[fdfd.waveguide_cyl] Improve documentation and add auxiliary functions (e.g. exy2exyz)

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 $\imath \beta H_y$:
 
 $$
 \begin{aligned}
-\gamma \tilde{\partial}_x \imath \omega E_z &= \gamma \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x H_y
+\imath \beta \tilde{\partial}_x \imath \omega E_z &= \imath \beta \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x H_y
-                                             - \gamma \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y H_x \\
+                                                   - \imath \beta \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y H_x \\
         &= \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x ( \imath \omega \epsilon_{xx} E_x - \hat{\partial}_y H_z)
          - \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y (-\imath \omega \epsilon_{yy} E_y - \hat{\partial}_x H_z)  \\
         &= \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x ( \imath \omega \epsilon_{xx} E_x)
          - \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y (-\imath \omega \epsilon_{yy} E_y)  \\
-\gamma \tilde{\partial}_x E_z &= \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
+\imath \beta \tilde{\partial}_x E_z &= \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
-                               + \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y) \\
+                                     + \tilde{\partial}_x \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y) \\
 \end{aligned}
 $$
 
-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
 
 $$
 \begin{aligned}
-\gamma \tilde{\partial}_y E_z &= \tilde{\partial}_y \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
+\imath \beta \tilde{\partial}_y E_z &= \tilde{\partial}_y \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
-                               + \tilde{\partial}_y \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y) \\
+                                     + \tilde{\partial}_y \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y) \\
 \end{aligned}
 $$
 
-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)
                                     )\\
@@ -100,25 +101,24 @@ 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
 
 $$
 \begin{aligned}
--\imath \omega \mu_{xx} (\gamma H_x) &= -\imath \omega \mu_{xx} (-\imath \omega \epsilon_{yy} E_y - \hat{\partial}_x H_z) \\
+-\imath \omega \mu_{xx} (\imath \beta H_x) &= -\imath \omega \mu_{xx} (-\imath \omega \epsilon_{yy} E_y - \hat{\partial}_x H_z) \\
-                                     &= -\omega^2 \mu_{xx} \epsilon_{yy} E_y
+                 &= -\omega^2 \mu_{xx} \epsilon_{yy} E_y + \imath \omega \mu_{xx} \hat{\partial}_x (
-                                        +\imath \omega \mu_{xx} \hat{\partial}_x (
+                         \frac{1}{-\imath \omega \mu_{zz}} (\tilde{\partial}_x E_y - \tilde{\partial}_y E_x)) \\
-                                           \frac{1}{-\imath \omega \mu_{zz}} (\tilde{\partial}_x E_y - \tilde{\partial}_y E_x)) \\
+                 &= -\omega^2 \mu_{xx} \epsilon_{yy} E_y
-                                     &= -\omega^2 \mu_{xx} \epsilon_{yy} E_y
+                         -\mu_{xx} \hat{\partial}_x \frac{1}{\mu_{zz}} (\tilde{\partial}_x E_y - \tilde{\partial}_y E_x) \\
-                                        -\mu_{xx} \hat{\partial}_x \frac{1}{\mu_{zz}} (\tilde{\partial}_x E_y - \tilde{\partial}_y E_x) \\
 \end{aligned}
 $$
 
@@ -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
 
 $$
 \begin{aligned}
--\gamma^2 E_x - \tilde{\partial}_x (
+\beta^2 E_x - \tilde{\partial}_x (
     \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
   + \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y)
     ) &= \omega^2 \mu_{yy} \epsilon_{xx} E_x
         +\mu_{yy} \hat{\partial}_y \frac{1}{\mu_{zz}} (\tilde{\partial}_x E_y - \tilde{\partial}_y E_x) \\
-\gamma^2 E_y + \tilde{\partial}_y (
+-\beta^2 E_y + \tilde{\partial}_y (
     \frac{1}{\epsilon_{zz}} \hat{\partial}_x (\epsilon_{xx} E_x)
   + \frac{1}{\epsilon_{zz}} \hat{\partial}_y (\epsilon_{yy} E_y)
     ) &= -\omega^2 \mu_{xx} \epsilon_{yy} E_y
@@ -165,14 +165,13 @@ $$
                     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.
 
 
 """
@@ -531,10 +530,37 @@ def exy2e(
         dxes: dx_lists_t,
         epsilon: vfdfield_t,
         ) -> sparse.spmatrix:
-    """
+    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`
@@ -905,7 +931,7 @@ def inner_product(    # TODO documentation
         prop_phase: float = 0,
         conj_h: bool = False,
         trapezoid: bool = False,
-        ) -> tuple[vcfdfield_t, vcfdfield_t]:
+        ) -> complex:
 
     shape = [s.size for s in dxes[0]]
 
@@ -920,7 +946,7 @@ def inner_product(    # TODO documentation
 
     # Find time-averaged Sz and normalize to it
     dxes_real = [[numpy.real(dxyz) for dxyz in dxeh] for dxeh in dxes]
-    if integrate:
+    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:

meanas/fdfd/waveguide_cyl.py

@@ -1,14 +1,66 @@
-"""
+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 typing import Any
 from collections.abc import Sequence
 import logging
 
