Validate chiral circular-Bragg stopband against literature

[garrekstemo]

Summary

Add a validation example/test that reproduces the circular-polarization Bragg stopband of a structurally chiral (helicoidal) medium and compares it against the expected behavior from the literature. Optionally add a graded-index antireflection-coating check against an analytic result. This is the physics-validation capstone of the graded/chiral feature set.

Scope: this issue validates the chiral physics — stopband center/depth and handedness selection. Correctness of the linear→circular basis transform itself is owned by issue #84 (circular-polarization basis output); this issue consumes that output rather than re-testing it.

Context

Depends on the helicoidal/chiral preset constructor issue (which builds a chiral medium on top of the graded-layer piecewise-uniform expansion) and on issue #84 ("Add circular-polarization basis output to transfer()"). The defining signature of a chiral film is that it strongly reflects one circular polarization (matching the structure's handedness) within a wavelength band centered near λ ≈ n̄ · pitch (with n̄ an average in-plane index) and bandwidth set by the in-plane birefringence — while transmitting the opposite handedness. Demonstrating this requires reporting reflectance/transmittance in a circular basis, which is what issue #84 adds; today transfer returns only linear p/s (and cross) terms in TransferResult (src/general_TMM.jl:99-108, calculate_tr at src/general_TMM.jl:460-491).

The project already ships worked examples in docs/src/guide/ (quickstart.md, tutorial.md, validation.md) and validation tests under test/ — this issue follows that pattern.

Why

Unit tests in the preceding split issues confirm the graded/chiral machinery runs and that degenerate cases reduce correctly, but they do not confirm the headline physics. A literature-grounded circular-Bragg check is what proves the implementation is physically correct and gives users a citable, reproducible reference. The graded-index AR-coating check provides a second, analytic cross-validation for non-chiral graded media.

Proposed approach

1. Add a chiral / cholesteric validation example under docs/src/guide/ (e.g. a new section in validation.md or a dedicated page) that constructs a helicoidal medium with the preset, sweeps wavelength at normal incidence, and plots/tabulates reflectance for both circular polarizations using the circular-basis output from issue Add circular-polarization basis output to transfer() #84.
2. Assert the expected qualitative and quantitative signatures: near-unity reflection of the co-handed circular polarization across the Bragg band, near-unity transmission of the cross-handed polarization, the band centered at the predicted λ_Bragg and with the predicted bandwidth from the chosen indices and pitch. Use a sufficiently large nslices (per the convergence-control issue) so the result is converged.
3. Add a corresponding regression test in test/ (new test/graded_validation.jl wired into runtests.jl) checking the stopband center, depth, and that handedness selection flips when the preset's handedness flips.
4. (Optional) Add a graded-index AR-coating example: a smooth index ramp between two media should suppress reflectance over a broad band; compare against the analytic expectation for a known profile (e.g. a quintic/Klopfenstein-style taper) and assert the reflectance is below a stated bound.
5. For non-absorbing test cases, exercise the existing validate=true energy-conservation check (transfer(...; validate=true), _validate_physics at src/general_TMM.jl:606-667) to confirm R + T ≈ 1.

Acceptance criteria

• A chiral/helicoidal example reproduces the circular-polarization Bragg stopband: co-handed reflectance is high across the band, cross-handed reflectance is low, band center matches the literature prediction within a stated tolerance.
• Flipping the preset handedness flips which circular polarization is reflected.
• A regression test in test/ encodes the stopband center/depth and handedness selection.
• (If included) the graded-index AR example reproduces the analytic broadband reflectance suppression within a stated bound.

Caveats / risks

• Re-derive the Bragg-band center and bandwidth from Mackay & Lakhtakia and validate; do not transcribe expected numbers or formulas from any third-party implementation. Cite the exact relations used (e.g. λ_Bragg ≈ n̄·pitch, bandwidth from the in-plane index contrast) so the assertions are auditable.
• Quantitative agreement depends on adequate slicing — ensure nslices is converged (per the convergence-control issue) before tightening tolerances, otherwise a failing assertion may reflect under-resolution rather than a bug.
• This issue is blocked until issue Add circular-polarization basis output to transfer() #84 lands, since the cleanest stopband demonstration is in the circular basis; if Add circular-polarization basis output to transfer() #84 slips, an interim version can show the effect via the linear cross-polarization terms already in TransferResult, but that is harder to interpret.
• Keep CI lean: validation sweeps can be wavelength-heavy. Use a modest grid sufficient to resolve the band, consistent with the project's Ubuntu-only CI policy.

References

• Mackay & Lakhtakia, The Transfer-Matrix Method in Electromagnetics and Optics (2020), DOI 10.1007/978-3-031-02022-3 — circular-Bragg phenomenon of structurally chiral mediums.
• Lakhtakia, "Chiral sculptured thin films," Nature 384, 616 (1996): https://www.nature.com/articles/384616a0
• Depends on the helicoidal/chiral preset constructor split issue and on issue Add circular-polarization basis output to transfer() #84 (circular-polarization basis output).
• Existing validation pattern: docs/src/guide/validation.md, test/, and the validate=true path (src/general_TMM.jl:606-667).