Lesson 3 of 5The Wave Nature of Light

Unit 4 · Lesson 3 of 5

Young's Double-Slit Experiment and Gratings

How two narrow slits turn one beam of light into a ripple-tank pattern, the equations that pull a wavelength out of a fringe spacing, why a grating with thousands of slits makes sharper spectra, and what a single slit does on its own.

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Key things to remember

1Double slit or grating, bright fringes: $d sin θ_{m} = mλ$ ; dark fringes: $d sin θ_{n} = (n - \frac{1}{2}) λ$ .
2Small-angle fringe spacing: $Δ y = \frac{λ L}{d}$ , so $λ = \frac{d Δ y}{L}$ .
3Grating spacing: $d = \frac{1}{lines per unit length}$ ; the highest visible order is the largest whole number $m$ with $m \leq d / λ$ .
4Single slit, dark fringes: $w sin θ_{n} = nλ$ ; the central maximum is about $2 λ L / w$ wide.
5Young's interference fringes showed that light is a wave and gave a way to measure its wavelength directly.

For over a century there were two models of light. Newton favoured a stream of tiny particles; others argued for a wave. The particle model explained straight-line shadows and reflection well, and the wave model seemed to fail one obvious test: light does not visibly bend around corners the way sound does. The last topic explains why — light's wavelength is tiny, so its diffraction and interference are hidden at everyday scales. Build an experiment small enough, and the hidden pattern appears. Thomas Young did exactly that in the early 1800s, and the stripes he saw on a screen were the ripple-tank pattern made of light.

The set-up and the pattern

Fig 1Two slits act as in-phase sources. Bright fringes appear where the path difference is a whole number of wavelengths. Not to scale.

Light from a single source (today a laser) falls on two very narrow slits a distance $d$ apart. Because both slits are lit by the same wavefront, they act as two in-phase sources — the job the two dippers did in the ripple tank. Each slit diffracts the light so that the two beams spread out and overlap, and on a screen a distance $L$ away you see a row of evenly spaced bright fringes (antinodal lines) separated by dark fringes (nodal lines). A particle model predicts two bright bands behind the two slits and nothing in between. Darkness appearing where light from two openings overlaps can only be explained by crests meeting troughs.

Locating the fringes

bright: d sin θ_{m} = mλ dark: d sin θ_{n} = (n - \frac{1}{2}) λ

Exactly the ripple-tank conditions, with d sin theta as the path difference from the two slits.

Here $m = 0$ is the central maximum straight ahead, $m = 1$ the first bright fringe on either side, and so on. For light the angles are tiny, because $λ$ is a few hundred nanometres while $d$ is a fraction of a millimetre. For small angles $sin θ \approx tan θ = y / L$ , where $y$ is the distance along the screen from the centre. Then the $m$ th bright fringe sits at $y_{m} = mλ L / d$ , and neighbouring bright fringes are all the same distance apart:

Δ y = \frac{λ L}{d}

Fringe spacing for small angles. Every length must be in the same unit, usually metres.

Why this was such a big deal

Δ y

L

and

d

are all measurable with a ruler and a microscope, so the fringe spacing gives the wavelength of light directly:

λ = \frac{d Δ y}{L}

. The experiment did more than show light is a wave — it measured the wave, and different colours gave different spacings, with red fringes wider apart than blue. Colour is wavelength.

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Change the wavelength, slit separation and screen distance and watch the fringe spacing follow lambda L over d.

Diffraction gratings

A diffraction grating is a surface ruled with thousands of equally spaced slits, usually quoted as lines per millimetre or per centimetre. The spacing is $d = 1/ (lines per unit length)$ . The bright maxima fall at the same angles as for two slits, $d sin θ = mλ$ , because any two neighbouring slits still need a whole-wavelength path difference. What changes is everything in between. With thousands of sources, light at any angle even slightly off a maximum is cancelled by some other slit in the grating, so the maxima become very narrow and very bright and the dark gaps between them become wide. Gratings also have tiny $d$ , so the angles are large and small wavelength differences are easy to separate.

