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14.5Multiple slits
As the number of slits increases, the interference pattern increases in complexity. Consider the case of four slits. The intensity pattern is shown in Figure 14.54. In the following graphs units on the vertical axis are arbitrary.We see that there are now two secondary maxima in-between the primary maxima.

Figure 14.54: Intensity pattern for four slits.

Figure 14.55 shows the case of six slits. There are now four secondary maxima between primary maxima. As the number of slits increases the secondary maxima become unimportant. (With N slits, there are N â 2 secondary maxima. The intensity of the central maximum is N2 times the intensity of just one slit by itself.)

Figure 14.55: The intensity distribution for six slits. Note how the width of the maxima decreases but their position stays the same. Note also how the relative importance of the secondary maxima decreases with increasing slit number.

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Figure 14.56 shows the case with 20 slits. The secondary maxima have all but disappeared. We also observe that as the number of slits increases, the width of the primary maxima decreases; the bright fringes become very sharp and easily identifiable.

Figure 14.56: a The intensity distribution for 20 slits. The secondary maxima are completely unimportant and the primary maxima are very narrow. In b the slit width is much smaller so that the single-slit diffraction pattern is very wide. The primary maxima now have roughly the same intensity.

Notice that the slit separation d and the slit width b have been kept the same in all graphs from Figure 14.52 to Figure 14.56a.
So, in summary, we observe that as the number of slits increases:
âThe primary maxima are observed at the same angles
âThe primary maxima get narrower and brighter
âThe secondary maxima become unimportant.
WORKED EXAMPLE 14.8
Look at Figure 14.56a which shows the intensity pattern for 20 slits. Verify that the slit width is three times the wavelength and determine the separation of two consecutive slits in terms of the wavelength.
Answer
The minimum of the single-slit diffraction pattern (the blue curve) is about 19Â° â 0.33 rad.
Using the approximate single-slit diffraction formula Î¸âÎ»b we deduce that:
bâÎ»Î¸=Î»0.33â3Î»
So the slit width is three times the wavelength, as claimed previously.
The first primary maximum away from the central is at Î¸ = 7Â°. Since:
dsinÎ¸=1ÃÎ»
âd=Î»sin7Â°â8Î»
The diffraction grating
The diffraction grating is an important device in spectroscopy (the analysis of light). It is mainly used to measure the wavelength of light. A diffraction grating consists of a large number of parallel slits of very small width. Instead of actual slits, modern gratings consist of a transparent slide on which rulings or grooves have been precisely cut.
The advantage of a large number of slits is that the maxima in the interference pattern are sharp and bright and can easily be distinguished from their neighbours (Figure 14.57). These bright fringes are called âlinesâ. Because the fringes are very narrow the measurement of their separation is more accurate.
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Figure 14.57: The intensity distribution for a diffraction grating. The maxima have roughly the same intensity and are very narrow.

The maxima of the pattern are observed at angles that can be found by an argument similar to that for just two slits: Figure 14.58 shows the path difference (in light blue) between rays leaving the slits. The smallest path difference is Î´: that between the top two rays. By similar triangles, any other path difference is an integral multiple of Î´. So if Î´ is an integral multiple of Î», all other path differences will also be an integral multiple of Î». But Î´ = d sin Î¸. Hence the condition for constructive interference is:
dsinÎ¸=nÎ»,n=0,Â±1,Â±2â¦
In practice, a diffraction grating is stated by its manufacturer to have âx lines per millimetreâ. This means that the separation of the slits is d=1xÂ mm. So a diffraction grating with 600 lines per mm corresponds to a slit separation of d = 1.67 Ã 10â6 m. A maximum observed at say n = 2 is called the second order maximum, that at n = 3 is called the third order maximum and so on.

Figure 14.58: The path difference between any two rays is an integral multiple of the smallest path difference Î´ between the top two rays. The red dashed line is normal to the rays, which makes the angle between this line and the grating Î¸.

