @@ def cylindrical_operator(
     which can then be solved for the eigenmodes of the system
     (an `exp(-i * wavenumber * theta)` theta-dependence is assumed for the fields).
 
+    (NOTE: See module docs and 10.1364/OL.33.001848)
+
     Args:
         omega: The angular frequency of the system
         dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types` (2D)
         epsilon: Vectorized dielectric constant grid
-        r0: Radius of curvature for the simulation. This should be the minimum value of
+        rmin: Radius at the left edge of the simulation domain (at minimum 'x')
-            r within the simulation domain.
 
     Returns:
         Sparse matrix representation of the operator
@@ -52,46 +121,34 @@ def cylindrical_operator(
     Dfx, Dfy = deriv_forward(dxes[0])
     Dbx, Dby = deriv_back(dxes[1])
 
-    rx = r0 + numpy.cumsum(dxes[0][0])
+    Ta, Tb = dxes2T(dxes=dxes, rmin=rmin)
-    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_x = sparse.diags_array(eps_parts[0])
-    eps_y = sparse.diags(eps_parts[1])
+    eps_y = sparse.diags_array(eps_parts[1])
-    eps_z_inv = sparse.diags(1 / eps_parts[2])
+    eps_z_inv = sparse.diags_array(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
 
     omega2 = omega * omega
+    diag = sparse.block_diag
 
-    op = (
+    sq0 = omega2 * diag((Tb @ Tb @ eps_x,
-        (omega2 * diag((Tx, Ty)) + pa) @ diag((a0, a1))
+                         Ta @ Ta @ eps_y))
-        - (sparse.bmat(((None, Ty), (Tx, None))) + pb / omega2) @ diag((b0, b1))
+    lin0 = sparse.vstack((-Tb @ Dby, Ta @ Dbx)) @ Tb @ sparse.hstack((-Dfy, Dfx))
-        )
+    lin1 = sparse.vstack((Dfx, Dfy)) @ Ta @ eps_z_inv @ sparse.hstack((Dbx @ Tb @ eps_x,
+                                                                       Dby @ Ta @ eps_y))
+    op = sq0 + lin0 + lin1
     return op
 
 
-def solve_mode(
+def solve_modes(
-        mode_number: int,
+        mode_numbers: Sequence[int],
-        omega: complex,
+        omega: float,
-        dxes: dx_lists_t,
+        dxes: dx_lists2_t,
-        epsilon: vfdfield_t,
+        epsilon: vfdslice,
-        r0: float,
+        rmin: float,
-        ) -> dict[str, complex | cfdfield_t]:
+        mode_margin: int = 2,
+        ) -> tuple[NDArray[numpy.complex128], NDArray[numpy.complex128]]:
     """
-    TODO: fixup
     Given a 2d (r, y) slice of epsilon, attempts to solve for the eigenmode
      of the bent waveguide with the specified mode number.
 
@@ -101,48 +158,336 @@ def solve_mode(
         dxes: Grid parameters [dx_e, dx_h] as described in meanas.fdmath.types.
               The first coordinate is assumed to be r, the second is y.
         epsilon: Dielectric constant
-        r0: Radius of curvature for the simulation. This should be the minimum value of
+        rmin: Radius of curvature for the simulation. This should be the minimum value of
                r within the simulation domain.
 
     Returns:
-        ```
+        e_xys: NDArray of vfdfield_t specifying fields. First dimension is mode number.
-        {
+        angular_wavenumbers: list of wavenumbers in 1/rad units.
-            'E': list[NDArray[numpy.complex_]],
-            'H': list[NDArray[numpy.complex_]],
-            'wavenumber': complex,
-        }
-        ```
     """
 
-    '''
+    #
-    Solve for the largest-magnitude eigenvalue of the real operator
+    # 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 = cylindrical_operator(numpy.real(omega), dxes_real, numpy.real(epsilon), r0)
+    A_r = cylindrical_operator(numpy.real(omega), dxes_real, numpy.real(epsilon), rmin=rmin)
-    eigvals, eigvecs = signed_eigensolve(A_r, mode_number + 3)
+    eigvals, eigvecs = signed_eigensolve(A_r, max(mode_numbers) + mode_margin)
-    e_xy = eigvecs[:, -(mode_number + 1)]
+    keep_inds = -(numpy.array(mode_numbers) + 1)
+    e_xys = eigvecs[:, keep_inds].T
+    eigvals = eigvals[keep_inds]
 
-    '''
+    #
-    Now solve for the eigenvector of the full operator, using the real operator's
+    # Now solve for the eigenvector of the full operator, using the real operator's
-     eigenvector as an initial guess for Rayleigh quotient iteration.
+    #  eigenvector as an initial guess for Rayleigh quotient iteration.
-    '''
+    #
-    A = cylindrical_operator(omega, dxes, epsilon, r0)
+    A = cylindrical_operator(omega, dxes, epsilon, rmin=rmin)
-    eigval, e_xy = rayleigh_quotient_iteration(A, e_xy)
+    for nn in range(len(mode_numbers)):
+        eigvals[nn], e_xys[nn, :] = rayleigh_quotient_iteration(A, e_xys[nn, :])
 
     # Calculate the wave-vector (force the real part to be positive)
-    wavenumber = numpy.sqrt(eigval)
+    wavenumbers = numpy.sqrt(eigvals)
-    wavenumber *= numpy.sign(numpy.real(wavenumber))
+    wavenumbers *= numpy.sign(numpy.real(wavenumbers))
 
-    # TODO: Perform correction on wavenumber to account for numerical dispersion.
+    # Wavenumbers assume the mode is at rmin, which is unlikely
+    # Instead, return the wavenumber in inverse radians
+    angular_wavenumbers = wavenumbers * cast('complex', rmin)
 
-    shape = [d.size for d in dxes[0]]
+    order = angular_wavenumbers.argsort()[::-1]
-    e_xy = numpy.hstack((e_xy, numpy.zeros(shape[0] * shape[1])))
+    e_xys = e_xys[order]
-    fields = {
+    angular_wavenumbers = angular_wavenumbers[order]
-        'wavenumber': wavenumber,
-        'E': unvec(e_xy, shape),
-        # 'E': unvec(e, shape),
-        # 'H': unvec(h, shape),
-    }
 
