Analytical Fresnel Treatment of Double-Slit Diffraction with Multiple Coherent Waves

TL;DR

This paper develops an analytic-numeric framework for double-slit diffraction under coherent multi-beam illumination, enforcing an edge-zero boundary condition to obtain compact closed-form Fresnel expressions in terms of the Fresnel integrals $C(u)$ and $S(u)$. Starting from a three-beam setup and extending to an arbitrary number of beams, the authors show how to tailor the aperture’s angular spectrum and derive unified expressions for the resulting field, while incorporating partial coherence and Gaussian-beam illumination. Numerical validation confirms high accuracy (RMSE of a few percent in key regions) and demonstrates how slit geometry, wavelength, coherence, and beam width influence the diffraction pattern. The work also outlines an experimentally feasible validation strategy and highlights applications in aperture apodization, pseudo-invariant beams, structured illumination, interference lithography, and coherent multiplexing, presenting a versatile route to propagation-robust, engineered diffraction profiles.

Abstract

We present an analytical and numerical investigation of double-slit diffraction under coherent illumination by three plane waves: one normally incident and two symmetrically angled at plus/minus theta. By imposing an edge-zero condition on the incident field, we derive compact closed-form Fresnel expressions written solely in terms of standard Fresnel integrals. The framework generalizes straightforwardly to an arbitrary number of incident plane-wave components, enabling intuitive control of the transmitted angular spectrum through interference engineered at the aperture. Numerical simulations confirm the accuracy of the closed-form model and characterize the influence of slit geometry, wavelength, partial coherence, and Gaussian beam width. We also discuss experimental feasibility and highlight potential applications in apodization, structured illumination, beam shaping, and multiplexed sensing. Overall, the results show that multi-wave coherent illumination provides a simple and tunable route to tailoring diffraction patterns and generating propagation-robust field profiles.

Figures (5)

• Figure 1: Normalized intensity profile for three-plane-wave illumination of a double slit at $z=1m$. The parameters are chosen such that one bright fringe is confined within each slit under the edge-zero condition.
• Figure 2: Effect of the scalar coherence parameter $\mu$ on the normalized intensity profile. Curves for $\mu=0,0.2,\dots,1.0$ are shown. As $\mu$ decreases, fringe visibility is reduced while the envelope remains similar.
• Figure 3: Effect of the transverse coherence length $\sigma_c$ on the normalized intensity profile in a Gaussian-decay coherence model. Smaller $\sigma_c$ values lead to reduced fringe visibility.
• Figure 4: Comparison between plane-wave illumination and Gaussian-beam illumination for two waist values $w_0$. Gaussian beams reduce sidelobes and slightly broaden the main lobe.
• Figure 5: Normalized intensity map $I(x_0;z)$ for three-beam illumination as a function of propagation distance $z$. The near field exhibits strong interference fringes, which evolve into smoother patterns in the far field.