[7/n][Adjoint Module] Evaluate the dispersive adjoint operator at the discrete lineshape - #3287
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`get_chi1_tensor_disp` built the dispersive permittivity from the continuum
model -- `lorentzian_susceptibility::chi1` and `1 + i*sigma/(2*pi*f)` -- while
the FDTD timesteps a discrete recurrence. The adjoint gradient was therefore
the exact derivative of an operator slightly different from the one being
simulated, and disagreed with a finite difference of the discrete objective at
`O((freq*dt)^2)`.
Matching the recurrences that are actually run:
Conductivity. The D update is
D' = ((1 - sigma dt/2) D + dcurl) / (1 + sigma dt/2)
(step_db.cpp, condinv from structure.cpp). With D ~ exp(-i w t), dividing
through by exp(-i w dt/2) gives
D [ -(2/dt) sin(w dt/2) i + sigma cos(w dt/2) ] = curl
against the continuum D [-i w + sigma] = curl, so the permittivity factor
1 + i sigma / w becomes 1 + i sigma (dt/2) cot(pi freq dt).
Lorentzian. From update_P(),
P'(1 + g dt/2) = P (2 - w0^2 dt^2) - P_prev (1 - g dt/2) + w0^2 dt^2 sigma E
the same substitution replaces freq^2 by [sin(pi freq dt)/(pi dt)]^2 and
gamma*freq by gamma sin(2 pi freq dt)/(2 pi dt), leaving the numerator alone.
Drude is the no_omega_0_denominator case and drops the resonance term.
Both reduce to the continuum expressions as dt -> 0. Relative disagreement
between the adjoint gradient and a central difference of the objective, at
resolution 20 / 40 / 80:
D-conductivity 0.455% 0.107% 0.027% -> 0.001% 0.000% 0.001%
Drude pole 0.232% 0.058% 0.015% -> 0.000% 0.000% 0.000%
The 4x-per-doubling on the left is the second-order signature being removed;
the right-hand columns sit at the same roundoff floor as a non-dispersive
structure.
The discrete lineshape is a separate `chi1_discrete` rather than a change to
`chi1`, because reporting material properties to the user is the opposite
case: `get_epsilon(frequency=...)` must keep describing the material, not the
discretization, or it would return resolution-dependent numbers for a
resolution-independent medium. A test pins that. The base implementation
falls back to `chi1`, so gyrotropic and multilevel susceptibilities -- which
derive from `susceptibility` directly and have their own update equations --
are unaffected.
`dt` reaches the material model through `geom_epsilon`, alongside the `tol` and
`maxeval` already stashed there for the gradient calculation; zero means "not
known" and selects the continuum form. The copy constructor was dropping all
of these, which would not fail so much as quietly compute against a different
operator than the original was built with.
The coarse-resolution test is the sharpest of the four: the error is largest
where dt is largest, so a formula that were merely closer than the continuum
one would still converge and still pass at resolution 20.
smartalecH
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