Interferometers Physics And Engineering Physics-PDF Free Download

Interferometers Physics and Engineering Physics

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The Michelson interferometer is one of the most useful of all optical instru Figure 1: Michelson Interferometer ments. It was ...



EP421 Lab Interferometers
Advanced Optics Lab Manual Michael P Bradley 2013
The apparatus basically consists of a half silvered beam splitting mirror M3 from which half of the light
travels to mirror M1 and is reflected while the other half of the light goes to mirror M2 and is reflected
Figure 2 Michelson Interferometer with compensating plate
1 Michelson Interference when light of a single wavelength is used
Suppose the light source produces light waves of a given wavelength These incident waves are
incident on the beam splitter and can be written as
E0 A sin kx t a
A sin 2 x 2 f t a
where k 2 is the propagation constant propagation assumed to be in air so the refractive index 1
and is the angular frequency
E0 A sin kx f t
Let us define the origin x 0 to be at the position of the beam splitting mirror x 0 we can do this
without loss of generality
E0 A sin f t
When the incident light encounters the beampslitter mounted at 45 assumed ideal for now half the
beam is reflected and as a consequence its path changes by 90 and it travels a distance l1 to fully
reflecting mirror M1 where it is reflected reverse direction and returns back to the beamsplitter having
traveled a total distance 2l1 for the moment we are ignoring the compensating plate Meanwhile the
transmitted half of the beam has likewise travelled a distance l2 to fully reflecting mirror M2 undergone
a reflection and reversed course travelling a total distance 2l2 to arrive back at the beamsplitter location
At the beamsplittter the electric field amplitudes of the two returned light waves are then
Please note that some of this material is taken from previous iterations of the U of S Engineering Physics Optics Lab
Manuals In general however the material has been extensively revised corrected and updated by MB and this revision
process is ongoing If you note any errors please contact MB at michael bradley usask ca
EP421 Lab Interferometers
Advanced Optics Lab Manual Michael P Bradley 2013
E1 a sin 2k l1 f t
E2 a sin 2k l2 f t
The difference of phase of between the two returned waves arises because half of the incident beam
reflects externally from the beam splitting mirror after travelling to M1 while the other half reflects
internally at the beam splitting mirror after travelling to M2 In the first case the beam is travelling in
air and reflecting at the air glass interface in the second case the beam is travelling in glass and
reflecting at the glass air interface A little consideration of the Fresnel equations for the phase shifts
experienced by the light waves upon reflection will hopefully make this clear This difference in
reflection boundary conditions experienced by the two beams is what leads to the net phase difference of
With our eye or another photodetector we view the intensity or irradiance associated with the total
electric field Etot E1 E2 As we saw in class the intensity of the light beam is given by
2 where Z0 is the characteristic impedance of free space So first we need to add the electric field
amplitudes and then square the result Using the trigonometric identity
sin a sin b 2 cos a b 2 sin a b 2
Etot E1 E2 2a cos k l1 l2 2 sin k l1 l2 2 f t
The eye detects the intensity of the wave which as we have seen is proportional to the time average of
the square of the electric field Etot
I E2 4a2 cos2 k l1 l2 2 sin k l1 l2 2 f t
In the time average only the last term on the right enters and since the time average of sin2 is
I E2 4a2 cos2 k l1 l2 2 1 2 2a2 sin2 k l1 l2
where we have also used the identity cos 2 sin
The maxima of observed optical intensity I thus occur when sin k l1 l2 1 Since k 2
Maxima of intensity occur when l1 l2 4 3 4 5 4
This is shown in Fig 3 below
Please note that some of this material is taken from previous iterations of the U of S Engineering Physics Optics Lab
Manuals In general however the material has been extensively revised corrected and updated by MB and this revision
process is ongoing If you note any errors please contact MB at michael bradley usask ca
EP421 Lab Interferometers
Advanced Optics Lab Manual Michael P Bradley 2013
Figure 3 Optical intensity irradiance showing intensity variation as a function of optical path length difference between the two
interferometer arms The repeating array of bright peaks is referred to as a set of interference fringes
