Optics
Fraunhofer Diffraction
In the far field, the diffraction pattern is the spatial Fourier transform of the aperture function
Fraunhofer diffraction (Joseph von Fraunhofer, 1820s) is the regime of wave diffraction where the source and observation point are effectively at infinity (or at the focal plane of a lens). The diffracted intensity pattern is the squared modulus of the Fourier transform of the aperture function: I(θ) ∝ |F[t(x)]|², where t(x) is the aperture transmittance. Examples: single slit width a → sinc²(πa sinθ/λ) — central peak with first zero at sin θ = λ/a; double slit (Young's experiment) → cos² envelope × sinc² of single slit; diffraction grating with N slits → sharp principal maxima. Applications: spectroscopy (gratings spread wavelengths), telescope diffraction limit (resolution = 1.22 λ/D Airy disc), x-ray crystallography (Fraunhofer of crystal lattice gives reciprocal lattice). The condition: source distance d >> a²/λ (a aperture size, λ wavelength).
- Far-field conditiond >> a²/λ
- Single slitI ∝ sinc²(πa sinθ/λ)
- First zerosin θ = λ/a
- Pattern|FT of aperture|²
- Diffraction limit1.22 λ/D (circular aperture)
- AuthorFraunhofer 1820s
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Why Fraunhofer diffraction matters
- Spectroscopy. Diffraction gratings — a Fraunhofer instrument par excellence — are how we measure the chemical composition of stars, the rotation of galaxies, the redshift of quasars, and the absorption lines of molecules. From sodium-D doublets to exoplanet atmospheres, Fraunhofer is the language.
- Microscope and telescope resolution. The diffraction limit 1.22 λ/D is a hard ceiling: a 100x lens can't resolve features smaller than ~λ/2 no matter the magnification. Super-resolution microscopy (STED, PALM, STORM) circumvents — but doesn't break — this limit by clever fluorescence tricks; SIM operates by sampling beyond cutoff.
- Antenna design. Radio antennas obey the same Fraunhofer relationship: the far-field radiation pattern is the Fourier transform of the aperture excitation. Phased arrays steer beams by adjusting the aperture phase. Synthetic aperture radar synthesizes a kilometer-scale aperture from a moving meter-scale antenna.
- X-ray crystallography. Atomic-resolution structures of proteins, viruses, ribosomes — including the work that won countless Nobel prizes — come from inverting Fraunhofer diffraction patterns of crystallized samples. Modern free-electron lasers extend this to single-particle imaging.
- Lithography. The 5 nm and 3 nm semiconductor nodes operate at the edge of EUV (13.5 nm) Fraunhofer limits. Resolution enhancement (off-axis illumination, optical proximity correction) is applied Fourier optics: shape the aperture so the pattern matches the desired chip features.
- Holography and Fourier optics. Holograms record both amplitude and phase of a Fraunhofer pattern by interference with a reference beam. Reconstructing illuminates the hologram and inverse-Fourier-transforms back to the original wavefront. Optical computers performed Fourier transforms at the speed of light long before silicon could.
- Particle sizing. Suspended particles produce Fraunhofer scattering. Laser diffraction granulometers measure particle-size distributions from cement to pharmaceuticals by inverting the angular pattern.
Common misconceptions
- "Fraunhofer requires infinite distance." Practically, d >> a²/λ is enough. For a 1 mm aperture at 500 nm, that's d >> 2 m. A focusing lens places the Fraunhofer pattern at its focal plane regardless of distance — that's how every imaging spectrometer works.
- "You always need a lens for Fraunhofer." Not strictly. A laser beam in space, or a star imaged through a small aperture, is naturally in the Fraunhofer regime. The lens is a convenience that brings the pattern from infinity to the focal plane.
- "Only visible light diffracts." Diffraction is a wave phenomenon. Fraunhofer applies to radio, microwaves, x-rays, electrons (de Broglie), neutrons, sound, water waves — anything with a wavelength. Electron microscopy and neutron diffraction are major Fraunhofer-imaging modalities.
