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engines/design23_v1/package/design23_recentered_optimizer/fryett_q_series_phase2/STATUS.md · assembled 2026-07-29 15:57 UTC.

Phase-2 numerical status¶

Date: 2026-07-16

Internal identities¶

Nine tests pass:

the homogeneous dyadic symbol exactly inverts the Fourier Maxwell operator;
normalized Legendre transforms have the correct zero-momentum limits;
tangent-map and panelized momentum rules integrate smooth reference functions correctly;
the guided-sector Galerkin matrix is Hermitian;
the Dyson-resummed Green matrix satisfies its defining equation;
the explicit inverse-beta quadrature has the expected algebra;
the projector and combined forms of the homogeneous Green symbol agree;
the outgoing-minus-incoming radiation spectral density is Hermitian positive semidefinite.

These tests validate the operator algebra. They do not validate the physical accuracy of a truncated waveguide or cavity calculation.

Initial guided-pole sequence¶

For the unperforated Fryett beam at 740 nm,

\[ W=450\ \mathrm{nm},\quad T=330\ \mathrm{nm},\quad n_b=2.0,\quad n_c=1.47, \]

the tangent-map Galerkin calculation gives:

Max Legendre degrees `(Ny,Nz)`	Momentum order	Matrix size	TE-like `n_eff`	TM-like `n_eff`
(1,1)	18	12	1.715947	1.700470
(2,2)	24	27	1.749542	1.728640
(3,3)	30	48	1.748922	1.727407

At the last level, the TE-like branch has component weights

\[ (|E_x|^2,|E_y|^2,|E_z|^2) \approx(0.0740,0.9250,0.0010), \]

and the TM-like branch has

\[ (|E_x|^2,|E_y|^2,|E_z|^2) \approx(0.0974,0.0021,0.9005). \]

The Fryett cavity field reported in the paper is E_y-like, so the first branch is the background guided branch relevant to the cavity.

Two weakly guided higher branches also appear close to the cladding light line, near n_eff=1.49. They are not yet accepted because branch tracking becomes delicate at the continuum threshold.

Explicit high-momentum cutoff sweep¶

For the (1,1) basis, a panelized transverse-momentum calculation gives:

Dimensionless cutoff `qL/2`	Nodes per transverse axis	TE-like `n_eff`	TM-like `n_eff`
8	64	1.721013	1.704457
12	96	1.718196	1.701638
16	128	1.716897	1.700382
24	192	1.715030	1.698605

This approaches the same neighborhood as the tangent-map (1,1) result, but the tail is still not converged at cutoff 24. The longitudinal projector in the Maxwell Green tensor produces slow high-momentum convergence. This is now a visible numerical axis rather than an unmeasured PML error.

First explicit radiation-continuum matrix¶

At beta=0.5 k_cladding, the constant-vector (0,0) cross-section basis gives the following eigenvalues of the Dyson-resummed radiation spectral density:

Radial cutoff	Angular order	Spectral-density eigenvalues	Trace
8	16	(228.82, 333.91, 409.23)	971.96
12	24	(226.53, 331.78, 409.49)	967.80
16	32	(225.50, 330.61, 409.55)	965.65
24	48	(224.26, 329.54, 409.59)	963.39

The absolute units depend on the normalized basis and length units. The important facts at this stage are that all eigenvalues are positive, the on-shell imaginary term is obtained analytically rather than from artificial absorption, and the result approaches a stable limit as the explicit momentum cutoff and angular order grow.

Claims permitted now¶

We have a working vector spectral-Galerkin representation of the bare Fryett waveguide Green tensor in both the nonsingular guided sector and at real frequencies inside the radiation cone. It finds physically classified TE-like and TM-like guided poles and a positive radiation spectral density without a transverse box, PML, or assigned radiation linewidth.

We do not yet have:

an accepted numerical effective index;
the analytically continued outgoing radiation contour;
the hole-to-waveguide Green matrix;
a cavity pole or cavity Q.

Immediate next implementation¶

The next layer is the beta-plane deformation:

locate and track the guided poles by symmetry and eigenvector overlap;
isolate their residues analytically;
parameterize the two radiation branch cuts starting at beta=+/- k_cladding;
continue the contour to Im(omega)<0 without crossing a pole or cut;
project the phase-1 elliptical-hole basis through that contour.