@@ -19,7 +71,7 @@ from scipy import sparse
 from ..fdmath import vec, unvec, dx_lists_t, vfdfield_t, vcfdfield_t
 from ..fdmath.operators import deriv_forward, deriv_back
 from ..eigensolvers import signed_eigensolve, rayleigh_quotient_iteration
-
+from . import waveguide_2d
 
 logger = logging.getLogger(__name__)
 
@@ -30,13 +82,21 @@ def cylindrical_operator(
         epsilon: vfdfield_t,
         rmin: float,
         ) -> sparse.spmatrix:
-    """
+    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]`.
 
@@ -46,11 +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
-        rmin: Radius at the left edge of the simulation domain (minimum 'x')
+        rmin: Radius at the left edge of the simulation domain (at minimum 'x')
 
     Returns:
         Sparse matrix representation of the operator
@@ -62,9 +124,9 @@ def cylindrical_operator(
     Ta, Tb = dxes2T(dxes=dxes, rmin=rmin)
 
     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])
 
     omega2 = omega * omega
     diag = sparse.block_diag
@@ -74,18 +136,6 @@ def cylindrical_operator(
     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 = (
-    #     # E
-    #     omega * omega * mu_yx @ eps_xy
-    #     + mu_yx @ sparse.vstack((-Dby, Dbx)) @ mu_z_inv @ sparse.hstack((-Dfy, Dfx))
-    #     + sparse.vstack((Dfx, Dfy)) @ eps_z_inv @ sparse.hstack((Dbx, Dby)) @ eps_xy
-
-    #     # H
-    #     omega * omega * eps_yx @ mu_xy
-    #     + eps_yx @ sparse.vstack((-Dfy, Dfx)) @ eps_z_inv @ sparse.hstack((-Dby, Dbx))
-    #     + sparse.vstack((Dbx, Dby)) @ mu_z_inv @ sparse.hstack((Dfx, Dfy)) @ mu_xy
-    #     )
-
     op = sq0 + lin0 + lin1
     return op
 
@@ -99,7 +149,6 @@ def solve_modes(
         mode_margin: int = 2,
         ) -> tuple[vcfdfield_t, 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.
 
@@ -142,7 +191,7 @@ def solve_modes(
 
     # Wavenumbers assume the mode is at rmin, which is unlikely
     # Instead, return the wavenumber in inverse radians
-    angular_wavenumbers = wavenumbers * rmin
+    angular_wavenumbers = wavenumbers * cast(complex, rmin)
 
     order = angular_wavenumbers.argsort()[::-1]
     e_xys = e_xys[order]
@@ -168,7 +217,7 @@ def solve_mode(
         (e_xy, angular_wavenumber)
     """
     kwargs['mode_numbers'] = [mode_number]
-    e_xys, wavenumbers = solve_modes(*args, **kwargs)
+    e_xys, angular_wavenumbers = solve_modes(*args, **kwargs)
     return e_xys[0], angular_wavenumbers[0]
 
 
@@ -184,11 +233,13 @@ def linear_wavenumbers(
       and the mode's energy distribution.
 
     Args:
-        e_xys: Vectorized mode fields with shape [num_modes, 2 * x *y)
+        e_xys: Vectorized mode fields with shape (num_modes, 2 * x *y)
-        angular_wavenumbers: Angular wavenumbers corresponding to the fields in `e_xys`
+        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 (minimum 'x')
+        rmin: Radius at the left edge of the simulation domain (at minimum 'x')
 
     Returns:
         NDArray containing the calculated linear (1/distance) wavenumbers
@@ -211,3 +262,241 @@ def linear_wavenumbers(
     return lin_wavenumbers
 
 
+def exy2h(
+        angular_wavenumber: complex,
+        omega: float,
+        dxes: dx_lists_t,
+        rmin: float,
+        epsilon: vfdfield_t,
+        mu: vfdfield_t | None = None
+        ) -> sparse.spmatrix:
+    """
+    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_lists_t,
+        rmin: float,
+        epsilon: vfdfield_t,
+        ) -> sparse.spmatrix:
+    """
+    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))
+    keep_x = sparse.block_array([[sparse.eye_array(n_pts), None], [None, zeros]])
+    keep_y = sparse.block_array([[zeros, None], [None, sparse.eye_array(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: complex,
+        dxes: dx_lists_t,
+        rmin: float,
+        mu: vfdfield_t | None = None
+        ) -> sparse.spmatrix:
+    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_lists_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: ArrayLike,
+        angular_wavenumber: complex,
+        omega: complex,
+        dxes: dx_lists_t,
+        rmin: float,
+        epsilon: vfdfield_t,
+        mu: vfdfield_t | None = None,
+        prop_phase: float = 0,
+        ) -> tuple[vcfdfield_t, vcfdfield_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: vcfdfield_t,
+        h: vcfdfield_t,
+        omega: complex,
+        dxes: dx_lists_t,
+        rmin: float,
+        epsilon: vfdfield_t,
+        mu: vfdfield_t | None = None,
+        prop_phase: float = 0,
+        ) -> tuple[vcfdfield_t, vcfdfield_t]:
+    h *= -1
+    # TODO documentation for normalized_fields
+    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)]
+
+    # 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
+    phase = numpy.exp(-1j * -prop_phase / 2)
+    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)
+
+    print('\nAAA\n', waveguide_2d.inner_product(e, h, dxes, prop_phase=prop_phase))
+    e *= norm_factor
+    h *= norm_factor
+    print(f'{sign=} {norm_amplitude=} {norm_angle=} {prop_phase=}')
+    print(waveguide_2d.inner_product(e, h, dxes, prop_phase=prop_phase))
+
+    return e, h