CHECK YOURSELF 6
Light consisting of two wavelengths, 400 nm and 500 nm, is incident on a diffraction grating.
Find the order (other than the zero order) of the 400 nm wavelength that coincides with an order of the 500 nm wavelength.
WORKED EXAMPLE 14.9
Light of wavelength 680 nm falls normally on a diffraction grating that has 600 lines per mm.
aWhat is the angle separating the first order (n = 1) from the second (n = 2)?
bHow many maxima can be seen?
Answer
aThe separation between slits is:
d=1600Ã10â3Â m
With n = 1 we find: (sinÎ¸=nÎ»d)
sinÎ¸=1Ã(680Ã10â9)Ã(600Ã103)=0.408
Hence Î¸ = 24.1Â°.
For n = 2 we find Î¸ = 54.7Â°. The angle separating the two is therefore 30.6Â°.
bThe largest Î¸ can get is 90Â° and so 1600Ã10â3Ã1=nÃ680Ã10â9, giving n = 2.45.
Since n is an integer we must take n = 2 as the highest order that can be observed, i.e. 5 in total, 2 on each side plus the central maximum.
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Figure 14.59 shows what happens when white light is incident on a diffraction grating.
In each order, the white light breaks up into its component wavelengths (only three colours are shown here for clarity) except of course for the n = 0 order which appears white. In Figure 14.59a the orders are well separated. In Figure 14.59b the n = 1 and n = 2 orders are beginning to overlap (see problems at the end of this chapter).

Figure 14.59: a Well separated orders for all wavelengths. b The first and second orders are beginning to overlap.

NATURE OF SCIENCE
Curiosity and serendipity
The shimmering colours of a peacockâs tail feathers, the colours inside the paua shell and the glistening surface of a golden stag beetleâs coat are all examples of iridescence in nature. Similar colours are seen in soap bubbles and oil films. Curious as to what caused these beautiful colours and effects in nature, scientists speculated on different mechanisms. One suggestion was that the surface pigments of iridescent feathers appeared to reflect different colours when viewed from different angles.
Another suggestion was that the colours were formed as a result of the structure of the surface, not anything to do with the pigments. With the acceptance of the wave theory of light, scientists in the late nineteenth century developed a theory of interference in thin films that could explain these colours. Developments in microscopy meant that finer and finer structure could be seen, and using an electron microscope in the twentieth century it was possible to see complex thin-film structures in iridescent bird feathers.

Figure 14.60: The iridescent âeyeâ of a peacockâs tail feather.

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In 1817 Joseph Fraunhofer accidentally produced a thin film coating when he left nitric acid on a polished glass surface. Experimenting with different glasses, he was able to produce a surface with the same vivid colours seen in soap bubbles. When he observed the same colours appearing as a coating of alcohol evaporated from a polished metal surface, he concluded that such colours would appear in any transparent thin film. This was the start of the technology of thin films, developed in the twentieth century to make optical coatings for glass.
TEST YOUR UNDERSTANDING
39aDraw a graph to show the variation with angle of the intensity of light observed on a screen some distance from two very narrow, parallel slits when coherent monochromatic light falls on the slits.
bDescribe how the graph you drew in a changes when:
ithe number of slits increases but their separation stays the same
iithe number of slits stays at two but their separation decreases.
40A diffraction grating with 400 lines per mm is illuminated normally with light of wavelength 620 nm.
aDetermine the angle of the first order maximum.
bDetermine the largest order that can be seen with this grating and this wavelength.
41A diffraction grating has 300 lines per mm. Light consisting of two wavelengths, 500 nm and 600 nm, is incident normally on the diffraction grating. What is the angle separating the second order maxima for each wavelength?
42A diffraction grating has 150 lines per mm. Light consisting of two wavelengths, 400 nm and 500 nm, is incident normally on a diffraction grating. What is the smallest angle for which maxima of the two wavelengths coincide?
43A diffraction grating produces two consecutive maxima at angles 17.46Â° and 23.58Â° with a wavelength of 656 nm. How many lines per mm does the grating have?
44Light (with wavelengths in the range of 400 nm to 700 nm) is incident normally on a diffraction grating with 400 lines per mm.
aShow that there are only seven maxima in which all wavelengths are present.
bWhat is the largest wavelength present in the fourth order?
45In a multiple slit arrangement the slit width is 1.20 Ã 10â6 m. The n = 4 interference maximum is missing. What is the separation of the slits?
46Light containing wavelengths in the range 420 nm to 630 nm is incident normally on a diffraction grating with 160 lines per mm. What is the smallest angle at which there is overlap between consecutive orders?
47A diffraction grating has 300 lines per mm. Visible light (400 nm to 700 nm) is incident normally on the grating.
aWhat is the highest order for which all wavelengths are present?
bWhat is the angular width of the order in part a?
