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Optics

Two-Slit Interference

Light plus light makes darkness, and the entire pattern follows from one number: how much further one wave had to travel.

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Two-Slit Interference

Light plus light makes darkness, and the entire pattern follows from one number: how much further one wave had to travel.

25 min5 checkpoints5 simulation tasks

Introduction

Send light through two narrow slits and you might expect two bright bands. What you get is a row of them, with dark gaps where light from both slits adds up to darkness.

Predict

Coherent light of one wavelength falls on two narrow slits, 10 µm apart, with a screen 40 cm behind them. What appears on the screen?

Where the fringes come from

Try it

The marker sits on the first dark fringe, where one wave travels half a wavelength further than the other. Move it out to the first bright fringe.

GoalThe marker on the first bright fringe.

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The two waves reach any point on the screen after travelling different distances, and the difference is the ticked segment:

δ=r2−r1≈dsin⁡θ\pos{\delta} = r_2 - r_1 \approx d\sin\theta

A whole number of wavelengths of extra path brings them crest with crest: bright. A half-integer number brings a crest with a trough: they cancel.

Moving the slits apart

The bright fringes are not images of the slits. Each slit radiates on its own, and the slits never interact with each other at all. The only thing that varies from one point on the screen to the next is how much further one wave had to travel to get there.

Predict

You move the slits apart, from 8 µm to 16 µm, changing nothing else. What happens to the pattern of fringes?

Try it

Set the slit separation so the bright fringes sit 20 mm apart.

GoalBright fringes 20 mm apart.

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The second length, and the orders that go missing

A slit has a width aa, and every part of that width sends its own wavelet, so even one slit alone makes a pattern: a broad central lobe whose first zero sits at sin⁡θ=λ/a\sin\theta = \lambda/a. That is the dashed envelope. The fringes ride inside it:

II0=cos⁡2 ⁣(πdsin⁡θλ)⏟two slits  ×  sinc2 ⁣(πasin⁡θλ)⏟one slit’s width\frac{I}{I_0} = \underbrace{\cos^2\!\left(\frac{\pi d \sin\theta}{\lambda}\right)}_{\text{two slits}} \; \times \; \underbrace{\mathrm{sinc}^2\!\left(\frac{\pi a \sin\theta}{\lambda}\right)}_{\text{one slit's width}}

If d/ad/a is a whole number, an envelope zero lands on a bright fringe, and that order vanishes.

Try it

Here d/a=4d/a = 4. Count outward from the centre and drag the marker onto the missing fourth bright fringe above it.

GoalThe marker on the missing fourth bright fringe.

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Predict

Keeping the separation fixed, you make each slit narrower. What happens?

Where the light at the dark fringes goes

At a dark fringe, two waves arrive and nothing comes out. Was the energy destroyed?

Try it

Move the marker to a dark fringe and run it. Watch the waves arriving from each slit, and the light they make together.

GoalA run with the marker on a dark fringe.

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Predict

At a dark fringe, what has become of the energy the two waves were carrying toward that point?

The formula everyone remembers, and the approximation inside it

The result you will be asked to memorise is the fringe spacing, Δy=λL/d\Delta y = \lambda L/d. It is a different kind of statement from dsin⁡θ=mλd\sin\theta = m\lambda.

Try it

Thirteen orders reach this screen, and the short ticks beside it are the evenly spaced mλL/dm\lambda L/d prediction. Drag the marker out to the thirteenth bright fringe.

GoalThe marker on the thirteenth bright fringe.

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Practice

Out at the thirteenth order, the λL/d prediction places the maximum most of a centimetre closer to the centre than it really is. Which step went wrong?

What comes next

Two slits give a smooth cos². Thousands of slits ruled into one barrier sharpen each maximum into a line so narrow that the wavelength can be read from its position, which is how we learned what stars are made of from nothing but their light.

Ideas this lesson took on

Each of these is a common intuition the lesson put to the test. If one still feels right, the section that tests it is worth another look.

  • “Two slits give two bright bands”
  • “Moving the slits apart spreads the fringes apart”
  • “Narrowing each slit narrows the whole pattern”
  • “At a dark fringe the energy is destroyed”

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Research behind this lesson

  • Ambrose et al. 1999: University students who had studied physical optics at introductory level and beyond often could not account for the pattern light makes on a screen after a single or double slit; many did not know whether to apply geometrical or physical optics to a given situation and inappropriately combined elements of both, and some difficulties were serious enough to preclude even a qualitative wave model for light.
  • Wosilait et al. 1999: Evidence that university students frequently lack a functional understanding of the wave model for light, together with the design of research-based tutorials intended to build that model rather than supply more formulas.