# Fresnel Diffraction Zones — Half-Period Bands & the Poisson Spot

Near-field diffraction from half-period zone decomposition. Toggle between circular aperture and opaque disk — watch the Poisson bright spot appear at the center behind an obstruction.

> Canonical page: https://elysiatools.com/en/visualizations/fresnel-diffraction-zones

- **Category:** Physics

## Overview

Interactive Fresnel near-field diffraction visualization based on Augustin-Jean Fresnel's half-period zone decomposition (1818). A wavefront is divided into concentric annular regions where each successive boundary differs in optical path by λ/2 from the observation point, so neighboring zones arrive roughly in antiphase and nearly cancel — the unobstructed sum telescopes down to half the first zone. Zone radii follow rₙ = √(n·λ·z) for plane-wave illumination, and the dimensionless Fresnel number Nf = a²/(λz) classifies the regime: Nf ≫ 1 gives near-field Fresnel diffraction with oscillatory rings, Nf ≪ 1 collapses to the Fraunhofer far-field Airy pattern. The field is computed exactly via the Lommel integral U(ρ) ∝ ∫₀ᵃ r·J₀(kρr/z)·exp(ikr²/2z) dr (radial symmetry reduces the 2D Fresnel-Kirchhoff integral to a Bessel-weighted 1D integral); the opaque-disk case uses Babinet's principle U_disk = U_unblocked − U_aperture, which analytically guarantees the Poisson (Arago) bright spot at the shadow center. Three visualization panels: (1) Half-period zone map showing concentric bands with alternating +/− phase shading and the obstacle edge overlaid (circular aperture vs opaque disk); (2) Radial intensity I(r)/I₀ plot with zone-band overlay shading and the on-axis center marker (analytic I(0) = 4sin²(πNf/2) for aperture, =1 for disk); (3) Diffraction image on the screen rendered via offscreen low-resolution sampling upscaled, colored by the current wavelength λ. Adjustable parameters: wavelength λ (380–700 nm with wavelength→RGB mapping), obstacle/aperture radius a (0.1–5 mm), screen distance z (0.1–10 m), obstacle geometry (circular aperture / opaque disk), overlays (zones, bands, center marker). Auto-sweep animation ramps z logarithmically through the Fresnel-to-Fraunhofer crossover. Four presets: Classic (a=1mm, z=1m), Poisson Spot (opaque disk, a=2.5mm), Many Zones (a=4mm, z=0.3m, large Nf), Single Zone (a=0.3mm, z=4m, small Nf). Real-time statistics: first zone radius r₁, visible zone count N, Fresnel number Nf, center intensity ratio I(0)/I₀. Educational content covers Fresnel half-period zone theory, the 1818 Poisson-Arago experiment confirming the wave nature of light, the Fresnel number regime classification, and applications (Fresnel zone plates for X-ray focusing, lithographic shadow limits, optical alignment, lighthouse/solar Fresnel lenses). Multi-language support (zh, en, es, fr, de, ru, pt).

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