Interference & Diffraction
When light waves pass through two narrow slits, each slit sends out spreading waves that overlap on a screen behind them. Where a crest from one slit meets a crest from the other, they add up into a bright fringe; where a crest meets a trough, they cancel into a dark gap. Bright fringes only show up at the special angles where the extra distance one wave travels is a whole number of wavelengths.
The formula
d · sin θ = m · λ
- d — slit spacing (m): distance between the two slits (or lines on a grating)
- θ — angle (°): angle from the center of the pattern out to a bright fringe
- m — order number (unitless): which bright fringe it is, counting 0, 1, 2… out from the center
- λ — wavelength (m): the length of one full wave cycle of the light
Worked example
Light of wavelength 500 nanometers shines through two slits spaced 20 micrometers apart. What is sin θ for the second bright fringe (m = 2)?
- d = 20 μm
- λ = 500 nm
- m = 2
- sin θ = m · λ / d
- sin θ = (2 × 500 nm) / 20 000 nm
sin θ = 0.05
Test yourself
Same slits (d = 20 micrometers) and same light (500 nanometers) — what is sin θ for the first bright fringe, m = 1?
- Correct answer: 0.025
- 0.05
- 0.1
Correct! sin θ = m·λ/d = (1 × 500 nm) / 20 000 nm = 0.025 — half of the m = 2 answer.
If you slide the two slits farther apart (increase d) while keeping the wavelength and order the same, what happens to the fringe spacing?
- Spacing increases
- Correct answer: Spacing decreases
- Spacing stays the same
Right! Since sin θ = m·λ/d, a bigger d makes sin θ smaller, so the fringes squeeze closer together.
Where you see this
The rainbow sheen sliding across a soap bubble and the colors reflecting off an old CD's grooves are interference at work: light waves reflecting from top and bottom of a thin film add or cancel by wavelength, sorting white light into bands. Two narrow slits and a screen make the same adding-and-cancelling visible as stripes of light and dark.
Common mistakes
The spacing direction trips almost everyone: spreading the slits farther apart (bigger d) makes the fringes move CLOSER together — d sits under the fraction in sin θ = m·λ/d, so bigger denominator, smaller angle. And each order m is its own angle, not a reuse of the last one: with d = 20 micrometers and 500 nm light, the first bright fringe sits at sin θ = 0.025.
How it connects
This is the optics module's crossing point into wave territory, and it leans on the waves module's foundation: v = f · λ governs light like any wave, and it is the wavelength that decides where the fringes land — which is how double-slit experiments measured light's wavelength in the first place. Polarization, next, completes the wave picture by showing which way the wave vibrates.