Extended Data Fig. 1: Optimized holography with inverse-designed, vertically-coupled microcavity arrays.

(a) Silicon L3 slab defect cavity design (hexagonal lattice constant a = 0.4μm; hole radius r/a = 0.25; slab thickness t = 220 nm) with overlaid midplane magnetic field profile H_z after Q optimization by displacing (δx_i, δy_i) and resizing (δr_i) the shaded holes in the \(16a\times 16(\sqrt{3}/2)a\) periodic unit cell. Hole shifts are magnified by 3 × for visualization. The confined cavity mode radiates into the broad far-field profile in (b, background), violating (C1) and yielding a zero-order diffraction efficiency η₀ ≪ 1. As a result, simulated trial holograms (c) from a 64 × 64 cavity array with optimized detunings (Supplement Section D) have minimal overall diffraction efficiency η. Inverse design (b) solves these problems. Guided mode expansion (GME) approximates the mode’s Q and far-field profile by sampling the losses {c} at the array’s diffraction orders (white × s) displaced by Bloch boundary conditions \({\overrightarrow{k}}_{i}\) (that is at the coloured dots). An objective function f that maximizes Q, confines \(\overrightarrow{H}\), and minimizes {c} at any non-zero diffraction order can then be efficiently optimized with respect to all hole parameters using reverse-mode automatic differentiation (b). The resulting devices with high-Q, efficient coupling, and directional emission enable high-performance (η ~ 1) resonant holography (d).