Interferometry U Of T Physics-PDF Free Download

Interferometry U of T Physics

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Figure 1: A schematic diagram of the Michelson interferometer. The purpose of the compensating plate G 2 is to render the path in glass of the two

Figure 1 A schematic diagram of the Michelson interferometer
The purpose of the compensating plate G2 is to render the path in glass of the two
rays equal 1 This is not essential for producing effective sharp and clear fringes
in monochromatic light but it is crucial for producing such fringes in white light a
reason will be given in the White Light Fringes section The mirror A1 is mounted
on a carriage whose position can be adjusted with a micrometer To obtain fringes
the mirrors A1 and A2 are made exactly perpendicular to each other by means of the
calibration screws seen in fig 1 controlling the tilt of A2
There are two very important requirements that need to be satisfied along with the
above set up in order for interference fringes to appear
1 One is advised to use an extended light source The point here is purely one of
illumination if the source is a point there is not much space for you to see the
fringes on You can convince yourself of the usefulness of using an extended source
by positioning a variable size aperture in front of an extended source and shrinking
its radius to the minimum possible thus effectively converting it to a point source
As you can see the field of view over which you can see the fringes shrinks right
with it Hence it is in your best interest to use as big of a source as possible a
diffusion screen is of further great aid here
2 The light must be monochromatic or nearly so This is especially important if the
distances of A1 and A2 from G1 are appreciably different The spacing of fringes for
a given colour of light is linearly proportional to the wavelength of that light hence
the fringes will only coincide near the region where the path difference is zero
The solid line here corresponds to the intensity of interference pattern of green
light and the dashed curve to that of red light We can see that only around
zero path difference will the colours remain relatively pure as we move farther
away from that region colours will start to mix and become impure unsaturated
already about 8 10 fringes away the colours mix back into white light making
fringes indistinguishable Hence the region where fringes are visible is very narrow
and hard to find with non monochromatic light
Some of the light sources suitable for the Michelson interferometer are a sodium flame
or a mercury arc If you use a small source bulb instead a ground glass diffusing screen
in front of the source will do the job looking at the mirror A1 through the plate G1 you
then see the whole field of view filled with light
Circular Fringes
To view circular fringes with monochromatic light the mirrors must be almost per
fectly perpendicular to each other The origin of the circular fringes is understood from
fig 23 Since light in the interferometer gets reflected many times we can think of the
extended source as being at L where L is behind the observer as seen in fig 2 L forms 2
virtual images L1 and L2 in mirrors A1 and A2 respectively The virtual sources in L1
and L2 are said to be in phase with each other such sources are called coherent sources
in that the phases of corresponding points in the two are exactly the same at all times
If d is the separation of A1 and A2 the virtual sources are then separated by 2d as can
be seen in the diagram fig 2
Figure 2 Virtual images from the two mirrors created by the light source and the beam
splitter in the Michelson interferometer
When d is exactly an integer number of half wavelengths4 every ray that is reflected
The real mirror A2 has been replaced by its virtual image A2 formed by the reflection in G1 hence
A2 is parallel to A1
The path difference 2d must then be an integer number of wavelengths
normal to the mirrors A1 and A2 will always be in phase Rays of light that are reflected
at other angles will not in general be in phase This means that the path difference
between two incoming rays from points P and P will be 2d cos where is the angle
between the viewing axis and the incoming ray We can say that is the same for the two
rays when A1 and A2 are parallel which implies that the rays themselves are parallel5
The parallel rays will interfere with each other creating a fringe pattern of maxima and
minima for which the following relation is satisfied
2d cos m 1
where d is the separation of A1 and A2 m is the fringe order is the wavelength of the
source of light used is as above if the two are nearly collinear we of course have
0 this is the case for the fringes in the very centre of the field of view
Since for a given m and d the angle is constant the maxima and minima lie
on a circular plane about the foot of the perpendicular axis stretching from the eye to
the mirrors As was mentioned before the Michelson interferometer uses division by
amplitude scheme hence the resultant amplitudes of the waves a1 and a2 are fractions
of the original amplitude A with respective phases 1 and 2 We can calculate the
phase difference between the two beams based on the respective mirror separation If
the path difference is 2d cos then the difference in phase for light of wavelength is
