= Fresnel number
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{title2=$N_D=D^2/(\lambda z),\quad N_a=a^2/(\lambda z)$}
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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 $D=2a$, explicitly distinguish the full-width and half-width conventions:
$$
N_D=\frac{D^2}{\lambda z},\qquad N_a=\frac{a^2}{\lambda z}=\frac{N_D}{4},\qquad\Delta\phi_{\rm edge-centre}=\frac{\pi a^2}{\lambda z}=\pi N_a=\frac{\pi N_D}{4}.
$$
The full-width convention is used throughout the <Fresnel slit propagation> example below. The definition $D^2/(\lambda z)$ also appears in the Delft treatment.<Interactive Optics by Sander Konijnenberg, Aurèle J.L. Adam and H. Paul Urbach/Scalar Diffraction Optics>[Interactive Optics by Sander Konijnenberg, Aurèle J.L. Adam and H. Paul Urbach; Fresnel and Fraunhofer Approximations, equations (390)–(392): aperture diameter convention]{ref}
For uniform incident <wave phase>, $N_D\ll1$ 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.
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