The Double-Slit Pattern, Measured
By Lucian — builder & engineer, LK Forge
Two narrow slits turn a single beam of light into a row of evenly spaced bright bands. The spacing follows a formula so simple it fits on a fingernail — and it behaves backwards from what you would guess. Then, hiding in the same pattern, whole fringes go missing. We computed the interference from the double-slit simulator’s own optics and measured all of it.
· 6 min read · exact Fraunhofer intensity, every number reproducible
The pattern itself
The intensity on the screen is the product of two things: the fast cos² interference fringes from the two slits, and a slow sinc² diffraction envelope from each slit’s finite width. Here they are together, at the simulator’s default settings. Count the bright fringes in the central hump: nine. The tenth and eleventh — orders ±5 — should be next, but the envelope is exactly zero there, so they are simply gone.
Intensity = interference fringes under the dashed diffraction envelope. The gaps at ±27.5 mm are the missing 5th orders.
Closer slits, wider fringes
The spacing is Δy = λL/d, and the d in the denominator is the surprise: halve the gap between the slits and the fringes get twice as wide. It is why the effect is visible at all — you need very fine, very close slits to spread the fringes far enough to see. Below, the measured spacing against slit separation traces the hyperbola exactly.
Fringe spacing against slit separation at 550 nm, 1 m screen — a clean 1/d hyperbola. Wavelength scales it too: red (700 nm) fringes run 7.0 mm against violet’s (400 nm) 4.0 mm.
The missing orders
The interference fringes are spaced by the slit separation d; the diffraction envelope is set by the slit width a. Their ratio decides everything about which fringes survive. A bright interference order sits at d·sinθ = nλ; an envelope zero sits at a·sinθ = mλ. When those coincide — when n = m·(d/a) — the fringe falls into a dark gap and disappears. Since d/a = 5 here, every 5th order is a missing order: ±5, ±10, ±15. It is a real, testable fingerprint of the width-to-separation ratio, not a rendering quirk.
One last subtlety the measurement turned up: the envelope also nudges the surviving fringes. A fringe’s brightness peak is pulled slightly toward the centre from its exact interference maximum, because the envelope is sloping across it — we measured the first-order peak at 5.43 mm against the 5.50 mm maximum, about 1.35% inward. The spacing law Δy = λL/d is a statement about the interference maxima; the visible peaks carry this small envelope-induced shift on top.
Reproduce it yourself
The whole pattern is one line of intensity, and the two headline results fall straight out of it.
const d = 1e-4, a = 2e-5, lambda = 550e-9, L = 1; // metres
// Fraunhofer double-slit intensity at screen angle theta
function I(theta) {
const s = Math.sin(theta);
const beta = Math.PI * a * s / lambda; // single-slit
const gamma = Math.PI * d * s / lambda; // two-slit
const env = beta === 0 ? 1 : (Math.sin(beta) / beta) ** 2;
return env * Math.cos(gamma) ** 2;
}
const fringeSpacing = lambda * L / d; // 5.5e-3 m = 5.50 mm
const missingOrders = n => n % (d / a) === 0; // d/a = 5 -> 5, 10, 15 ...The I(θ) and Δy = λL/d are lifted from the double-slit solver; the sweeps and peak-finding are the small script behind this post.
Make some fringes
Open the simulator, slide the slits closer, change the colour, and watch the fringes widen — then narrow the slit width and watch orders drop out.