Single-slit diffraction
The intensity pattern I(θ) = I₀ sinc²(πa sin θ/λ) produced when light of wavelength λ passes through a slit of width a. First minimum at sin θ = λ/a; central maximum carries most of the energy.
Definition
Single-slit diffraction is the intensity pattern produced when a plane wave of wavelength λ passes through an aperture of width a and is observed on a distant screen (Fraunhofer regime) or via a lens. The intensity as a function of angle θ from the straight-through direction is I(θ) = I₀ [sin(πa sin θ/λ) / (πa sin θ/λ)]² — the squared sinc function. The central maximum at θ = 0 contains about 91% of the total energy; minima occur at sin θ = mλ/a for integer m ≠ 0; subsidiary maxima between the minima carry progressively less energy.
The first-minimum condition sin θ ≈ λ/a (for small a, small θ) is the cornerstone of diffraction-limited optics. A telescope of aperture D observes point sources as Airy discs of angular radius 1.22 λ/D — the circular-aperture version of the single-slit rule, with the 1.22 coming from the first zero of the Bessel function J₁. A laser beam of wavelength λ emerging from a beam-waist of diameter 2w₀ diverges into a far-field cone of half-angle θ ≈ λ/(πw₀). A 100 m-diameter radio dish at 21 cm (the hydrogen line) has an angular resolution of ~ 8 arcmin, comparable to the best unaided human eye in the visible. Single-slit diffraction is the simplest quantitative demonstration of wave behaviour and is used in every introductory optics laboratory to measure λ, a, or both.