Interference and Diffraction of Light

Lessons

Senior 5 · Physics

Interference and Diffraction of Light

4 lessons

Lesson 2 of 4

Young's Double-Slit Experiment

25 min

The experiment that brought the wave back

For a century after Newton, most scientists believed light was a stream of tiny particles. Between 1801 and 1803 the English physician Thomas Young showed otherwise. He let sunlight through a pinhole, split the beam with a thin card (later, two narrow slits) and saw on a screen a row of bright and dark bands, fringes, which no particle theory could explain. From their spacing he calculated the wavelength of light: under a millionth of a metre. His experiment is still a standard way to measure the wavelength of light.

The apparatus

A source of nearly monochromatic light (one wavelength), such as a sodium lamp (589 nm) or a filtered lamp, illuminates a single narrow slit. Light spreading from it reaches two narrow parallel slits a distance $d$ apart (typically 0.1 mm to 1 mm), which therefore act as two coherent sources: every random change in the light reaches both at the same moment. A school laser can replace the lamp and single slit, because its light is already coherent across the beam. The light from the two slits overlaps on a screen a distance $D$ away (1 m to 2 m), where equally spaced fringes appear.

Warning

Laser safety. Use only a low-power laser (class 2, under 1 mW) and only under supervision. Never look into the beam or at its reflection from shiny surfaces, never point it at anyone, and keep the beam below eye level.

Deriving the fringe spacing

Consider a point P on the screen at height $y$ above the central point O, in a direction at angle $\theta$ to the centre line. The screen is so far away ( $D \gg d$ ) that the rays $S_{1} P$ and $S_{2} P$ are practically parallel. Drop a perpendicular from $S_{1}$ onto $S_{2} P$ , meeting it at N. Then $S_1P \approx NP$ , and the path difference is

\Delta = S_2N = d\sin\theta

P is bright if the path difference is a whole number of wavelengths, and dark if it is an odd number of half wavelengths:

\text{bright: } d\sin\theta = n\lambda, \qquad \text{dark: } d\sin\theta = \left(n + \tfrac{1}{2}\right)\lambda, \qquad n = 0, \pm 1, \pm 2, \ldots

The angles are small (a few milliradians), so $\sin\theta \approx \tan\theta = \frac{y}{D}$ . The $n$ th bright fringe is therefore at

y_n = \frac{n\lambda D}{d}

and the distance between neighbouring bright (or dark) fringes, the fringe spacing or fringe width, is

w = \frac{\lambda D}{d}

The fringes are equally spaced. They spread further apart for longer wavelengths, for a more distant screen and for slits closer together. The centre, $n = 0$ , is always bright, because the two paths are equal there.

Definition

The fringe spacing $w$ is the distance between the centres of two neighbouring bright fringes (or dark fringes): $w = \frac{\lambda D}{d}$ , valid when $D \gg d$ and the angles are small. The integer $n$ is the order of a fringe. Example: with $\lambda = 600$ nm, $D = 1.5$ m and $d = 0.25$ mm, $w = \frac{(600 \times 10^{-9})(1.5)}{0.25 \times 10^{-3}} = 3.6$ mm.

Worked example 1: predicting the pattern

Light of wavelength 600 nm falls on two slits 0.25 mm apart, and the screen is 1.5 m away. Find (a) the fringe spacing, (b) the distance from the centre to the third dark fringe, (c) the fringe spacing if the whole apparatus is immersed in water (refractive index 1.33).

Solution

(a) $w = \frac{\lambda D}{d} = 3.6$ mm (as above).

(b) The dark fringes are at $\left(n + \tfrac{1}{2}\right)w$ ; the third is at $n = 2$ : $2.5 \times 3.6 = 9.0$ mm.

(c) In water the wavelength becomes $\frac{600}{1.33} = 451$ nm, so $w = \frac{3.6}{1.33} = 2.7$ mm: the fringes move closer together.

Worked example 2: measuring a wavelength

In a school experiment with slits 0.50 mm apart and a screen 2.00 m away, a student measures 22.5 mm across ten fringe spacings. Find the wavelength, and explain why she measured ten spacings and not one.

Solution

$w = \frac{22.5}{10} = 2.25$ mm. $\lambda = \frac{wd}{D} = \frac{(2.25 \times 10^{-3})(0.50 \times 10^{-3})}{2.00} = 5.6 \times 10^{-7}$ m $= 563$ nm.

A ruler reading has an uncertainty of about ±0.5 mm. Over one spacing (2.25 mm) that is over 20%; over ten spacings (22.5 mm) it is about 2%.