-    return fields
+    return e_xys, angular_wavenumbers
+
+
+def solve_mode(
+        mode_number: int,
+        *args: Any,
+        **kwargs: Any,
+        ) -> tuple[vcfdslice, complex]:
+    """
+    Wrapper around `solve_modes()` that solves for a single mode.
+
+    Args:
+       mode_number: 0-indexed mode number to solve for
+       *args: passed to `solve_modes()`
+       **kwargs: passed to `solve_modes()`
+
+    Returns:
+        (e_xy, angular_wavenumber)
+    """
+    kwargs['mode_numbers'] = [mode_number]
+    e_xys, angular_wavenumbers = solve_modes(*args, **kwargs)
+    return e_xys[0], angular_wavenumbers[0]
+
+
+def linear_wavenumbers(
+        e_xys: list[vcfdfield2_t],
+        angular_wavenumbers: ArrayLike,
+        epsilon: vfdslice,
+        dxes: dx_lists2_t,
+        rmin: float,
+        ) -> NDArray[numpy.complex128]:
+    """
+    Calculate linear wavenumbers (1/distance) based on angular wavenumbers (1/rad)
+      and the mode's energy distribution.
+
+    Args:
+        e_xys: Vectorized mode fields with shape (num_modes, 2 * x *y)
+        angular_wavenumbers: Wavenumbers assuming fields have theta-dependence of
+            `exp(-i * angular_wavenumber * theta)`. They should satisfy
+            `operator_e() @ e_xy == (angular_wavenumber / rmin) ** 2 * e_xy`
+        epsilon: Vectorized dielectric constant grid with shape (3, x, y)
+        dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types` (2D)
+        rmin: Radius at the left edge of the simulation domain (at minimum 'x')
+
+    Returns:
+        NDArray containing the calculated linear (1/distance) wavenumbers
+    """
+    angular_wavenumbers = numpy.asarray(angular_wavenumbers)
+    mode_radii = numpy.empty_like(angular_wavenumbers, dtype=float)
+
+    shape2d = (len(dxes[0][0]), len(dxes[0][1]))
+    epsilon2d = unvec(epsilon, shape2d)[:2]
+    grid_radii = rmin + numpy.cumsum(dxes[0][0])
+    for ii in range(angular_wavenumbers.size):
+        efield = unvec(e_xys[ii], shape2d, 2)
+        energy = numpy.real((efield * efield.conj()) * epsilon2d)
+        energy_vs_x = energy.sum(axis=(0, 2))
+        mode_radii[ii] = (grid_radii * energy_vs_x).sum() / energy_vs_x.sum()
+
+    logger.info(f'{mode_radii=}')
+    lin_wavenumbers = angular_wavenumbers / mode_radii
+    return lin_wavenumbers
+
+
+def exy2h(
+        angular_wavenumber: complex,
+        omega: float,
+        dxes: dx_lists2_t,
+        rmin: float,
+        epsilon: vfdslice,
+        mu: vfdslice | None = None
+        ) -> sparse.sparray:
+    """
+    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
+
+    Args:
+        angular_wavenumber: Wavenumber assuming fields have theta-dependence of
+            `exp(-i * angular_wavenumber * theta)`. It should satisfy
+            `operator_e() @ e_xy == (angular_wavenumber / rmin) ** 2 * e_xy`
+        omega: The angular frequency of the system
+        dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types` (2D)
+        rmin: Radius at the left edge of the simulation domain (at minimum 'x')
+        epsilon: Vectorized dielectric constant grid
+        mu: Vectorized magnetic permeability grid (default 1 everywhere)
+
+    Returns:
+        Sparse matrix representing the operator.
+    """
+    e2hop = e2h(angular_wavenumber=angular_wavenumber, omega=omega, dxes=dxes, rmin=rmin, mu=mu)
+    return e2hop @ exy2e(angular_wavenumber=angular_wavenumber, omega=omega, dxes=dxes, rmin=rmin, epsilon=epsilon)
+
+
+def exy2e(
+        angular_wavenumber: complex,
+        omega: float,
+        dxes: dx_lists2_t,
+        rmin: float,
+        epsilon: vfdslice,
+        ) -> sparse.sparray:
+    """
+    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
+
+    Unlike the straight waveguide case, the H_z components do not cancel and must be calculated
+    from E_x and E_y in order to then calculate E_z.
+
+    Args:
+        angular_wavenumber: Wavenumber assuming fields have theta-dependence of
+            `exp(-i * angular_wavenumber * theta)`. It should satisfy
+            `operator_e() @ e_xy == (angular_wavenumber / rmin) ** 2 * e_xy`
+        omega: The angular frequency of the system
+        dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types` (2D)
+        rmin: Radius at the left edge of the simulation domain (at minimum 'x')
+        epsilon: Vectorized dielectric constant grid
+
+    Returns:
+        Sparse matrix representing the operator.
+    """
+    Dfx, Dfy = deriv_forward(dxes[0])
+    Dbx, Dby = deriv_back(dxes[1])
+    wavenumber = angular_wavenumber / rmin
+
+    Ta, Tb = dxes2T(dxes=dxes, rmin=rmin)
+    Tai = sparse.diags_array(1 / Ta.diagonal())
+    #Tbi = sparse.diags_array(1 / Tb.diagonal())
+
+    epsilon_parts = numpy.split(epsilon, 3)
+    epsilon_x, epsilon_y = (sparse.diags_array(epsi) for epsi in epsilon_parts[:2])
+    epsilon_z_inv = sparse.diags_array(1 / epsilon_parts[2])
+
+    n_pts = dxes[0][0].size * dxes[0][1].size
+    zeros = sparse.coo_array((n_pts, n_pts))
+
+    mu_z = numpy.ones(n_pts)
+    mu_z_inv = sparse.diags_array(1 / mu_z)
+    exy2hz = 1 / (-1j * omega) * mu_z_inv @ sparse.hstack((Dfy, -Dfx))
+    hxy2ez = 1 / (1j * omega) * epsilon_z_inv @ sparse.hstack((Dby, -Dbx))
+
+    exy2hy = Tb / (1j * wavenumber) @ (-1j * omega * sparse.hstack((epsilon_x, zeros)) - Dby @ exy2hz)
+    exy2hx = Tb / (1j * wavenumber) @ ( 1j * omega * sparse.hstack((zeros, epsilon_y)) - Tai @ Dbx @ Tb @ exy2hz)
+
+    exy2ez = hxy2ez @ sparse.vstack((exy2hx, exy2hy))
+
+    op = sparse.vstack((sparse.eye_array(2 * n_pts),
+                        exy2ez))
+    return op
+
+
+def e2h(
+        angular_wavenumber: complex,
+        omega: float,
+        dxes: dx_lists2_t,
+        rmin: float,
+        mu: vfdslice | None = None
+        ) -> sparse.sparray:
+    r"""
+    Returns an operator which, when applied to a vectorized E eigenfield, produces
+     the vectorized H eigenfield.
+
+    This operator is created directly from the initial coordinate-transformed equations:
+    $$
+    \begin{aligned}
+    \imath \omega \epsilon_{xx} E_x &= \hat{\partial}_y H_z + \imath \beta \frac{H_y}{\hat{T}} \\
+    \imath \omega \epsilon_{yy} E_y &= -\imath \beta H_x - \{1}{\tilde{t}_x} \hat{\partial}_x \hat{t}_x} H_z \\
+    \imath \omega \epsilon_{zz} E_z &= \hat{\partial}_x H_y - \hat{\partial}_y H_x \\
+    \end{aligned}
+    $$
+
+    Args:
+        angular_wavenumber: Wavenumber assuming fields have theta-dependence of
+            `exp(-i * angular_wavenumber * theta)`. It should satisfy
+            `operator_e() @ e_xy == (angular_wavenumber / rmin) ** 2 * e_xy`
+        omega: The angular frequency of the system
+        dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types` (2D)
+        rmin: Radius at the left edge of the simulation domain (at minimum 'x')
+        mu: Vectorized magnetic permeability grid (default 1 everywhere)
+
+    Returns:
+        Sparse matrix representation of the operator.
+    """
+    Dfx, Dfy = deriv_forward(dxes[0])
+    Ta, Tb = dxes2T(dxes=dxes, rmin=rmin)
+    Tai = sparse.diags_array(1 / Ta.diagonal())
+    Tbi = sparse.diags_array(1 / Tb.diagonal())
+
+    jB = 1j * angular_wavenumber / rmin
+    op = sparse.block_array([[    None, -jB * Tai,           -Dfy],
+                             [jB * Tbi,      None, Tbi @ Dfx @ Ta],
+                             [     Dfy,      -Dfx,           None]]) / (-1j * omega)
+    if mu is not None:
+        op = sparse.diags_array(1 / mu) @ op
+    return op
+
+
+def dxes2T(
+        dxes: dx_lists2_t,
+        rmin: float,
+        ) -> tuple[NDArray[numpy.float64], NDArray[numpy.float64]]:
+    r"""
+    Returns the $T_a$ and $T_b$ diagonal matrices which are used to apply the cylindrical
+      coordinate transformation in various operators.
+
+    Args:
+        dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types` (2D)
+        rmin: Radius at the left edge of the simulation domain (at minimum 'x')
+
+    Returns:
+        Sparse matrix representations of the operators Ta and Tb
+    """
+    ra = rmin + numpy.cumsum(dxes[0][0])                      # Radius at Ey points
+    rb = rmin + dxes[0][0] / 2.0 + numpy.cumsum(dxes[1][0])   # Radius at Ex points
+    ta = ra / rmin
+    tb = rb / rmin
+
+    Ta = sparse.diags_array(vec(ta[:, None].repeat(dxes[0][1].size, axis=1)))
+    Tb = sparse.diags_array(vec(tb[:, None].repeat(dxes[1][1].size, axis=1)))
+    return Ta, Tb
+
+
+def normalized_fields_e(
+        e_xy: vcfdfield2,
+        angular_wavenumber: complex,
+        omega: float,
+        dxes: dx_lists2_t,
+        rmin: float,
+        epsilon: vfdslice,
+        mu: vfdslice | None = None,
+        prop_phase: float = 0,
+        ) -> tuple[vcfdslice_t, vcfdslice_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.
+
+    Args:
+        e_xy: Vector containing E_x and E_y fields
+        angular_wavenumber: Wavenumber assuming fields have theta-dependence of
+            `exp(-i * angular_wavenumber * theta)`. It should satisfy
+            `operator_e() @ e_xy == (angular_wavenumber / rmin) ** 2 * e_xy`
+        omega: The angular frequency of the system
+        dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types` (2D)
+        rmin: Radius at the left edge of the simulation domain (at minimum 'x')
+        epsilon: Vectorized dielectric constant grid
+        mu: Vectorized magnetic permeability grid (default 1 everywhere)
+        prop_phase: Phase shift `(dz * corrected_wavenumber)` over 1 cell in propagation direction.
+                    Default 0 (continuous propagation direction, i.e. dz->0).
+
+    Returns:
+        `(e, h)`, where each field is vectorized, normalized,
+        and contains all three vector components.
+    """
+    e = exy2e(angular_wavenumber=angular_wavenumber, omega=omega, dxes=dxes, rmin=rmin, epsilon=epsilon) @ e_xy
+    h = exy2h(angular_wavenumber=angular_wavenumber, omega=omega, dxes=dxes, rmin=rmin, epsilon=epsilon, mu=mu) @ e_xy
+    e_norm, h_norm = _normalized_fields(e=e, h=h, omega=omega, dxes=dxes, rmin=rmin, epsilon=epsilon,
+                                        mu=mu, prop_phase=prop_phase)
+    return e_norm, h_norm
+
+
+def _normalized_fields(
+        e: vcfdslice,
+        h: vcfdslice,
+        omega: complex,
+        dxes: dx_lists2_t,
+        rmin: float,        # Currently unused, but may want to use cylindrical poynting
+        epsilon: vfdslice,
+        mu: vfdslice | None = None,
+        prop_phase: float = 0,
+        ) -> tuple[vcfdslice_t, vcfdslice_t]:
+    h *= -1
+    # TODO documentation for normalized_fields
+    shape = [s.size for s in dxes[0]]
+
+    # Find time-averaged Sz and normalize to it
+    # H phase is adjusted by a half-cell forward shift for Yee cell, and 1-cell reverse shift for Poynting
+    Sz_tavg = waveguide_2d.inner_product(e, h, dxes=dxes, prop_phase=prop_phase, conj_h=True).real   # Note, using linear poynting vector
+    assert Sz_tavg > 0, f'Found a mode propagating in the wrong direction! {Sz_tavg=}'
+
+    energy = numpy.real(epsilon * e.conj() * e)
+
+    norm_amplitude = 1 / numpy.sqrt(Sz_tavg)
+    norm_angle = -numpy.angle(e[energy.argmax()])       # Will randomly add a negative sign when mode is symmetric
+
+    # Try to break symmetry to assign a consistent sign [experimental]
+    E_weighted = unvec(e * energy * numpy.exp(1j * norm_angle), shape)
+    sign = numpy.sign(E_weighted[:,
+                                 :max(shape[0] // 2, 1),
+                                 :max(shape[1] // 2, 1)].real.sum())
+    assert sign != 0
+
+    norm_factor = sign * norm_amplitude * numpy.exp(1j * norm_angle)
+
+    e *= norm_factor
+    h *= norm_factor
+    return vcfdslice_t(e), vcfdslice_t(h)

meanas/fdmath/__init__.py

@@ -741,8 +741,46 @@ the true values can be multiplied back in after the simulation is complete if no
 normalized results are needed.
 """
 
