[11/13][Adjoint Module] Subpixel-smooth dispersive materials and differentiate their shape - #3293
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`structure_chunk::set_chi1inv` volume-averages the instantaneous permittivity while `add_susceptibility` point-samples sigma, so a dispersive material switches on a whole pixel at a time as an object moves. That is a staircase, and a staircase has no derivative -- the same wall the geometry gradients hit, reached through the material rather than the shape. Add `material_function::eff_sigma_row`, standing to `sigma_row` as `eff_chi1inv_row` stands to `chi1p1`, and call it from the two sigma loops with the pixel volume the chi1inv loop already uses. The default implementation samples the volume centre, so this is behaviour-preserving on its own: nothing overrides it yet. The sigma loops now index in ivec like the chi1inv loop above them, since the half-pixel offdiagonal shift and the smoothing diameter have to agree between the two for the eventual averaging to be consistent. Verified neutral: a Drude MaterialGrid's transmittance is unchanged bit for bit, and test_adjoint_dispersion, test_material_grid, test_material_dispersion, test_dispersive_eigenmode, test_faraday_rotation, test_multilevel_atom, test_geometry_gradient and test_conductivity all pass.
Taper each medium's pole amplitude by the pixel's filling fraction, using the
same front object, shift and fill that eff_chi1inv_matrix uses, so the
permittivity and the susceptibility describe the same interface.
Swept sub-pixel across one pixel, a Drude slab whose eps_infinity matches the
background -- so that the only position dependence there is comes from sigma:
point sampled constant to the last digit at 10 of 11 offsets, then a
single 20% jump
averaged smooth, symmetric about the pixel centre, and 60x less
total variation across the sweep
Linear in fill rather than Kottke, deliberately. Kottke's normal component is
a harmonic average of a permittivity; there is no analogous exact result for a
resonant amplitude, and a harmonic average against a medium that lacks the pole
would be zero in every boundary pixel. Scaling each medium's whole sigma
tensor by a non-negative weight and adding keeps the result positive
semidefinite, which is what passivity depends on, and makes the amplitude
exactly linear in fill, so the shape derivative is a constant rather than
something to be differenced.
Multilevel atoms and gyrotropic media are excluded and stay point sampled. A
multilevel atom carries per-pixel level populations with their own nonlinear
dynamics, so a partially filled pixel is not a weaker gain medium -- without
this exclusion test_multilevel_atom fails. A gyrotropic sigma is
antisymmetric, so the positive-semidefiniteness argument does not hold for it
either. Material grids and user materials also stay point sampled: their
material already varies continuously in space, and mixing a level set with a
geometric filling fraction is a separate question.
The pole lookup is factored out of sigma_row into material_sigma_row so both
paths ask the same question of a material.
`geometry_addgradient` contracted only the real, non-dispersive `eff_chi1inv_matrix`, so a Drude or Lorentz object took a branch that dropped its poles entirely: the gradient described a structure the forward solve never ran. That is the piece a metal reflector's position needs. Add `eff_chi1inv_row_disp_fill`, the dispersive counterpart of the `fill_override` on `eff_chi1inv_matrix`. `eff_chi1inv_row_disp` cannot serve, because it resolves the material at a point and so knows nothing about a filling fraction. The construction matches what the forward solve now runs: the instantaneous part is Kottke-smoothed, and each pole's amplitude is taken linearly in fill from whichever medium carries it, exactly as `eff_sigma_row` writes into the sigma arrays. Since chi1 is linear in sigma, a pole contributes `lineshape(freq) * weight * sigma` and the two media's pole lists never have to be matched against each other -- the same reason `epsilon_material_grid` can concatenate rather than blend. The branch is chosen from the materials at *this* interface rather than from `md->trivial` of the front object. That distinction matters: `trivial` is a property of a material, but under mixing it becomes a property of a pixel, so a plain dielectric sitting in front of a metal would otherwise take the non-dispersive branch and drop the whole contribution at precisely the boundary pixels that carry the derivative. `interface_fill` now hands back both materials, since the caller needs them to build the mixture. Against a central difference of the objective, at resolution 20: dielectric (control) 0.139% Drude metal 0.017% both inside the 2% the existing shape-derivative tests use, which is set by the O(dx) accuracy of a shape derivative rather than by anything dispersive.
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Stacked on #3287.
Second-order subpixel smoothing for dispersive and conductive materials, and the matching shape derivative, which is what a metal reflector's position needs.
geometry_addgradientcontracted only the real non-dispersive tensor, so a Drude block's poles contributed nothing and the gradient described a structure that was never simulated.Agrees with a finite difference to 0.017%. The union-of-poles construction is preserved throughout -- each medium's sigma is scaled by its own weight and poles are never blended, which keeps the tensor positive semidefinite and avoids the non-passive
[[0,s],[s,0]]form behind #666 / #3229.