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.
Augustin-Jean Fresnel decomposed a wavefront into concentric annular regions called half-period zones. The path length from the zone edge to the observation point differs by λ/2 from one boundary to the next, so neighboring zones arrive roughly in antiphase and nearly cancel. The outer radius of the n-th zone follows rₙ = √(n·λ·z) for a plane wave. Because annular area grows with radius while obliquity shrinks, each zone carries nearly the same amplitude — so the unobstructed sum telescopes down to roughly half the first zone's contribution.
In 1818 Siméon Poisson attacked Fresnel's wave theory by deriving an absurd prediction: a bright spot should appear at the center of the shadow of a perfectly circular opaque disk. Dominique Arago then performed the experiment and observed exactly that spot — turning the apparent refutation into one of the strongest confirmations of the wave nature of light. The disk simply masks the first N zones; the unblocked zones still sum coherently at the center, and because the obstruction is symmetric, the center sees the same bright maximum as if the disk were absent.
The dimensionless Fresnel number Nf = a²/(λz) classifies the regime. When Nf ≫ 1 many half-period zones fit inside the aperture — near-field Fresnel diffraction with oscillatory I(r). When Nf ≪ 1 only a fraction of the first zone contributes and the pattern collapses to the Fraunhofer (far-field) Airy pattern. The boundary between near-field and far-field is conventionally z ≈ a²/λ. This visualization covers the entire Fresnel-to-Fraunhofer crossover as you sweep z.
Fresnel zone theory underpins the Fresnel zone plate — a flat lens where alternating zones are blocked (amplitude) or phase-shifted (phase), focusing light by constructive summation of the surviving zones. Zone plates focus X-rays that ordinary glass lenses cannot. The Poisson spot limits the sharpness of lithographic shadows and is used to align precision optics (the spot marks the optical axis). Fresnel diffraction also governs the near-field of laser beams, camera bokeh edges, knife-edge tests, and the design of Fresnel lenses in lighthouses and solar concentrators.
Drag Wavelength λ, obstacle radius a, and screen distance z. The Zone Map shows the concentric half-period bands with alternating ± phase shading; the obstacle edge sits among them. The Radial Intensity plot shows I(r) on the screen with band overlays — watch bright/dark rings march inward or outward as z changes. Toggle Aperture ↔ Opaque Disk: in Disk mode a bright Poisson spot appears at r = 0 even though that point is geometrically shadowed. The Fresnel Number readout tells you which regime you are in — large Nf means many oscillating rings (Fresnel), small Nf means a single broad Airy-like peak (Fraunhofer).