- "More slits make the pattern brighter." Not exactly. N slits make principal maxima taller (∝ N²) but narrower (∝ 1/N), so total energy in each peak scales as N — proportional to total open area. The benefit is sharpness, not raw brightness.
- "The first zero is always at λ/a." Only for a single slit. For a circular aperture (Airy pattern), the first zero is at 1.22 λ/D. For arbitrary apertures, it's wherever the Fourier transform first vanishes — find by computing F[t(x)].
- "Fraunhofer diffraction is the same as interference." Two-source interference is a special case (two delta functions in t(x)). Diffraction is the general continuous-aperture case. Young's two-slit pattern is famously called "interference" but the single-slit envelope is "diffraction" — the same Fourier transform handles both.
Frequently asked questions
What's the difference between Fresnel and Fraunhofer diffraction?
Both come from the Huygens-Fresnel integral over an aperture. Fresnel (near-field) keeps the full quadratic phase term in the path-length expansion — patterns depend on distance and look like a partially-shrunk geometric image with rings. Fraunhofer (far-field) drops the quadratic term because d >> a²/λ — the pattern is angle-only, scale-invariant in distance, and equal to the Fourier transform of the aperture. The same aperture-pattern relationship gets restored in the focal plane of a lens, regardless of distance.
Why is the pattern a Fourier transform?
Each point in the aperture acts as a Huygens secondary source. The contribution to angle θ from position x in the aperture has phase exp(i k x sin θ). Summing across the aperture amplitude t(x): U(θ) ∝ ∫ t(x) exp(−i k x sin θ) dx — exactly a Fourier transform with conjugate variable k sin θ ≈ kx_screen/d. Intensity is |U|². The far-field condition is what justifies dropping the higher-order phase terms; below it you're in Fresnel territory and the relationship breaks.
Why is the first zero of single slit at sin θ = λ/a?
Pair off Huygens sources from opposite halves of a width-a slit. A source at position 0 and one at position a/2 differ in path length by (a/2) sin θ. They cancel when this equals λ/2 — i.e., sin θ = λ/a. Every source in the top half pairs with one in the bottom half, all canceling simultaneously. The Fourier-transform statement is equivalent: F[rect(x/a)] = a sinc(πa f), with first zero at f = 1/a (i.e., sin θ = λ/a).
How does a grating sharpen peaks (N slits)?
N slits at spacing d produce I ∝ [sin(N π d sinθ/λ) / sin(π d sinθ/λ)]² × sinc² of single-slit envelope. The numerator-over-denominator factor has principal maxima of height N² at d sinθ = mλ, with first zeros at distance Δ(sinθ) ∝ 1/(Nd). So peaks scale as 1/N in width while their height scales as N², making them sharper and brighter as N grows. The angular dispersion λ → angle stays the same; only the resolving power R = λ/Δλ = mN improves.
What's the diffraction limit of a telescope?
A circular aperture of diameter D in the far field produces an Airy pattern: J₁²(πD sinθ/λ)/(πD sinθ/λ)² — the 2D Fourier transform of a disc. The first zero is at sin θ = 1.22 λ/D. Two point sources are 'just resolved' when the peak of one falls on the first zero of the other (Rayleigh criterion): θ_min = 1.22 λ/D. For a 10 m telescope at 550 nm, that's 0.067 arcsec. Atmospheric seeing usually limits ground telescopes to ~1 arcsec; adaptive optics or space telescopes recover the diffraction limit.
How does x-ray crystallography use Fraunhofer?
X-rays of wavelength comparable to atomic spacing (~0.1 nm) diffract from a crystal lattice. In the far field, the scattered intensity is the squared Fourier transform of the electron density — sampled on the reciprocal lattice (Bragg peaks at q = 2π/d). Measuring intensities and phases (or solving the phase problem) and inverse-Fourier-transforming reconstructs the atomic structure. Watson and Crick determined DNA's double helix from Rosalind Franklin's Photograph 51, a Fraunhofer diffraction pattern.