add sensitivity calculation
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@ -185,7 +185,7 @@ from numpy.linalg import norm
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import scipy.sparse as sparse # type: ignore
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from ..fdmath.operators import deriv_forward, deriv_back, cross
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from ..fdmath import unvec, dx_lists_t, vfdfield_t, vcfdfield_t
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from ..fdmath import vec, unvec, dx_lists_t, vfdfield_t, vcfdfield_t
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from ..eigensolvers import signed_eigensolve, rayleigh_quotient_iteration
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@ -718,6 +718,111 @@ def e_err(
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return float(norm(op) / norm(e))
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def sensitivity(
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e_norm: vcfdfield_t,
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h_norm: vcfdfield_t,
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wavenumber: complex,
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omega: complex,
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dxes: dx_lists_t,
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epsilon: vfdfield_t,
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mu: vfdfield_t | None = None,
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) -> vcfdfield_t:
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r"""
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Given a waveguide structure (`dxes`, `epsilon`, `mu`) and mode fields
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(`e_norm`, `h_norm`, `wavenumber`, `omega`), calculates the sensitivity of the wavenumber
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$\beta$ to changes in the dielectric structure $\epsilon$.
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The output is a vector of the same size as `vec(epsilon)`, with each element specifying the
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sensitivity of `wavenumber` to changes in the corresponding element in `vec(epsilon)`, i.e.
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$$sens_{i} = \frac{\partial\beta}{\partial\epsilon_i}$$
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An adjoint approach is used to calculate the sensitivity; the derivation is provided here:
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Starting with the eigenvalue equation
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$$\beta^2 E_{xy} = A_E E_{xy}$$
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where $A_E$ is the waveguide operator from `operator_e()`, and $E_{xy} = \begin{bmatrix} E_x \\
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E_y \end{bmatrix}$,
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we can differentiate with respect to one of the $\epsilon$ elements (i.e. at one Yee grid point), $\epsilon_i$:
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$$
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(2 \beta) \partial_{\epsilon_i}(\beta) E_{xy} + \beta^2 \partial_{\epsilon_i} E_{xy}
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= \partial_{\epsilon_i}(A_E) E_{xy} + A_E \partial_{\epsilon_i} E_{xy}
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$$
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We then multiply by $H_{yx}^\star = \begin{bmatrix}H_y^\star \\ -H_x^\star \end{bmatrix}$ from the left:
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$$
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(2 \beta) \partial_{\epsilon_i}(\beta) H_{yx}^\star E_{xy} + \beta^2 H_{yx}^\star \partial_{\epsilon_i} E_{xy}
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= H_{yx}^\star \partial_{\epsilon_i}(A_E) E_{xy} + H_{yx}^\star A_E \partial_{\epsilon_i} E_{xy}
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$$
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However, $H_{yx}^\star$ is actually a left-eigenvector of $A_E$. This can be verified by inspecting
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the form of `operator_h` ($A_H$) and comparing its conjugate transpose to `operator_e` ($A_E$). Also, note
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$H_{yx}^\star \cdot E_{xy} = H^\star \times E$ recalls the mode orthogonality relation. See doi:10.5194/ars-9-85-201
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for a similar approach. Therefore,
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$$
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H_{yx}^\star A_E \partial_{\epsilon_i} E_{xy} = \beta^2 H_{yx}^\star \partial_{\epsilon_i} E_{xy}
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$$
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and we can simplify to
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$$
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\partial_{\epsilon_i}(\beta)
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= \frac{1}{2 \beta} \frac{H_{yx}^\star \partial_{\epsilon_i}(A_E) E_{xy} }{H_{yx}^\star E_{xy}}
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$$
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This expression can be quickly calculated for all $i$ by writing out the various terms of
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$\partial_{\epsilon_i} A_E$ and recognizing that the vector-matrix-vector products (i.e. scalars)
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$sens_i = \vec{v}_{left} \partial_{\epsilon_i} (\epsilon_{xyz}) \vec{v}_{right}$, indexed by $i$, can be expressed as
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elementwise multiplications $\vec{sens} = \vec{v}_{left} \star \vec{v}_{right}$
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Args:
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e_norm: Normalized, vectorized E_xyz field for the mode. E.g. as returned by `normalized_fields_e`.
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h_norm: Normalized, vectorized H_xyz field for the mode. E.g. as returned by `normalized_fields_e`.
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wavenumber: Propagation constant for the mode. The z-axis is assumed to be continuous (i.e. without numerical dispersion).
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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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epsilon: Vectorized dielectric constant grid
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mu: Vectorized magnetic permeability grid (default 1 everywhere)
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Returns:
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Sparse matrix representation of the operator.
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"""
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if mu is None:
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mu = numpy.ones_like(epsilon)
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Dfx, Dfy = deriv_forward(dxes[0])
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Dbx, Dby = deriv_back(dxes[1])
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eps_x, eps_y, eps_z = numpy.split(epsilon, 3)
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eps_xy = sparse.diags(numpy.hstack((eps_x, eps_y)))
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eps_z_inv = sparse.diags(1 / eps_z)
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mu_x, mu_y, mu_z = numpy.split(mu, 3)
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mu_yx = sparse.diags(numpy.hstack((mu_y, mu_x)))
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mu_z_inv = sparse.diags(1 / mu_z)
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da_exxhyy = vec(dxes[1][0][:, None] * dxes[0][1][None, :])
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da_eyyhxx = vec(dxes[1][1][None, :] * dxes[0][0][:, None])
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ev_xy = numpy.concatenate(numpy.split(e_norm, 3)[:2]) * numpy.concatenate([da_exxhyy, da_eyyhxx])
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hx, hy, hz = numpy.split(h_norm, 3)
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hv_yx_conj = numpy.conj(numpy.concatenate([hy, -hx]))
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sens_xy1 = (hv_yx_conj @ (omega * omega * mu_yx)) * ev_xy
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sens_xy2 = (hv_yx_conj @ sparse.vstack((Dfx, Dfy)) @ eps_z_inv @ sparse.hstack((Dbx, Dby))) * ev_xy
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sens_z = (hv_yx_conj @ sparse.vstack((Dfx, Dfy)) @ (-eps_z_inv * eps_z_inv)) * (sparse.hstack((Dbx, Dby)) @ eps_xy @ ev_xy)
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norm = hv_yx_conj @ ev_xy
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sens_tot = numpy.concatenate([sens_xy1 + sens_xy2, sens_z]) / (2 * wavenumber * norm)
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return sens_tot
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def solve_modes(
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mode_numbers: list[int],
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omega: complex,
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