Second Year Laboratory - University of Sheffield/file/L9.pdf · difference between the mirrors ... mechanical lever connected to a the micrometer screw gauge as shown ... University

Second Year Laboratory

The Michelson Interferometer

L9

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Health and Safety Instructions.

There is no significant risk to this experiment when performed under controlled laboratory conditions. Take care however, when measuring the thickness of the microscope cover slide and the mica (part (c)) to avoid breaking either. The final part of the experiment you will use a Class 2 diode laser. This has a low power, visible, red beam. Eye damage can occur if the laser beam is shone directly into the eye. The laser/detector module is designed to prevent this and to operate only when it is correctly installed into the interferometer and the protective microswitch is depressed. Do not attempt to use the laser in any other circumstances.

________________________________________________________________ 17-03-06

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Second Year Laboratory Department of Physics and Astronomy, University of Sheffield

1 Aims

The main aims of this experiment are first, to use the Michelson interferometer to measure the refractive index of samples of glass and mica and air and second, to identify three unknown gases on the basis of the value of their refractive index.

2 Apparatus

• Michelson interferometer,

• mercury and white light sources,

• self contained laser source/detector with photodiode,

• green filter,

• aluminium plate with pinhole,

• microscope cover slip with holder,

• mica sheet with holder,

• gas cell and equivalent compensator plates,

• pumping line and gas lines with manometer,

• taps and needle valve,

• digital micrometer.

Note: All the minor components for this experiment are contained in a multi compartment plastic box at the interferometer. Use the same box throughout the experiment, check the components at the beginning and end of each afternoon’s work and make sure the box does not leave your particular instrument.

3 Introduction

(a) General

This experiment with the Michelson interferometer will involve precision measurements on the scale of the wavelength of light*. Note that the refractive index of helium given below differs from that of a vacuum, by only 3 parts in 105! Michelson used his interferometer to establish the first absolute measurements of length, when the standard metre was compared with the wavelengths of the spectral lines from cadmium. The Michelson-Morley experiment using the interferometer, was also of crucial importance in the development of the subject of relativity by demonstrating the absence of the so-called ‘aether drift’.

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The first half of this experiment will be performed in the conventional way, using the eye to detect and count fringe shifts. In the second part of the experiment a more modern version will use a self contained laser source/detector with photodiode, whose output is interfaced to a computer. Michelson believed that fractional parts of a fringe spacing could be measured, - one-tenth of a fringe easily, down to one-fifteenth with care! It’s not clear that any of us can today work to that kind of precision, - but the computer interfaced detector should allow fractional parts of a fringe to be measured from a graph of the data obtained.

(b) The Michelson interferometer

Figure 1 The Michelson interferometer

In the Michelson Interferometer, two beams obtained by amplitude division are sent in two directions at 90° and are reflected from plane mirrors so that they recombined to form interference fringes. The intensity pattern is proportional to ( )cos2 2π λd where d is the path difference between the mirrors and λ is the wavelength.

The arrangement of the interferometer and the optical paths are shown schematically in Figure 1. Two highly polished plane mirrors, M1 and M2, and two plane-parallel glass plates form the main optical parts. The rear side of the first plate is lightly silvered so that the light coming from the source S is divided into (1) reflected and (2) transmitted beams of equal intensity (amplitude division). Light reflected normally from mirror M1 passes through this first glass plate a third time and reaches the eye as shown. Light reflected from the mirror M2 passes through a second glass plate twice, once on the outward journey and once after reflection from M2. This second glass plate is known as the compensating plate, and its function is to make the path of the two rays in glass equal.

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In the interferometer, the two glass plates are combined into a single compensator unit. This compensator is not necessary for producing fringes with monochromatic light, but it is essential when white light is being used because of the shorter length of the wavetrains.

The mirror M1, is mounted on a carriage and because during the experiment it is moved only multiples of the wavelength of light (≈ 0.5µm) these distances must be amplified by a mechanical lever connected to a the micrometer screw gauge as shown in Figure 1. In the current experiment, a digital micrometer with a remote output is used. To obtain any kind of fringes with the interferometer, the mirrors, M1 and M2 must be made perpendicular to each other by means of screws shown on mirror M2, in Figure 1.

This alignment is a crucial part of the experiment.

(c) Circular and localised fringe systems

Two difference systems of fringes can be observed for different settings of the interferometer

Figure 2 Circular fringes of equal inclination

Circular fringes are produced with monochromatic light when the mirrors are in exact adjustment as shown in Figure 2. The real mirror, M2, has been replaced by its virtual image, M2', formed by reflection in G1. Mirror M2' is then parallel to M1. These virtual sources are coherent because the phases of corresponding points in the two sources are exactly the same at all instants. If d is the separation M1M2', the virtual sources will be separated by 2d. When d is exactly an integral number of half wavelengths, all light reflected normally to the mirrors will be in phase. The path difference between the two rays coming to the eye from corresponding points P' and P'' is 2dcosθ = nλ, as shown in Figure 2. Hence, when the eye is focused to receive parallel rays, the rays will reinforce each other to produce maxima for those angles θ which satisfy the equation,

2dcosθ = nλ (1)

Since, for a given n, λ and d the angle θ is constant, the maxima will lie in the form of circles centred on the perpendicular from the eye to the mirrors. Fringes of this type are called fringes of equal inclination.