-from .types import fdfield_t, vfdfield_t, cfdfield_t, vcfdfield_t, dx_lists_t, dx_lists_mut
+from .types import (
-from .types import fdfield_updater_t, cfdfield_updater_t
+      fdfield_t as    fdfield_t,
-from .vectorization import vec, unvec
+     vfdfield_t as   vfdfield_t,
-from . import operators, functional, types, vectorization
+     cfdfield_t as   cfdfield_t,
+    vcfdfield_t as  vcfdfield_t,
+      fdfield2_t as   fdfield2_t,
+     vfdfield2_t as  vfdfield2_t,
+     cfdfield2_t as  cfdfield2_t,
+    vcfdfield2_t as vcfdfield2_t,
+      fdfield   as    fdfield,
+     vfdfield   as   vfdfield,
+     cfdfield   as   cfdfield,
+    vcfdfield   as  vcfdfield,
+      fdfield2   as   fdfield2,
+     vfdfield2   as  vfdfield2,
+     cfdfield2   as  cfdfield2,
+    vcfdfield2   as vcfdfield2,
+      fdslice_t as   fdslice_t,
+     vfdslice_t as  vfdslice_t,
+     cfdslice_t as  cfdslice_t,
+    vcfdslice_t as vcfdslice_t,
+      fdslice   as   fdslice,
+     vfdslice   as  vfdslice,
+     cfdslice   as  cfdslice,
+    vcfdslice   as vcfdslice,
+    dx_lists_t  as dx_lists_t,
+    dx_lists2_t as dx_lists2_t,
+    dx_lists_mut  as dx_lists_mut,
+    dx_lists2_mut as dx_lists2_mut,
+    fdfield_updater_t as fdfield_updater_t,
+    cfdfield_updater_t as cfdfield_updater_t,
+    )
+from .vectorization import (
+    vec as vec,
+    unvec as unvec,
+    )
+from . import (
+    operators as operators,
+    functional as functional,
+    types as types,
+    vectorization as vectorization,
+    )
 

meanas/fdmath/functional.py

@@ -3,16 +3,18 @@ Math functions for finite difference simulations
 
 Basic discrete calculus etc.
 """
-from typing import Sequence, Callable
+from typing import TypeVar
+from collections.abc import Sequence, Callable
 
 import numpy
 from numpy.typing import NDArray
+from numpy import floating, complexfloating
 
 from .types import fdfield_t, fdfield_updater_t
 
 
 def deriv_forward(
-        dx_e: Sequence[NDArray[numpy.float_]] | None = None,
+        dx_e: Sequence[NDArray[floating | complexfloating]] | None = None,
         ) -> tuple[fdfield_updater_t, fdfield_updater_t, fdfield_updater_t]:
     """
     Utility operators for taking discretized derivatives (backward variant).
@@ -36,7 +38,7 @@ def deriv_forward(
 
 
 def deriv_back(
-        dx_h: Sequence[NDArray[numpy.float_]] | None = None,
+        dx_h: Sequence[NDArray[floating | complexfloating]] | None = None,
         ) -> tuple[fdfield_updater_t, fdfield_updater_t, fdfield_updater_t]:
     """
     Utility operators for taking discretized derivatives (forward variant).
@@ -59,9 +61,12 @@ def deriv_back(
     return derivs
 
 
+TT = TypeVar('TT', bound='NDArray[floating | complexfloating]')
+
+
 def curl_forward(
-        dx_e: Sequence[NDArray[numpy.float_]] | None = None,
+        dx_e: Sequence[NDArray[floating | complexfloating]] | None = None,
-        ) -> fdfield_updater_t:
+        ) -> Callable[[TT], TT]:
     r"""
     Curl operator for use with the E field.
 
@@ -75,7 +80,7 @@ def curl_forward(
     """
     Dx, Dy, Dz = deriv_forward(dx_e)
 
-    def ce_fun(e: fdfield_t) -> fdfield_t:
+    def ce_fun(e: TT) -> TT:
         output = numpy.empty_like(e)
         output[0] = Dy(e[2])
         output[1] = Dz(e[0])
@@ -89,8 +94,8 @@ def curl_forward(
 
 
 def curl_back(
-        dx_h: Sequence[NDArray[numpy.float_]] | None = None,
+        dx_h: Sequence[NDArray[floating | complexfloating]] | None = None,
-        ) -> fdfield_updater_t:
+        ) -> Callable[[TT], TT]:
     r"""
     Create a function which takes the backward curl of a field.
 
@@ -104,7 +109,7 @@ def curl_back(
     """
     Dx, Dy, Dz = deriv_back(dx_h)
 
-    def ch_fun(h: fdfield_t) -> fdfield_t:
+    def ch_fun(h: TT) -> TT:
         output = numpy.empty_like(h)
         output[0] = Dy(h[2])
         output[1] = Dz(h[0])
@@ -118,7 +123,7 @@ def curl_back(
 
 
 def curl_forward_parts(
-        dx_e: Sequence[NDArray[numpy.float_]] | None = None,
+        dx_e: Sequence[NDArray[floating | complexfloating]] | None = None,
         ) -> Callable:
     Dx, Dy, Dz = deriv_forward(dx_e)
 
@@ -131,7 +136,7 @@ def curl_forward_parts(
 
 
 def curl_back_parts(
-        dx_h: Sequence[NDArray[numpy.float_]] | None = None,
+        dx_h: Sequence[NDArray[floating | complexfloating]] | None = None,
         ) -> Callable:
     Dx, Dy, Dz = deriv_back(dx_h)
 

meanas/fdmath/operators.py

@@ -3,10 +3,11 @@ Matrix operators for finite difference simulations
 
 Basic discrete calculus etc.
 """
-from typing import Sequence
+from collections.abc import Sequence
 import numpy
 from numpy.typing import NDArray
-import scipy.sparse as sparse   # type: ignore
+from numpy import floating, complexfloating
+from scipy import sparse
 
 from .types import vfdfield_t
 
@@ -15,7 +16,7 @@ def shift_circ(
         axis: int,
         shape: Sequence[int],
         shift_distance: int = 1,
-        ) -> sparse.spmatrix:
+        ) -> sparse.sparray:
     """
     Utility operator for performing a circular shift along a specified axis by a
      specified number of elements.
@@ -33,8 +34,8 @@ def shift_circ(
     if axis not in range(len(shape)):
         raise Exception(f'Invalid direction: {axis}, shape is {shape}')
 
-    shifts = [abs(shift_distance) if a == axis else 0 for a in range(3)]
+    shifts = [abs(shift_distance) if a == axis else 0 for a in range(len(shape))]
-    shifted_diags = [(numpy.arange(n) + s) % n for n, s in zip(shape, shifts)]
+    shifted_diags = [(numpy.arange(n) + s) % n for n, s in zip(shape, shifts, strict=True)]
     ijk = numpy.meshgrid(*shifted_diags, indexing='ij')
 
     n = numpy.prod(shape)
@@ -43,7 +44,7 @@ def shift_circ(
 
     vij = (numpy.ones(n), (i_ind, j_ind.ravel(order='C')))
 
-    d = sparse.csr_matrix(vij, shape=(n, n))
+    d = sparse.csr_array(vij, shape=(n, n))
 
     if shift_distance < 0:
         d = d.T
@@ -55,7 +56,7 @@ def shift_with_mirror(
         axis: int,
         shape: Sequence[int],
         shift_distance: int = 1,
-        ) -> sparse.spmatrix:
+        ) -> sparse.sparray:
     """
     Utility operator for performing an n-element shift along a specified axis, with mirror
     boundary conditions applied to the cells beyond the receding edge.
@@ -81,8 +82,8 @@ def shift_with_mirror(
         v = numpy.where(v < 0, - 1 - v, v)
         return v
 
-    shifts = [shift_distance if a == axis else 0 for a in range(3)]
+    shifts = [shift_distance if a == axis else 0 for a in range(len(shape))]
-    shifted_diags = [mirrored_range(n, s) for n, s in zip(shape, shifts)]
+    shifted_diags = [mirrored_range(n, s) for n, s in zip(shape, shifts, strict=True)]
     ijk = numpy.meshgrid(*shifted_diags, indexing='ij')
 
     n = numpy.prod(shape)
@@ -91,13 +92,13 @@ def shift_with_mirror(
 
     vij = (numpy.ones(n), (i_ind, j_ind.ravel(order='C')))
 
-    d = sparse.csr_matrix(vij, shape=(n, n))
+    d = sparse.csr_array(vij, shape=(n, n))
     return d
 
 
 def deriv_forward(
-        dx_e: Sequence[NDArray[numpy.float_]],
+        dx_e: Sequence[NDArray[floating | complexfloating]],
-        ) -> list[sparse.spmatrix]:
+        ) -> list[sparse.sparray]:
     """
     Utility operators for taking discretized derivatives (forward variant).
 
@@ -113,18 +114,18 @@ def deriv_forward(
 
     dx_e_expanded = numpy.meshgrid(*dx_e, indexing='ij')
 
-    def deriv(axis: int) -> sparse.spmatrix:
+    def deriv(axis: int) -> sparse.sparray:
-        return shift_circ(axis, shape, 1) - sparse.eye(n)
+        return shift_circ(axis, shape, 1) - sparse.eye_array(n)
 
-    Ds = [sparse.diags(+1 / dx.ravel(order='C')) @ deriv(a)
+    Ds = [sparse.diags_array(+1 / dx.ravel(order='C')) @ deriv(a)
           for a, dx in enumerate(dx_e_expanded)]
 
     return Ds
 
 
 def deriv_back(
-        dx_h: Sequence[NDArray[numpy.float_]],
+        dx_h: Sequence[NDArray[floating | complexfloating]],
-        ) -> list[sparse.spmatrix]:
+        ) -> list[sparse.sparray]:
     """
     Utility operators for taking discretized derivatives (backward variant).
 
