Physics › 🔦 Optics & Light

Diffraction & Interference of Light

How splitting one beam of light into two paths carves a shadow into stripes — and proves light is a wave.

AP Physics 2 levelUni Year 1

💡

The big idea: Light doesn't just travel in straight-line rays — it spreads through narrow openings (diffraction) and, when two spreading waves overlap, they reinforce in some places and cancel in others (interference). Young's double-slit experiment turns this into a strikingly simple, measurable pattern of bright and dark bands, and the spacing of those bands is fully predictable from wavelength, slit separation, and screen distance.

🎯 By the end, you'll be able to

explain why Young's double-slit pattern is direct evidence that light behaves as a wave, not a stream of particles
predict where bright and dark fringes fall using the interference condition \(d\sin\theta = m\lambda\)
calculate fringe spacing with \(\Delta y = \lambda L/d\), and solve for any one variable given the other three
distinguish single-slit diffraction from two-slit interference and know when the small-angle approximation is (and isn't) valid

📎 You should already know

Waves & Wave Superposition
Basic Trigonometry (small-angle approximation)
Electromagnetic Spectrum Basics

A shadow that shouldn't be there

Shine light through two narrow slits onto a screen, and simple geometry says you should see two bright stripes — one behind each slit — with darkness everywhere else. That's what would happen if light traveled as a stream of tiny bullets.

That is not what happens. Instead you get a whole row of evenly spaced bright and dark bands, fading in and out like ripples on a pond. The only way to explain that is if light is a wave — and this one experiment, first performed by Thomas Young in 1801, remains one of the most convincing demonstrations in all of physics.

🔑 Interference requires coherence

Two waves only build a steady, visible pattern if they hold a fixed phase relationship — this is called coherence. Splitting a single wavefront into two paths (two slits, or two mirrors) ensures this automatically, because both halves came from the exact same wave at the exact same instant. Two separate light bulbs never visibly interfere, because their phases jitter randomly relative to each other billions of times per second.

\[ d \sin\theta = m\lambda, \quad m = 0, \pm1, \pm2, \ldots \]

Double-slit bright-fringe condition: constructive interference occurs where the path-length difference between the two slits equals a whole number of wavelengths.

\[ \Delta y = \frac{\lambda L}{d} \]

Fringe spacing on a screen a distance \(L\) from slits separated by \(d\) (valid for small angles, i.e. \(d \ll L\)).

\[ a \sin\theta = m\lambda, \quad m = \pm1, \pm2, \ldots \]

Single-slit diffraction: this condition locates dark bands from one slit of width \(a\) — the opposite of the double-slit case, where the analogous condition gives bright bands.

🎮 Interactive: Double-Slit Interference Pattern LIVE

Adjust the wavelength, slit separation, and screen distance and watch the fringe spacing respond in real time.

Two patterns, stacked together

In a real double-slit setup you're actually seeing two effects layered on top of each other: each slit diffracts light outward on its own, spreading it into a broad glow, and then the two spreading waves interfere with each other, carving that glow into fine stripes. The single-slit diffraction pattern acts like an envelope that controls how bright the interference fringes are allowed to get, while the interference condition controls exactly where those fringes sit.

The same wave behavior explains why a CD's surface flashes rainbow colors and why a spider web glints with color in morning light — closely and regularly spaced structures diffracting different wavelengths by different amounts.

📝 Worked example: In a double-slit experiment, the slits are separated by 0.25 mm and the screen is 2.0 m away. If the light has a wavelength of 600 nm, how far apart are the bright fringes on the screen?

Identify the values: \(\lambda = 600\text{ nm} = 6.0\times10^{-7}\text{ m}\), \(L = 2.0\text{ m}\), \(d = 0.25\text{ mm} = 2.5\times10^{-4}\text{ m}\).
Use the fringe-spacing formula: \(\Delta y = \dfrac{\lambda L}{d}\).
Substitute: \(\Delta y = \dfrac{(6.0\times10^{-7})(2.0)}{2.5\times10^{-4}} = \dfrac{1.2\times10^{-6}}{2.5\times10^{-4}}\).
\(\Delta y = 4.8\times10^{-3}\text{ m} = 4.8\text{ mm}\).

