SPH3UW/SPH4UI
Notes
Unit 9.4 Multi-Slit (Diffraction Grating)
Page 1 of 5
Physics Tool box

The surface of a diffraction grating consists of a large number of closely
spaced parallel slits.
The diffraction gratings deliver brighter interference patterns that typical
double slit set-ups. With maxima that are narrower and more widely
separated.

Diffraction gratings are governed by the relationship

d sin m   m , where
d is the distance between adjacent gratings, and m is the order of the
maxima.
When you have a large number of equally spaced parallel slits is called a diffraction grating.
Bet better name for this apparatus would be a Multi-slit interference grating. A typical
diffraction grating would have on the order of 10,000 lines per centimetre. The analysis of
a diffraction grating will be very similar to the analysis technique we used on the Thomas
Young Double Slit.
The major difference between the pattern produced by a diffraction grating and a double
slit is that the maximum (bright bands) bands are more sharper and narrower for a grating
than for a double slit.



The surface of a diffraction grating consists of a large number of closely spaced
parallel slits.
The diffraction gratings deliver brighter interference patterns that typical double slit
set-ups. With maxima that are narrower and more widely separated.
Diffraction gratings are governed by the relationship
d sin m   m , where d is the
distance between adjacent gratings, and m is the order of the maxima.
Note:
This formula is identical to the one we used for double Slit interference. We need only
determine d, the slit difference by dividing the grating width by the number of slits.
d
width
# gratings
Example
Determine the distance d between the gratings ifwe have a 1 cm gratings square with 10,000 wires.
Solution
width
# gratings
1cm

10000
d
 0.001cm
SPH3UW/SPH4UI
Unit 9.4 Multi-Slit (Diffraction Grating)
Page 2 of 5
The intensity of the interference bands goes as the square of the number of slits (more light is hitting
the screen). That is, a 3 slit grating will produce bands 9 ( 32 ) times more intense than a single slit.
For a grating, the waves from two adjacent will not significantly be out of phase. But the phase from slit
1 and slit 3 will be slightly more out of phase, and so on until we obtain a slit number (say 1000) that
will be completely out of phase. This will provide us with more of a cancelling (destructive interference)
effect between each constructive interference patter (see figure above).
SPH3UW/SPH4UI
Double Slit
Unit 9.4 Multi-Slit (Diffraction Grating)
Page 3 of 5
10 slits
Example
At what angle will 652–nm light produce a third order maximum when passing through a
grating of 1000 lines/cm?
Solution:
  6.52 107 m
We ensured that the distance
and the wavelength are in the
same units, metres.
1
 1.00 105 m
lines
1000
cm
m3
d
sin  m  
sin 3  
m
d
 3  6.52 107 m 
1.00 105 m
3  11.3
Suppose that the light that passes through the diffraction grating is not a single colour
(monochromatic), but rather consists of two or more distinct colours (wavelengths) . Then for all orders
other than m=0, we would have a different colour producing a maximum at a different angle. This will
provide us a distinct spectrum of colours all spread out over the screen.
SPH3UW/SPH4UI
Unit 9.4 Multi-Slit (Diffraction Grating)
Page 4 of 5
Example
Determine the angular positions of the first and second order maxima for light of wavelength
750nm incident on a diffraction grating containing 10,000 lines/cm.
Solution
First, let’s determine the distance between each line.
d
1cm
 1.00 104 cm  1.00 106 m
10000
For first maximum, m=1
d sin  450   m
sin  450  
m
d
 m 

 d 
 450  sin 1 
 1  7.50 107 m  

 sin 
 1.00 106 m 


 48.6
1
For first maximum, m=2
d sin  750   m
sin  750  
m
d
 m 

 d 
750  sin 1 
  2   7.50 107 m  

 sin 1 
 1.00 106 m 


 error
There no higher orders than 1 will appear for this wavelength of light.
SPH3UW/SPH4UI
Unit 9.4 Multi-Slit (Diffraction Grating)
Extra Notes and Comments
Page 5 of 5