Here the ratio of the path difference to the wavelength tells you what fraction of a
wavelength have you passed and multiplication by 2 makes it a fraction of the full
period of a sinusoid thus giving you exactly the sought phase difference
By starting with A1 a few centimetres beyond A2 the fringe system will have the
general appearance which is shown in fig 3 where the rings of the system are very
closely spaced As the distance between A1 and A2 decreases the fringe pattern evolves
growing at first until the point of zero path difference is reached and then shrinking
again as that point is passed
Since the eye focused to receive the parallel rays it is more convenient to use a telescope lens
especially for looking at interference patterns with large values of d
Figure 3 The circular fringe interference pattern produced by a Michelson interferometer
This implies that a given ring characterized by a given value of the fringe order m
must have a decreasing radius in order for 2 to remain true The rings therefore shrink
and vanish at the centre where a ring will disappear each time 2d decreases by This
is because at the centre cos 1 and so we have the simplified version of equation 2
From here we see that the fringe order changes by 1 precisely when 2d changes by
hence for a fringe to disappear we need to decrease 2d by as claimed above
Localized fringes
In case when the mirrors are not exactly parallel fringes can still be observed for
path differences not much greater than a few millimetres using monochromatic light
of course The space between the mirrors is wedge shaped as can be seen in fig 4
thus the two rays reaching the eye from the mirrors are no longer parallel and appear to
Figure 4 Formation of localized fringes with non perpendicular mirrors the air wedge
is clearly seen
Hence the interference picture will be more like that of fig 5 the fringes are now
semi circles with the centre lying outside the field of view such fringes are often called
localized fringes The reason these fringes are almost straight is primarily because of
the variation of the thickness of air in the wedge as that is now the main reason for the
variation of the path difference between the two beams across the field of view One would
expect all fringes to be perfectly straight parallel to the edge of the wedge however that
is not the case as the path difference still does vary somewhat with the angle especially
if d is large Depending on the magnitude of d we can observe different interference
patterns as we change the path difference the fringes become straighter until we hit
point of zero path difference At that point if we were looking at circular fringes they
would fill the whole field of view become very large circles that means that localized
fringes would become parallel lines as if there were small sections of the circumferences
of very large circles
Figure 5 The localized fringe interference patterns produced by a Michelson interfer
ometer a and c are depictions of curved fringes implying the mirror is far from the
region of zero path difference and b shows straight parallel fringes this must be at
or very near the point of zero path difference
The association large circular fringes parallel localized fringes will be important
in the next section when we use it to locate white light fringes
White Light Fringes
If instead of using monochromatic light we wish to study the fringes created by white
light no fringes will be seen at all except for when the path difference between A1 and A2
does not exceed a couple of wavelengths6 This is well demonstrated in fig 6 the dashed
line corresponds to the intensity of the interference pattern of green light while the solid
line to that of yellow light As you can see from the diagram the patterns only overlap
over the narrow range of zero path difference between the two incoming beams now if
there are many different wavelengths involved in an interference process as is the case
for white light one can conclude that anywhere too far away from the region of zero path
difference the colours will mix back up into white light and no fringes will be visible
Figure 6 The intensity curves of interference of green light dashed line and yellow light
solid line The dispersion of the interference patterns away from the region of zero path
difference is readily observed
With white light there will be a central dark fringe bordered on either side by 8 or
10 coloured fringes Since the region over which the white light fringes are visible is so
narrow trying to search for it with white light alone is too time consuming Instead you
They is extremely difficult to find so have patience they do exist
can first approximate its location by using monochromatic light and finding the region
of zero path difference It will correspond to one of two regions a if the mirrors are
perfectly parallel and we are observing circular fringes the region with the largest circular
fringes is the region of zero path difference b if the mirrors are almost parallel and we
are observing localized fringes then the region with straight parallel fringes will be the
region of zero path difference Once we have approximately found the right region we
switch back to white light and move VERY slowly through the region the bright fringes