ADouble slit

Two sources, $d$ usually a fraction of a millimetre
Broad, fuzzy bright fringes of similar width to the dark ones
Small angles, so $Δ y = λ L / d$ works well
Good for showing interference and measuring one wavelength

BDiffraction grating

Thousands of sources, $d$ of order a micrometre
Sharp, bright lines separated by wide dark regions
Large angles, so use $d sin θ = mλ$ , not the small-angle formula
Spreads white light into a clean spectrum; the basis of spectroscopes

A single slit on its own

Even one slit of width $w$ makes a pattern: a wide, bright central maximum with fainter maxima on each side. Think of the slit as a row of Huygens wavelets. At the angle where the wavelets from the top half of the slit are each half a wavelength out of step with matching wavelets in the bottom half, they cancel in pairs and the screen is dark. That happens when $w sin θ = λ$ , and again at $2 λ$ , $3 λ$ and so on. Note that for a single slit the whole-number condition gives dark fringes, the opposite of the double slit.

single-slit minima: w sin θ_{n} = nλ (n = 1, 2, 3, \dots) central maximum width \approx \frac{2 λ L}{w}

A narrower slit gives a wider central maximum, which is diffraction getting stronger as the opening shrinks.

Worked examples

Reveal one step at a time and try to beat it to the next line.

Example 1

Red light of wavelength

633 nm

from a helium-neon laser passes through two slits

0.25 mm

apart. The screen is

3.0 m

away. Find the spacing between bright fringes.

Example 2

Green light of wavelength

520 nm

falls on a grating with

600

lines per millimetre. Find the angles of the first- and second-order maxima, and the highest order that can be seen.

Example 3

Yellow light of wavelength

580 nm

passes through a single slit

0.10 mm

wide onto a screen

2.0 m

away. How far from the centre is the first dark fringe, and how wide is the central maximum?

Example 4

In a double-slit experiment with slits

0.15 mm

apart and a screen

2.50 m

away, a student measures

4.5 cm

across five fringe spacings. What is the wavelength of the light?

Where marks get lost

Mixing units in $Δ y = λ L / d$ : nanometres for $λ$ , millimetres for $d$ and metres for $L$ . Convert everything to metres first.
Using $Δ y = λ L / d$ for a diffraction grating. Grating angles are large, so use $d sin θ = mλ$ and then trigonometry if a screen distance is needed.
Using the line density as $d$ . A grating with 600 lines/mm has $d = 1/600 mm$ , not $600$ .
Applying the double-slit bright-fringe rule to a single slit. For one slit, $w sin θ = nλ$ gives dark fringes.
Thinking a wider slit separation spreads the fringes out. $Δ y$ is inversely proportional to $d$ , so closer slits give wider spacing.

Flashcards 8

Fringe spacing (double slit)

Bright fringes for a double slit or grating

Dark fringes for a single slit

Grating spacing from line density

+ 4 more

Practice

Light of wavelength

480 nm

passes through two slits

0.30 mm

apart onto a screen

2.0 m

away. What is the fringe spacing in millimetres?

In a double-slit experiment, the slit separation is halved while everything else stays the same. What happens to the fringe spacing?

Light of wavelength

600 nm

falls on a grating ruled with

5000

lines per centimetre. At what angle is the first-order maximum? Give degrees to one decimal place.

For the same grating (

5000

lines/cm) and light (

600 nm

), what is the highest order maximum that can appear?

A grating with

300

lines per millimetre produces a first-order maximum at

10. 0^{\circ}

. What is the wavelength of the light, in nanometres?

Laser light of wavelength

633 nm

passes through a single slit onto a screen

1.5 m

away. The first dark fringe is

8.0 mm

from the centre. How wide is the slit, in millimetres? Give three significant figures.

Why are diffraction gratings, rather than double slits, used in spectroscopes?

What is the name of the bright fringe at the centre of a double-slit pattern, where the path difference is zero?

White light passes through a diffraction grating. Order these colours in the first-order spectrum from smallest angle to largest angle.

1Red
2Yellow
3Green
4Violet
5Blue

Match each equation to the pattern feature it locates.

d sin θ = mλ

d sin θ = (n - \frac{1}{2}) λ

w sin θ = nλ

Δ y = λ L / d

Which observation from the double-slit experiment most directly contradicts a simple particle model of light?

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