Conditions for Interference
Recall that light, which obeys the wave equation, must also obey the principle of
superposition. In fact, both the electric and magnetic fields of the light in a region where two light
waves overlap must be the vector sums of the individual electric and magnetic fields. Interference is
collective name for the effects that occur when two or more light waves overlap to give an irradiance
that is different from the sum of the component irradiances. The irradiance is defined as the
average energy per area per unit time falling on a surface. It is proportional to the square of the
amplitude of the light wave:
I = < S > = = ε
_{0}
c < E
^{2} > =


where
< F >
denotes the time average of
F
, and
E
_{0}
is the amplitude of the electric field vector. The
irradiance is a very useful concept to use because it can be easily measured by a range of optical
instruments (including our eyeirradiance is a measure of the 'brightness' we perceive).
When we have a single light wave the value of
< E
^{2} >
is just given by
< E.E >
. However, if
two electric field vectors
E
_{
1
}
and
E
_{
2
}
overlap at a point we have
< E
^{2} > = < (E
_{
1
} + E
_{
2
}).(E
_{
1
} + E
_{
2
}) >
from the principle of superposition. Since
< (E
_{
1
} + E
_{
2
}).(E
_{
1
} + E
_{
2
}) = E
_{
1
}
^{2} + E
_{
2
}
^{2} +2E
_{
1
}
.E
_{
2
}
, we can write:
< E
^{2} > = < E
_{
1
}
^{2} > + < E
_{
2
}
^{2} > +2 < E
_{
1
}
.E
_{
2
} >


The last term is called the interference term. We can evaluate that term in the case of
harmonic waves as:
E
_{
1
}
.E
_{
2
}


= E
_{01}.E
_{02}cos(k
_{1}
x  σt + ε
_{1})cos(k
_{2}
x  σt + ε
_{2}) 



= E
_{01}.E
_{02}[cos(k
_{1}
x + ε
_{1})cos(σt) + sin(k
_{1}
x + ε
_{1})sin(σt)] 



ƒ[cos(k
_{2}
x  ε
_{2})cos(σt) + sin(k
_{2}
x  σt)sin(σt)] 

Where
ε
_{1}
and
ε
_{2}
are arbitrary phase constants. Taking the time average
and using the fact that
< cos^{2}(σt) > = < sin^{2}(σt) > = 1/2
and
< sin(σt)cos(σt) > = 0
we find:
< E
_{
1
}
.E
_{
2
} > = 1/2×E
_{01}.E
_{02}cos(k
_{1}
x + ε
_{1}  k
_{2}
x  ε
_{2})


But the argument of the cosine is just the phase difference between the two waves, arising from their
initial phase offset (
ε
s) and their different path lengths (the distance they took to reach the
point). Thus we can write this as:
2 < E
_{
1
}
.E
_{
2
} > = E
_{01}.E
_{02}cosδ


Since
< E
_{
1
}
^{2} > =
and recalling that irradiance is proportional to
< E
^{2} >
we can write an expression for the total irradiance at a point as:
I = I
_{1} + I
_{2} +2cosδ


since
I
_{1} = εc < E
_{
1
}
^{2} >
and
2 < E
_{
1
}
.E
_{
2
} > = 2εc
.
Thus at various points in space the irradiance can be greater than, equal to or less that
I
_{1} + I
_{2}
depending on
δ
. Maxima (total constructive interference) occur where
cosδ = 1
, or where
the component waves are completely inphase. Minima (total destructive interference) occur
where
cosδ =  1
and the phase difference is an odd multiple of
±Π
. When the
amplitudes of the two overlapping waves are equal (
E
_{01} = E
_{02}
) we have
I
_{1} = I
_{2} = I
_{0}
and:
I = 2I
_{0} +2I
_{0}cosδ = 4I
_{0}cos^{2}(δ/2)