@@ -140,18 +141,18 @@ def deriv_back(
 
     dx_h_expanded = numpy.meshgrid(*dx_h, indexing='ij')
 
-    def deriv(axis: int) -> sparse.spmatrix:
+    def deriv(axis: int) -> sparse.sparray:
-        return shift_circ(axis, shape, -1) - sparse.eye(n)
+        return shift_circ(axis, shape, -1) - sparse.eye_array(n)
 
-    Ds = [sparse.diags(-1 / dx.ravel(order='C')) @ deriv(a)
+    Ds = [sparse.diags_array(-1 / dx.ravel(order='C')) @ deriv(a)
           for a, dx in enumerate(dx_h_expanded)]
 
     return Ds
 
 
 def cross(
-        B: Sequence[sparse.spmatrix],
+        B: Sequence[sparse.sparray],
-        ) -> sparse.spmatrix:
+        ) -> sparse.sparray:
     """
     Cross product operator
 
@@ -163,13 +164,14 @@ def cross(
         Sparse matrix corresponding to (B x), where x is the cross product.
     """
     n = B[0].shape[0]
-    zero = sparse.csr_matrix((n, n))
+    zero = sparse.csr_array((n, n))
-    return sparse.bmat([[zero, -B[2], B[1]],
+    return sparse.block_array([
+            [zero, -B[2], B[1]],
             [B[2], zero, -B[0]],
             [-B[1], B[0], zero]])
 
 
-def vec_cross(b: vfdfield_t) -> sparse.spmatrix:
+def vec_cross(b: vfdfield_t) -> sparse.sparray:
     """
     Vector cross product operator
 
@@ -181,11 +183,11 @@ def vec_cross(b: vfdfield_t) -> sparse.spmatrix:
         Sparse matrix corresponding to (b x), where x is the cross product.
 
     """
-    B = [sparse.diags(c) for c in numpy.split(b, 3)]
+    B = [sparse.diags_array(c) for c in numpy.split(b, 3)]
     return cross(B)
 
 
-def avg_forward(axis: int, shape: Sequence[int]) -> sparse.spmatrix:
+def avg_forward(axis: int, shape: Sequence[int]) -> sparse.sparray:
     """
     Forward average operator `(x4 = (x4 + x5) / 2)`
 
@@ -200,10 +202,10 @@ def avg_forward(axis: int, shape: Sequence[int]) -> sparse.spmatrix:
         raise Exception(f'Invalid shape: {shape}')
 
     n = numpy.prod(shape)
-    return 0.5 * (sparse.eye(n) + shift_circ(axis, shape))
+    return 0.5 * (sparse.eye_array(n) + shift_circ(axis, shape))
 
 
-def avg_back(axis: int, shape: Sequence[int]) -> sparse.spmatrix:
+def avg_back(axis: int, shape: Sequence[int]) -> sparse.sparray:
     """
     Backward average operator `(x4 = (x4 + x3) / 2)`
 
@@ -218,8 +220,8 @@ def avg_back(axis: int, shape: Sequence[int]) -> sparse.spmatrix:
 
 
 def curl_forward(
-        dx_e: Sequence[NDArray[numpy.float_]],
+        dx_e: Sequence[NDArray[floating | complexfloating]],
-        ) -> sparse.spmatrix:
+        ) -> sparse.sparray:
     """
     Curl operator for use with the E field.
 
@@ -234,8 +236,8 @@ def curl_forward(
 
 
 def curl_back(
-        dx_h: Sequence[NDArray[numpy.float_]],
+        dx_h: Sequence[NDArray[floating | complexfloating]],
-        ) -> sparse.spmatrix:
+        ) -> sparse.sparray:
     """
     Curl operator for use with the H field.
 

meanas/fdmath/types.py

@@ -1,26 +1,65 @@
 """
 Types shared across multiple submodules
 """
-from typing import Sequence, Callable, MutableSequence
+from typing import NewType
-import numpy
+from collections.abc import Sequence, Callable, MutableSequence
 from numpy.typing import NDArray
+from numpy import floating, complexfloating
 
 
 # Field types
-fdfield_t = NDArray[numpy.float_]
+fdfield_t = NewType('fdfield_t', NDArray[floating])
+type fdfield = fdfield_t | NDArray[floating]
 """Vector field with shape (3, X, Y, Z) (e.g. `[E_x, E_y, E_z]`)"""
 
-vfdfield_t = NDArray[numpy.float_]
+vfdfield_t = NewType('vfdfield_t', NDArray[floating])
+type vfdfield = vfdfield_t | NDArray[floating]
 """Linearized vector field (single vector of length 3*X*Y*Z)"""
 
-cfdfield_t = NDArray[numpy.complex_]
+cfdfield_t = NewType('cfdfield_t', NDArray[complexfloating])
+type cfdfield = cfdfield_t | NDArray[complexfloating]
 """Complex vector field with shape (3, X, Y, Z) (e.g. `[E_x, E_y, E_z]`)"""
 
-vcfdfield_t = NDArray[numpy.complex_]
+vcfdfield_t = NewType('vcfdfield_t', NDArray[complexfloating])
+type vcfdfield = vcfdfield_t | NDArray[complexfloating]
 """Linearized complex vector field (single vector of length 3*X*Y*Z)"""
 
 
-dx_lists_t = Sequence[Sequence[NDArray[numpy.float_]]]
+fdslice_t = NewType('fdslice_t', NDArray[floating])
+type fdslice = fdslice_t | NDArray[floating]
+"""Vector field slice with shape (3, X, Y) (e.g. `[E_x, E_y, E_z]` at a single Z position)"""
+
+vfdslice_t = NewType('vfdslice_t', NDArray[floating])
+type vfdslice = vfdslice_t | NDArray[floating]
+"""Linearized vector field slice (single vector of length 3*X*Y)"""
+
+cfdslice_t = NewType('cfdslice_t', NDArray[complexfloating])
+type cfdslice = cfdslice_t | NDArray[complexfloating]
+"""Complex vector field slice with shape (3, X, Y) (e.g. `[E_x, E_y, E_z]` at a single Z position)"""
+
+vcfdslice_t = NewType('vcfdslice_t', NDArray[complexfloating])
+type vcfdslice = vcfdslice_t | NDArray[complexfloating]
+"""Linearized complex vector field slice (single vector of length 3*X*Y)"""
+
+
+fdfield2_t = NewType('fdfield2_t', NDArray[floating])
+type fdfield2 = fdfield2_t | NDArray[floating]
+"""2D Vector field with shape (2, X, Y) (e.g. `[E_x, E_y]`)"""
+
+vfdfield2_t = NewType('vfdfield2_t', NDArray[floating])
+type vfdfield2 = vfdfield2_t | NDArray[floating]
+"""2D Linearized vector field (single vector of length 2*X*Y)"""
+
+cfdfield2_t = NewType('cfdfield2_t', NDArray[complexfloating])
+type cfdfield2 = cfdfield2_t | NDArray[complexfloating]
+"""2D Complex vector field with shape (2, X, Y) (e.g. `[E_x, E_y]`)"""
+
+vcfdfield2_t = NewType('vcfdfield2_t', NDArray[complexfloating])
+type vcfdfield2 = vcfdfield2_t | NDArray[complexfloating]
+"""2D Linearized complex vector field (single vector of length 2*X*Y)"""
+
+
+dx_lists_t = Sequence[Sequence[NDArray[floating | complexfloating]]]
 """
  'dxes' datastructure which contains grid cell width information in the following format:
 
@@ -31,9 +70,23 @@ dx_lists_t = Sequence[Sequence[NDArray[numpy.float_]]]
    and `dy_h[0]` is the y-width of the `y=0` cells, as used when calculating dH/dy, etc.
 """
 
-dx_lists_mut = MutableSequence[MutableSequence[NDArray[numpy.float_]]]
+dx_lists2_t = Sequence[Sequence[NDArray[floating | complexfloating]]]
+"""
+ 2D 'dxes' datastructure which contains grid cell width information in the following format:
+
+     [[[dx_e[0], dx_e[1], ...], [dy_e[0], ...]],
+      [[dx_h[0], dx_h[1], ...], [dy_h[0], ...]]]
+
+   where `dx_e[0]` is the x-width of the `x=0` cells, as used when calculating dE/dx,
+   and `dy_h[0]` is the y-width of the `y=0` cells, as used when calculating dH/dy, etc.
+"""
+
+dx_lists_mut = MutableSequence[MutableSequence[NDArray[floating | complexfloating]]]
 """Mutable version of `dx_lists_t`"""
 
+dx_lists2_mut = MutableSequence[MutableSequence[NDArray[floating | complexfloating]]]
+"""Mutable version of `dx_lists2_t`"""
+
 
 fdfield_updater_t = Callable[..., fdfield_t]
 """Convenience type for functions which take and return an fdfield_t"""

meanas/fdmath/vectorization.py

@@ -4,11 +4,16 @@ 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.
 """
 
-from typing import overload, Sequence
+from typing import overload
+from collections.abc import Sequence
 import numpy
-from numpy.typing import ArrayLike
+from numpy.typing import ArrayLike, NDArray
 
-from .types import fdfield_t, vfdfield_t, cfdfield_t, vcfdfield_t
+from .types import (
+    fdfield_t, vfdfield_t, cfdfield_t, vcfdfield_t,
+    fdslice_t, vfdslice_t, cfdslice_t, vcfdslice_t,
+    fdfield2_t, vfdfield2_t, cfdfield2_t, vcfdfield2_t,
+    )
 
 
 @overload
@@ -24,17 +29,35 @@ def vec(f: cfdfield_t) -> vcfdfield_t:
     pass
 
 @overload
-def vec(f: ArrayLike) -> vfdfield_t | vcfdfield_t:
+def vec(f: fdfield2_t) -> vfdfield2_t:
     pass
 
-def vec(f: fdfield_t | cfdfield_t | ArrayLike | None) -> vfdfield_t | vcfdfield_t | None:
+@overload
+def vec(f: cfdfield2_t) -> vcfdfield2_t:
+    pass
+
+@overload
+def vec(f: fdslice_t) -> vfdslice_t:
+    pass
+
+@overload
+def vec(f: cfdslice_t) -> vcfdslice_t:
+    pass
+
+@overload
+def vec(f: ArrayLike) -> NDArray:
+    pass
+
+def vec(
+        f: fdfield_t | cfdfield_t | fdfield2_t | cfdfield2_t | fdslice_t | cfdslice_t | ArrayLike | None,
+        ) -> vfdfield_t | vcfdfield_t | vfdfield2_t | vcfdfield2_t | vfdslice_t | vcfdslice_t | NDArray | None:
     """
-    Create a 1D ndarray from a 3D vector field which spans a 1-3D region.
+    Create a 1D ndarray from a vector field which spans a 1-3D region.
 
     Returns `None` if called with `f=None`.
 