meanas/fdmath/functional.py

@@ -14,7 +14,7 @@ from .types import fdfield_t, fdfield_updater_t
 
 
 def deriv_forward(
-        dx_e: Sequence[NDArray[floating]] | 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).
@@ -38,7 +38,7 @@ def deriv_forward(
 
 
 def deriv_back(
-        dx_h: Sequence[NDArray[floating]] | 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).
@@ -65,7 +65,7 @@ TT = TypeVar('TT', bound='NDArray[floating | complexfloating]')
 
 
 def curl_forward(
-        dx_e: Sequence[NDArray[floating]] | None = None,
+        dx_e: Sequence[NDArray[floating | complexfloating]] | None = None,
         ) -> Callable[[TT], TT]:
     r"""
     Curl operator for use with the E field.
@@ -94,7 +94,7 @@ def curl_forward(
 
 
 def curl_back(
-        dx_h: Sequence[NDArray[floating]] | None = None,
+        dx_h: Sequence[NDArray[floating | complexfloating]] | None = None,
         ) -> Callable[[TT], TT]:
     r"""
     Create a function which takes the backward curl of a field.
@@ -123,7 +123,7 @@ def curl_back(
 
 
 def curl_forward_parts(
-        dx_e: Sequence[NDArray[floating]] | None = None,
+        dx_e: Sequence[NDArray[floating | complexfloating]] | None = None,
         ) -> Callable:
     Dx, Dy, Dz = deriv_forward(dx_e)
 
@@ -136,7 +136,7 @@ def curl_forward_parts(
 
 
 def curl_back_parts(
-        dx_h: Sequence[NDArray[floating]] | None = None,
+        dx_h: Sequence[NDArray[floating | complexfloating]] | None = None,
         ) -> Callable:
     Dx, Dy, Dz = deriv_back(dx_h)
 

meanas/fdmath/operators.py

@@ -6,7 +6,7 @@ Basic discrete calculus etc.
 from collections.abc import Sequence
 import numpy
 from numpy.typing import NDArray
-from numpy import floating
+from numpy import floating, complexfloating
 from scipy import sparse
 
 from .types import vfdfield_t
@@ -97,7 +97,7 @@ def shift_with_mirror(
 
 
 def deriv_forward(
-        dx_e: Sequence[NDArray[floating]],
+        dx_e: Sequence[NDArray[floating | complexfloating]],
         ) -> list[sparse.spmatrix]:
     """
     Utility operators for taking discretized derivatives (forward variant).
@@ -124,7 +124,7 @@ def deriv_forward(
 
 
 def deriv_back(
-        dx_h: Sequence[NDArray[floating]],
+        dx_h: Sequence[NDArray[floating | complexfloating]],
         ) -> list[sparse.spmatrix]:
     """
     Utility operators for taking discretized derivatives (backward variant).
@@ -219,7 +219,7 @@ def avg_back(axis: int, shape: Sequence[int]) -> sparse.spmatrix:
 
 
 def curl_forward(
-        dx_e: Sequence[NDArray[floating]],
+        dx_e: Sequence[NDArray[floating | complexfloating]],
         ) -> sparse.spmatrix:
     """
     Curl operator for use with the E field.
@@ -235,7 +235,7 @@ def curl_forward(
 
 
 def curl_back(
-        dx_h: Sequence[NDArray[floating]],
+        dx_h: Sequence[NDArray[floating | complexfloating]],
         ) -> sparse.spmatrix:
     """
     Curl operator for use with the H field.

meanas/fdmath/vectorization.py

@@ -49,15 +49,15 @@ def vec(
 
 
 @overload
-def unvec(v: None, shape: Sequence[int], nvdim: int) -> None:
+def unvec(v: None, shape: Sequence[int], nvdim: int = 3) -> None:
     pass
 
 @overload
-def unvec(v: vfdfield_t, shape: Sequence[int], nvdim: 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], nvdim: int) -> cfdfield_t:
+def unvec(v: vcfdfield_t, shape: Sequence[int], nvdim: int = 3) -> cfdfield_t:
     pass
 
 def unvec(