Equations · equation
Airy Diffraction Pattern
Waves & Optics · 1835
- Equation
- I(θ) = I₀·[2J₁(x)/x]², x = πD sin θ / λ; first zero at x = 3.8317 ⇒ sin θ = 1.22 λ/D
- Field
- Waves & Optics
- Named After
- George Biddell Airy
- Year
- 1835
Symbols
I(θ) intensity at angle θ from the axis · I₀ the intensity at the centre · J₁ the first-order Bessel function · D the aperture diameter · λ the wavelength · x = πD sin θ/λ · the first zero of J₁ lands at x = 3.8317, and 3.8317/π = 1.2197 — which is the 1.22 in Rayleigh's criterion
Significance
A circular aperture cannot make a point. It makes a bright disc ringed by faint ones, and every star in every telescope image is this pattern rather than a star. About 84 % of the light lands in the central disc, the rest in the rings, and the first dark ring sits at 1.22 λ/D — a number that is not a fudge factor but the first zero of a Bessel function divided by π. The shape of the hole sets the constant: a long slit gives sin x / x and its first zero at exactly 1.00 λ/D, a square aperture the same, and only the circle pays 1.22 for being round. Airy worked it out in 1835 for telescope object-glasses, forty-four years before Rayleigh turned it into a rule for when two stars are two.
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