Diffraction is the spreading out of a wave when passing through a gap in a boundary. It is optimised when the width of the gap is similar in magnitude to the wavelength of the wave. With a small enough gap we can diffract visible light.
X-rays have an even smaller wavelength than visible light. These diffract on an atomic scale, enabling materials scientists to determine the structure of crystals.
Key Concepts
A laser is shone through a rectangular slit onto a screen.
We might expect the screen to be brightest in the centre and for the laser light to reduce in intensity outwards. In fact, the pattern is not so simple: the brightness minimises before increasing again. We can plot a graph of intensity against distance from the centre.
The diffraction pattern is formed by summing together to vector components of all parts of the wave, which have travelled different distances to each point on the screen.
We can determine the position of the first minimum using the approximation equation:
\(\theta = {\lambda \over a}\)
- \(\theta\) is the angle subtended at the slit by the first minimum and the centre (rad)
- \(\lambda\) is the wavelength of the light (m)
- \(a\) is the width of the slit (m)
Since the angle is small, we can also state that it is approximately equal to the ratio of the distance to the minimum and the distance from the slit to the screen:
\(\theta = {y\over D}\)
- \(y\) is the distance between the centre and the first minimum (on either side since symmetrical) (m)
- \(D\) is the distance from the slit to the screen (m)
The equation for the first minimum indicates that the angle is increased by increasing the wavelength of light. Red light produces a broader pattern than green and blue light when passing through the same slit.
White light contains all the wavelengths of visible light. Because the angle to the first minimum is affected by wavelength, the colours of light separate.
This is not evident in the central maxium as all wavelengths have travelled the same distance and so the vector components add to give the original combination. However, outward from the centre, the colours of light separate continuously as their wavelengths form maxima at slightly different angles. Blue light is nearest the centre and red outermost.
Use quizzes to practise application of theory.
START QUIZ!
Red light is diffracted by a single slit.
If the amplitude of the source is reduced by a factor \(1\over 2\) the intenity of the peak will be reduced by a factor:
Intensity is proportional to the square of the amplitude:
\({I_1\over I_2}={A_1\over A_2}^2\)
Light is diffracted by a single slit.
\(D\) = 1 m
\(y\) = 5 mm
What is the angle subtended by the first minimum?
\(\theta={5\times 10^{-3}\over 1}\)
This equation relies on the small angle approximation. Therefore the angle is measured in radians.
Light is diffracted by a single slit.
\(D\) = 50 cm
\(y\) = 20 cm
What is the angle subtended by the first minimum?
In this case the small angle approximation does not apply. Instead we use trigonometry:
\(\tan \theta={y\over D}={20\over 50}\)
\(\tan^{-1}0.4=22^\text{o}\)
Light is diffracted by a single slit.
\(D\) = 1 m
\(\theta\) = 0.01 rad
The width of the central maximum is:
The small angle approximation applies:
\(y=D\times \theta=0.01 \text{ m}\)
The width of central maxmum (see diagram) is \(2y=0.02 \text{ m}\)
Light is diffracted by a single slit.
\(D\) = 1 m
\(λ\) = 400 nm
\(a\) = 0.2 mm
What is the angle subtended by the first minimum?
In this case we are not given the distance \(y\). Therefore, we use the small angle approximation at the slit for the path difference to cause destructive interference:
\(\theta={\lambda\over a}={400 \times 10^{-9}\over 0.2 \times 10^{-3}} = 0.002 \text{ rad}\)
Light is diffracted by a single slit.
\(D\) = 1 m
\(y\) = 1 mm
The distance \(D\) is doubled. To keep \(y\) constant, the slit width must be changed by a factor:
The small angle approximation applies, so \(\sin\theta =\tan\theta\)
\({y\over D} = {λ\over a}\)
Since \(y\) and \(\lambda\) are constant, \(D\over a\) must be constant.
Diffraction pattern A is produced with a blue laser then pattern B is produced with a red laser.
What change must have taken place?
Red light has a longer wavelength than blue light so the pattern should be more spread. This means that \(\theta\) has been reduced between the two productions:
\(\theta={\lambda \over a}\) so \(a\) could have been increased.
Alternatively, reducing \(D\) would give less distance for the same angle to spread out.
The image is from a GeoGebra simulation showing wavelet amplitudes adding across a single slit.
At the position of the first minimum, the phase difference between the first and last wavelet is:
When the resultant amplitude is zero, the wavelet vectors would have formed a closed loop. The phase difference is \(2π\).
How much of Single slit diffraction have you understood?
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