Historical derivation — preserved in full

Source: engines/quan_loncar_v1/lineage/phases_01_28/quan_loncar_complete_lineage_phases_01_28/quan_loncar_series_phase22/DERIVATION_PHASE20_22.md
Snapshot: Quan–Lončar complete lineage, phases 1–28. The body below is unabridged.

Longitudinal extraction and the surviving Q series

1. Pole condition

For every finite hole and waveguide basis, the cavity pole is a zero of

A_N(omega) = I - k0^2 Delta-epsilon G_holes,N(omega).

The quality factor uses the exp(-i omega t) convention,

Q_N = -Re(omega_N)/(2 Im(omega_N)), with Im(omega_N) < 0.

The target is the dielectric-centered Ey-like fundamental in exact mirror sector (+1,-1,+1). Its retained hole coefficients are approximately 98.16% Ey throughout the final sequence.

2. Why the old transverse series was slow

For k=(beta,q_y,q_z), the homogeneous Maxwell Green symbol is

G_c(k) = P_T/(k.k-k_c^2) - P_L/k_c^2,

where P_L = k k^T/(k.k) and P_T = I-P_L. The second term does not acquire the extra high-momentum denominator carried by the transverse term. A square cutoff therefore approximates a noncompact longitudinal/self contribution at the same time that it tries to resolve the radiation shell. A very small cavity linewidth becomes the difference of much larger cutoff-dependent matrix contributions.

3. Exact q_z reduction for the current basis

The retained waveguide basis is constant through thickness T, so

Phi_0(q_z) = sqrt(T) sinc(q_z T/2).

Put A = beta^2 + q_y^2 and choose a = sqrt(A) on the branch with nonnegative real part. Then

I0(A) = integral dq_z/(2pi) |Phi_0|^2/(A+q_z^2)

is exactly

I0(A) = 1/A - (1-exp(-aT))/(T a^3),

and

I2(A) = integral dq_z/(2pi) |Phi_0|^2 q_z^2/(A+q_z^2)

is exactly

I2(A) = (1-exp(-aT))/(T a).

Consequently the rapidly convergent blocks of the projected longitudinal operator reduce to one-dimensional q_y integrals:

• P_xx: beta^2 I0;
• P_xy=P_yx: beta q_y I0;
• P_zz: I2.

The only slow diagonal block is then supplied without a cutoff:

P_yy = I - P_xx - P_zz.

This is an operator identity, not a fitted tail model.

4. Regularized finite sequence

Let L_N(beta) be the extracted full longitudinal projector and let T_K,r(beta,omega) be the panel approximation to the compact transverse term. The bare-cladding matrix is evaluated as

G_c,N,K,r = T_K,r - L_N/k_c^2.

The waveguide Dyson step and the longitudinal beta-contour integral are then unchanged. The complete current approximant is therefore

A_(N,K,r,b)(omega) = I - k0^2 Delta-epsilon

* integral_Cbeta d beta/(2pi) C_N(-beta)^T

* [I-k0^2 chi_b G_c,N,K,r(beta)]^-1

* G_c,N,K,r(beta) C_N(beta).

The pole series is the sequence of zeros of this matrix. Phase 22 shows that the longitudinal part is controlled, but the Cartesian realization of T_K,r is not efficient enough for the desired linewidth.

5. Required next representation

For each beta-contour node define kappa^2 = k_c^2-beta^2. In polar transverse momentum the compact term has radial denominator q^2-kappa^2. The next rule must subtract its on-shell numerator before radial quadrature and add the logarithmic shell contribution analytically. This is the complex-frequency continuation of the real-axis shell extraction already derived in Phase 2.

Only the compact transverse evaluation changes. The nonlinear determinant, symmetry projector, physical branch template, longitudinal extraction, and geometry dependence all remain part of the convergent series.