Young's Double Slit Experiment: Interference

In 1801, Thomas Young performed an experiment that changed physics forever. He shone light through two closely-spaced narrow slits and observed a pattern of alternating bright and dark bands on a distant screen — an interference pattern. This was the definitive proof that light is a wave, because only waves can interfere (particles would just pile up behind each slit in two blobs).

The double slit experiment is arguably the most important experiment in all of physics. It's simple enough to draw on a napkin, yet it touches the deepest questions about the nature of reality (in quantum mechanics, single particles fired one at a time still produce interference patterns — but that's another story for Problem 154's quantum cousin).

The wave picture

Each slit acts as a source of light waves. These waves spread out (diffract) from each slit and overlap on the far screen. At any point on the screen, the two waves arrive with some phase difference depending on how far each wave traveled to get there.

• If the two waves arrive in phase (crest meets crest), they add up: constructive interference → bright fringe.
• If they arrive out of phase (crest meets trough), they cancel: destructive interference → dark fringe.

The key to the pattern is the path difference — the difference in distance traveled by the two waves to reach a given point on the screen.

Deriving the fringe positions

Let the two slits be separated by a distance $d$ (center to center), the screen be at distance $L$ from the slits, and consider a point on the screen at height $y$ above the central axis.

The path from the top slit to the point is (by the Pythagorean theorem):

$r_{1} = \sqrt{L^{2} + \left(y - \frac{d}{2}\right)^{2}}$

The path from the bottom slit:

$r_{2} = \sqrt{L^{2} + \left(y + \frac{d}{2}\right)^{2}}$

The path difference is $\Delta = r_{2} - r_{1}$. For $L \gg d$ and $L \gg y$ (the usual far-field approximation), this simplifies to:

$\Delta \approx d \sin\theta \approx \dfrac{d\,y}{L}$

where $\theta$ is the angle from the central axis.

Bright fringes (constructive interference) occur when the path difference is a whole number of wavelengths:

$\Delta = m\lambda \quad \Rightarrow \quad y_{m} = \dfrac{m\lambda L}{d}, \quad m = 0, \pm 1, \pm 2, \ldots$

Dark fringes (destructive interference) occur at half-integer multiples:

$\Delta = \left(m + \frac{1}{2}\right)\lambda \quad \Rightarrow \quad y_{m} = \dfrac{(m + \frac{1}{2})\lambda L}{d}$

Solving the problem

$\lambda = 600$ nm $= 6 \times 10^{-7}$ m, $d = 0.1$ mm $= 10^{-4}$ m, $L = 1.5$ m.

Fringe spacing (distance between adjacent bright fringes):

$\Delta y = \dfrac{\lambda L}{d} = \dfrac{(6 \times 10^{-7})(1.5)}{10^{-4}} = \dfrac{9 \times 10^{-7}}{10^{-4}} = 9 \times 10^{-3} \text{ m} = 9 \text{ mm}$

So bright fringes appear every 9 mm — easily visible to the naked eye.

Positions of the first few bright fringes:

• $m = 0$ (central maximum): $y_{0} = 0$ (dead center)
• $m = 1$: $y_{1} = 9$ mm
• $m = 2$: $y_{2} = 18$ mm
• $m = -1$: $y_{-1} = -9$ mm

First dark fringe: $y = \lambda L/(2d) = 4.5$ mm from center.

The intensity pattern (method 2 — wave superposition)

The intensity at angle $\theta$ from the center is:

$I(\theta) = I_{0} \cos^{2}\!\left(\dfrac{\pi d \sin\theta}{\lambda}\right)$

or equivalently, at position $y$ on the screen:

$I(y) = I_{0} \cos^{2}\!\left(\dfrac{\pi d\,y}{\lambda L}\right)$

This is a smooth cosine-squared pattern — the intensity oscillates between maximum ($I_{0}$) and zero. The maxima are the bright fringes; the zeros are the dark fringes.

Note: this idealized pattern assumes infinitely narrow slits. Real slits have finite width, which introduces a single-slit diffraction envelope that modulates the two-slit pattern (see Problem 155).

What changes what

• Increase $\lambda$ (longer wavelength, redder light): Fringe spacing $\Delta y = \lambda L/d$ increases. Red light has wider fringes than blue.
• Increase $d$ (wider slit separation): Fringe spacing decreases. More closely packed fringes.
• Increase $L$ (farther screen): Fringe spacing increases (pattern spreads out).
• Use white light instead of monochromatic: Each wavelength creates its own pattern at slightly different spacing, so the colors overlap. The central fringe is still white (all colors line up at $m = 0$), but the higher-order fringes show rainbow edges as the colors separate.

Historical significance

Before Young's experiment, Newton's particle theory of light was dominant (Newton was a formidable authority). Young showed that only a wave theory could explain interference. His experiment was initially met with hostility — the Royal Society rejected his paper and one reviewer mocked him — but the evidence was irrefutable. Combined with later work by Fresnel and Maxwell, the wave nature of light was established beyond doubt.

In the 20th century, the double slit was revisited in quantum mechanics. Single electrons, single photons, and even single molecules can be sent through the double slit one at a time, and over many trials, an interference pattern builds up. This means each particle somehow "interferes with itself" — one of the deepest mysteries in quantum physics. Feynman called it "the only mystery."

Common mistakes

• Confusing slit separation $d$ with slit width $a$. $d$ is the center-to-center distance between the two slits. $a$ is the width of each individual slit. They affect different parts of the pattern ($d$ determines fringe spacing, $a$ determines the single-slit diffraction envelope).
• Forgetting unit conversions. Wavelength is in nanometers ($10^{-9}$), slit separation often in millimeters ($10^{-3}$), and screen distance in meters. Mixing these produces wrong fringe spacings.
• Treating the pattern as sharp lines. The bright fringes are smooth cosine peaks, not infinitely sharp lines. They have a finite width that depends on the number of slits and the slit width.

Try it in the visualization

Drag the wavelength slider from violet (400 nm) to red (700 nm) and watch the fringe spacing change. Increase the slit separation and watch the fringes pack closer together. Toggle the intensity graph to see the smooth $\cos^{2}$ pattern. Switch to white light to see the rainbow halos around the central white fringe.