HoloFNO: A Physics-Supervised Neural Operator for Low-Latency Holographic Phase Synthesis

Fast synthesis of phase-only holograms is critical for programmable optical trapping, for which classic iterative phase-retrieval algorithms need repeated forward-backward propagation and constraint-projection steps. We formulate holographic phase synthesis as a physics-supervised inverse-design problem: the network predicts a spatial light modulator (or SLM) phase mask and is optimized through a differentiable Fourier-optics model against the reconstructed target-plane intensity, without requiring Gerchberg-Saxton (GS) generated phase labels. We investigate this problem at a fixed native discretization resolution of $128\times 128$ pixels across both the SLM aperture and the focal plane, and introduce HoloFNO, a Fourier neural operator augmented with a harmonic-oscillator-inspired multi-frequency sinusoidal bypass. To uncover the internal representation mechanics and validate that the operator learns genuine diffraction physics rather than superficial heuristics, we perform fixed-resolution mechanistic evaluations consisting of branch-scrubbing interventions, low-order Zernike-residual decomposition, and mechanistic diagnostics across 3 independent training runs and 5 shared evaluation draws per run. Diagnostic probes demonstrate that the selected topology exhibits a reproducible phase content. At its 60-iteration quality-matched point, HoloFNO leads GS on reconstruction fidelity within the region of interest, quantified as the relative root-mean-square error (Rel-RMSE) between the reconstructed and ideal target intensity patterns ($Rel-RMS{E}_{ROI}=0.1621\pm 0.0253$ vs. $0.1494\pm 0.0213$ for HoloFNO), while HoloFNO's fixed one-shot cost eliminates the per-sample runtime variability inherent to GS's convergence-dependent iteration count.