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engines/design23_v1/package/design23_recentered_optimizer/multilayer_nanobeam_phase8/DERIVATION_PHASE8.md · assembled 2026-07-29 15:57 UTC.

Exact 50-hole feed projection

The supplied design has 25 independent positive-side holes. Its packed geometry is

p = (g_0,...,g_24, rx_0,...,rx_24, ry_0,...,ry_24),

where g_0 is the full central separation and subsequent g_j are positive center-to-center gaps. Reflection creates the 25 negative-side holes, leaving 75 differentiable physical coordinates.

The FDTD builder's actual ellipse convention is retained: rx=47 nm is the longitudinal semi-axis and ry=83...143.2 nm is the tapered transverse semi-axis.

For the target Ey-like cavity branch, the physical mirror sector is x=+1, y=-1. No z eigenvalue is imposed. With a constant disk mode and z degrees 0 and 1, each positive hole therefore retains two Ey coefficients and the reduced determinant has dimension 50. A three-axis projector would discard half of these coefficients and is invalid for the film stack.

The feed-pole Fourier integral is restored exactly in real space. For two different holes the +beta_f residue propagates when x>x', and the -beta_f residue propagates when x<x'. Inside the same finite ellipse, disk quadrature nodes are sorted by local x and cumulative sums impose the same ordering. Coincident nodes receive the symmetric Heaviside value ½. This finite-hole self term changes the benchmark guided matrix by about 8%, so a center-point approximation is not adequate.