     Args:
-        f: A vector field, `[f_x, f_y, f_z]` where each `f_` component is a 1- to
+        f: A vector field, e.g. `[f_x, f_y, f_z]` where each `f_` component is a 1- to
            3-D ndarray (`f_*` should all be the same size). Doesn't fail with `f=None`.
 
     Returns:
@@ -42,37 +65,62 @@ def vec(f: fdfield_t | cfdfield_t | ArrayLike | None) -> vfdfield_t | vcfdfield_
     """
     if f is None:
         return None
-    return numpy.ravel(f, order='C')
+    return numpy.ravel(f, order='C')        # type: ignore
 
 
 @overload
-def unvec(v: None, shape: Sequence[int]) -> None:
+def unvec(v: None, shape: Sequence[int], nvdim: int = 3) -> None:
     pass
 
 @overload
-def unvec(v: vfdfield_t, shape: Sequence[int]) -> fdfield_t:
+def unvec(v: vfdfield_t, shape: Sequence[int], nvdim: int = 3) -> fdfield_t:
     pass
 
 @overload
-def unvec(v: vcfdfield_t, shape: Sequence[int]) -> cfdfield_t:
+def unvec(v: vcfdfield_t, shape: Sequence[int], nvdim: int = 3) -> cfdfield_t:
     pass
 
-def unvec(v: vfdfield_t | vcfdfield_t | None, shape: Sequence[int]) -> fdfield_t | cfdfield_t | None:
+@overload
+def unvec(v: vfdfield2_t, shape: Sequence[int], nvdim: int = 3) -> fdfield2_t:
+    pass
+
+@overload
+def unvec(v: vcfdfield2_t, shape: Sequence[int], nvdim: int = 3) -> cfdfield2_t:
+    pass
+
+@overload
+def unvec(v: vfdslice_t, shape: Sequence[int], nvdim: int = 3) -> fdslice_t:
+    pass
+
+@overload
+def unvec(v: vcfdslice_t, shape: Sequence[int], nvdim: int = 3) -> cfdslice_t:
+    pass
+
+@overload
+def unvec(v: ArrayLike, shape: Sequence[int], nvdim: int = 3) -> NDArray:
+    pass
+
+def unvec(
+        v: vfdfield_t | vcfdfield_t | vfdfield2_t | vcfdfield2_t | vfdslice_t | vcfdslice_t | ArrayLike | None,
+        shape: Sequence[int],
+        nvdim: int = 3,
+        ) -> fdfield_t | cfdfield_t | fdfield2_t | cfdfield2_t | fdslice_t | cfdslice_t | NDArray | None:
     """
-    Perform the inverse of vec(): take a 1D ndarray and output a 3D field
+    Perform the inverse of vec(): take a 1D ndarray and output an `nvdim`-component field
-     of form `[f_x, f_y, f_z]` where each of `f_*` is a len(shape)-dimensional
+     of form e.g. `[f_x, f_y, f_z]` (`nvdim=3`) where each of `f_*` is a len(shape)-dimensional
      ndarray.
 
     Returns `None` if called with `v=None`.
 
     Args:
-        v: 1D ndarray representing a 3D vector field of shape shape (or None)
+        v: 1D ndarray representing a vector field of shape shape (or None)
         shape: shape of the vector field
+        nvdim: Number of components in each vector
 
     Returns:
         `[f_x, f_y, f_z]` where each `f_` is a `len(shape)` dimensional ndarray (or `None`)
     """
     if v is None:
         return None
-    return v.reshape((3, *shape), order='C')
+    return v.reshape((nvdim, *shape), order='C')  # type: ignore
 

meanas/fdtd/__init__.py

@@ -159,8 +159,22 @@ Boundary conditions
 # TODO notes about boundaries / PMLs
 """
 
-from .base import maxwell_e, maxwell_h
+from .base import (
-from .pml import cpml_params, updates_with_cpml
+    maxwell_e as maxwell_e,
-from .energy import (poynting, poynting_divergence, energy_hstep, energy_estep,
+    maxwell_h as maxwell_h,
-                     delta_energy_h2e, delta_energy_j)
+    )
-from .boundaries import conducting_boundary
+from .pml import (
+    cpml_params as cpml_params,
+    updates_with_cpml as updates_with_cpml,
+    )
+from .energy import (
+    poynting as poynting,
+    poynting_divergence as poynting_divergence,
+    energy_hstep as energy_hstep,
+    energy_estep as energy_estep,
+    delta_energy_h2e as delta_energy_h2e,
+    delta_energy_j as delta_energy_j,
+    )
+from .boundaries import (
+    conducting_boundary as conducting_boundary,
+    )

meanas/fdtd/energy.py

@@ -1,6 +1,6 @@
 import numpy
 
-from ..fdmath import dx_lists_t, fdfield_t
+from ..fdmath import dx_lists_t, fdfield_t, fdfield
 from ..fdmath.functional import deriv_back
 
 
@@ -8,8 +8,8 @@ from ..fdmath.functional import deriv_back
 
 
 def poynting(
-        e: fdfield_t,
+        e: fdfield,
-        h: fdfield_t,
+        h: fdfield,
         dxes: dx_lists_t | None = None,
         ) -> fdfield_t:
     r"""
@@ -84,14 +84,14 @@ def poynting(
     s[0] = numpy.roll(ey, -1, axis=0) * hz - numpy.roll(ez, -1, axis=0) * hy
     s[1] = numpy.roll(ez, -1, axis=1) * hx - numpy.roll(ex, -1, axis=1) * hz
     s[2] = numpy.roll(ex, -1, axis=2) * hy - numpy.roll(ey, -1, axis=2) * hx
-    return s
+    return fdfield_t(s)
 
 
 def poynting_divergence(
-        s: fdfield_t | None = None,
+        s: fdfield | None = None,
         *,
-        e: fdfield_t | None = None,
+        e: fdfield | None = None,
-        h: fdfield_t | None = None,
+        h: fdfield | None = None,
         dxes: dx_lists_t | None = None,
         ) -> fdfield_t:
     """
@@ -122,15 +122,15 @@ def poynting_divergence(
 
     Dx, Dy, Dz = deriv_back()
     ds = Dx(s[0]) + Dy(s[1]) + Dz(s[2])
-    return ds
+    return fdfield_t(ds)
 
 
 def energy_hstep(
-        e0: fdfield_t,
+        e0: fdfield,
-        h1: fdfield_t,
+        h1: fdfield,
-        e2: fdfield_t,
+        e2: fdfield,
-        epsilon: fdfield_t | None = None,
+        epsilon: fdfield | None = None,
-        mu: fdfield_t | None = None,
+        mu: fdfield | None = None,
         dxes: dx_lists_t | None = None,
         ) -> fdfield_t:
     """
@@ -150,15 +150,15 @@ def energy_hstep(
         Energy, at the time of the H-field `h1`.
     """
     u = dxmul(e0 * e2, h1 * h1, epsilon, mu, dxes)
-    return u
+    return fdfield_t(u)
 
 
 def energy_estep(
-        h0: fdfield_t,
+        h0: fdfield,
-        e1: fdfield_t,
+        e1: fdfield,
-        h2: fdfield_t,
+        h2: fdfield,
-        epsilon: fdfield_t | None = None,
+        epsilon: fdfield | None = None,
-        mu: fdfield_t | None = None,
+        mu: fdfield | None = None,
         dxes: dx_lists_t | None = None,
         ) -> fdfield_t:
     """
@@ -178,17 +178,17 @@ def energy_estep(
         Energy, at the time of the E-field `e1`.
     """
     u = dxmul(e1 * e1, h0 * h2, epsilon, mu, dxes)
-    return u
+    return fdfield_t(u)
 
 
 def delta_energy_h2e(
         dt: float,
-        e0: fdfield_t,
+        e0: fdfield,
-        h1: fdfield_t,
+        h1: fdfield,
-        e2: fdfield_t,
+        e2: fdfield,
-        h3: fdfield_t,
+        h3: fdfield,
-        epsilon: fdfield_t | None = None,
+        epsilon: fdfield | None = None,
-        mu: fdfield_t | None = None,
+        mu: fdfield | None = None,
         dxes: dx_lists_t | None = None,
         ) -> fdfield_t:
     """
@@ -211,17 +211,17 @@ def delta_energy_h2e(
     de = e2 * (e2 - e0) / dt
     dh = h1 * (h3 - h1) / dt
     du = dxmul(de, dh, epsilon, mu, dxes)
-    return du
+    return fdfield_t(du)
 
 
 def delta_energy_e2h(
         dt: float,
-        h0: fdfield_t,
+        h0: fdfield,
-        e1: fdfield_t,
+        e1: fdfield,
-        h2: fdfield_t,
+        h2: fdfield,
-        e3: fdfield_t,
+        e3: fdfield,
-        epsilon: fdfield_t | None = None,
+        epsilon: fdfield | None = None,
-        mu: fdfield_t | None = None,
+        mu: fdfield | None = None,
         dxes: dx_lists_t | None = None,
         ) -> fdfield_t:
     """
@@ -244,12 +244,12 @@ def delta_energy_e2h(
     de = e1 * (e3 - e1) / dt
     dh = h2 * (h2 - h0) / dt
     du = dxmul(de, dh, epsilon, mu, dxes)
-    return du
+    return fdfield_t(du)
 
 
 def delta_energy_j(
-        j0: fdfield_t,
+        j0: fdfield,
-        e1: fdfield_t,
+        e1: fdfield,
         dxes: dx_lists_t | None = None,
         ) -> fdfield_t:
     """
@@ -267,14 +267,14 @@ def delta_energy_j(
           * dxes[0][0][:, None, None]
           * dxes[0][1][None, :, None]
           * dxes[0][2][None, None, :])
-    return du
+    return fdfield_t(du)
 
 
 def dxmul(
-        ee: fdfield_t,
+        ee: fdfield,
-        hh: fdfield_t,
+        hh: fdfield,
-        epsilon: fdfield_t | float | None = None,
+        epsilon: fdfield | float | None = None,
-        mu: fdfield_t | float | None = None,
+        mu: fdfield | float | None = None,
         dxes: dx_lists_t | None = None,
         ) -> fdfield_t:
     if epsilon is None:
@@ -292,4 +292,4 @@ def dxmul(
               * dxes[1][0][:, None, None]
               * dxes[1][1][None, :, None]
               * dxes[1][2][None, None, :])
-    return result
+    return fdfield_t(result)