should come into view These fringes will only occur over a very narrow range of path
difference values corresponding to about a 20 degree turn on the micrometer hence
the need to move slowly otherwise you can miss them We advise that upon finding the
fringes you mark the approximate position of the micrometer to simplify future search
as the position of white light fringes will be needed for other experiments
Fabry Pero t interferometer theory7
Another commonplace division of amplitude interferometer is the Fabry Pero t inter
ferometer which uses a principle similar to that of the Michelson interferometer to pro
duce interference fringes The core of such a device consists of two parallel flat glass
plates one movable one fixed the inner surfaces of which are coated with a partially
reflective metallic layer see fig 7
Figure 7 The reflected and transmitted beams of light going through the two
glass surfaces of a Fabry Pero t interferometer letters indicate points of reflec
tion refraction Source http what when how com radial velocities in the zodiacal
dust cloud preparations and experimental details 1971 1974 zodiacal dust cloud part 2
Due to the coating a beam of light incident on the first plate at an angle to
the horizontal produces a series of beams passing through to the other side as each
The experiments related to the Fabry Pero t interferometer consitute one 1 weight
continuously gets either transmitted through the second plate to go on to the observer
or bounces back and forth between the inner surfaces until it does it could also potentially
come back out from the side the original beam entered the arrangement but those rays
are of no consequence to us Each of the beams arrives at the point of observation with
a path difference of with the one before and after it thus they reinforce each other and
produce an interference pattern Let the distance between the plates be t We see from
fig 7 that the path difference between the rays exiting at B and D is exactly
In the diagram the line BK is normal to CD The angle between BC and CK is 2
and the triangle BCK is a right angle one Hence we may write
CK BC cos 2
Moreover we can relate the hypotenuse BC to the distance between the plates via
Thus we have for the path difference
BC CK BC 1 cos 2 2BC cos2 2t cos 4
The condition for constructive interference is as always
n 2t cos 5
where n is the fringe order and is of course the wavelength We can vary the separation
between the glass plates and watch the fringes disappear in the centre of the field of
view thus allowing us to do almost exactly the same measurements as we could with a
Michelson interferometer The advantage of the Fabry Pero t is its high resolving power
it makes it a valuable tool in the study of the Zeeman effect and the hyperfine structure
of certain spectral lines
One point has to be made concerning this device since the interference only occurs
for light incident on the plate as an angle a perfectly parallel beam of light may not
produce fringes hence we must once again use an extended light source to remedy this
NOTE Most mirrors in the apparatus are front surface aluminized Do not
touch the surfaces nor wipe them They can easily be permanently damaged
Michelson interferometer setup
The Michelson Interferometer is a fundamental design of a large variety of two beam
interferometer configurations In this experiment we will use the most basic apparatus
see fig 8 Light from a source unit N a mercury or sodium lamp in this experiment
passing through a diffusing screen filter holder unit D8 is incident on the plane parallel
beam splitter plate with compensating plate they are together in one whole unit C and
is divided into two beams the axes of which 1 and 2 fall normally on the mirrors A and
B respectively The returned beams re unite at the semi reflecting surface of C The
interference pattern can be viewed directly with the naked eye or by means of a telescope
at the viewing position
The diffusing screen is a piece of ground glass used to spread out or diffuse the light across the field
of view to get soft light It is important for the same reason it is to make use of an extended light source
one needs to illuminate as large a part of the field of view as possible to simplify fringe observation
Figure 8 The Michelson interferometer setup used in this experiment letters indicate
units as noted in the explanation above
The compensating plate at C is identical in thickness to the beam splitter plate and
is set accurately parallel to it Its insertion then equalizes the glass paths in the two
beams as mentioned earlier When the mirrors A and B are perpendicular and A is
slightly closer than B the image from A will fall in front of that from B and a series of
interference fringes will be seen When the mirrors are equidistant and perpendicular the
interference field will be covered by one large circular fringe When the surfaces B and A
are not precisely parallel and the separation distance is very small a series of fringes in
shapes of approximately straight lines will be seen For a non laser source fringe contrast
increases as distance apart is reduced
Fabry Pero t interferometer setup
The Fabry Pero t interferometer used in this experiment is depicted in fig 9 Light
originates from an extended source in the back of the setup in the diagram a sodium