Fringes in white light

With white light, each colour makes its own set of fringes, with spacing proportional to its wavelength. At the centre all colours have zero path difference, so the central fringe is white. Away from the centre the patterns separate: each bright fringe is coloured, with violet on the inside and red on the outside, and after a few orders the colours overlap so much that the screen looks evenly lit. A white central fringe is a useful way to locate the zero-order fringe.

Worked example 3: overlapping fringes

A double slit is lit with light containing two wavelengths. The third bright fringe of the 450 nm light falls exactly on the second bright fringe of the other wavelength. Find that wavelength.

Solution

The two fringes are at the same $y$ , so $\frac{3(450)D}{d} = \frac{2\lambda D}{d}$ , giving $\lambda = \frac{3 \times 450}{2} = 675$ nm (red light).

Warning

Units and the small-angle condition. Convert $d$ from millimetres and $\lambda$ from nanometres before substituting into $w = \frac{\lambda D}{d}$ . The formula assumes $D \gg d$ and small angles; for large angles use $d\sin\theta = n\lambda$ directly.

Tip

"Wider when Distant, narrower when Divided far apart": $w$ grows with $\lambda$ and $D$ , and shrinks as $d$ grows.

Young's Double Slit Experiment - A Level Physics, Physics Online. Watch how the path difference, d sin θ, leads to the fringe-spacing formula.

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Practice

Explain the purpose of the single slit in Young's experiment with a lamp, and why it is not needed with a laser.
Sodium light (589 nm) passes through slits 0.40 mm apart onto a screen 1.2 m away. Find the fringe spacing.
In Question 2, find the angle of the second-order bright fringe.
Fringes 1.8 mm apart are produced with slits 0.30 mm apart and a screen 0.90 m away. Find the wavelength.
State and explain the effect on the fringe spacing of (a) doubling the slit separation, (b) moving the screen further away, (c) using blue light instead of red.
Why is the central fringe white when white light is used, and why are the other fringes coloured?
A student finds the fringes too close to measure. Suggest two changes to the apparatus and evaluate any drawback of each.

Show answersHide answers

The single slit acts as one narrow source, so the light reaching the two slits comes from the same point and is coherent. A laser beam is already coherent across its width.
$w = \frac{(589 \times 10^{-9})(1.2)}{0.40 \times 10^{-3}} = 1.8 \times 10^{-3}$ m $= 1.8$ mm.
$\sin\theta = \frac{2\lambda}{d} = \frac{2(589 \times 10^{-9})}{0.40 \times 10^{-3}} = 2.9 \times 10^{-3}$ , $\theta = 0.17^\circ$ .
$\lambda = \frac{wd}{D} = \frac{(1.8 \times 10^{-3})(0.30 \times 10^{-3})}{0.90} = 6.0 \times 10^{-7}$ m $= 600$ nm.
(a) Halves it ( $w \propto \frac{1}{d}$ ); (b) increases it in proportion ( $w \propto D$ ); (c) decreases it, because blue light has a shorter wavelength.
At the centre the path difference is zero for every wavelength, so all colours are bright together and add to white. Elsewhere the bright positions $\frac{n\lambda D}{d}$ differ for each wavelength, so the colours separate, with violet nearest the centre.
Move the screen further away (larger $D$ ): wider fringes, but dimmer, since the light spreads over a bigger area. Use slits closer together (smaller $d$ ): wider fringes, but the slits must be finer and make less light, and they are harder to make. Measuring across many fringes or using a travelling microscope also helps.

Key points

Two slits lit by one source are coherent sources; their waves form equally spaced fringes.
Path difference $d\sin\theta$ ; bright where $d\sin\theta = n\lambda$ , dark where $d\sin\theta = \left(n + \frac{1}{2}\right)\lambda$ .
Small angles: $y_n = \frac{n\lambda D}{d}$ , fringe spacing $w = \frac{\lambda D}{d}$ .
White light gives a white central fringe with coloured fringes either side.

Sources and credits

Image: Young Diffraction, drawing by Thomas Young (1807), Wikimedia Commons, public domain.
Video: Young's Double Slit Experiment - A Level Physics, Physics Online, embedded from YouTube.

Written by Ganzaa. The two diagrams, the worked examples and the practice exercises are by Ganzaa.

1The experiment that brought the wave back

2The apparatus

3Deriving the fringe spacing

4Fringes in white light

The experiment that brought the wave back

The apparatus

Deriving the fringe spacing

Fringes in white light