meanas/fdtd/misc.py

@@ -0,0 +1,166 @@
+from collections.abc import Callable
+import logging
+
+import numpy
+from numpy.typing import NDArray, ArrayLike
+from numpy import pi
+
+
+logger = logging.getLogger(__name__)
+
+
+pulse_fn_t = Callable[[int | NDArray], tuple[float, float, float]]
+
+
+def gaussian_packet(
+        wl: float,
+        dwl: float,
+        dt: float,
+        turn_on: float = 1e-10,
+        one_sided: bool = False,
+        ) -> tuple[pulse_fn_t, float]:
+    """
+    Gaussian pulse (or gaussian ramp) for FDTD excitation
+
+    exp(-a*t*t) ==> exp(-omega * omega / (4 * a))   [fourier, ignoring leading const.]
+
+    FWHM_time is 2 * sqrt(2 * log(2)) * sqrt(2 / a)
+    FWHM_omega is 2 * sqrt(2 * log(2)) * sqrt(2 * a)  = 4 * sqrt(log(2) * a)
+
+    Args:
+        wl: wavelength
+        dwl: Gaussian's FWHM in wavelength space
+        dt: Timestep
+        turn_on: Max allowable amplitude at t=0
+        one_sided: If `True`, source amplitude never decreases after reaching max
+
+    Returns:
+        Source function: src(timestep) -> (envelope[tt], cos[... * tt], sin[... * tt])
+        Delay: number of initial timesteps for which envelope[tt] will be 0
+    """
+    logger.warning('meanas.fdtd.misc functions are still very WIP!')    # TODO
+    # dt * dw = 4 * ln(2)
+
+    omega = 2 * pi / wl
+    freq = 1 / wl
+    fwhm_omega = dwl * omega * omega / (2 * pi)         # dwl -> d_omega (approx)
+    alpha = (fwhm_omega * fwhm_omega) * numpy.log(2) / 8
+    delay = numpy.sqrt(-numpy.log(turn_on) / alpha)
+    delay = numpy.ceil(delay * freq) / freq     # force delay to integer number of periods to maintain phase
+    logger.info(f'src_time {2 * delay / dt}')
+
+    def source_phasor(ii: int | NDArray) -> tuple[float, float, float]:
+        t0 = ii * dt - delay
+        envelope = numpy.sqrt(numpy.sqrt(2 * alpha / pi)) * numpy.exp(-alpha * t0 * t0)
+
+        if one_sided and t0 > 0:
+            envelope = 1
+
+        cc = numpy.cos(omega * t0)
+        ss = numpy.sin(omega * t0)
+
+        return envelope, cc, ss
+
+    # nrm = numpy.exp(-omega * omega / alpha) / 2
+
+    return source_phasor, delay
+
+
+def ricker_pulse(
+        wl: float,
+        dt: float,
+        turn_on: float = 1e-10,
+        ) -> tuple[pulse_fn_t, float]:
+    """
+    Ricker wavelet (second derivative of a gaussian pulse)
+
+    t0 = ii * dt - delay
+    R = w_peak * t0 / 2
+    f(t) = (1 - 2 * (pi * f_peak * t0) ** 2) * exp(-(pi * f_peak * t0)**2
+         = (1 - (w_peak * t0)**2 / 2 exp(-(w_peak * t0 / 2) **2)
+         = (1 - 2 * R * R) * exp(-R * R)
+
+    # NOTE: don't use cosine/sine for J, just for phasor readout
+
+    Args:
+        wl: wavelength
+        dt: Timestep
+        turn_on: Max allowable amplitude at t=0
+
+    Returns:
+        Source function: src(timestep) -> (envelope[tt], cos[... * tt], sin[... * tt])
+        Delay: number of initial timesteps for which envelope[tt] will be 0
+    """
+    logger.warning('meanas.fdtd.misc functions are still very WIP!')    # TODO
+    omega = 2 * pi / wl
+    freq = 1 / wl
+    # r0 = omega / 2
+
+    from scipy.optimize import root_scalar
+    delay_results = root_scalar(lambda tt: (1 - omega * omega * tt * tt / 2) * numpy.exp(-omega * omega / 4 * tt * tt) - turn_on, x0=0, x1=-2 / omega)
+    delay = delay_results.root
+    delay = numpy.ceil(delay * freq) / freq     # force delay to integer number of periods to maintain phase
+
+    def source_phasor(ii: int | NDArray) -> tuple[float, float, float]:
+        t0 = ii * dt - delay
+        rr = omega * t0 / 2
+        ff = (1 - 2 * rr * rr) * numpy.exp(-rr * rr)
+
+        cc = numpy.cos(omega * t0)
+        ss = numpy.sin(omega * t0)
+
+        return ff, cc, ss
+
+    return source_phasor, delay
+
+
+def gaussian_beam(
+        xyz: list[NDArray],
+        center: ArrayLike,
+        waist_radius: float,
+        wl: float,
+        tilt: float = 0,
+        ) -> NDArray[numpy.complex128]:
+    """
+    Gaussian beam
+    (solution to paraxial Helmholtz equation)
+
+    Default (no tilt) corresponds to a beam propagating in the -z direction.
+
+    Args:
+        xyz: List of [[x0, x1, ...], [y0, ...], [z0, ...]] positions specifying grid
+            locations at which the field will be sampled.
+        center: [x, y, z] location of beam waist
+        waist_radius: Beam radius at the waist
+        wl: Wavelength
+        tilt: Rotation around y axis. Default (0) has beam propagating in -z direction.
+    """
+    logger.warning('meanas.fdtd.misc functions are still very WIP!')    # TODO
+    w0 = waist_radius
+    grids = numpy.asarray(numpy.meshgrid(*xyz, indexing='ij'))
+    grids -= numpy.asarray(center)[:, None, None, None]
+
+    rot = numpy.array([
+        [ numpy.cos(tilt), 0, numpy.sin(tilt)],
+        [               0, 1,               0],
+        [-numpy.sin(tilt), 0, numpy.cos(tilt)],
+        ])
+
+    xx, yy, zz = numpy.einsum('ij,jxyz->ixyz', rot, grids)
+    r2 = xx * xx + yy * yy
+    z2 = zz * zz
+
+    zr = pi * w0 * w0 / wl
+    zr2 = zr * zr
+    wz2 = w0 * w0 * (1 + z2 / zr2)
+    wz = numpy.sqrt(wz2)        # == fwhm(z) / sqrt(2 * ln(2))
+
+    kk = 2 * pi / wl
+    Rz = zz * (1 + zr2 / z2)
+    gouy = numpy.arctan(zz / zr)
+
+    gaussian = w0 / wz * numpy.exp(-r2 / wz2) * numpy.exp(1j * (kk * zz + kk * r2 / 2 / Rz - gouy))
+
+    row = gaussian[:, :, gaussian.shape[2] // 2]
+    norm = numpy.sqrt((row * row.conj()).sum())
+    return gaussian / norm

meanas/fdtd/pml.py

@@ -7,12 +7,13 @@ PML implementations
 """
 # TODO retest pmls!
 
-from typing import Callable, Sequence, Any
+from typing import Any
+from collections.abc import Callable, Sequence
 from copy import deepcopy
 import numpy
 from numpy.typing import NDArray, DTypeLike
 
-from ..fdmath import fdfield_t, dx_lists_t
+from ..fdmath import fdfield, fdfield_t, dx_lists_t
 from ..fdmath.functional import deriv_forward, deriv_back
 
 
@@ -96,7 +97,7 @@ def updates_with_cpml(
         cpml_params: Sequence[Sequence[dict[str, Any] | None]],
         dt: float,
         dxes: dx_lists_t,
-        epsilon: fdfield_t,
+        epsilon: fdfield,
         *,
         dtype: DTypeLike = numpy.float32,
         ) -> tuple[Callable[[fdfield_t, fdfield_t, fdfield_t], None],
@@ -111,7 +112,7 @@ def updates_with_cpml(
     params_H: list[list[tuple[Any, Any, Any, Any]]] = deepcopy(params_E)
 
     for axis in range(3):
-        for pp, polarity in enumerate((-1, 1)):
+        for pp, _polarity in enumerate((-1, 1)):
             cpml_param = cpml_params[axis][pp]
             if cpml_param is None:
                 psi_E[axis][pp] = (None, None)
@@ -184,7 +185,7 @@ def updates_with_cpml(
     def update_H(
             e: fdfield_t,
             h: fdfield_t,
-            mu: fdfield_t = numpy.ones(3),
+            mu: fdfield_t | tuple[int, int, int] = (1, 1, 1),
             ) -> None:
         dyEx = Dfy(e[0])
         dzEx = Dfz(e[0])

meanas/test/conftest.py

@@ -3,7 +3,8 @@
 Test fixtures
 
 """
-from typing import Iterable, Any
+# ruff: noqa: ARG001
+from typing import Any
 import numpy
 from numpy.typing import NDArray
 import pytest       # type: ignore
@@ -20,18 +21,18 @@ FixtureRequest = Any
                         (5, 5, 5),
                         # (7, 7, 7),
                        ])
-def shape(request: FixtureRequest) -> Iterable[tuple[int, ...]]:
+def shape(request: FixtureRequest) -> tuple[int, ...]:
-    yield (3, *request.param)
+    return (3, *request.param)
 
 
 @pytest.fixture(scope='module', params=[1.0, 1.5])
-def epsilon_bg(request: FixtureRequest) -> Iterable[float]:
+def epsilon_bg(request: FixtureRequest) -> float:
-    yield request.param
+    return request.param
 
 
 @pytest.fixture(scope='module', params=[1.0, 2.5])
-def epsilon_fg(request: FixtureRequest) -> Iterable[float]:
+def epsilon_fg(request: FixtureRequest) -> float:
-    yield request.param
+    return request.param
 
