Which orders can propagate?
Predict an order set, change one parameter, and check the result. These are analytical channel directions; their amplitudes and efficiencies require a scattering solution.
Outgoing directions
Equal-length arrows mark directions. They do not represent intensity. Grazing channels run along the interface; closed orders are listed in the table.
Tangential wavevectors
The shaded interval is |kₓ/k₀| < n for that side. A larger index widens it. Every dot shares the same Bloch multiplier.
Order-by-order result
| Order m | kₓ/k₀ | State | Signed angle | Outward κ/k₀ |
|---|
Equations, signs, and limits
Use exp(−iωt), spatial exp(ik·r), incidence from z < 0, and periodicity along x. With qᵢ = nᵢ sin θᵢ, the tangential coordinate is qₘ = qᵢ + mλ₀/Λ. An order is open if |qₘ| < n; equality is grazing, and larger magnitude is evanescent.
The displayed normal root is κ/k₀ = √(n² − qₘ²) in the outward coordinate. In laboratory coordinates kz = +κ for transmission and −κ for reflection. Closed outgoing fields decay in both half-spaces. The dimensionless cutoff tolerance is 10⁻¹⁰ on |qₘ/n| − 1. Closed channels have no real far-field angle.
All possibly open orders are enumerated automatically, with two closed neighbors beyond conservative bounds. Indices are positive and real; the model has kᵧ = 0. It does not classify lossy/anisotropic media or calculate a metasurface's amplitudes, absorption, or efficiency.