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Global Self-Attention with Exact Fourier Propagation for Phase-Only Far-Field Holography

Dilawer Singh, Antoni J. Wojcik, Timothy D. Wilkinson

TL;DR

This work tackles phase-only hologram synthesis in the Fraunhofer regime, where far-field propagation is a global Fourier transform and local phase updates can influence the entire reconstruction. It introduces a physics-in-the-loop framework that trains a transformer-based generator end-to-end with exact FFT-based propagation, outputting a unit-modulus phase field from a target intensity I^T. The approach demonstrates stable training, good generalization to unseen digits and hand-drawn targets, and scalability to higher resolutions via a controlled coarse-to-fine tokenization strategy, suggesting transformers are well-suited for nonlocal diffraction mappings. Practically, it enables single-pass hologram generation after training, offering fast, physics-consistent phase synthesis with potential hardware calibration extensions.

Abstract

Phase-only computer-generated holography (CGH) seeks a phase pattern for a spatial light modulator (SLM) whose propagated optical field reproduces a desired intensity distribution. In the far-field (Fraunhofer) regime, optical propagation reduces to a Fourier transform, such that each hologram pixel contributes to the entire reconstructed intensity distribution. When restricted to phase-only modulation, intensity must be shaped through global phase interference effects, making the inverse mapping from target intensity to phase highly non-linear and sensitive to local minima. We present a proof-of-concept physics-in-the-loop approach in which a transformer maps a target intensity image to a phase-only SLM field and is trained end-to-end through exact FFT-based propagation embedded directly within optimization. We further observe that patch tokenization strongly shapes the optimization geometry: coarse tokenization acts as an implicit spectral regularizer that stabilizes training and suppresses checkerboard-like attractors, while finer tokenization increases spatial degrees of freedom but benefits from curriculum or hierarchical refinement. Despite training on limited primitives and a single digit class (only digit 6), the learned generator exhibits out-of-distribution (OOD) generalization to unseen digits and hand-drawn target patterns. These results suggest that transformer architectures, whose self-attention enables global token interactions, are a natural fit for far-field holography and provide a viable foundation for scalable physics-grounded hologram generation.

Global Self-Attention with Exact Fourier Propagation for Phase-Only Far-Field Holography

TL;DR

This work tackles phase-only hologram synthesis in the Fraunhofer regime, where far-field propagation is a global Fourier transform and local phase updates can influence the entire reconstruction. It introduces a physics-in-the-loop framework that trains a transformer-based generator end-to-end with exact FFT-based propagation, outputting a unit-modulus phase field from a target intensity I^T. The approach demonstrates stable training, good generalization to unseen digits and hand-drawn targets, and scalability to higher resolutions via a controlled coarse-to-fine tokenization strategy, suggesting transformers are well-suited for nonlocal diffraction mappings. Practically, it enables single-pass hologram generation after training, offering fast, physics-consistent phase synthesis with potential hardware calibration extensions.

Abstract

Phase-only computer-generated holography (CGH) seeks a phase pattern for a spatial light modulator (SLM) whose propagated optical field reproduces a desired intensity distribution. In the far-field (Fraunhofer) regime, optical propagation reduces to a Fourier transform, such that each hologram pixel contributes to the entire reconstructed intensity distribution. When restricted to phase-only modulation, intensity must be shaped through global phase interference effects, making the inverse mapping from target intensity to phase highly non-linear and sensitive to local minima. We present a proof-of-concept physics-in-the-loop approach in which a transformer maps a target intensity image to a phase-only SLM field and is trained end-to-end through exact FFT-based propagation embedded directly within optimization. We further observe that patch tokenization strongly shapes the optimization geometry: coarse tokenization acts as an implicit spectral regularizer that stabilizes training and suppresses checkerboard-like attractors, while finer tokenization increases spatial degrees of freedom but benefits from curriculum or hierarchical refinement. Despite training on limited primitives and a single digit class (only digit 6), the learned generator exhibits out-of-distribution (OOD) generalization to unseen digits and hand-drawn target patterns. These results suggest that transformer architectures, whose self-attention enables global token interactions, are a natural fit for far-field holography and provide a viable foundation for scalable physics-grounded hologram generation.
Paper Structure (14 sections, 8 equations, 4 figures, 4 tables)

This paper contains 14 sections, 8 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Methods
  3. Physics-in-the-Loop Far-Field Generation
  4. Attention-Based Phase Generator
  5. Loss Function and Energy Scaling
  6. Experimental Overview
  7. Training Data and Curriculum
  8. Optimization and Hyperparameters
  9. Results
  10. Quantitative Results
  11. Qualitative Results
  12. Generalization to Custom Hand-Drawn Targets
  13. High-Resolution Single-Target Demonstration
  14. Discussion and Conclusion

Figures (4)

  • Figure 1: OOD MNIST digits: one example per class. Top: target intensity. Bottom: predicted far-field intensity obtained from the phase-only hologram produced by the model and propagated via FFT-based Fraunhofer diffraction.
  • Figure 2: Training loss (MSE) versus epoch for the final curriculum stage (1000 epochs).
  • Figure 3: Generalization to unseen $28\times28$ custom targets. Each subfigure (a)-(d) corresponds to a different target. Within each subfigure (left to right): predicted phase hologram [-$\pi$,$\pi$], predicted far-field intensity, and target intensity.
  • Figure 4: Optimization progression under coarse-to-refinement training. For each panel (left to right): (a) target intensity, (b) initialization (MSE=8.2567), (c) end of coarse training (MSE=0.4803), and (d) end of refinement (MSE=0.1032). Each image in (b)-(d) shows the predicted far-field from the model's predicted phase hologram. Progressive emergence of coherent structure demonstrates stable physics-constrained optimization.