Fresnel number

The Fresnel number is a dimensionless quantity used in optics to characterize the diffraction of light through an aperture or around obstacles. It is named after the French engineer and physicist Augustin-Jean Fresnel, who made significant contributions to the understanding of wave optics.

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A Fresnel number compares squared aperture size with the product of wavelength and propagation distance. For an optical slit of full width

D = 2 a

, explicitly distinguish the full-width and half-width conventions:

N_{D} = \frac{D ^{2}}{λ z}, N_{a} = \frac{a ^{2}}{λ z} = \frac{N _{D}}{4}, Δ ϕ_{edge - centre} = \frac{π a ^{2}}{λ z} = π N_{a} = \frac{π N _{D}}{4} .

The full-width convention is used throughout the Fresnel slit propagation example below. The definition

D^{2} / (λ z)

also appears in the Delft treatment.^[ref]

For uniform incident wave phase,

N_{D} ≪ 1

makes the omitted aperture wave phase small and gives the Fraunhofer diffraction regime. When

N_{D}

is of order one or larger, retain it and use Fresnel diffraction, provided the paraxial approximation still holds. These are continuous changes in accuracy, rather than a physical boundary at one selected number. For a rectangular aperture, check both transverse dimensions. Incident wavefront curvature contributes its own wave phase through

U_{0}

; the aperture-to-screen Fresnel number alone does not describe a spherical source's illumination.

The scaling is revealing: doubling the optical slit width at fixed wavelength and distance multiplies the Fresnel number by four. Maintaining the same Fresnel number then requires four times the distance. Two systems can consequently have the same dimensionless pattern despite very different physical lengths.