[waveguide_cyl] adjust operators to match waveguide_2d in the large-radius limit
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1 changed files with 42 additions and 58 deletions
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@ -43,39 +43,9 @@ T_b &= \operatorname{diag}(r_b / r_{\min}).
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$$
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With the same forward/backward derivative notation used in `waveguide_2d`, the
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coordinate-transformed discrete curl equations used here are
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$$
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\begin{aligned}
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-\imath \omega \mu_{rr} H_r &= \tilde{\partial}_y E_z + \imath \beta T_a^{-1} E_y, \\
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-\imath \omega \mu_{yy} H_y &= -\imath \beta T_b^{-1} E_r
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- T_b^{-1} \tilde{\partial}_r (T_a E_z), \\
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-\imath \omega \mu_{zz} H_z &= \tilde{\partial}_r E_y - \tilde{\partial}_y E_r, \\
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\imath \beta H_y &= -\imath \omega T_b \epsilon_{rr} E_r - T_b \hat{\partial}_y H_z, \\
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\imath \beta H_r &= \imath \omega T_a \epsilon_{yy} E_y
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- T_b T_a^{-1} \hat{\partial}_r (T_b H_z), \\
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\imath \omega E_z &= T_a \epsilon_{zz}^{-1}
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\left(\hat{\partial}_r H_y - \hat{\partial}_y H_r\right).
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\end{aligned}
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$$
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The first three equations are the cylindrical analogue of the straight-guide
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relations for `H_r`, `H_y`, and `H_z`. The next two are the metric-weighted
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versions of the straight-guide identities for `\imath \beta H_y` and
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`\imath \beta H_r`, and the last equation plays the same role as the
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longitudinal `E_z` reconstruction in `waveguide_2d`.
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Following the same elimination steps as in `waveguide_2d`, apply
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`\imath \beta \tilde{\partial}_r` and `\imath \beta \tilde{\partial}_y` to the
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equation for `E_z`, substitute for `\imath \beta H_r` and `\imath \beta H_y`,
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and then eliminate `H_z` with
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$$
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H_z = \frac{1}{-\imath \omega \mu_{zz}}
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\left(\tilde{\partial}_r E_y - \tilde{\partial}_y E_r\right).
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$$
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This yields the transverse electric eigenproblem implemented by
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implementation treats the transverse electric eigenproblem as the canonical
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cylindrical discretization. It reduces to `waveguide_2d.operator_e(...)` in the
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large-radius limit `T_a, T_b \to I`. The eigenproblem implemented by
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`cylindrical_operator(...)`:
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$$
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@ -111,6 +81,33 @@ T_a \epsilon_{zz}^{-1}
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\begin{bmatrix} E_r \\ E_y \end{bmatrix}.
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$$
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The full fields reconstructed by `exy2e(...)` and `e2h(...)` use the matching
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large-radius-compatible identities
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$$
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E_z =
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\frac{1}{\imath \beta} T_a \epsilon_{zz}^{-1}
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\begin{bmatrix}
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\hat{\partial}_r T_b \epsilon_{rr} &
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\hat{\partial}_y T_a \epsilon_{yy}
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\end{bmatrix}
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\begin{bmatrix} E_r \\ E_y \end{bmatrix},
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$$
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and
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$$
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\begin{bmatrix} H_r \\ H_y \\ H_z \end{bmatrix}
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=
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\frac{1}{-\imath \omega}\mu^{-1}
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\begin{bmatrix}
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0 & \imath\beta T_a^{-1} & \tilde{\partial}_y \\
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-\imath\beta T_b^{-1} & 0 & -T_b^{-1}\tilde{\partial}_r T_a \\
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-\tilde{\partial}_y & \tilde{\partial}_r & 0
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\end{bmatrix}
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\begin{bmatrix} E_r \\ E_y \\ E_z \end{bmatrix}.
