Mathematical conventions

The time dependence is exp(-i omega t). With normalized epsilon_0 = mu_0 = 1, the driven electric field satisfies

[K(rho) - omega^2 M_epsilon(rho)] e = i omega j.

The outer boundary is PEC. The PML uses fixed complex coordinate stretches with positive imaginary stretch for this time convention. Physical outgoing poles therefore have Im(omega_star) < 0, and

Q = -Re(omega_star) / [2 Im(omega_star)].

The emitted power convention is

P = -1/2 Re(j^H e),

and normalized LDOS is the design power divided by a vacuum solve on the same mesh, source, PML, and frequency.

For the real LDOS objective, the implementation solves the conjugate-transpose adjoint and differentiates both the Maxwell operator and the frequency-dependent source analytically. QNM sensitivities use the reciprocal complex-symmetric transpose bilinear form, not a Hermitian energy inner product. All reported residuals are recomputed against the physical, unshifted operator in double precision.

The topology variables are piecewise constant on design tetrahedra. A normalized compact centroid filter precedes a smooth Heaviside projection. No design variables exist in the PML or outside the design mask; the configured emitter exclusion removes intersecting design cells.