Fresnel number and Fraunhofer vs Fresnel diffraction

Light passing an aperture spreads by diffraction, and the pattern it forms depends on how far away it is observed. Close to the aperture the light still looks like a sharp-edged shadow with ripples along the edges; far away it settles into a pattern whose shape no longer changes, only its size, which grows in proportion to distance. The two regimes are called Fresnel and Fraunhofer diffraction, and a single number, the Fresnel number, says which applies:

where $a$ is the aperture radius (or half-width), $L$ the distance to the observation plane and $\lambda$ the wavelength. Physically, $F$ is the number of Fresnel zones, rings across which the path to the observation point changes by half a wavelength, that fit in the aperture.

The two regimes

When $F\ll 1$, the path differences across the aperture are small, the field in the observation plane is the Fourier transform of the aperture field, and the result is the Fraunhofer pattern: the sinc² pattern of a slit or the Airy pattern of a circular hole. When $F$ is of order 1 or more, the quadratic phase across the aperture matters and the pattern is Fresnel diffraction, computed with the Fresnel integrals; on axis behind a circular hole the intensity oscillates between bright and dark as $F$ passes through even and odd integers. At very large $F$ the pattern approaches geometric shadow.

For a 1 mm radius aperture at 632.8 nm, $F$ = 1 at 1.58 m and $F$ = 0.1 at 15.8 m, so observing a clean far-field pattern needs a distance of more than ten meters. A lens avoids that: its back focal plane shows the Fraunhofer pattern at any distance, which is why far-field patterns, point spread functions and the focal-plane method for beam divergence all use a lens.

Relation to beams

For a Gaussian beam, the analogous boundary is the Rayleigh range, $z_R=\pi w_0^2\mathrm{/}\lambda$, which is $F=1\mathrm{/}\pi$ written with the waist radius $w_0$ in place of $a$. Within a Rayleigh range the beam is in its near field and grows by at most a factor of $\sqrt{2}$; many Rayleigh ranges away it expands at its far-field divergence. Antenna engineers use a more conservative boundary, $2D^2\mathrm{/}\lambda$ for an aperture of diameter $D$, which is $F=1\mathrm{/}8$ and about 2.5 Rayleigh ranges for a beam whose waist diameter is $D$.

Practice

The regime decides which tool models a setup correctly: Fraunhofer optics for spectrometer gratings, focal spots and far-field measurements; Fresnel propagation, usually computed numerically by the angular-spectrum method, for apertures close to detectors, lithography gaps and holography. A quick check of $F$ is the way to decide whether a "far-field" measurement really is one.

References: J. W. Goodman, Introduction to Fourier Optics, 4th ed. (W. H. Freeman, 2017), Ch. 4; M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 1999), Ch. 8; E. Hecht, Optics, 5th ed. (Pearson, 2017), Ch. 10.