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Diffraction - acknowledgements to R. S. Bocking

Diffraction is the spreading of waves as they pass through an opening or round an obstacle into regions where we would not expect them.

Diffraction of water waves is seen using a ripple tank.

 

 

The greatest effect is seen when the gap is comparable to the wavelength of the waves and diffraction of the wavefronts extends completely into the geometrical shadow (i.e. the slit behaves as a point source).

Diffraction of light waves

If a point source of light is observed through a single narrow slit a pattern is observed consisting of a central band of light with bands of light on either side. The bands become dimmer the further they are from the centre. The bands of light get closer together as the colour of the light changes from red to green to blue.

1.        How does the separation of the bands vary with frequency of light?

The pattern can be accounted for by considering Huygens construction. We consider each point in the aperture through which the waves pass to be made up of point sources each of which produces secondary wavelets, which then spread out and interfere with each other.

DIFFRACTION INVOLVES SUPERPOSITION OF WAVELETS FROM DIFFERENT PARTS OF THE SAME WAVEFRONT.

The diagram below helps to explain the single slit pattern of light and dark bands. 

 

 Consider the single slit to be made up of two strips, each half the width of the slit. A secondary source in one half has a corresponding source in the other half (A1 and A2 in the diagram are a corresponding pair). Wavelets from corresponding pairs of sources always arrive in phase at O and so a bright fringe is observed. Now consider point P. For a dark fringe to be seen at P wavelets from corresponding pairs of sources must arrive at P exactly half a wavelength out of phase so that they cancel each other.

 If q is the direction of the minima, l is the wavelength and a is the width of the slit it can be shown that 

                   sin q  =  l

                                  a

 Background reading : Breithaupt p269-276  

Intensity Distribution

 

The intensity of the bands on either side of the central band are approximately 5% or less of the intensity of the central band.

If the number of slits is increased then each slit produces a diffraction pattern and also there is interference between the diffraction patterns of each slit. The resulting pattern is one of bright and dark fringes which decrease in intensity with distance from the central maximum.

The Diffraction Grating

In principle the diffraction grating is a large number of narrow, parallel, equidistant slits. It is usually made by scoring lines on glass using a fine diamond point. The lines scatter the incident light and behave as though they are opaque while the spaces between them transmit light and behave as slits. 

Diffraction gratings are usually quoted in terms of the number of lines per metre. The grating spacing is therefore the inverse of this value.

Example : A grating has 2000 lines per millimetre. Calculate the grating spacing.

A fine grating (2000mm-1) produces more sharply defined spectra which are further apart than does a coarse grating (300mm-1).

Theory (for the diffraction grating equation) 

 

In the diagram d is the distance between corresponding points, A and B, in two adjacent slits (ie. d is the grating spacing).

Consider a wavelet leaving the top slit from point A and the bottom slit from point B.

For constructive interference to occur in the direction q, the distance AC must be equal to a whole number of wavelengths.

                   AC = nl          where n is an integer corresponding to the order of the spectrum

Angle ACB = 900 and so

                   Sin q  = AC

                                 d

Thus, for constructive interference

                   d sin q = nl          This is the diffraction grating equation 

Since sin q can never be greater than 1, then nl / d can never be greater than 1. This means that the order number n can never be greater than d/l.

Example calculation

 A diffraction grating with 500 lines mm-1 is illuminated normally by yellow light of wavelength 5.89 x 10-7 m.

a)      at what angle to the normal is the first order image ?

b)      what is the highest order in which an image can be obtained ?

c)       what is the total number of diffracted beams that can be observed with this grating ?  

Missing orders

A diffraction grating produces an interference pattern with a single slit diffraction pattern superimposed upon it. For a single slit the position of the first minimum is given by

                      Sin q = l / a        where a is the slit width

Thus when

                      Sin q = l / a   =   nl /d        no order will be observed

The order is said to be missing and the number of the missing order is given by

                   n  = d / a

Diffraction Questions

1. Monochromatic light of wavelength 5.0 x 10-7 m falls normally on a diffraction grating having 600 lines mm-1. Calculate the angular position of the second order image.

2. A diffraction grating has 800 lines mm-1 and is illuminated normally by monochromatic light of wavelengths 5.6 x 10-7 m and 5.9 x 10-7 m. Calculate the angular difference between the two first order images.

3. A parallel beam of sodium light of wavelength 5.89 x 10-7 m is incident normally on a grating, and the first order diffraction image is formed at an angle of of 26 13 with the normal. Calculate the number of lines per centimetre ruled on the grating.

4. A diffraction grating has 400 lines mm-1 and is illuminated normally by monochromatic light of wavelengths 4.5 x 10-7 m and 6.0 x 10-7 m. In what respective order of image do the colours overlap, and what is the corresponding angle of diffraction ?

Email Mr. Lindley if you'd like me to upload the answers to these questions