lamp is used passes through the first mirror A installed in the top position and into the
second mirror E It continues along the viewing axis and into the telescope L clamped
in a holder H with a screw Adjust the telescope magnifying unit until the light source
is in focus Do not use any collector lenses as they only obstruct the view
Figure 9 The Fabry Pero t interferometer setup used in this experiment letters indicate
units as noted in the explanation above
Vacuum pump
For one part of this experiment you will need to use a vacuum pump to find the index
of refraction of a gas at normal atmospheric temperature and pressure Specifically you
will be using an Edward High Vacuum Ltd rotary vacuum pump parts of the technical
manual are available in the resource room 229 The pump apparatus consists of several
1 The pump itself housing the rotor and the oil chamber attached to its vacuum
connection see the manual for a diagram is the main access valve separating the
gas cell from the insides of the pump
2 A gas cell connected to the pump via three reinforced plastic tubes
3 A tall dial gauge indicating the pressure inside the gas cell in mm Hg operates
between 1 and 760 mm Hg
4 A release valve also connected to the gas cell to control the pressure level
One operates the pump as follows
1 Ensure the gas cell is properly connected to the pump
2 Connect the pump to a wall outlet
3 Slowly open the main access valve and establish a pressure of 760 mm Hg in the
4 Use the release valve to vary the pressure inside the gas cell
Be sure to keep the release valve open when you unplug the vacuum pump to let the
pressure slowly leak from the pump When you turn on the pump it will make ungodly
noise that is to be expected
Experiment and Procedure
With the Michelson interferometer 1 3 wt
In this part of the experiment you will learn on specific examples the standard
measurement techniques which utilize the Michelson interferometer in particular you
1 Learn how to set up the device and align the mirrors
2 Observe the interference fringes with both monochromatic and white light
3 How to use it to determine the index of refraction of transparent solids and of gases
The arrangement of the interferometer outlined in the section Apparatus Michelson
interferometer setup will be the arrangement used for the entire experiment except for
when you will need to switch between the light sources used
Initial adjustments observation of fringes and calibration 1 wt
Dim the room lights Position the sodium lamp it will take some time for it to warm
up after you turn it on in front of the diffusing screen holder D and insert the diffusing
screen into the holder Looking through the opening at the viewing position usually the
closer the better the view you will observe dark fringes on a yellow orange background
they will most likely be localized fringes if you do not see any fringes at all try rotating
the micrometer screw until they appear Adjust the calibration screws on mirror B to
make it perpendicular to mirror A you will know they are perpendicular when you see
complete circular fringes with the centre of the fringes right in the centre of your field of
Once you have observed the fringes locate the region where the path difference be
tween the two beams of light is close to zero Recall that when viewing circular fringes
this region is the region where the fringes observed are largest covering the entire field
of view whereas when viewing localized fringes this region corresponds to the region
where the fringes are parallel to each other It is advisable to use the latter to locate the
region of zero path difference Mark the approximate location of the region by noting the
micrometer reading to speed up the procedure for next time
Switch the source of light to white light By rotating the micrometer and moving the
mirror carriage very slowly through this region you can observe the elusive white fringes
As was noted earlier these fringes are only observable over a range of about a 20 degree
rotation of the micrometer head the range is only about 20 fringes wide so be sure to
rotate the micrometer very slowly The fringes in white light can only be viewed when
the path difference 2d cos 0
Now switch the light source back to the sodium lamp and adjust mirror B until you
see circular fringes of medium to large size You are going to set up a calibration curve
between the motion of the micrometer screw and the actual displacement of mirror A
Since there is a rather non trivial system of levers connecting the mirror carriage with
the micrometer screw head not all of the motion of the micrometer is translated into
the motion of the carriage we need to determine the exact relationship To do so we
will make use of equation 3 and our earlier observation that if the distance 2d changes
by the wavelength then one fringe passes out of the field of view Hence if we count
the number of fringes that disappeared from the field of view in a given distance moved
by the micrometer we can directly relate the two as using the number of fringes and
equation 3 we can calculate how much the mirror actually moved and relate that to
what we took down for the motion of the micrometer
NOTE The mirror carriage should always be driven towards the observer

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