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If on the other hand, the mirrors M1 and M2' are not exactly parallel, fringes will still be seen with monochromatic light for path differences which do not exceed a few millimetres. In this case the space between the mirrors is wedge-shaped. These localised fringes are almost straight because the path difference across the field of view is due primarily to the variation of the thickness of the "air film" between the mirrors.

Figure 3 shows schematically the fringes of equal inclination and the localised fringes and shows that the spacing of the fringes varies as the path difference between the two beams increases.

→→→→→→ increasing path difference d →→→→→→

Figure 3 Examples of the fringe systems observed

If a source of white light is used white light fringes are only seen when the path difference d between the two beams is so small that it does not exceed a few wavelengths. A central dark fringe, bordered on either side by 8 or 10 coloured fringes will then be observed. Few fringes are observed with white light which contains all wavelengths between 400 and 700 nm. The fringes in different colours will only coincide for the central uncoloured fringe at d = 0 and will begin to separate on either side of the centre so that after a few fringes, so many colours are present that the resultant is essentially white.

4. Experimental procedure

NB. When working to the level of precision required by this experiment all the parts of the apparatus need to be perfectly aligned and also in 100% working condition. The experiment will require you to mount and dismount various parts of the apparatus. Always do this with the maximum care and never touch the optical surfaces. In the event of any problem or accident with the apparatus, consult a demonstrator immediately.

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WARNING: DO NOT, UNDER ANY CIRCUMSTANCES, ALTER THE MICROMETER ZERO POSITION YOURSELF.

(a) Adjustment of the interferometer

1. The interferometer will be set up as shown in Figure 1. You do not need either the gas cell or the two small, circular compensator plates (in a Terry Clip) in the beams for the first part of the experiment. Remove them if they are in place, taking care not to touch the optical surfaces.

2. To align the mirror M2 perpendicular to the beams to observe the fringes of equal inclination, place the aluminium plate with the pinhole in front of the diffuser of the mercury lamp. Three images of the pinhole in the aluminium plate will be seen and the mirror M2 should be gently adjusted by its screws so that two of the images of the pinhole are superimposed. Remove the aluminium plate and make very fine adjustments to screws on the back of the mirror M2 to optimise the fringes observed and centralise them in the field of view. Next make a long traverse of the mirror M1 using the micrometer screw gauge and observe the fringe pattern changes as shown in the lower part of Figure 3. Make a note of the mirror position on the micrometer screw gauge when these fringes have maximum width, since this corresponds to the minimum path difference d between the two beams.

3. Now displace the mirror M2 (sideways) slightly by fine movements of the screws on the back of the mirror mount and observe the localised fringes for a similar range of values of path difference. The fringes will now correspond to those shown in the upper part of Figure 3.

4. The position of the white light fringes can be more difficult to find, so it is best to repeat the alignment for the circular fringes of equal inclination very carefully with the mercury lamp and then to switch back and forwards between mercury/white light (to verify the continued alignment) while slowly displacing mirror M1 by the micrometer screw gauge. In case of real difficulty the micrometer position for white light fringes is probably near to zero on the remote display of the digital micrometer. DO NOT, UNDER ANY CIRCUMSTANCES, ALTER THIS ZERO POSITION YOURSELF.

To summarise: When the mirrors are perfectly perpendicular, the fringes are circular and when the two arms of the interferometer are equal, these fringes are very broad. In this case, the central fringe will virtually fill the eye piece. This is the most sensitive condition for measurement. Conversely, when the lengths of the two arms are very different, then the circular fringes are narrow as shown in Figure 3. The same general rules apply for the localised fringes.

(b) Calibration of the drive to the mirror M1

To make proper measurements it is first necessary to obtain the calibration factor or reduction ratio R, which gives the correspondence between the distance D moved by the micrometer screw and the actual motion d of the mirror M1. At the centre of the fringe system, cosθ = 1,

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so that equation (1) becomes, 2d n= λ . This equation shows that if the fringe number n changes by unity then the equivalent change to 2d must be λ.