We see that movement of M1 by 2 causes one complete interference fringe to pass by i e the
observed intensity goes from a maximum to a minimum and then back to a maximum again Thus by
counting the number of fringes that pass by when the micrometer screw changes the position of M1 by a
given amount we can determine the wavelength of the light used
By determining the mirror movement between the individual fringes the average wavelength can be
calculated
Michelson Procedure With reference to the Michelson configuration in Fig 2 the reflected light
beams from the two mirrors then recombine at M3 and are examined by eye as shown Whether the
interference between the two beams will be constructive or destructive depends upon the path lengths in
the two arms Notice that movement of mirror M1 by one half wavelength will cause the beams to
undergo a net path difference of one whole wavelength The purpose of the compensating plate is to
ensure that both beams travel through equal path lengths in glass The compensating plate is exactly
equal in thickness to mirror M3 In the diagram shown you can see that each beam passes through 3
thicknesses of glass in going from the source to your eye
Mirror M2 has two tilt adjustment screws which can be used to align M2 with mirror M1 mounted on
the carriage The carriage is movable by means of a micrometer screw which actuates a pivoted beam
coupled to the carriage The beam provides a 5 1 reduction from the indicated micrometer reading
to the actual length traversed by the carriage The micrometer itself has 25 mm of movement and
vernier graduations for reading to 0 01 mm hence the carriage has 5 mm movement which can be read
to 0 002 mm
Please note that some of this material is taken from previous iterations of the U of S Engineering Physics Optics Lab
Manuals In general however the material has been extensively revised corrected and updated by MB and this revision
process is ongoing If you note any errors please contact MB at michael bradley usask ca
EP421 Lab Interferometers
Advanced Optics Lab Manual Michael P Bradley 2013
As shown in Fig 3 above the movement of mirror M1 by a distance 2 causes one complete
interference fringe to pass by i e from bright to dark to bright again Count the number of fringes that
pass by when the micrometer screw changes the position of mirror M1 by a given amount you will need
to record the initial and final micrometer settings for the screw position Use this measurement to
determine the wavelength of the light used Note that because the yellow light from the Na lamp is a
doublet and actually consists of two very close but different atomic emission wavelengths what you
are actually measuring this way is the average wavelength
The fact that the sodium lamp produces two closely spaced wavelengths a doublet results in a variation
in fringe visibility as moveable mirror M1 is moved over larger distances This variation can easily be
observed It is described in more detail in the Appendix
We will use the Fabry Perot interferometer in the next Part to measure the doublet separation
Part 2 Fabry Perot Interferometer
The Fabry Perot configuration see Fig 4 consists of two partially reflecting mirrors separated by a
distance L This widely used instrument was first constructed in the early 1800s by Charles Fabry and
Alfred Perot The Fabry Perot interferometer has an extremely high resolving power about 10 times
better than a grating spectrometer which is already at least an order of magnitude better than a prism
spectrometer As such it has many applications in precision measurement and is often referred to as an
etalon the French word for stallion which has come to mean a standard of measurement for
reasons that are not entirely clear The Fabry Perot etalon configuration is widely used in precision
spectroscopy precision distance measurement and it also serves as the optical resonator cavity for
most lasers Thus it is worth studying in some detail
Figure 4 Fabry Perot Etalon
Fig 4 shows a schematic diagram of a Fabry Perot etalon configuration The complete interferometer
consists of a Fabry Perot etalon and a lens system or eyepiece to focus the light either onto a screen or
Please note that some of this material is taken from previous iterations of the U of S Engineering Physics Optics Lab
Manuals In general however the material has been extensively revised corrected and updated by MB and this revision
process is ongoing If you note any errors please contact MB at michael bradley usask ca
EP421 Lab Interferometers
Advanced Optics Lab Manual Michael P Bradley 2013