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$$
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Since `\beta = m / r_{\min}`, the solver implemented in this file returns the
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angular wavenumber `m`, while the operator itself is most naturally written in
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terms of the linear quantity `\beta`. The helpers below reconstruct the full
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@ -367,7 +364,6 @@ def exy2h(
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def exy2e(
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angular_wavenumber: complex,
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omega: float,
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dxes: dx_lists2_t,
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rmin: float,
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epsilon: vfdslice,
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@ -383,7 +379,6 @@ def exy2e(
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angular_wavenumber: Wavenumber assuming fields have theta-dependence of
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`exp(-i * angular_wavenumber * theta)`. It should satisfy
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`operator_e() @ e_xy == (angular_wavenumber / rmin) ** 2 * e_xy`
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omega: The angular frequency of the system
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dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types` (2D)
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rmin: Radius at the left edge of the simulation domain (at minimum 'x')
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epsilon: Vectorized dielectric constant grid
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@ -391,30 +386,22 @@ def exy2e(
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Returns:
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Sparse matrix representing the operator.
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"""
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Dfx, Dfy = deriv_forward(dxes[0])
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Dbx, Dby = deriv_back(dxes[1])
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wavenumber = angular_wavenumber / rmin
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Ta, Tb = dxes2T(dxes=dxes, rmin=rmin)
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Tai = sparse.diags_array(1 / Ta.diagonal())
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#Tbi = sparse.diags_array(1 / Tb.diagonal())
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epsilon_parts = numpy.split(epsilon, 3)
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epsilon_x, epsilon_y = (sparse.diags_array(epsi) for epsi in epsilon_parts[:2])
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epsilon_z_inv = sparse.diags_array(1 / epsilon_parts[2])
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n_pts = dxes[0][0].size * dxes[0][1].size
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zeros = sparse.coo_array((n_pts, n_pts))
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mu_z = numpy.ones(n_pts)
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mu_z_inv = sparse.diags_array(1 / mu_z)
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exy2hz = 1 / (-1j * omega) * mu_z_inv @ sparse.hstack((Dfy, -Dfx))
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hxy2ez = 1 / (1j * omega) * epsilon_z_inv @ sparse.hstack((Dby, -Dbx))
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exy2hy = Tb / (1j * wavenumber) @ (-1j * omega * sparse.hstack((epsilon_x, zeros)) - Dby @ exy2hz)
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exy2hx = Tb / (1j * wavenumber) @ ( 1j * omega * sparse.hstack((zeros, epsilon_y)) - Tai @ Dbx @ Tb @ exy2hz)
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exy2ez = hxy2ez @ sparse.vstack((exy2hx, exy2hy))
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exy2ez = (
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Ta @ epsilon_z_inv
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@ sparse.hstack((Dbx @ Tb @ epsilon_x,
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Dby @ Ta @ epsilon_y))
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/ (1j * wavenumber)
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)
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op = sparse.vstack((sparse.eye_array(2 * n_pts),
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exy2ez))
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@ -460,9 +447,9 @@ def e2h(
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Tbi = sparse.diags_array(1 / Tb.diagonal())
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jB = 1j * angular_wavenumber / rmin
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op = sparse.block_array([[ None, -jB * Tai, -Dfy],
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[jB * Tbi, None, Tbi @ Dfx @ Ta],
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[ Dfy, -Dfx, None]]) / (-1j * omega)
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op = sparse.block_array([[ None, jB * Tai, Dfy],
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[-jB * Tbi, None, -Tbi @ Dfx @ Ta],
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[ -Dfy, Dfx, None]]) / (-1j * omega)
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if mu is not None:
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op = sparse.diags_array(1 / mu) @ op
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return op
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@ -565,19 +552,16 @@ def _normalized_fields(
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The normalization procedure is:
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1. Flip the magnetic field sign so the reconstructed `(e, h)` pair follows
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the same forward-power convention as `waveguide_2d`.
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2. Compute the time-averaged forward power with
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1. Compute the time-averaged forward power with
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`waveguide_2d.inner_product(..., conj_h=True)`.
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3. Scale by `1 / sqrt(S_z)` so the resulting mode has unit forward power.
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4. Remove the arbitrary complex phase and apply a quadrant-sum sign heuristic
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2. Scale by `1 / sqrt(S_z)` so the resulting mode has unit forward power.
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3. Remove the arbitrary complex phase and apply a quadrant-sum sign heuristic
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so symmetric modes choose a stable representative.
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`prop_phase` has the same meaning as in `waveguide_2d`: it compensates for
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the half-cell longitudinal staggering between the E and H fields when the
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propagation direction is itself discretized.
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"""
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h *= -1
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shape = [s.size for s in dxes[0]]
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# Find time-averaged Sz and normalize to it
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