Extensions of the Multiple Slit Diffraction Model: Nonlinear and Near-Field Effects

Numerical Simulation Techniques for Multiple Slit Diffraction Models

Overview

Numerical simulation of multiple-slit diffraction computes the propagated electromagnetic field (intensity and phase) for arbitrary slit geometries, illumination, and propagation distances. Common goals: model near-field (Fresnel) and far-field (Fraunhofer) patterns, include material/surface effects, and handle subwavelength features or vectorial fields.

Main numerical methods (when to use)

• Fourier optics / FFT-based Fraunhofer — fast for far-field patterns and large-scale parameter sweeps; treat aperture plane as complex transmission function and compute far-field via FFT of aperture field. Use when z ≫ (slit size)^2/λ.
• Fresnel (convolution) integral via FFT — efficient for near-to-intermediate propagation distances (paraxial). Implement using angular spectrum or Fresnel transfer function with padding to avoid wrap-around.
• Angular Spectrum Method (ASM) — exact within scalar, homogeneous-medium model; good for intermediate and near fields and for oblique incidence. Compute spatial Fourier transform of aperture field, multiply by propagation phase factor exp(i k_z z), inverse FFT.
• Beam Propagation Method (BPM) — split-step (paraxial) propagation with refractive-index variations; use for waveguides or slowly varying media profiles (not accurate for large-angle scattering).
• Finite-Difference Time-Domain (FDTD) — full Maxwell-vectorial, time-domain solver; use for subwavelength slits, metallic films, plasmonic effects, and when polarization/dispersion/near-field details matter. Computationally intensive.
• Finite Element Method (FEM) — frequency-domain vector solver for arbitrary geometries, material dispersion, and boundary conditions; suitable for high-accuracy results in complex structures.
• Method of Moments / Boundary Integral Methods — efficient for open-region scattering and thin-screen problems; reduces problem to integrals over apertures/edges.

Practical implementation steps (reasonable defaults)

1. Represent slit geometry and complex aperture field E(x,y,0) (uniform plane wave or Gaussian illumination).
2. Choose model: scalar FFT/Fresnel/ASM for λ ≲ feature sizes; vectorial (FDTD/FEM) if feature sizes ≲ λ or metals present.
3. Discretize with sampling Δx satisfying Nyquist for highest spatial frequency: Δx ≤ min(λ/ (2 NA), slit_width/10). Ensure array size large enough to capture diffraction envelope (zero-padding).
4. For FFT methods, apply windowing/apodization to reduce edge artifacts and use adequate zero-padding to avoid aliasing.
5. Implement propagation:
   • Fraunhofer: far-field pattern ∝ |FFT{E(x)}|^2 (apply correct k-scaling).
   • Fresnel (transfer function): FFT aperture → multiply by H(fx) = exp(-iπλz f_x^2) → inverse FFT.
   • ASM: FFT aperture → multiply by exp(i k_z z) with k_z = sqrt(k^2 – (2π f)^2) → inverse FFT.
6. Post-process intensity, phase, and compute metrics (fringe visibility, order positions). For multi-slit arrays include coherent sum across slits with relative phase/offsets.
7. Validate: compare Fresnel → Fraunhofer limits and, when possible, against analytical formulas (single/double-slit) or convergence study vs. grid refinement.

Numerical considerations & tips

• Use double precision for FDTD/FEM; single/double both fine for FFT methods unless extreme dynamic range needed.
• Boundary conditions: use perfectly matched layers (PML) for FDTD/FEM to absorb outgoing waves.
• For periodic multi-slit arrays consider Bloch/ Floquet boundary conditions or compute a unit cell with phase ramp.
• GPU-accelerated FFTs and FDTD libraries (e.g., MEEP, Lumerical, OpenEMS) speed large simulations.
• Sampling in k-space for ASM: avoid evanescent truncation if near-field evanescent waves matter (include |f| > k/2π components).
• When modeling partially coherent sources, propagate mutual intensity or use Monte Carlo ensemble of random-phase realizations.

Common pitfalls

• Insufficient padding → aliasing/wrap-around artifacts.
• Grid too coarse → missed high-frequency fringes.
• Using paraxial BPM outside its validity (large-angle diffraction).
• Ignoring vector effects for slit widths comparable to λ (causes polarization-dependent results).

Recommended tools & libraries

• FFT: NumPy/FFTW, cuFFT (GPU)
• Angular spectrum / Fresnel: custom FFT code or MATLAB/NumPy implementations
• FDTD: MEEP (open-source), Lumerical FDTD
• FEM: COMSOL, ANSYS HFSS, open-source GetFEM / FreeFEM (for optics, use vector EM modules)
• Optical simulation packages: LightPipes (Python/Matlab) for scalar diffraction experiments

If you want, I can generate a ready-to-run example (Python + NumPy) for: (A) Fraunhofer FFT multi-slit, (B) Angular spectrum propagation, or © a comparison table of accuracy vs. cost for these methods.