## Specifically light 831 Diffraction of light

To try the experiment with light we change the set-up slightly by shining a beam of light through a thin vertical slit. Some distance behind the slit we place a screen at right angles to the light beam. If the light is diffracted as it passes through the slit and produces an interference pattern in the same way as the water waves, we should see the evidence in the final image on the screen.

We can predict the form of the image using Huygens' method in the same way as we did for water waves. Looking at Figure 8.5, let us imagine a screen on the right side of the picture. Bright bands occur when the overlapping wavelets

intensity

intensity position

Figure 8.6 Distribution of intensity in the image of light diffracted from a slit.

interfere constructively as they arrive at the screen. Dark areas on the screen correspond to regions of calm water. The central bright band is brighter than the others because more light goes forwards than in any other direction. According to the rules of diffraction we have a bright band in the middle. The secondary maxima to either side are much less bright and about half the width of the central maximum.

The width of the diffraction pattern depends on the ratio of the wavelength to the width of the slit. The pattern of intensity variation represented in Figure 8.6 is taken from the mathematical treatment in Appendix 8.1 where Huygens' principle is used to predict the form of the image.

8.3.2 The experiment with light

The experimental set-up is quite simple. The slit and laser beam are schematic drawings, but the image is an

Single slit diffraction experiment.

actual photograph of what appears on the screen. There is a central bright spot of maximum intensity with smaller peaks to the left and right. These peaks occur where the secondary wavelets interfere constructively and are separated by nodes where there is destructive interference. Note that whereas the slit is vertical, the diffraction pattern is spread out horizontally.

Contrary to common instinct, as the slit width is made narrower the angular width of the central maximum increases. Eventually we reach a limit where we effectively have a point source and the central maximum expands to the whole width of the screen. The three images show how the image expands laterally as the slit width is decreased (top to bottom in the picture).

Diffraction patterns for different slit widths.

8.3.3 Other apertures

The diffraction pattern of a square aperture is a superposition of vertical and horizontal single slit patterns. The symmetrical pattern reflects the proportions of the aperture.

Diffraction pattern of a square aperture: real (left) and computer reconstruction (right).

The diffraction pattern of a circular aperture (hole) is a series of concentric bright and dark rings, as we might expect from symmetry.

Diffraction images on pp. 248-250, Courtesty of James Ellis, USD School of Physics, UCD.

### 8.3.4 The curious case of the opaque disc

It would be perfectly reasonable to expect the diffraction pattern of a small opaque disc of about the same size as the hole to have the same symmetry as the diffraction pattern from the hole with a dark spot at the centre, but that is not what wave theory predicts.

In 1818, Augustin Fresnel (1788-1827) submitted a paper on the wave theory of diffraction for a competition sponsored by the French Academy of Sciences. Simeon Poisson (1781-1840), a member of the judging committee, applied Fresnel's theory to the diffraction of light by a small circular disc. It follows from symmetry that the wavelets from all sources around the circumference of the disc will all be in phase when they meet at some central point downstream from the disc. Thus the theory predicts that at the very centre of the diffraction pattern there should be a bright spot! Poisson, who did not believe in the wave theory of light, deemed this to be absurd. Fortunately Dominique Arago (1786-1853), also a member of the judging committee, soon verified the existence of the 'absurd' spot with a

Diffraction pattern of a circular aperture.
Poisson's Spot: Shadow of a ball bearing suspended on a needle. Courtesy of Chris Jones, Union College, Schenectady, NY, USA.

simple experiment. Fresnel won the competition and the wave theory triumphed again. It is ironic that the spot is generally referred to as Poisson's spot.

8.4 Is there a limit to what we can distinguish?

The extent to which optical systems can resolve closely spaced objects is limited by the wave nature of light.

### 8.4.1 Images may overlap

The resolving power of an optical instrument is its ability to produce separate images of adjacent points. Light entering an optical instrument through some sort of aperture (the pupil of an eye, the lens of a camera and the objective lens of a telescope are some examples) is diffracted. The image of a point source of light is never itself a point but is a pool of light accompanied by concentric bright rings. This is clearly illustrated in the image taken with the Nicmos camera on the Hubble Space Telescope.

Images of stars at the galactic centre. Courtesy of NASA JPL-Caltech.

8.4.2 The Rayleigh criterion

John William Strutt, Lord Rayleigh (1842-1919), received the Nobel Prize for Physics in 1904 for his discovery of the rare gas argon. His work, both experimental and theoretical, spanned almost the entire field of physics. Rayleigh developed the theory which explains the blue colour of the sky as the result of scattering of sunlight by small particles in the atmosphere. Craters on Mars and the moon are named after him.

Lord Rayleigh

When a telescope is pointed at a group of stars, light from each star passes through the aperture and is independently diffracted. The extent to which the images of individual stars can be resolved depends on how much their diffraction patterns overlap.

The amount of overlap depends on the size of the angle between the beams of light from the two stars. As this angle becomes smaller the individual diffraction patterns merge into one.

Rayleigh's empirical criterion for resolving two point sources of equal brightness says that two objects are 'just resolved' when the central peak of the diffraction pattern from the first object coincides with the first minimum of the pattern from the second.

In Figure 8.7 the images of the two stars are just resolved according to Rayleigh's criterion. At that point, the angle between the light beams from the two stars has a value equal to one half of the angular width of the main maximum of the diffraction pattern (Appendix 8.1). This then is the

Figure 8.7 Image of a pair of distant stars.

minimum angular separation of objects which are just resolved by the telescope. It is called the angular resolution or diffraction limit.

F or circular apertures Q =-

To improve the angular resolution we can either increase the size of the aperture or decrease the value of the wavelength.

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