Diffraction

What it draws

The pattern a square hole casts on a distant screen. Light through a slit of width a spreads into sinc, and a rectangular hole is two slits at right angles, so the two directions multiply and the answer is sinc(3x) · sinc(3y). The 3 stands in for the hole: a wider hole means a larger number here and a tighter pattern.

There is a bright square at the centre and a cross of weaker lobes running out along both axes. The dark lines between them sit where either factor is zero, at 3x = π and its multiples, so the first is at x ≈ 1.05. Off the axes the two factors are both small and the corners stay dim: the lobe at (1.5, 1.5) is 4.7% of the centre where the one at (1.5, 0) is 21.7%.

Amplitude, not brightness

Height here is the amplitude of the wave, which is signed, so the lobes beside the peak hang below the plane rather than standing on it. Each one is half a wavelength out of step with its neighbour, and that sign is the reason the dark lines are dark.

What an eye or a sensor measures is the square of this, and the square is a poor picture. It would put the first side lobe at 4.7% of the peak instead of 21.7%, and the corner lobe at 0.2%, so the cross that carries the structure would disappear against the central square.

Try

• Take the intensity with sinc(3x)² · sinc(3y)². Everything is positive now, and the side lobes all but vanish, which is what the eye actually sees.
• Open the hole wider with sinc(6x) · sinc(6y). The zeros halve to x ≈ 0.52 and the pattern tightens: a bigger hole diffracts less.
• Use a single slit, sinc(3x) alone. The pattern becomes stripes, since only one direction is confined.
• Press Top and raise Mesh. The cross and the dark lines between the lobes read most clearly from directly above.

Read more

• Diffraction
• Fraunhofer diffraction
• Sinc function
• Angular resolution