for the instrument you will use in this lab at the observation point your eye will be effectively at the
position of the screen in the diagram
When a broad monochromatic light source is used as the input to the interferometer a portion of the
light ray entering at an angle to the axis normal to the etalon will also leave the etalon at the same
angle as shown in Fig 4 Because the etalon mirrors are partially reflecting a portion of the light ray
will also be reflected two times and will then leave the etalon parallel to the first transmitted ray This
pattern of multiple refletions will be repeated and as shown will lead to multiple transmitted beams All
the rays that are parallel and in the same plane of incidence will combine at a point P on the screen
Since the individual rays are not coherent with each other the intensity at P will simply be the sum of
the intensities of the individual waves The resulting interference pattern is a series of concentric light
and dark rings
The fringe system of a Fabry Perot Interferometer is the same as the basic equation for the cavity modes
in the resonator but is generalized to include light rays at an angle q to the normal The path of the ray
is resolved into components parallel and perpendicular to the normal at the mirror face so that the
parallel component which contributes to the fringe intensity is given by kcos with k 2 The
resulting equation is
m 2 n L cos
where m is the fringe index fringe order
is the free space wavelength of light used
n is the refractive index of the material inside the etalon 1 in our case but may
be very different from 1 for laser cavities filled with an active medium for example
L is the separation between the etalon mirror surfaces inside the cavity
Experimental Procedure Fabry Perot
Measuring the Sodium Doublet Separation on the Fabry Perot Interferometer
The length L of the Fabry Perot interferometer is adjusted by using a micrometer screw to move one of
the parallel mirrors forming the etalon The mirror position can be read on the micrometer which is
calibrated in millimeters The mirror is moved by a lever connected to the micrometer screw so the
ratio of the micrometer reading to the actual movement of the mirror is 1 5
Then with air for the medium between the mirrors we have n 1 and at the center of the fringe pattern
cos 1 The fringe system equation becomes
As noted above the resolving power of a Fabry Perot etalon is extremely high Thus it is well suited to
be used to measure closely space wavelengths such as the Na doublet presents Using our Na light
source a set of two superimposed fringe patterns from the Na doublet can be observed The Na doublet
consists of two spectral lines in the yellow having wavelengths of 5890 and 5896 Angstroms In a Na
Please note that some of this material is taken from previous iterations of the U of S Engineering Physics Optics Lab
Manuals In general however the material has been extensively revised corrected and updated by MB and this revision
process is ongoing If you note any errors please contact MB at michael bradley usask ca
EP421 Lab Interferometers
Advanced Optics Lab Manual Michael P Bradley 2013
discharge lamp these two wavelengths are emitted incoherently and thus each will present its own fringe
pattern which our eye will see as being superimposed one on top of the other We can distinguish the
patterns due to the two wavelengths by their intensities the 5890 spectral emission line is twice as
intense as the 5896 spectral emission line this has to do with the so called oscillator strengths and
transition matrix elements of the quantum states of the outer shell electron in the sodium atom which
would be covered in an atomic physics course Because of the differing intensities of the two emission
lines the movable mirror can be adjusted so that the ultrafine fringes due to the weaker 5896 line will
appear to be exactly halfway between the heavier fringes due to the 5890 line Adjust the Fabry
Perot interferometer mirror spacing using the micrometer to achieve this condition Record the
micrometer setting at which you achieve this
Now our task is to measure the two separate wavelengths call them 1 and 2 of the sodium doublet
This is colloquially referred to as resolving the doublet
The first micrometer reading taken above corresponds to a mirror spacing L L1 such that
2 L1 m1 1 m2 p 1 2 2
where 1 is greater than 2 i e we choose 1 to represent the longer of the two wavelengths we can do
this without loss of generality The factor of in the last term on the right hand side means that the
fringe order of the shorter wavelength ring system differs from that of the longer wavelength ring system