 
 @pytest.fixture(scope='module', params=['center', '000', 'random'])
@@ -40,7 +41,7 @@ def epsilon(
         shape: tuple[int, ...],
         epsilon_bg: float,
         epsilon_fg: float,
-        ) -> Iterable[NDArray[numpy.float64]]:
+        ) -> NDArray[numpy.float64]:
     is3d = (numpy.array(shape) == 1).sum() == 0
     if is3d:
         if request.param == '000':
@@ -60,17 +61,17 @@ def epsilon(
                                   high=max(epsilon_bg, epsilon_fg),
                                   size=shape)
 
-    yield epsilon
+    return epsilon
 
 
 @pytest.fixture(scope='module', params=[1.0])  # 1.5
-def j_mag(request: FixtureRequest) -> Iterable[float]:
+def j_mag(request: FixtureRequest) -> float:
-    yield request.param
+    return request.param
 
 
 @pytest.fixture(scope='module', params=[1.0, 1.5])
-def dx(request: FixtureRequest) -> Iterable[float]:
+def dx(request: FixtureRequest) -> float:
-    yield request.param
+    return request.param
 
 
 @pytest.fixture(scope='module', params=['uniform', 'centerbig'])
@@ -78,7 +79,7 @@ def dxes(
         request: FixtureRequest,
         shape: tuple[int, ...],
         dx: float,
-        ) -> Iterable[list[list[NDArray[numpy.float64]]]]:
+        ) -> list[list[NDArray[numpy.float64]]]:
     if request.param == 'uniform':
         dxes = [[numpy.full(s, dx) for s in shape[1:]] for _ in range(2)]
     elif request.param == 'centerbig':
@@ -90,5 +91,5 @@ def dxes(
         dxe = [PRNG.uniform(low=1.0 * dx, high=1.1 * dx, size=s) for s in shape[1:]]
         dxh = [(d + numpy.roll(d, -1)) / 2 for d in dxe]
         dxes = [dxe, dxh]
-    yield dxes
+    return dxes
 

meanas/test/test_fdfd.py

@@ -1,4 +1,4 @@
-from typing import Iterable
+# ruff: noqa: ARG001
 import dataclasses
 import pytest       # type: ignore
 import numpy
@@ -6,7 +6,7 @@ from numpy.typing import NDArray
 #from numpy.testing import assert_allclose, assert_array_equal
 
 from .. import fdfd
-from ..fdmath import vec, unvec
+from ..fdmath import vec, unvec, vcfdfield, vfdfield, dx_lists_t
 from .utils import assert_close  # , assert_fields_close
 from .conftest import FixtureRequest
 
@@ -61,24 +61,24 @@ def test_poynting_planes(sim: 'FDResult') -> None:
 # Also see conftest.py
 
 @pytest.fixture(params=[1 / 1500])
-def omega(request: FixtureRequest) -> Iterable[float]:
+def omega(request: FixtureRequest) -> float:
-    yield request.param
+    return request.param
 
 
 @pytest.fixture(params=[None])
-def pec(request: FixtureRequest) -> Iterable[NDArray[numpy.float64] | None]:
+def pec(request: FixtureRequest) -> NDArray[numpy.float64] | None:
-    yield request.param
+    return request.param
 
 
 @pytest.fixture(params=[None])
-def pmc(request: FixtureRequest) -> Iterable[NDArray[numpy.float64] | None]:
+def pmc(request: FixtureRequest) -> NDArray[numpy.float64] | None:
-    yield request.param
+    return request.param
 
 
 #@pytest.fixture(scope='module',
 #                params=[(25, 5, 5)])
-#def shape(request):
+#def shape(request: FixtureRequest):
-#    yield (3, *request.param)
+#    return (3, *request.param)
 
 
 @pytest.fixture(params=['diag'])        # 'center'
@@ -86,7 +86,7 @@ def j_distribution(
         request: FixtureRequest,
         shape: tuple[int, ...],
         j_mag: float,
-        ) -> Iterable[NDArray[numpy.float64]]:
+        ) -> NDArray[numpy.float64]:
     j = numpy.zeros(shape, dtype=complex)
     center_mask = numpy.zeros(shape, dtype=bool)
     center_mask[:, shape[1] // 2, shape[2] // 2, shape[3] // 2] = True
@@ -96,22 +96,22 @@ def j_distribution(
     elif request.param == 'diag':
         j[numpy.roll(center_mask, [1, 1, 1], axis=(1, 2, 3))] = (1 + 1j) * j_mag
         j[numpy.roll(center_mask, [-1, -1, -1], axis=(1, 2, 3))] = (1 - 1j) * j_mag
-    yield j
+    return j
 
 
 @dataclasses.dataclass()
 class FDResult:
     shape: tuple[int, ...]
-    dxes: list[list[NDArray[numpy.float64]]]
+    dxes: dx_lists_t
-    epsilon: NDArray[numpy.float64]
+    epsilon: vfdfield
     omega: complex
-    j: NDArray[numpy.complex128]
+    j: vcfdfield
-    e: NDArray[numpy.complex128]
+    e: vcfdfield
-    pmc: NDArray[numpy.float64] | None
+    pmc: vfdfield | None
-    pec: NDArray[numpy.float64] | None
+    pec: vfdfield | None
 
 
-@pytest.fixture()
+@pytest.fixture
 def sim(
         request: FixtureRequest,
         shape: tuple[int, ...],
@@ -141,11 +141,11 @@ def sim(
     j_vec = vec(j_distribution)
     eps_vec = vec(epsilon)
     e_vec = fdfd.solvers.generic(
-        J=j_vec,
+        J = j_vec,
-        omega=omega,
+        omega = omega,
-        dxes=dxes,
+        dxes = dxes,
-        epsilon=eps_vec,
+        epsilon = eps_vec,
-        matrix_solver_opts={'atol': 1e-15, 'tol': 1e-11},
+        matrix_solver_opts = dict(atol=1e-15, rtol=1e-11),
         )
     e = unvec(e_vec, shape[1:])
 

meanas/test/test_fdfd_pml.py

@@ -1,11 +1,11 @@
-from typing import Iterable
+# ruff: noqa: ARG001
 import pytest       # type: ignore
 import numpy
 from numpy.typing import NDArray
 from numpy.testing import assert_allclose
 
 from .. import fdfd
-from ..fdmath import vec, unvec, dx_lists_mut
+from ..fdmath import vec, unvec, dx_lists_mut, vfdfield, cfdfield_t
 #from .utils import assert_close, assert_fields_close
 from .test_fdfd import FDResult
 from .conftest import FixtureRequest
@@ -44,41 +44,41 @@ def test_pml(sim: FDResult, src_polarity: int) -> None:
 # Also see conftest.py
 
 @pytest.fixture(params=[1 / 1500])
-def omega(request: FixtureRequest) -> Iterable[float]:
+def omega(request: FixtureRequest) -> float:
-    yield request.param
+    return request.param
 
 
 @pytest.fixture(params=[None])
-def pec(request: FixtureRequest) -> Iterable[NDArray[numpy.float64] | None]:
+def pec(request: FixtureRequest) -> NDArray[numpy.float64] | None:
-    yield request.param
+    return request.param
 
 
 @pytest.fixture(params=[None])
-def pmc(request: FixtureRequest) -> Iterable[NDArray[numpy.float64] | None]:
+def pmc(request: FixtureRequest) -> NDArray[numpy.float64] | None:
-    yield request.param
+    return request.param
 
 
 @pytest.fixture(params=[(30, 1, 1),
                         (1, 30, 1),
                         (1, 1, 30)])
-def shape(request: FixtureRequest) -> Iterable[tuple[int, ...]]:
+def shape(request: FixtureRequest) -> tuple[int, int, int]:
-    yield (3, *request.param)
+    return (3, *request.param)
 
 
 @pytest.fixture(params=[+1, -1])
-def src_polarity(request: FixtureRequest) -> Iterable[int]:
+def src_polarity(request: FixtureRequest) -> int:
-    yield request.param
+    return request.param
 
 
-@pytest.fixture()
+@pytest.fixture
 def j_distribution(
         request: FixtureRequest,
         shape: tuple[int, ...],
-        epsilon: NDArray[numpy.float64],
+        epsilon: vfdfield,
         dxes: dx_lists_mut,
         omega: float,
         src_polarity: int,
-        ) -> Iterable[NDArray[numpy.complex128]]:
+        ) -> cfdfield_t:
     j = numpy.zeros(shape, dtype=complex)
 
     dim = numpy.where(numpy.array(shape[1:]) > 1)[0][0]    # Propagation axis
@@ -106,18 +106,18 @@ def j_distribution(
 
     j = fdfd.waveguide_3d.compute_source(E=e, wavenumber=wavenumber_corrected, omega=omega, dxes=dxes,
                                          axis=dim, polarity=src_polarity, slices=slices, epsilon=epsilon)
-    yield j
+    return j
 
 
-@pytest.fixture()
+@pytest.fixture
 def epsilon(
         request: FixtureRequest,
         shape: tuple[int, ...],
         epsilon_bg: float,
         epsilon_fg: float,
-        ) -> Iterable[NDArray[numpy.float64]]:
+        ) -> NDArray[numpy.float64]:
     epsilon = numpy.full(shape, epsilon_fg, dtype=float)
-    yield epsilon
+    return epsilon
 
 
 @pytest.fixture(params=['uniform'])
@@ -127,7 +127,7 @@ def dxes(
         dx: float,
         omega: float,
         epsilon_fg: float,
-        ) -> Iterable[list[list[NDArray[numpy.float64]]]]:
+        ) -> list[list[NDArray[numpy.float64]]]:
     if request.param == 'uniform':
         dxes = [[numpy.full(s, dx) for s in shape[1:]] for _ in range(2)]
     dim = numpy.where(numpy.array(shape[1:]) > 1)[0][0]    # Propagation axis
@@ -141,10 +141,10 @@ def dxes(
                 epsilon_effective=epsilon_fg,
                 thickness=10,
                 )
-    yield dxes
+    return dxes
 
