= Fraunhofer diffraction
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<Fraunhofer diffraction> is the far-field approximation for <diffraction>: contributions from an <aperture> reach a selected observation direction with approximately linear path-dependent <wave phases>. Adding their <complex amplitudes> gives a <Fourier transform> of the field across the <aperture>. It also describes the appropriate rear <focal plane> of an ideal <optical lens> under coherent plane-wave illumination. The <diffraction grating> examples use this regime. <Oscillations and Waves by Richard Fitzpatrick/Two-Dimensional Fourier Optics>[Richard Fitzpatrick, Wave Optics, §10.9: Two-dimensional Fourier optics]{ref}; https://web.mit.edu/2.710/Fall06/2.710-wk10-a-sl.pdf[MIT 2.71/2.710 Optics, The ideal thin lens as a Fourier transform engine, slides 12–14: back focal plane diffraction]{ref}.
For a slit of width $a$, <wavelength> $\lambda$ and screen distance $L$, sufficient conditions near the forward direction are
$$
L\gg a,\qquad L\gg\frac{a^2}{\lambda},\qquad \frac{a^2}{\lambda L}\ll1.
$$
The last two statements are equivalent: the important comparison is with $a^2/\lambda$, rather than just with the slit width. For a finite rectangular <aperture>, check the corresponding condition for both transverse dimensions. The <single-slit diffraction> calculation treats the transverse cut associated with width $a$. https://farside.ph.utexas.edu/teaching/315/Waves/node99.html[Richard Fitzpatrick, Wave Optics, §10.10, equations 10.93–10.96: far-field conditions]{ref}.
A short explanation is that the distance from slit coordinate $x$ to an observation point at distance $R$ and <angle> $\theta$ is
$$
r(x)=\sqrt{R^2-2Rx\sin\theta+x^2}=R-x\sin\theta+\frac{x^2\cos^2\theta}{2R}+O(a^3/R^2).
$$
The linear term supplies the <phase difference>; the quadratic term supplies a <wave phase> error of order $a^2/(\lambda R)$, since the <wavenumber> is $2\pi/\lambda$. Near the forward direction $R\approx L$, so the stated condition makes this error small. This is why a distant screen permits the simple <single-slit diffraction> formula. <Oscillations and Waves by Richard Fitzpatrick/Two-Dimensional Fourier Optics>[Fitzpatrick, §10.9, equations 10.70–10.74: path expansion and aperture superposition]{ref}.
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