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engines/majumdar_lab/package/multilayer_autonomous_optimizer/multilayer_nanobeam_phase1/DERIVATION_PHASE1.md · assembled 2026-07-29 15:57 UTC.

Phase 1 derivation: the stratified reference Green function

Why the old reference problem fails

The confirmed Quan engine treats the uniform beam as a rectangular dielectric in one homogeneous cladding. Fourier transformation along the invariant beam axis then leaves a homogeneous three-dimensional Green symbol. The Majumdar device instead has, from low to high z,

  1. a substrate half-space, n = 1.45;
  2. a 300 nm air source layer containing the localized Si3N4 strip, n = 2.0;
  3. 200 nm anthracene, n = 1.8;
  4. 200 nm PVA, n = 1.5;
  5. an air half-space.

The planar films are invariant in x and y. The correct reference is therefore the planar stack with the Si3N4 beam treated as a localized susceptibility in the 300 nm air layer. Replacing the films by an effective cladding would lose their guided and leaky poles and is not an acceptable basis for Q optimization.

The stack also breaks z reflection. The old z = +1 sector is not closed under the Maxwell operator. Exact x reflection of the symmetric cavity and y reflection of the cross-section survive, so the new projector retains a fixed (x,y) sector while carrying both z parities.

Generalized reflections

For fixed in-plane momentum

\[ k_\parallel^2 = \beta^2 + q_y^2, \]

the vertical wave number in layer j is

\[ k_{zj}=\sqrt{(n_j k_0)^2-k_\parallel^2}, \qquad \operatorname{Im} k_{zj}\ge 0. \]

TE and TM Fresnel reflections are recursively composed through each finite layer. If R_j is the reflection beyond a layer of thickness d_j, then the reflection seen from the preceding medium is

\[ R_{j-1}= \frac{r_{j-1,j}+R_j e^{2 i k_{zj}d_j}} {1+r_{j-1,j}R_j e^{2 i k_{zj}d_j}}. \]

This retains the exact film poles and is holomorphic in thickness and material parameters away from branch points.

Scalar vertical Green kernels

Let the source layer occupy 0 < z < d. The generalized reflections at its bottom and top are r_b and r_t. Solutions satisfying the two outgoing reflection conditions are

\[ u_L(z)=e^{-ik_z z}+r_b e^{ik_z z}, \qquad u_R(z)=e^{ik_z(z-d)}+r_t e^{-ik_z(z-d)}. \]

Their Wronskian is

\[ W=2ik_z\left(e^{-ik_zd}-r_b r_t e^{ik_zd}\right). \]

Therefore the reciprocal scalar kernel normalized by \((\partial_z^2+k_z^2)g=-\delta\) is

\[ g(z,z')=-\frac{u_L(z_<)u_R(z_>)}{W}. \]

Phase 1 implements and tests this expression for TE and TM. Phase 2 must reconstruct the full vector dyadic, project q_y and z on an asymmetric cross-section basis, solve the multilayer uniform-waveguide poles, and then feed that Green operator to the existing longitudinal cavity determinant.