by an odd half integer This is by design remember we adjusted the etalon mirror spacing so that the
ring patterns have been adjusted to fall midway between each other with the dark part of one
wavelength s fringe pattern overlaying the bright part of the other wavelength s pattern
The mirror is then adjusted and the fringe pattern will seem to move outwards from the center of the
pattern When the fine rings are once again halfway between the heavier rings a second reading of the
micrometer is taken to determine the new mirror spacing L L2 is taken
2 L2 m2 1 m2 p 3 2 2
Note that in general we do not start with the plates in contact i e we never have the condition L 0 In
fact we cannot since the physical contact would damage the delicate partially reflecting surfaces The
integer p is introduced in the above two equations to account for the non zero mirror starting separation
Subtracting these two equations gives us
2 L2 L1 m2 m1 1 m2 m1 1 2
m2 m1 1 2 2
1 2 2 m2 m1 1 2 2 L2 L1
Since 1 and 2 are so close for the sodium doublet we may take them to be approximately equal to the
average wavelength Na yellow light wavelength i e 1 2 This average wavelength may be
determined fringe contrast counting for either ring pattern separately or it may bet determined
separately using the Michelson interferometer for example Writing our above result in terms of the
average wavelength in the numerator we get
Please note that some of this material is taken from previous iterations of the U of S Engineering Physics Optics Lab
Manuals In general however the material has been extensively revised corrected and updated by MB and this revision
process is ongoing If you note any errors please contact MB at michael bradley usask ca
EP421 Lab Interferometers
Advanced Optics Lab Manual Michael P Bradley 2013
1 2 2 2 L2 L1
This expresses our desired doublet wavelength separation in terms of the measured mirrors position L1
and L2 The mirror separation L2 L1 is evaluated from micrometer readings as
L2 L1 0 10 D2 D1 K where D2 D1 is the change of the micrometer reading as read in
millimeters and K is the ratio of the mirror carriage movement to micrometer reading for our system K
0 20 because there is a lever arm with a 5 1 lever ratio connecting the micrometer lead screw to the
moveable mirror carriage
Using the above calculate the separation of the Na doublet spectral lines Express your final result
in ngstroms Compare the values to the accepted value of 6 Angstroms D1 5896 A D2 5890
A How close is your result to the expected value
PART 3 Shearing Interferometer Set up the
PART 4 Holographic Interferometer
Follow the instructions in the holographic interferometer manual separate manaual Satify yourself
that you can see fringe shifts when the aluminum block is subjected to stress and undergoes strain
APPENDIX Theory of Fringe Contrast in the Michelson Interferometer
The sodium doublet consists of two spectral lines in the yellow having wavelengths of 5890 and 5896
Angstroms The 5890 A line is twice as intense as the 5896 A line Therefore we have to consider the
interference pattern when the incident light consists of wavetrains with two different but closely
separated wavelengths We can write the electric field of the incident light beam as
E0 A1 sin kAx g t A2 sin kBx h t
In the incoming light there are two wavelengths A and B with electric field amplitudes amplitudes
A and B respectively In the following discussion it will be assumed that the amplitudes of these two
waves are approximately equal i e that A1 A2 It is very important to realize that the time terms g t
and h t are random with respect to one another These wavelengths arise when an outer electron of an
atom undergoes a quantum jumps from a higher energy state to a lower state The quantum states
involved in the emission of light with wavelength 1 are different from those involved in the emission of
light with wavelength 2 and the electron transitions between the first two levels are independent of the
transitions between the other two levels because the jumps occur in different atoms Hence there is no
fixed relationship in time between the appearance of the two waves i e they are random in time with
respect to each other When this is the case we way the wavetrains are incoherent
Because of the incoherence of the optical emission at two wavelengths 1 and 2 their behavior in the
Michelson interferometer must be treated individually i e there can be no interference between the
two waves of different wavelength only wavetrains which are coherent with each other can interfere