 
-@pytest.fixture()
+@pytest.fixture
 def sim(
         request: FixtureRequest,
         shape: tuple[int, ...],
@@ -162,7 +162,7 @@ def sim(
         omega=omega,
         dxes=dxes,
         epsilon=eps_vec,
-        matrix_solver_opts={'atol': 1e-15, 'tol': 1e-11},
+        matrix_solver_opts={'atol': 1e-15, 'rtol': 1e-11},
         )
     e = unvec(e_vec, shape[1:])
 

meanas/test/test_fdtd.py

@@ -1,4 +1,5 @@
-from typing import Iterable, Any
+# ruff: noqa: ARG001
+from typing import Any
 import dataclasses
 import pytest       # type: ignore
 import numpy
@@ -150,8 +151,8 @@ def test_poynting_planes(sim: 'TDResult') -> None:
 
 
 @pytest.fixture(params=[0.3])
-def dt(request: FixtureRequest) -> Iterable[float]:
+def dt(request: FixtureRequest) -> float:
-    yield request.param
+    return request.param
 
 
 @dataclasses.dataclass()
@@ -168,8 +169,8 @@ class TDResult:
 
 
 @pytest.fixture(params=[(0, 4, 8)])  # (0,)
-def j_steps(request: FixtureRequest) -> Iterable[tuple[int, ...]]:
+def j_steps(request: FixtureRequest) -> tuple[int, ...]:
-    yield request.param
+    return request.param
 
 
 @pytest.fixture(params=['center', 'random'])
@@ -177,7 +178,7 @@ def j_distribution(
         request: FixtureRequest,
         shape: tuple[int, ...],
         j_mag: float,
-        ) -> Iterable[NDArray[numpy.float64]]:
+        ) -> NDArray[numpy.float64]:
     j = numpy.zeros(shape)
     if request.param == 'center':
         j[:, shape[1] // 2, shape[2] // 2, shape[3] // 2] = j_mag
@@ -185,10 +186,10 @@ def j_distribution(
         j[:, 0, 0, 0] = j_mag
     elif request.param == 'random':
         j[:] = PRNG.uniform(low=-j_mag, high=j_mag, size=shape)
-    yield j
+    return j
 
 
-@pytest.fixture()
+@pytest.fixture
 def sim(
         request: FixtureRequest,
         shape: tuple[int, ...],
@@ -199,8 +200,7 @@ def sim(
         j_steps: tuple[int, ...],
         ) -> TDResult:
     is3d = (numpy.array(shape) == 1).sum() == 0
-    if is3d:
+    if is3d and dt != 0.3:
-        if dt != 0.3:
         pytest.skip('Skipping dt != 0.3 because test is 3D (for speed)')
 
     sim = TDResult(

meanas/test/utils.py

@@ -1,5 +1,3 @@
-from typing import Any
-
 import numpy
 from numpy.typing import NDArray
 
@@ -10,22 +8,25 @@ PRNG = numpy.random.RandomState(12345)
 def assert_fields_close(
         x: NDArray,
         y: NDArray,
-        *args: Any,
-        **kwargs: Any,
-        ) -> None:
-    numpy.testing.assert_allclose(
-        x, y, verbose=False,            # type: ignore
-        err_msg='Fields did not match:\n{}\n{}'.format(numpy.moveaxis(x, -1, 0),
-                                                       numpy.moveaxis(y, -1, 0)),
         *args,
         **kwargs,
+        ) -> None:
+    x_disp = numpy.moveaxis(x, -1, 0)
+    y_disp = numpy.moveaxis(y, -1, 0)
+    numpy.testing.assert_allclose(
+        x,           # type: ignore
+        y,           # type: ignore
+        *args,
+        verbose=False,
+        err_msg=f'Fields did not match:\n{x_disp}\n{y_disp}',
+        **kwargs,
         )
 
 def assert_close(
         x: NDArray,
         y: NDArray,
-        *args: Any,
+        *args,
-        **kwargs: Any,
+        **kwargs,
         ) -> None:
     numpy.testing.assert_allclose(x, y, *args, **kwargs)
 

pyproject.toml

@@ -39,9 +39,10 @@ include = [
     ]
 dynamic = ["version"]
 dependencies = [
-    "numpy~=1.21",
+    "gridlock",
-    "scipy",
+    "numpy>=2.0",
-    ]
+    "scipy~=1.14",
+]
 
 
 [tool.hatch.version]
@@ -49,5 +50,55 @@ path = "meanas/__init__.py"
 
 [project.optional-dependencies]
 dev = ["pytest", "pdoc", "gridlock"]
-examples = ["gridlock"]
+examples = [
+    "gridlock>=2.1",
+    "matplotlib>=3.10.8",
+]
 test = ["pytest"]
+
+
+[tool.ruff]
+exclude = [
+    ".git",
+    "dist",
+    ]
+line-length = 245
+indent-width = 4
+lint.dummy-variable-rgx = "^(_+|(_+[a-zA-Z0-9_]*[a-zA-Z0-9]+?))$"
+lint.select = [
+    "NPY", "E", "F", "W", "B", "ANN", "UP", "SLOT", "SIM", "LOG",
+    "C4", "ISC", "PIE", "PT", "RET", "TCH", "PTH", "INT",
+    "ARG", "PL", "R", "TRY",
+    "G010", "G101", "G201", "G202",
+    "Q002", "Q003", "Q004",
+    ]
+lint.ignore = [
+    #"ANN001",   # No annotation
+    "ANN002",   # *args
+    "ANN003",   # **kwargs
+    "ANN401",   # Any
+    "SIM108",   # single-line if / else assignment
+    "RET504",   # x=y+z; return x
+    "PIE790",   # unnecessary pass
+    "ISC003",   # non-implicit string concatenation
+    "C408",     # dict(x=y) instead of {'x': y}
+    "PLR09",    # Too many xxx
+    "PLR2004",  # magic number
+    "PLC0414",  # import x as x
+    "TRY003",   # Long exception message
+    "TRY002",   # Exception()
+    ]
+
+
+[[tool.mypy.overrides]]
+module = [
+    "scipy",
+    "scipy.optimize",
+    "scipy.linalg",
+    "scipy.sparse",
+    "scipy.sparse.linalg",
+    ]
+ignore_missing_imports = true
+
+[tool.uv.sources]
+gridlock = { path = "../gridlock", editable = true }

uv.lock

@@ -0,0 +1,806 @@
+version = 1
+revision = 3
+requires-python = ">=3.11"
+
+[[package]]
+name = "colorama"
+version = "0.4.6"
+source = { registry = "https://pypi.org/simple" }
+sdist = { url = "https://files.pythonhosted.org/packages/d8/53/6f443c9a4a8358a93a6792e2acffb9d9d5cb0a5cfd8802644b7b1c9a02e4/colorama-0.4.6.tar.gz", hash = "sha256:08695f5cb7ed6e0531a20572697297273c47b8cae5a63ffc6d6ed5c201be6e44", size = 27697, upload-time = "2022-10-25T02:36:22.414Z" }
+wheels = [
+    { url = "https://files.pythonhosted.org/packages/d1/d6/3965ed04c63042e047cb6a3e6ed1a63a35087b6a609aa3a15ed8ac56c221/colorama-0.4.6-py2.py3-none-any.whl", hash = "sha256:4f1d9991f5acc0ca119f9d443620b77f9d6b33703e51011c16baf57afb285fc6", size = 25335, upload-time = "2022-10-25T02:36:20.889Z" },
+]
+
+[[package]]
+name = "contourpy"
+version = "1.3.3"
+source = { registry = "https://pypi.org/simple" }
+dependencies = [
+    { name = "numpy" },
+]
+sdist = { url = "https://files.pythonhosted.org/packages/58/01/1253e6698a07380cd31a736d248a3f2a50a7c88779a1813da27503cadc2a/contourpy-1.3.3.tar.gz", hash = "sha256:083e12155b210502d0bca491432bb04d56dc3432f95a979b429f2848c3dbe880", size = 13466174, upload-time = "2025-07-26T12:03:12.549Z" }
+wheels = [
+    { url = "https://files.pythonhosted.org/packages/91/2e/c4390a31919d8a78b90e8ecf87cd4b4c4f05a5b48d05ec17db8e5404c6f4/contourpy-1.3.3-cp311-cp311-macosx_10_9_x86_64.whl", hash = "sha256:709a48ef9a690e1343202916450bc48b9e51c049b089c7f79a267b46cffcdaa1", size = 288773, upload-time = "2025-07-26T12:01:02.277Z" },
+    { url = "https://files.pythonhosted.org/packages/0d/44/c4b0b6095fef4dc9c420e041799591e3b63e9619e3044f7f4f6c21c0ab24/contourpy-1.3.3-cp311-cp311-macosx_11_0_arm64.whl", hash = "sha256:23416f38bfd74d5d28ab8429cc4d63fa67d5068bd711a85edb1c3fb0c3e2f381", size = 270149, upload-time = "2025-07-26T12:01:04.072Z" },
+    { url = "https://files.pythonhosted.org/packages/30/2e/dd4ced42fefac8470661d7cb7e264808425e6c5d56d175291e93890cce09/contourpy-1.3.3-cp311-cp311-manylinux_2_26_aarch64.manylinux_2_28_aarch64.whl", hash = "sha256:929ddf8c4c7f348e4c0a5a3a714b5c8542ffaa8c22954862a46ca1813b667ee7", size = 329222, upload-time = "2025-07-26T12:01:05.688Z" },
+    { url = "https://files.pythonhosted.org/packages/f2/74/cc6ec2548e3d276c71389ea4802a774b7aa3558223b7bade3f25787fafc2/contourpy-1.3.3-cp311-cp311-manylinux_2_26_ppc64le.manylinux_2_28_ppc64le.whl", hash = "sha256:9e999574eddae35f1312c2b4b717b7885d4edd6cb46700e04f7f02db454e67c1", size = 377234, upload-time = "2025-07-26T12:01:07.054Z" },
+    { url = "https://files.pythonhosted.org/packages/03/b3/64ef723029f917410f75c09da54254c5f9ea90ef89b143ccadb09df14c15/contourpy-1.3.3-cp311-cp311-manylinux_2_26_s390x.manylinux_2_28_s390x.whl", hash = "sha256:0bf67e0e3f482cb69779dd3061b534eb35ac9b17f163d851e2a547d56dba0a3a", size = 380555, upload-time = "2025-07-26T12:01:08.801Z" },
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