Accuracy assessment of scalar wave propagation methods for diffractive optics design: from thin elements to thick binary grating

Abstract: We present a systematic accuracy assessment of the thin-element approximation (TEA), the beam propagation method (BPM), and the wave propagation method (WPM) for binary diffractive gratings, using the rigorous Fourier modal method (FMM) as a reference. Random binary gratings are generated over a range of spatial frequency cutoffs and thicknesses, and the transmitted field overlap between each scalar method and the reference is measured. The results are summarized as accuracy maps in the spatial frequency-thickness parameter space, revealing the domain of validity of each method and providing practical guidelines for the choice of forward model in diffractive optics inverse design pipelines.

• The paper quantitatively compares TEA, BPM, and WPM using a normalized field overlap metric to benchmark modeling accuracy.
• It shows TEA’s limitations for thick gratings, BPM’s moderate performance, and WPM's superior fidelity for thick structures under moderate angles.
• The accuracy maps inform the selection of forward solvers in gradient-based inverse design workflows, enhancing both design efficiency and robustness.

Systematic Assessment of Scalar Wave Propagation Methods for Diffractive Optics Design

Overview

The paper "Accuracy assessment of scalar wave propagation methods for diffractive optics design: from thin elements to thick binary grating" (2605.09470) provides a quantitative evaluation of three scalar wave propagation methods—thin-element approximation (TEA), beam propagation method (BPM), and wave propagation method (WPM)—in modeling binary diffractive gratings. Using the rigorous Fourier modal method (FMM) as a reference, the study delineates their respective performance domains within the spatial frequency–thickness parameter space, thereby establishing practical guidelines for inverse design pipelines in computational optics.

Methodology

The assessment proceeds via systematic generation and simulation of random binary gratings covering a sweep of spatial frequency cutoffs and physical thicknesses. Gratings are delineated on a periodic grid, with variations controlled by the spatial frequency cutoff ($f_c$) and physical thickness ($h$), leading to a comprehensive (fc, h) test suite. For each configuration, the transmitted field calculated by each scalar method is compared to the FMM reference using the normalized field overlap metric. All scalar methods under test are implemented in FluxOptics.jl, leveraging GPU acceleration for scalable differentiation. The FMM reference is computed with FMMAX using a high Fourier truncation order, ensuring stable convergence and comparability.

Key aspects:

• TEA: Models the grating as an infinitely thin phase mask, omitting propagation effects.
• BPM: Slices the grating volume and applies the paraxial approximation; errors accumulate particularly for thick structures and wide-angle propagation.
• WPM: Uses the exact non-paraxial propagator, enabling accurate modeling of thick structures and large diffraction angles.

The study focuses exclusively on transmitted, co-polarized fields, aligning with the scalar approximation used in TEA/BPM/WPM and neglects cross-polarization and reflected components.

Numerical Results

Strong numerical results highlight the distinctions between the three scalar methods:

• TEA exhibits rapid accuracy degradation as thickness increases; at $h/λ = 31$ and moderate angles ($\theta_{max} \sim 16.5^\circ$), the overlap drops to $75.1\%$, and falls below $50\%$ for thick gratings.
• BPM shows improved performance over TEA, achieving overlap values around $89.6\%$ at moderate thickness/angles, but errors still accumulate due to the paraxial approximation.
• WPM achieves highest fidelity, maintaining $97.3\%$ overlap at the same benchmark, accurately reproducing both transmitted intensity and phase. Its domain of high accuracy (overlap $>90\%$) persists across the full explored thickness range for small diffraction angles, with the method outperforming BPM notably for thick structures ($h > 32$) and moderate angles ($h$0).

Accuracy maps in the (fc, h) domain sharply illustrate the boundaries of validity for each method, confirming that:

• TEA is only valid for ultra-thin structures irrespective of diffraction angle.
• BPM is reliable for moderate thicknesses at low angles and for thin gratings at large angles.
• WPM is optimal for thick structures at moderate angles, due to its exact dispersion handling.

Binary gratings present a stringent, conservative benchmark due to their sharp index discontinuities and broad spatial frequency spectrum. The reported accuracy bounds are lower than likely achievable for smoother, continuous-relief DOEs.

Practical and Theoretical Implications

The presented accuracy maps have direct practical relevance for gradient-based inverse design workflows in diffractive optics. Selection of forward models based on these maps enables efficient iterative optimization, avoiding the pitfalls of insufficient accuracy or excessive computational overhead. For thick diffractive structures, particularly under moderate angular regimes, WPM should be favored to mitigate phase error accumulation and ensure convergence to physically realizable designs.

From a theoretical perspective, this work quantitatively delineates the breakdown mechanisms of scalar approximations in the presence of sharp refractive index transitions and volumetric phase evolution. The results highlight the continued utility and limitations of paraxial-based approaches, while corroborating the necessity of non-paraxial methods for high-fidelity modeling.

Looking forward, extension of this systematic benchmarking to include split-step non-paraxial methods (SSNP) is indicated, alongside investigation of back-propagation and vectorial effects neglected by scalar approximations. As inverse design methods increasingly address freeform optics and metasurfaces with complex, high-index structures, the bounds established here will inform the development and selection of forward solvers that balance accuracy and tractability.

Future Developments

Future progress may involve:

• Incorporating SSNP and other advanced propagation methods to address scalar method deficiencies.
• Extending benchmarks to continuous-relief structures and vectorial models to more closely mirror real device physics.
• Integrating these accuracy maps into automated differentiable design frameworks, facilitating adaptive model selection as design progresses.

As computational diffractive optics moves towards more aggressive device miniaturization and wider angular regimes, non-paraxial propagation methods will likely become standard for robust, scalable inverse design.

Conclusion

This paper delivers a rigorous, parameterized accuracy assessment of TEA, BPM, and WPM for binary diffractive gratings, positioning WPM as the preferred method for thick structures at moderate angles in inverse design scenarios. The established accuracy maps provide actionable guidelines for forward model selection, underpinning both practical device design and theoretical understanding of model constraints in wave optics. Extension to more general structures and methods remains a promising direction for enhancing robustness and fidelity in computational optical design workflows.