AR Waveguide k-space Designer

Method: the guided-mode "two-circle" diagram after Levola (J. Soc. Inf. Display 14, 467, 2006) and Kress (Proc. SPIE 11062, 2019): a field is guided iff 1 < ρ ≤ n with ρ = |k_t|/k₀ = n·sin θ, and each diffraction translates the field footprint by λ/Λ along its grating vector. Fields map to k-space by direction cosines (k_x,k_y)/k₀ = (tan θ_x, tan θ_y)/√(1+tan²θ_x+tan²θ_y), so the binding constraint is the diagonal corner field, not sin(H/2)/sin(V/2). Grating vectors satisfy the Levola closure ΣK = 0 (zero net dispersion → achromatic virtual image); the fold period follows Λ_fold = Λ_in/(2·sin(turn/2)). Max-FOV limits use the annulus width (mono: 2·asin((n·sinθ_max−n·sinθ_min)/2)) and the single-set colour bound 2·asin((n·λ_B−λ_R)/(λ_B+λ_R)); bounces use plane-parallel geometry s = 2·t·tan θ, N_p = L_out/s + 1. This is the first-order layout step only: it fixes periods, orientations, FOV feasibility and plate count, and assumes ideal thin gratings, single order m = 1 and a wavelength-flat band. The max-FOV numbers are the conservative radial-annulus-width bounds; real 2D-expansion architectures (dual-channel, butterfly) beat them by routing horizontal and vertical fields azimuthally around the full annulus, at the cost of layout complexity — so treat those ceilings as the single-channel limit, not a hard wall. Diffraction efficiency, uniformity, ghost/rainbow orders and eyebox luminance need a rigorous coupled-wave (RCWA) solve in VirtualLab Fusion, RSoft or Lumerical — seed them with the prescription here.