Shine a laser pointer through a card with two fine scratches and project the result onto a wall. You do not get two bright spots, as a particle picture of light would suggest. You get a row of bright and dark bands, evenly spaced, stretching out symmetrically from the center. Cover one slit and the bands vanish. The pattern exists only when light can travel through both openings at once.
That experiment, first done carefully by Thomas Young in 1801, was the evidence that settled a long argument: light travels as a wave. This article explains how the fringe pattern forms, how to predict the spacing between bright bands, and why the result is one of the cleanest demonstrations in all of physics.
Why this calculation matters
Double-slit interference is far more than a teaching demonstration. The same principle underlies a wide range of measurement and imaging technology. Interferometers split a beam and recombine it to detect length changes smaller than a wavelength — the technique that lets gravitational-wave detectors sense a strain of one part in 10^21. Thin-film coatings on lenses and solar cells are designed by controlling interference between reflections. Diffraction gratings, which are simply many slits instead of two, are the heart of spectrometers that identify materials by their light.
The calculation matters because the fringe spacing is a direct, measurable link between geometry and wavelength. Measure how far apart the bands fall, know the slit separation and screen distance, and you can read off the wavelength of the light — or, run the other way, use a known wavelength to measure a tiny separation. The double-slit formula turns a length you can see with a ruler into a length far too small to see at all.
The core formula
When light passes through two slits, each slit acts as a source of waves spreading outward. At any point on a distant screen, two waves arrive — one from each slit — having traveled slightly different distances. That difference in path length is what decides whether they reinforce or cancel.
Where the path difference is a whole number of wavelengths, the waves arrive in step and add: a bright fringe. Where the path difference is a half-integer number of wavelengths, they arrive exactly out of step and cancel: a dark fringe. The condition for bright fringes is:
d * sin(theta) = m * lambda m = 0, 1, 2, ...
Here d is the separation between the slits, theta is the angle from the central axis to the fringe, lambda is the wavelength, and m is the fringe order — m = 0 is the central bright band, m = 1 the first on either side, and so on.
For a screen far away compared with the slit separation, the angles are small, so sin(theta) is close to theta, and the geometry simplifies to an evenly spaced pattern. The spacing between adjacent bright fringes becomes:
delta_y = lambda * L / d
where L is the distance from the slits to the screen. This compact result carries all the intuition. Fringe spacing grows with wavelength, so red light spreads wider than blue. It grows with screen distance, so moving the screen back stretches the pattern. And it shrinks as the slits move apart — closely spaced slits give a wide, easy-to-measure pattern, while widely spaced slits crowd the fringes together.
A worked example
Take a typical setup. Light of wavelength lambda = 600 nm (orange light) passes through two slits separated by d = 0.1 mm. The screen sits L = 2 m away. Find the spacing between adjacent bright fringes.
Step 1 — convert everything to consistent SI units.
lambda = 600 nm = 600e-9 m
d = 0.1 mm = 1e-4 m
L = 2 m
Mixing nanometers, millimeters, and meters is the fastest way to a wrong answer here, so put everything in meters first.
Step 2 — apply the fringe spacing formula.
delta_y = lambda * L / d
delta_y = (600e-9 * 2) / 1e-4
delta_y = 1.2e-6 / 1e-4
delta_y = 0.012 m = 12 mm
The bright fringes fall 12 mm apart on the screen. That is a wide, comfortable spacing — you could measure it with an ordinary ruler. And that is the quiet genius of the experiment. The wavelength of light, 600 nm, is far too small to observe directly. But by passing the light through two slits a tenth of a millimeter apart and catching it on a screen two meters away, the geometry magnifies that invisible length into a 12 mm pattern of bands. The wide, measurable fringe spacing is exactly what makes the double slit a clean, convincing demonstration that light behaves as a wave.
Common mistakes
Leaving units mismatched. The single most common slip. Wavelengths come in nanometers, slit separations in millimeters, distances in meters. Convert all three to meters before substituting, or the result will be off by powers of ten.
Confusing slit separation with slit width. The separation d in the fringe formula is the distance between the two slit centers. The width of each individual slit is a different quantity — it controls the broad diffraction envelope that modulates the brightness of the fringes, not their spacing.
Applying the small-angle formula too far out. delta_y = lambda*L/d assumes sin(theta) is approximately theta, which holds near the center. For high-order fringes at large angles, the spacing is no longer perfectly uniform, and the full equation d*sin(theta) = m*lambda is needed.
Forgetting that the source must be coherent. Stable fringes require light that is coherent across both slits — a laser, or a single small source with a single slit in front of it. Two independent bulbs will not produce a steady pattern, because their phase relationship drifts.
Reading the order number wrong. The central bright fringe is m = 0, not m = 1. Counting the center as the first order shifts every subsequent calculation by one fringe.
Try the interactive NovaSolver calculator
The formula is short, but seeing the pattern respond to each parameter is what makes it stick. The Young's Double-Slit Interference Simulator on NovaSolver lets you adjust the wavelength, the slit separation, and the screen distance and watch the fringe pattern redraw in real time, reporting the fringe spacing and the color of the light as you go. Pushing the sliders to their extremes is the quickest way to feel why red fringes spread wider than blue, and why moving the slits apart squeezes the bands together.
Related calculators
- Wave Interference Calculator — for the underlying principle of superposition, where two waves add to give constructive and destructive results.
- Single-Slit Diffraction Calculator — for the broad envelope a single opening produces, which shapes the brightness of the double-slit fringes.
- Diffraction Grating Calculator — for what happens when two slits become thousands, sharpening the fringes into spectral lines.
You can browse the rest in the optics and waves tools hub.
Closing note
The double-slit experiment endures because it does so much with so little: two narrow openings, a beam of light, and a screen produce a result that no particle picture can explain. The fringe spacing formula, delta_y = lambda*L/d, is the quantitative heart of it — a clean proportionality that connects an invisible wavelength to a pattern you can measure with a ruler. Keep your units consistent, distinguish slit separation from slit width, and remember the central fringe is order zero. Get those right and a row of bright bands becomes a precision instrument.
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