Multiple slit interference

We now understand the diffraction effect of a single slit on light passing through. But what about two slits?

When light passes through two or more slits, it interferes. A pattern can be observed if the waves are coherent. The effect continues with multiple slits or a diffraction grating.


Key Concepts

The phenonoma discussed on this page rely on coherent light. Coherent light beams have constant phase difference (and therefore the same frequency). They should also have the same amplitude.

Young's double slit experiment

The Young's double slit experiment consisted of a beam of coherent light passing through two identical slits before striking a screen.

The result is a pattern of evenly spaced bright fringes separated by dark fringes. It is caused by the interference of the light from the two slits as the two paths to each point on the screen vary in distance. The central maximum is most intense.

The fringe spacing can be calculated as follows:

\(s={D\lambda\over d}\)

  • \(s\) is the distance between consecutive maxima (m)
  • \(D\) is the distance from the slits to the screen (m)
  • \(\lambda\) is the wavelength of light (m)
  • \(d\) is the distance separating the centres of the two slits (m)

The fringe spacing is increaed by increasing the distance between the slits and the screen or by reducing the slit separation. It is also increased by increasing the wavelength of the light.

Modulation

When we look beyond the interference pattern shown above, we notice that the intensity of the maxima appears not to follow a straightforward pattern of reducing outward from the centre.

The two-slit interference pattern is modulated by the one-slit diffraction effect.

This affects the distribution of intensity against distance on the screen from the centre by capping the double slit pattern within the intensity of the diffraction pattern.

Essentials

Multiple slits

When light shines through multiple slits, an interference pattern forms similar to that through a double slit.

\(n\lambda = d \sin \theta\)

  • \(n\) is the number (order) of the maximum, where the central maximum is order 0
  • \(\lambda\) is the wavelength of the light (m)
  • \(d\) is the distance separating the slits (m)
  • \(\theta\) is the angle subtended by the maximum and the centre

Subsidiary maxima form in between the overall interference maxima.The number of subsidiary maxima produced is equal to the number of slits minus 2.

The ratio of \(\text{slit width}\over \text{separation}\) can be obtained using a combination of the single slit diffraction equation and the multiple slit interference equation. At the first diffraction minimum:

\(\theta = {\lambda\over b}={n\lambda\over d}\Rightarrow {b\over d}={1\over n}\)

Diffraction gratings

The more slits added, the more narrow and intense the peaks become.

A diffraction grating is an array of identical, equally-spaced slits. The equation used to determine the angles at which maxima are produced is the same as that for multiple slits.

You may have to perform an initial calculation to determine \(d\), the separation of the slits. If you are informed that the grating contains e.g. 100 slits per mm, you would convert this into 100 000 slits per m. The inverse gives the distance between slits in metres.

Summary

Interference patterns emerge from a variety of types of coherent sources such as electromagnetic waves and sound.

Interference can be modelled using a simulation.

Test Yourself

Use quizzes to practise application of theory.


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