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Fresnel diffraction

Near-field diffraction produces patterns that depend on distance. Close to the aperture (low Fresnel number), the pattern retains the aperture’s shape with rippling edges. Far away (high Fresnel number ≪ 1), it smoothly becomes the Fraunhofer far-field pattern. Slide the screen distance to watch the transition happen.

NF = a² / (λ z)   —   Fresnel number
Fresnel number: 10.0
Wavelength: 550 nm
Aperture: 0.5 mm
Screen distance: 0.45 m
Regime: Fresnel
Wavelength 550 nm
Aperture width 0.50 mm
Screen distance 0.45 m
Intensity scale 1.0

Augustin-Jean Fresnel developed his diffraction theory in 1818, showing that Huygens’ wavelets with proper phase accounting explain all diffraction phenomena. Near the aperture, the curvature of the wavefronts matters — this is Fresnel (near-field) diffraction. Far away, wavefronts become effectively flat and the simpler Fraunhofer (far-field) approximation applies.

The Fresnel number NF = a²/(λz) determines which regime you’re in. When NF ≫ 1, you see Fresnel diffraction with its characteristic edge ripples. When NF ≪ 1, the pattern transitions to Fraunhofer diffraction — a sinc² envelope for a single slit, an Airy disk for a circular aperture.

The computation uses Fresnel integrals C(u) and S(u), evaluated numerically. The Cornu spiral is the parametric plot of (C(u), S(u)) — amplitude at any screen point corresponds to the chord length between two points on this spiral.

For the straight edge, the beautiful result is that intensity at the geometric shadow edge is exactly 1/4 of the unobstructed intensity — not 1/2 as geometric optics would predict.