Interference
When two groups of waves (called wave trains) meet and overlap they interfere with each other.
The resulting amplitude will depend on the amplitudes of both the waves at that point.
If the crest of one wave meets the crest of the other the waves are said to be in phase and the
resulting intensity will be large. This is known as constructive interference. If the crest of one
wave meets the trough of the other (and the waves are of equal amplitude) they are said to be
out of phase by  then the resulting intensity will be zero. This is known as destructive
interference.
This phase difference may be produced by allowing the two sets of waves to travel different
distances - this difference in distance of travel is called the path difference between the two
waves.
There will be many intermediate conditions between these two extremes that will give a small
variation in intensity but we will confine ourselves to total constructive or total destructive
interference for the moment.
The diagrams in Figure 1 below show two waves of equal amplitudes with different phase and
path differences between them. The first pair have a phase difference of  or 180o and a path
difference of an odd number of half-wavelengths. The second pair have a phase difference of
zero and a path difference of a whole number of wavelengths, including zero.
+
=
(a) destructive interference
+
Figure 1
=
(b) constructive interference
Figure 1(a) shows destructive interference and Figure 1(b) constructive interference.
To obtain a static interference pattern at a point (that is, one that is constant with time) we must
have
(a) two sources of the same wavelength, and
(b) two sources which have a constant phase difference between them.
Sources with synchronised phase changes between them are called coherent sources and
those with random phase changes are called incoherent sources.
This condition is met by two speakers connected to a signal generator because the sound
waves that they emit are continuous – there are no breaks in the waves.
1
However two separate light sources cannot be used as sources for a static interference pattern
because although they may be monochromatic the light from them is emitted in a random series
of pulses of around 10-8 s duration. The phase difference that may exist between one pair of
pulses emitted from the source may well be quite different from that between the next pair of
pulses (Figure 2).
Pulses from source A
Pulses from source B
Figure 2
Therefore although an interference pattern still occurs, it changes so rapidly that you get the
impression of uniform illumination. Another problem is that the atoms emitting the light may
collide with each other so producing phase changes within one individual photon. We must
therefore use one light source and split the waves from it into two in some way.
There are two ways of doing this:
(a) division of amplitude, where the amplitude at all points along the wavefront is divided
between the two secondary waves, and
(b) division of wavefront, where the original wave-front is divided in two, half of it forming
each of the secondary waves.
However, the length of each pulse limits the path difference that we may obtain between
even these two waves from the same source. Since the pulses are only about 10-8 s long the
maximum path difference is 3 m, although in practice good results are only obtained with shorter
path differences than this.
Interference between two waves
The diagrams in Figure 3-7 show two sources S1 and S2 emitting waves - they could be light,
sound or microwaves.
Minimum – crest meets trough
The plan view of the waves in Figure 3
shows waves coming from two slits and
interfering with each other. The lines
Maximum – crest meets crest
along which the path differences will
give maxima or minima.
Minimum – crest meets trough
S1
Maximum – crest meets crest
S2
Minimum – crest meets trough
Maximum – crest meets crest
Figure 3
This type of arrangement is like that
produced in a ripple tank or in the
double slits experiment with light (see
later).
It should be realised that between the
maxima and minima the intensity will
change gradually from one extreme to
the other.
Minimum – crest meets trough
2
Figure 4 (a)
Figure 4(a) shows light interfering as it
passes through two slits. In Figure 4(b)
the appearance of the interference
pattern on a screen placed in the path of
the beam is shown.
You can see the maxima and minima
and the way in which the intensity
changes from one to the other.
Changing the wavelength of the light (its
wavelength), the separation of the slits or
the distance of the slits from the screen
will all give changes in the separation of
the maxima in the interference pattern.
Figure 4 (b)
minimum
maximum
maximum
Figure 5
maximum
Figure 5 shows the interference effects of two
speakers. The sound waves spread out all
round the speakers and a static interference
pattern is formed. (Not all the maxima and
minima are labelled). You can hear this by
setting up two speakers in the lab connected to
one signal generator and then simply walking
round the room. You will hear the sound go
from loud to soft as you pass from maximum to
minimum. (A frequency of around 400 Hz is
suitable).
minimum
3
In Figures 6 and 7 you can see that at the different points on the screen the waves from S1 have
travelled a different distance from those from S2. In Figure 6 the path difference is zero, in
Figure 7 it is half a wavelength
Path difference = /2
Path difference = 0
S1
S1
S2
screen
S2
screen
Figure 6
Figure 7
Path length in a material
When light passes through a material of refractive
index n it is slowed down, its velocity in the material
being 1/n times that in a vacuum.
For example, the velocity of light in glass is about
2.0x108 ms-1 compared with about 3.00 x 108 ms-1
in a vacuum.
Path length = nL
gas such as air
Path length = L
vacuum
L
vacuum
The time light takes to pass through a given length
of the material is therefore n times that which it
takes to pass through the same length of air.
Equivalent path length = nL
Figure 8
The path length in a material of length L and refractive index n is therefore nL (Figure 8). If one
part of a light beam travels a distance L in air and the other a distance L in the material then a
path difference will exist between them of L(n - 1) and if the two beams are made to overlap an
interference pattern will result.
Example problem
Calculate the path difference formed between two beams of yellow light (wavelength = 600 nm), one
of which passes through 1 m of air (refractive index 1.000298) and the other that passes through 1 m
of free space (a vacuum).
Path difference = 1(1.000298 - 1) = 0.000298 m = 0.298 mm = 0.298/0.0006 = 500 wavelengths
4