✓ The bright fringes are 4.8 mm apart — easily visible to the naked eye on a screen.

📝 Worked example: A student shines a laser through two slits 0.30 mm apart and measures the bright fringes on a screen 1.5 m away to be 3.0 mm apart. What is the wavelength of the laser?

Start from \(\Delta y = \dfrac{\lambda L}{d}\) and solve for \(\lambda\): \(\lambda = \dfrac{\Delta y\, d}{L}\).
Convert units: \(\Delta y = 3.0\times10^{-3}\text{ m}\), \(d = 3.0\times10^{-4}\text{ m}\), \(L = 1.5\text{ m}\).
Substitute: \(\lambda = \dfrac{(3.0\times10^{-3})(3.0\times10^{-4})}{1.5} = \dfrac{9.0\times10^{-7}}{1.5}\).
\(\lambda = 6.0\times10^{-7}\text{ m} = 600\text{ nm}\).

✓ 600 nm — in the orange-red part of the visible spectrum, consistent with a typical red laser pointer.

⚠️ Where the simple formula breaks down

The tidy relation \(\Delta y = \lambda L/d\) relies on the small-angle approximation — it only holds when the slit separation \(d\) is tiny compared to the screen distance \(L\), keeping the fringes close to the centerline. Push the slits far apart, or look at fringes far off-axis, and you need the exact relation \(d\sin\theta = m\lambda\) instead.

Also don't mix up the two conditions: for double-slit interference, \(d\sin\theta = m\lambda\) locates bright fringes. For single-slit diffraction, \(a\sin\theta = m\lambda\) locates dark fringes. Same-looking equation, opposite meaning — because the physical mechanism and the relevant opening (slit separation vs. slit width) are different.

Check your understanding

1. Why do two ordinary light bulbs never produce a visible interference pattern, while light from a single laser passed through two slits does?

Coherence — a fixed phase relationship between the two sources — is required for a stable, visible interference pattern. Splitting one wave into two paths preserves that relationship; two independent sources do not.

2. In a double-slit setup, the slit separation is 0.20 mm, the screen is 1.0 m away, and the wavelength is 500 nm. What is the fringe spacing?

\(\Delta y = \lambda L/d = (5.0\times10^{-7}\text{ m})(1.0\text{ m})/(2.0\times10^{-4}\text{ m}) = 2.5\times10^{-3}\text{ m} = 2.5\text{ mm}\).

3. If you double the slit separation \(d\) while keeping the wavelength and screen distance the same, what happens to the fringe spacing?

Fringe spacing \(\Delta y = \lambda L/d\) is inversely proportional to \(d\), so doubling \(d\) halves the spacing — the fringes crowd closer together.

4. What must happen at each individual slit before any interference pattern can form at all?

Each slit must diffract the light passing through it, spreading it outward. Without that spreading, the light from the two slits would never overlap on the screen, and no interference pattern could form.

✅ Key takeaways

Light bends and spreads through narrow openings (diffraction) because it is a wave — this spreading is what lets light from two slits overlap in the first place.
Where two coherent waves overlap, they interfere: crests aligning with crests gives bright fringes (constructive), crests aligning with troughs gives dark fringes (destructive).
The double-slit bright-fringe condition \(d\sin\theta = m\lambda\) and the fringe-spacing formula \(\Delta y = \lambda L/d\) let you predict or measure wavelength, slit separation, or screen distance from any of the others.
Single-slit diffraction and double-slit interference use similar-looking equations but opposite outcomes (dark vs. bright), and the simple spacing formula only holds under the small-angle approximation.