For two beams to interfere to produce a stable pattern, they must have very nearly the same frequency. If
this is not the case,
δ
would vary very rapidly in time and the interference term would average to zero
over time. However, two white light (polychromatic) beams will produce an interference pattern since blue
will interfere with blue, yellow with yellow, etc. For the clearest patterns waves should also have equal
amplitudes; this allows complete destructive interference when the waves are out of phase. The sources,
however, needn't be inphase with one another, however, the phase difference between them must remain constant;
such sources are said to be coherent.
Young's Experiment
The macroscopic result of the interference term is that interfering sources create a pattern of varying
brightness on an appropriately placed screen. This may be caused either by the two sources being
outofphase with one another or by the two sources traversing paths of different lengths to a
particular
point on the screen. Path lengths that vary by amounts of the order of a fraction of the wavelength of the
light cause arriving waves to be outofphase with one another (for example if the path lengths of in
phase sources differ by
λ/2
a peak will arrive with a trough and destructive interference
will result) and interfere. The light and dark zones created by an two of more interfering waves are called
fringes.
One of the simplest and most important examples of this effect is an experiment, first performed by Thomas
Young in 1805 to prove that light had a wave nature. He shined sunlight through a single pinhole to create
one coherent beam which he then shone through two slitshaped apertures. The cylindrical waves emerging
from these slits comprise coherent, inphase sources. Another alternative is to use plane waves
from a laser to illuminate the apertures. Young's arrangement is shown in .
Figure %: Young's Double Slit Experiment
The distance between the apertures and the screen must be many times (thousands) greater than the distance
between the slits. The approximate path difference to a point P is given by
S
_{1}
P  S
_{2}
P = S
_{1}
B
, where
S
_{2}
B
is perpendicular to
S
_{1}
P
. But from the diagram it is clear that
S
_{1}
B = d sinθ
_{m}
, where
d
is the distance between the slits. When the screen is very far away, the angle
θ
_{m}
will be very
small and we can use the approximation
sinθ
_{m}
θ
_{m}
to write:
S
_{1}
B
aθ
_{m}
. But
sinθ
_{m}
θ
_{m}
y
_{m}/L
and this implies
S
_{1}
B
ay
_{m}/L
.
Now, constructive interference will occur when
S
_{1}
B
is equal to a whole number of wavelengths. So for
an integer
m
,
S
_{1}
B = mλ
so the position of the bright fringes is given by:
y
_{m}


The angular position of the fringes is given by:
sinθ
_{m} =


or
θ
_{m}
mλ/d
. The spacing between fringes is given by:
Since red light has a greater wavelength than blue light, red fringes must be broader. Note that the
region in the center of the screen, between the slits is a bright maximum, called the zeroth order fringe
corresponding to
m = 0
. The phase difference at any point is given by
δ = k(S
_{1}
B)
where
k = 2Π/λ
. If the beams both have irradiance
I
_{0}
the irradiance at any point is
given by:
I = 4I
_{0}cos^{2}
= 4I
_{0}cos^{2}


It is worth bearing in mind that these results are an idealization assuming an infinitesimally small width of the
slits. A full treatment of the situation, accounting for the finite width of the slits requires us to take
diffraction into account. A plot of irradiance against position for Young's experiment is shown in
.
Figure %: Irradiance as a function of position for Young's slits.
Note that the condition for destructive interference (often called nodes) is
(m + 1/2)λ = d sinθ
_{m}
, such that the waves are exactly 180
^{
o
}
out of phase.
Interferometers
Interferometers are devices that employ the effects of interference to make very accurate measurements.
The most famous of these is called a Michelson interferometer, which was used to measure the wavelength
of lines in atomic spectra.
Figure %: A Michelson Interferometer
The light from a source is split into two by a beamsplitter, half traveling to the right towards the mirror
M
_{1}
and half upwards towards the mirror
M
_{2}
. Both beams reflect back from the mirrors to the
beamsplitter. Part of each of the two beams from
M
_{1}
and
M
_{2}
are reflected downwards and are
reunited as they head towards the detector. The interference pattern will depend on the difference in the
path length of the two beams, which can be adjusted by moving the mirror
M
_{1}
further away from or
closer to the beamsplitter. The nature of the interference pattern at the detector can be used to deduce the
wavelength of the light.