Please note that some of this material is taken from previous iterations of the U of S Engineering Physics Optics Lab
Manuals In general however the material has been extensively revised corrected and updated by MB and this revision
process is ongoing If you note any errors please contact MB at michael bradley usask ca
EP421 Lab Interferometers
Advanced Optics Lab Manual Michael P Bradley 2013
Thus each wavelength has its own intensity pattern in the interferometer as described in the Theory
section for the Michelson interferometer above
The two intensity patterns in the interferometer arising from the two wavelengths can be represented as
a function of mirror movement by the following bar diagram
Figure 5 Fringes for 2 wavelengths
The two patterns coincide for l1 l2 d 0 cancel each other as the mirror is moved from zero path
difference and coincide again as the mirror is moved further This coincidence and cancellation
continues as the mirror is moved The overall fringe visibility will thus vary with mirror position
When the patterns for the two wavelengths coincide the fringes will be very distinct minima black
maxima bright When the patterns are out of step such that the maxima for one wavelength fall on the
minima for the other wavelength then fringe visibility will be poor and the individual fringes will fade
out into the background because the fringe patterns for the two different wavelengths tend to cancel
This variation is shown in the diagram below
Please note that some of this material is taken from previous iterations of the U of S Engineering Physics Optics Lab
Manuals In general however the material has been extensively revised corrected and updated by MB and this revision
process is ongoing If you note any errors please contact MB at michael bradley usask ca
EP421 Lab Interferometers
Advanced Optics Lab Manual Michael P Bradley 2013
Figure 6 Fringe contrast
Recall that the distance the moveable mirror M1 must be moved between consecutive fringe maxima is
2 Also note from the previous bar diagram that if the number of fringe maxima between
coincidences of the two intensity patterns is N for 1 then it is N 1 for 2
Let dc be the distance the mirror M1 must be moved between consecutive positions of pattern
coincidence i e between consecutive high contrast high visibility fringe patterns
dc N 1 1 2 N 2 2
N 2 2 2 2 N 1 2
dc N 1 2 avg avg 2
where dc is the distance the mirror moves Recall also that for our interferometer the ratio of mirror
micrometer movement 1 5 because of the lever arm which transmits the micrometer motion to the
mirror carriage
Note also that the result is approximate because we approximated 1 by avg 5893 Angstroms
However the degree of approximation is very high for the sodium doublet because the spectral line
wavelengths are so close together only 6 apart
Finally the doublet separation is given by 2avg 2dc
The value of dc can be obtained from the change of the micrometer reading as read in millimetres
remember to account for the 5 1 ratio imposed by the lever arm The overall optical intensity pattern
Please note that some of this material is taken from previous iterations of the U of S Engineering Physics Optics Lab
Manuals In general however the material has been extensively revised corrected and updated by MB and this revision
process is ongoing If you note any errors please contact MB at michael bradley usask ca
EP421 Lab Interferometers
Advanced Optics Lab Manual Michael P Bradley 2013
observed when using the interferometer to analyze a doublet is shown schematically in the following
Figure 7 Michelson Interferometer Fringe Visibility variation when analyzing a doiublet
By determining the mirror movement between the individual fringes the average wavelength can be
calculated this was explained in detail in the Michelson Interferometer Theory section By determining
the mirror movement between two successive fringe visibility maxima positions of coincidence the
wavelength difference between the two wavelengths can be calculated Note that the wavelength
difference can also be obtained from the mirror movement between two successive visibility minima
positions of cancellation where the individual fringes disappear into the background light since
dc d coincidence d cancellation
References
Please note that some of this material is taken from previous iterations of the U of S Engineering Physics Optics Lab
Manuals In general however the material has been extensively revised corrected and updated by MB and this revision
process is ongoing If you note any errors please contact MB at michael bradley usask ca


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