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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