[docs] document scattering conventions and frozen-basis workflows

Update examples for sampled-interface tapers, explicit physical CPML spacing, copied SCPML widths, and the consolidated complex mode solver. Document breaking API changes, overlap measures, and PML convergence.

docs/api/waveguides.md

@@ -2,8 +2,90 @@
 
 ::: meanas.fdfd.waveguide_2d
 
-::: meanas.fdfd.waveguide_2d_complex
-
 ::: meanas.fdfd.waveguide_3d
 
 ::: meanas.fdfd.waveguide_cyl
+
+## Overlap measures and normalization
+
+`waveguide_2d.inner_product(..., measure="physical", conj_h=True)` uses real
+Yee widths for physical Poynting power. `measure="stretched"` keeps complex
+widths for coordinate reaction. `lorentz_metric` and `lorentz_overlap` default
+to the stretched measure. Use the same grid and measure when constructing and
+using dual fields. A real overlap grid still defines a consistent constructed
+dual basis, even when the eigenproblem uses a separate PML grid.
+
+`normalized_fields_e/h(..., normalization="lorentz")` normalizes unconjugated
+self-reaction to one. This fixes phase up to a sign and is distinct from unit
+forward power, which remains the default. Zero-power or self-orthogonal modes
+raise `ValueError`; open self-orthogonal modes require paired primal/dual
+normalization through `ModeBasis`.
+
+## Targeted and reproducible mode solving
+
+For modes near an effective index guess, use `solve_modes_near` with
+`target_wavenumbers=[omega * n_guess]`. It targets the electric beta-squared
+operator at `(omega * n_guess)**2`; `solve_modes_near_beta` instead targets the
+first-order signed-beta operator. Both use the LU shift-invert backend and
+support `eigensolver_opts={"v0": starting_vector}`. Factorization cost and
+memory depend on the grid, so this is not universally faster than indexed
+solving. `solve_modes` keeps its global mode-number semantics and now accepts
+`rng` or `v0` for reproducible initialization.
+
+`waveguide_3d.mode_from_2d` accepts a supplied transverse electric eigenvector
+and reuses field reconstruction, normalization, numerical-dispersion
+correction, Yee staggering, rotation, and full-grid embedding without solving
+an eigenproblem.
+
+## Interface and sampled-taper scattering
+
+All scattering matrices use outgoing rows and incoming columns, ordered
+`[left, right]`:
+
+```text
+[b_left ]   [R_left        T_right_to_left] [a_left ]
+[b_right] = [T_left_to_right       R_right] [a_right]
+```
+
+`get_s` and `get_abcd` require explicit `dxes`. Mixed Petrov-Galerkin testing is
+the default: left electric duals test H continuity, and right magnetic duals
+test E continuity. Both incidence directions are solved together with full
+projection matrices. Mixed scattering supports unequal mode counts.
+`formulation="galerkin"` selects equal-count left testing of both equations.
+Neither formulation silently clips transmission. The optional `force_nogain`
+and `force_reciprocal` flags are numerical projections, not general physical
+tests for arbitrary open or evanescent modal coordinates.
+
+`get_taper_s` replaces finite-difference CMT with half propagation in the left
+basis, a midpoint interface, and half propagation in the right basis for each
+sampled interval. Redheffer composition avoids growing passive backward
+transfer branches. Refine section sampling and modal/PML bases to establish
+convergence. The ineffective `rtol`, `atol`, and `max_step` arguments have been
+removed. The existing `eme_taper_cmt.py` example filename is retained.
+
+```python
+prepared = eme.prepare_taper(sections, dxes)
+s_original = eme.evaluate_taper_s(prepared)
+s_longer = eme.evaluate_taper_s(prepared, z_coords=prepared.z_coords * 1.2)
+s_changed_beta = eme.evaluate_taper_s(
+    prepared, wavenumber_matrices=beta_matrices,
+)
+```
+
+`PreparedTaper` owns read-only copies of phase-aligned interface data. Beta
+matrix overrides have shape `(section, mode, mode)` in the original supplied
+section bases; recorded basis transforms are applied automatically. Explicit
+backward branches require backward beta overrides when forward beta is changed.
+Evaluation performs no field overlaps or eigensolves. Length sweeps reuse the
+sampled model; wavelength sweeps with reused fields are explicitly frozen-basis
+approximations. Reprepare when fields, duals, grids, or formulations change.
+
+`cascade_s(left, right, internal_modes=...)` connects spatially ordered S
+matrices, including unequal external ports. Public `abcd_to_s` and `s_to_abcd`
+use `[f_right, b_right] = ABCD @ [f_left, b_left]` and require equal port counts.
+`get_taper_abcd` converts the stable S result; zero transmission or an
+unrepresentable transfer matrix raises `LinAlgError`. Prefer S for long lossy
+devices. Singular interface and cascade systems also raise contextual errors
+rather than silently substituting pseudoinverses.
+
+::: meanas.fdfd.eme