Investigate periodically corrugated (grating) interfaces (large / exploratory)

[garrekstemo]

Summary

(Large / exploratory) Investigate support for periodically corrugated (grating) interfaces — diffraction into multiple orders — extending the method beyond planar interfaces. This is the subject of a full chapter of the Mackay & Lakhtakia (2020) treatment and is currently out of scope for the package.

Context

TransferMatrix.jl (and the planar TMM formalism in general) assumes laterally homogeneous, planar interfaces: the in-plane wavevector ξ = √ε_inc sin θ is conserved and there is a single reflected/transmitted beam per polarization. Real corrugated/grating structures couple light into multiple diffraction orders, which the planar formalism cannot represent.

Why

Mackay & Lakhtakia (2020) devote a chapter to extending the transfer-matrix method to periodically corrugated interfaces, handling diffraction by expanding fields in spatial harmonics (a rigorous coupled-wave / C-method-style treatment) while retaining the 4×4 bianisotropic layer machinery. Supporting this would let the package model diffraction gratings, photonic surfaces, and grating couplers — a substantial capability expansion.

Proposed approach (scoping only at this stage)

1. Study the M&L corrugated-interface chapter and the relations between this approach and rigorous coupled-wave analysis (RCWA) / the differential (C) method.
2. Assess fit: this requires a Fourier/harmonic field expansion and per-order bookkeeping that does not reuse the current single-ξ pipeline directly — it is closer to a new solver than an incremental feature.
3. Decide: implement here behind a clearly separate API, defer to interop with an existing RCWA package, or mark as explicitly out of scope.

Caveats / risks

• Major undertaking — effectively a second solver (RCWA-class), with its own convergence (number of retained orders), stability, and validation concerns. This issue is a placeholder for scoping/decision, not a near-term implementation.
• Lowest priority; listed for completeness so the capability is tracked rather than forgotten.

References

• Mackay & Lakhtakia, The Transfer-Matrix Method in Electromagnetics and Optics (2020), DOI 10.1007/978-3-031-02022-3 — chapter on periodically corrugated interfaces.
• Background: rigorous coupled-wave analysis (RCWA) and the differential/C-method for gratings.