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Unformatted text preview: 19. Lecture 19
19.1 Interference
The principle of superposition implies than when two electromagnetic waves meet each
other, the total electric and magnetic ﬁelds are the sum of the electric and magnetic
ﬁeld of each wave. This is quite important since, for example, the electric ﬁelds can
add up or subtract depending on how they are oriented. When they are opposite and
of the same strength they cancel each other and we have the very surprising property
that two beams of light can cancel each other and give a dark spot. This is called
destructive interference (whereas constructive is when they add up). However it does
not seem to be something we see everyday!. In fact we know that two light bulbs always
illuminate more than one. So what is going on?. The situation is that in a light bulb
the atoms of the ﬁlament are emitting light independently, so from each light bulb we
have an enormous number of pulses or wave packets that are not in phase with each
other. So wave packets from diﬀerent sources sometimes add up and sometimes cancel
each other. In average the intensities just add up. Notice that this is not constructive
interference. In constructive interference the electric ﬁelds add up. But the intensity
is proportional to the square of the electric ﬁeld so two beams of equal intensity give
through constructive interference a beam of four times the intensity. On the other hand
random interference gives just twice the intensity.
In ﬁg.107 we see the case of two wave packets interfering constructively, destructively or an intermediate case.
The question is now if we can actually see interference and cancel light with light.
The trick is to split each wave packet in two parts and then join them back so essentially each packet interferes with itself. How to do so we know from when we studied
refraction, an incident beam is split into a reﬂected beam an a transmitted beam. This
is the principle used in the interferometer.
19.1.1 Interferometer
The interferometer (see ﬁg.108) works by splitting a beam into two using a partially
reﬂecting mirror (drawn in blue). The two beams are reﬂected from ﬂat mirrors (drawn
in green) and joined back together using the same partially reﬂective mirror. Part of
the beam will be sent back to the source and lost. After the two beams are joined
they are projected onto a screen. If the diﬀerence in the length of the two arms
of the interferometer is an integer multiple of the wavelength then the interference
in constructive. If it is a halfinteger multiple is destructive. Otherwise we have an
intermediate situation. The trick is that all wave packets emitted by the source interfere
in the same way provided that the source is monochromatic, namely has a single wave – 123 – Figure 107: Two wave packets can interfere constructively or destructively. Also there are
many intermediate situations as illustrated in the third plot. length. It is also important that the diﬀerence in length between the two arms of the
interferometer is smaller than the typical length of the wavepackets (also known as
coherence length). Otherwise a packet cannot interfere with itself. To summarize:
mλ
→ Constructive
∆L =
(19.1)
1
m + 2 λ → Destructive
where m is an arbitrary integer. The wave length for visible light is of the order of
500nm so by moving the mirror so slightly one can go from constructive to destructive – 124 – interference. Not only is the interferometer a clear proof of the wavelike nature of light
but it also provides a very sensitive instrument to measure length. Although this is
not practical to use in everyday life, for certain physics experiments it is very useful.
One such experiment which is under way is the detection of gravitational waves. Such
waves would make the mirrors move and such motion can be detected. It requires great
care since, for example, even small vibrations that change the position of the mirrors
are enough to destroy the interference pattern. Figure 108: Interferometer. 19.1.2 Thin ﬁlms
Other situation where interference appears and is actually very easy to see is in thin
ﬁlms. For example soapy water on glass can form a thin ﬁlm. It is known that when
illuminated with white light such ﬁlms appear colored as a rainbow (e.g. soap bubbles).
The reason is that we have interference as shown in ﬁg.109. The two reﬂected beams
will interfere constructively or destructively. The diﬀerence in path length is
2d
∆L =
(19.2)
cos θ – 125 – where d is the thickness of the ﬁlm and θ is the angle of refraction (given by Snell’s
law). There is a subtlety which is that, as we are going to see later, when a beam is
reﬂected from an interface there could be a phase shift of 180o equivalent to a halfwave length shift. The rule is that when coming from medium 1 and reﬂecting from an
interface with medium 2:
n1 < n2 → 180o phase shift
n1 > n2 → no phase shift (19.3)
(19.4)
(19.5) where n1,2 are the indices of refraction of the two media. The other point to take into
account is that the wavelength changes when going to a diﬀerent medium. The wave
length is related to the frequency through
λ= c
f (19.6) where c is the speed of light in the medium and f is the frequency which is the same
in all regions (since the boundary conditions enforce that the electric and magnetic
ﬁeld oscillate in unison in all media). If we use λ1 the wavelength in medium 1 as a
reference, the wavelength λ2 in medium two is given by
λ2 = c2
n1
λ1 = λ
c1
n2 (19.7) since the speed of light is inversely proportional to the refraction index. Of medium 1
is air we take n1 = 1 and obtain
If n1 < n2 < n3 , or n1 > n2 > n3 then
If n1 < n2 > n3 , or n1 > n2 < n3 then
∆L =
∆L = 2d
cos θ
2d
cos θ ∆L =
∆L = 2d
cos θ
2d
cos θ λ
= m n2
λ
= m + 1 n2
2
λ
= m n2
λ
= m + 1 n2
2 → Constructive
→ Destructive
→ Destructive
→ Constructive Since the angle at which there is constructive interference depends on the wavelength,
diﬀerent colors will be seen at diﬀerent angles giving rise to the rainbow eﬀect. The
same happens for example for two ﬂat glass surfaces separated by a thin layer of air.
In fact the eﬀect is more evident if the surfaces are not parallel but slightly at angle.
Then dark and bright bands appear. An example is when a lens is positioned over a
ﬂat surface as seen in ﬁg.111. – 126 – Figure 109: Thinﬁlm interference. Care should be taken to include the extra 180o phase
shift introduce when reﬂecting form a material of larger index of refraction. 19.1.3 Two slits
Another experiment is the two slit experiment also knows an Young’s experiment. Light
is shone toward an opaque surface with two very thin slits close to each other. A screen
on the other side shows alternative bright and dark bands produces by interference.
From ﬁg.112 we see that the bright bands appear whenever2 :
∆L = d sin θM = mλ (19.8) where d is the distance between slits and θM is the angle at which we ﬁnd a maximum.
Also the integer m is known as the order of the maximum. The bright and dark bands
are also called fringes.
2 Remember that in these formulas it is conventional to use m to denote an arbitrary integer. – 127 – Figure 110: Demo: Thinﬁlm interference. The rainbow colors are produced by interference
maxima located at diﬀerent positions for diﬀerent wavelengths. – 128 – Figure 111: A thin ﬁlm of air can be obtained between a lens and a ﬂat specular surface.
The interference pattern is known as Newton’s rings. – 129 – Figure 112: Young’s experiment, interference between light coming from two slits illuminated by the same source. – 130 – ...
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