Diffraction gratings
If the number of slits in an obstacle is now
increased we will see that the sharpness of the
pattern is improved, the maxima getting
narrower. Obstacles with a large number of slits
(more than, say, 20 to the millimetre) are called
diffraction gratings. These were first developed
by Fraunhofer in the late eighteenth century and
they consisted of fine silver wire wound on two
parallel screws giving about 30 obstacles to the
millimetre.
diffraction grating
surface of grating
Since then many improvements have been
Figure 1
made, in 1882 Rowland used a diamond to rule
fine lines on glass, the ridges acting as the slits and the rulings as the obstacles (Figure 1).
Using this method it is possible to obtain diffraction gratings with as many as 3000 lines per
millimetre although 'coarse' gratings with about 500 lines per millimetre are better for general
use.
In many schools two types are in common use, one with 300 lines per mm and the other with
80 lines per mm.
Reflection gratings are also used, where the diffracted image is viewed after reflection from a
ruled surface. A very good example of a reflection diffraction grating is a CD. A DVD with finer
rulings gives a much broader diffraction pattern.
The wave theory and the diffraction grating
Figure 2
New wave front
Figure 2 shows the Huygens construction for a grating. You can see how the circular diffracted
waves from each slit add together in certain directions to give a diffracted wave which has a
plane wave front just like the waves hitting the grating from the left. This plane wave is formed
by drawing the line that meets all the small circular waves and is called an envelope of all these
small secondary waves.
1
The diffraction grating formula
Consider a parallel beam of light incident normally
on a diffraction grating with a grating spacing e (the
grating spacing is the inverse of the number of lines
per unit length). Consider light that is diffracted at an
angle  to the normal and coming from
corresponding points on adjacent slits (Figure 3).
For a maximum the path difference = AC = m
But AC = e sin. Therefore for a maximum:
A

B
C


Diffraction grating maximum
m = e sin

where m = 0, 1, 2,3...
Figure 3
The number m is known as the order of the spectrum, that is, a first-order spectrum is formed
for m = 1, and so on.
If light of a single wavelength, such as that from a laser, is used, then a series of sharp lines
occur, one line to each order of the spectrum. With a white light source a series of spectra is
formed with the light of the shortest wavelength having the smallest angle of diffraction.
In deriving the formula above, we assumed that the incident beam is at right angles to the face
of the grating. Allowance must be made if this is not the case. The simplest way is to measure
the position of the first order spectrum on either side of the centre, record the angle between
these positions and then halve it, as shown in Figure 4.
The number of orders of spectra visible
with a given grating depends on the
grating spacing, more spectra being visible
with coarser gratings. The ruled face of the
grating should always point away from the
incident light to prevent errors due to
changes of direction because of refraction
in the glass. The diagram shows a central
white fringe with three spectra on either
side giving a total of seven images.
(See example problem )
m=0
m=3
m=2
m=1
m=1
m=2
m=3
Figure 4
2
Example problems
1. Calculate the wavelength of the monochromatic light where the second order image is diffracted
through an angle of 25o using a diffraction grating with 300 lines per millimetre.
Grating spacing (e) = 10-3/300 m = 3.3x10-6 m
Wavelength () = esin25/2 = [3.3x10-6 x 0.42]/2 = 6.97 x 10-7 m = 697 nm
2. Calculate the maximum number of orders visible with a diffraction grating of 500 lines per
millimetre, using light of wavelength 600 nm.
Maximum angle of diffraction = 90o e = 10-3/500 = 2x10-6 m
Therefore m = esin/ = 2x10-6/600x10-9 = 3.33
Therefore maximum number of orders = 3, and a total of seven images of the source can be seen
(three on each side of a central image).
Intensity
The intensity distribution for a
large number of slits is shown
in Figure 5. Notice that the
maxima become much sharper;
the greater the number of slits
per metre, the better defined
are the maxima.
m=1
m=1
m=2
m=2
Figure 5
Student investigation
The diffraction of cadmium or mercury light is
used to determine the separation of two lines
on an integrated circuit. The following results
were obtained for the second order diffracted
images for different wavelengths. Use them to
plot an appropriate linear graph and thence
determine the mean spacing of the wires on
the circuit.
Wavelength/nm
468
480
509
546
577
644
Angle of diffraction (o)
28.0
28.7
31.0
33.0
35.5
40.0
3