Diffraction & Resolution

Nothing about a distant star is spread out: its wavefronts arrive flat, and a perfect lens would bring them to a point. The aperture will not allow it. Only the part of each wavefront that passes the opening survives, and summing the Huygens wavelets across a circular opening of diameter D gives an image that is a bright central disk surrounded by faint rings — the Airy pattern. About 84 % of the light lands in the central disk and 7 % in the first ring. The edge of that disk, the first dark ring, sits at sin θ₁ = 1.22 λ/D, and because λ/D is minute for real instruments the sine is almost always dropped: θ₁ ≈ 1.22 λ/D radians. For a 0.20 m telescope at λ = 550 nm that is 3.4 × 10⁻⁶ rad, a blur of angular radius 0.69 arcsecond. The blur is not a flaw to be polished away. It is the price of having a finite opening at all, and it is the single-slit calculation wrapped around a circle.

A slit of width a puts its first zero at a sin θ = λ, so the number in front of λ/a is exactly 1. A circle gives 1.22, and the extra 22 % is pure geometry. Adding wavelets across a slit is a sum over one coordinate and produces sin u / u, whose first zero is at u = π. Adding them across a circle is a sum over a disc, and near the rim the chords of the opening are short, so the outer parts contribute less amplitude than the centre. The result is 2J₁(u)/u, where J₁ is a Bessel function of the first kind, and its first zero lies at u = 3.8317 rather than at π = 3.1416. Since u = πD sin θ/λ, the zero sits at sin θ₁ = (3.8317/π)(λ/D) = 1.2197 λ/D, which everyone rounds to 1.22. You will not be asked to evaluate a Bessel function. You should know that 1.22 is a property of the circle, and that an aperture of another shape carries a different number.

Two stars are mutually incoherent: their phases drift independently, so the interference between them averages to nothing and the detector adds intensities, not amplitudes. Slide the two Airy patterns together and their summed profile keeps a dip between the peaks — until it does not. Rayleigh's convention places the limit where the peak of one pattern falls on the first dark ring of the other, θₘᵢₙ ≈ 1.22 λ/D. At that spacing the valley between the maxima sits at about 74 % of a peak, a dip of 26 %: obvious on a good detector, marginal to the eye. Nothing physical happens at exactly that separation. Push closer and the dip shrinks smoothly to nothing near 0.95 λ/D, the Sparrow limit, where the profile first flattens; with a strong signal and a known point-spread function a separation well inside Rayleigh's line can still be measured. Treat θₘᵢₙ as an honest benchmark, not a wall.

θₘᵢₙ = 1.22 λ/D carries no magnification, no focal length and no exposure time. It falls when the aperture grows and when the wavelength shrinks, and for no other reason. Take an eye whose pupil is 4.0 mm wide in ordinary room light, at λ = 550 nm: θₘᵢₙ = 1.22 × 550 × 10⁻⁹ ÷ (4.0 × 10⁻³) = 1.7 × 10⁻⁴ rad, or 35 arcseconds. At a reading distance of 0.25 m that is s = L θₘᵢₙ = 42 μm, the right order for the finest print a reader can separate — although a real eye manages only about 1 arcminute, because the spacing of the retinal cones and the lens's own aberrations give up before diffraction does. An angle becomes a length only through the distance, so multiply by L, and remember that 1 rad = 206265 arcseconds. Moving from red light at 700 nm to blue at 400 nm sharpens any instrument by a factor of 1.75 for free, which is why microscopes reach further in the ultraviolet and an electron beam further still.

A camera does not hand you an angle; it hands you a spot at the focal plane. Multiply the angular diameter of the Airy disk, 2 × 1.22 λ/D, by the focal length f and the spot's diameter is d_Airy = 2.44 λ f/D, which depends on the two lengths only through their ratio — the f-number N = f/D. At λ = 550 nm an f/8 lens spreads every point into a disk 2.44 × 550 × 10⁻⁹ × 8 = 10.7 μm wide, and an f/2.8 lens into 3.8 μm. Set that against a sensor of 3.8 μm pixels: at f/8 one blur covers nearly three pixels, so the lens sets the detail and extra pixels buy nothing, while at f/2.8 the blur has shrunk to one pixel and the sensor takes over. This is why stopping a lens down stops helping. A smaller opening tames the aberrations and deepens the depth of field, but it widens every Airy disk in proportion.

The diffraction limit is a floor, and most instruments sit well above it. A ground-based telescope looks through turbulent air that smears a star into a blob about 1 arcsecond across on a good night. A 2.4 m mirror has θₘᵢₙ = 0.058 arcsecond, so the atmosphere throws away a factor of 17 and a 0.3 m telescope on the same site resolves nearly as well — which is the case for putting that mirror in orbit, or for reshaping the wavefront hundreds of times a second with adaptive optics. Polishing errors, focus errors and a central obstruction add blur of their own. Radio astronomy has the opposite trouble: at λ = 6 cm a 25 m dish gives θₘᵢₙ = 2.9 × 10⁻³ rad, about 10 arcminutes, which is why dishes are linked into arrays whose resolution goes as λ/B for a baseline B — 30 km of baseline at that wavelength reaches 0.4 arcsecond. Before quoting 1.22 λ/D, check that diffraction really is the largest error in the instrument.