Circular apertures & the diffraction limit

In the Fraunhofer limit the far-field amplitude is the phasor sum over the aperture, E(θ) ∝ ∫∫ exp(i k ξ sin θ) dA, with k = 2π/λ and ξ the coordinate across the aperture in the plane of θ. For a disc of radius a use polar coordinates, ξ = ρ cos φ. The angular integral ∫₀^(2π) exp(i k ρ sin θ cos φ) dφ equals 2π J₀(k ρ sin θ), a standard result, and the radial integral ∫₀^a J₀(k ρ sin θ) ρ dρ equals a J₁(k a sin θ)/(k sin θ) by the identity d[u J₁(u)]/du = u J₀(u). Dividing by the on-axis value π a² gives E(θ)/E(0) = 2 J₁(x)/x with x = k a sin θ = π D sin θ/λ, and the intensity is its square. Two checks: x is dimensionless, a length times a wavenumber, and since J₁(x) ≈ x/2 for small x, 2J₁(x)/x → 1 on axis as it must. Run the same procedure on a slit and sin(x)/x comes out instead. The slit's pattern is a sine because every strip of the slit has the same length, whereas a disc's chords shrink towards its rim, weighting the edges less and pushing the zeros outward into a Bessel function.

J₁ vanishes at x = 3.8317, 7.0156 and 10.1735, so the dark rings sit at sin θ = 1.220, 2.233 and 3.238 λ/D; the first is the famous 1.22. Between them the secondary maxima at x = 5.136 and 8.417 carry only 1.75% and 0.42% of the central intensity, and integrating the pattern puts 83.8% of the power inside the first dark ring and 91.0% inside the second, which is why the central disc is what an eye or detector registers. Put in the Hubble Space Telescope, D = 2.4 m at 550 nm: θ₁ = 1.22 × 5.50 × 10⁻⁷ ÷ 2.4 = 2.8 × 10⁻⁷ rad, and multiplying by 206 265 arcsec rad⁻¹ gives 0.058″. Halve λ or double D and the whole pattern shrinks by two; its shape never changes. That fixed shape is the point: a lens of any quality draws a point as an Airy disc of this size, and no amount of polishing removes it.

Two incoherent sources, different stars or different fluorescent molecules, have no fixed phase relation, so their intensities add: I(θ) = A(θ − Δθ/2) + A(θ + Δθ/2), with A the Airy profile. At Rayleigh's separation Δθ = θ₁ the midpoint lies x = 3.8317/2 = 1.916 from each peak, where 2J₁(x)/x = 0.6062, so the midpoint intensity is 2 × 0.6062² = 0.735 of a single peak: a 26.5% dip. Move the sources closer and the dip shallows; at Δθ = 0.947 λ/D the second derivative at the midpoint vanishes and the two humps merge into one flat-topped blob, the Sparrow limit. Dawes's empirical visual limit for double stars, 116″/D(mm), works out to 1.00 λ/D at 562 nm, in between. None of these is a wall. With enough photons a two-source model fitted to the profile separates a pair well below θ₁, and a single isolated emitter can be located to a small fraction of its disc, which is what localisation microscopy does. The criterion is also only for incoherent sources: two coherent in-phase sources add in amplitude and show no dip at all at θ₁, while an anti-phase pair shows a true zero between them.

For an eye with a 3 mm daylight pupil at 550 nm, θ₁ = 2.2 × 10⁻⁴ rad = 0.77 arcmin; measured acuity is about 1 arcmin, so the retina's cone spacing and the lens's aberrations, not diffraction, set the last factor. In a camera the angular limit becomes a length on the sensor, r₁ = f θ₁ = 1.22 λ N: at f/8 and 550 nm the Airy disc is 2 × 5.37 = 10.7 μm across, at f/5.6 it is 7.5 μm. The finest period the lens passes is λN, so a sensor with 3.8 μm pixels is pixel-limited at f/5.6 and diffraction-limited beyond about f/14, where λN/2 reaches the pixel pitch; that is why stopping down for depth of field softens fine detail. At radio wavelengths the same formula is punishing: a 100 m dish at λ = 21 cm has θ₁ = 2.6 mrad = 8.8 arcmin, worse than the unaided eye, so radio astronomy replaces D by the baseline between dishes and synthesises the aperture by interferometry. In every case check the limit: θ₁ → 0 as λ → 0 recovers geometrical optics, where a point images to a point.

Apply r₁ = 1.22 λ L/D in the image plane of an objective, L being the distance from the lens to that image, and divide by the magnification M to find the equivalent distance in the specimen. Abbe's sine condition, n sin α = M n′ sin α′, relates the object-side half-angle α to the small image-side angle α′ ≈ D/(2L), so 1.22 λ L/(D M) becomes dₘᵢₙ = 0.61 λ/(n sin α) = 0.61 λ/NA. The numerical aperture is a cone, not a diameter: a wide barrel with a long working distance collects a narrow cone and resolves poorly. Since sin α ≤ 1, a dry objective has NA < 1 (0.95 in practice, giving dₘᵢₙ = 0.61 × 550/0.95 = 353 nm), and only an immersion liquid matching the coverslip lets the steep rays through: oil at n = 1.515 allows NA = 1.40 and dₘᵢₙ = 240 nm, and violet 405 nm light in the same objective reaches 176 nm. Image brightness scales as NA², but that is a different statement from resolution, and a higher-power objective with the same NA shows nothing new.

Real images are blurred by several effects at once, and for roughly Gaussian blurs the widths add in quadrature. Through the atmosphere a wavefront stays flat only over a patch of size r₀ ≈ 10 cm at 500 nm, so a long-exposure star image has a seeing width near 0.98 λ/r₀ ≈ 1″ whatever the aperture; an 8 m telescope without adaptive optics resolves no better than a 12 cm one, it only collects some 4000 times more light. Aberration is bounded by the Maréchal criterion: an RMS wavefront error of λ/14 leaves the Strehl ratio, the peak intensity relative to the perfect Airy peak, at exp(−(2π/14)²) = 0.82, the usual threshold for calling a system diffraction limited. Noise is the last gate. With N photons in each peak the shot noise is √N, so the 26.5% Rayleigh dip is a 3σ detection only when 0.265 N ≥ 3√N, that is N ≥ 130 photons per source; below that the two sources are unresolved by any criterion. Diffraction is the floor beneath all three, but it is the floor you reach last.