If a bright fringe of order n1 corresponds to a position d1,

R D d n2 21 1 1= = λ

and a second a bright fringe of order n2, corresponds to a position d2,

R D d n2 22 2 2= = λ

then

( ) ( ) ( )R D D d d n n2 21 2 1 2 1 2− = − = − λ (2)

and this equation gives the correspondence between the actual distance moved by the mirror M1 in wavelengths of light and the reading on the micrometer screw gauge, R(D1 - D2)

5. Insert the green filter; (a wide band filter centred on a wavelength value of 546.1 nm) and find the widely spaced, low order circular fringes Count while about 50 or more fringes disappear through the centre of the fringe system as the mirror M1 is moved by the micrometer screw gauge and record the distance moved. Use equation (2) above to calibrate the micrometer screw gauge in terms of the wavelength of the mercury green line and determine the reduction ratio R of the micrometer movement. You should expect that the manufacturer of the interferometer will have made this reduction ratio a simple fraction. Remember that all of your subsequent measurements will be influenced by the accuracy of this calibration, so you should consider how many times it is necessary to repeat the calibration measurement to obtain concordant results and a reliable value of R. It is also a good idea if both partners do this (and every subsequent part of the experiment) independently, to obtain the most consistent results.

(c) Measurement of the Refractive Index of glass and mica

If a layer of material of thickness t and refractive index µG is placed in the fixed arm of the interferometer the optical path length of this beam is increased since the light travels more slowly in the material. The effective change in optical path length ∆d is given by,

( )R D d t G A2 2 2∆ ∆= = −µ µ

where µA is the refractive index of air and (see the next section) µA ≈ 1, so that,

( )R D d t G2 2 2 1∆ ∆= = −µ (3)

6. Replace the mercury lamp by the tungsten lamp and find the localised white light fringes and measure the position of the central maximum with the micrometer screw gauge. Measure the thickness of the microscope cover glass several times using the digital micrometer provided. Insert the microscope cover glass in its holder into the

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interferometer arm of fixed length. The white light fringes will now be shifted on account of the change produced the optical path length. Which direction must the mirror M1 be moved to relocate the white light fringes? Measure the new position of the central fringe, repeat as many times and necessary to obtain concordant results and hence find the value of the refractive index µG of the microscope cover slide using equation (3) above. Repeat this experiment using the mica sheet and hence obtain the refractive index µM of mica. Compare your results with the accepted values.

(d) Measurement of the Refractive Index of Air

The measurement of the refractive index of air is another variation of the experiment described in the section above. The special gas cell has length l, so if it is first evacuated and then refilled with air, the change in the optical path length will be given by,

( )R D d l Gas Vac2 2 2∆ ∆= = −µ µ (4)

where µGas and µVac are the refractive indices of the gas/air and the vacuum respectively (take µVac ≡ 1).

Thus if the interferometer is adjusted to observe low order circular fringes, a fringe shift of ∆n will be observed on refilling the gas cell, which is given by combining equations (2) and (4).

( )∆n l Gasλ µ= −2 1 (5)

7. To measure the refractive index of the air, the special cell provided should be placed in position using the holes in the variable arm of the interferometer. The compensating glass plates (in the Terry spring clip) which are equivalent to the two windows of the cell should be placed in the fixed arm and gently tightened in place with the knurled nut so that they are perpendicular to the light path and cannot twist during the experiment. The connecting tubes are connected to a manifold outside the experimental box, as shown in Figure 4. The farthest green tap connects the cell to the vacuum pump and the nearest green tap connects it back to the air. The flow of air back into the evacuated cell can be controlled by the needle valve and the rate can be adjusted so that the fringe shifts observed move at a conveniently slow rate as the gas cell is refilled.

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Figure 4 Arrangement of the gas and vacuum lines

8. No matter how carefully the gas cell and compensator plates are attached to the interferometer, this will have spoilt the previous alignment, which was correct on the scale of the wavelength of light. To measure the refractive indices of the air, the first task is therefore to re-adjust mirror M2 of the interferometer so that wide circular fringes can be seen again, using the mercury source with the green filter in place.

9. When this has been done, the gas cell can be evacuated completely (as indicated by the vacuum gauge) and the green valve closed to isolate the pump. Count the fringe shift as the air is slowly readmitted to the cell. Use equation 5 above to obtain the refractive index of air. The internal length of the gas cell is l = 50.6 mm and the green filter is centred on a wavelength value of 546.1 nm

10. Finally, compare the value obtained for the refractive index of air with the accepted value.

(e) Measurement of the Refractive Index of three unknown gases.

Caution: in this final part of the experiment you will use a Class 2 diode laser. This has a low power, visible, red beam. Eye damage can occur if the laser beam is shone directly into the eye. The laser/detector module is designed to prevent this and to operate only when it is correctly installed into the interferometer and the protective microswitch is depressed. Do not attempt to use the laser in any other circumstances.

11. In this final part of the experiment, the self contained laser/detector module, whose output is interfaced to a computer, will be used to try to measure the refractive indices of three unknown gases to the precision that Michelson believed was possible with his interferometer. However, before mounting the laser/detector module, it is wise to double check that the best possible alignment of the interferometer has been obtained. Then the L-shaped laser/detector module can be carefully installed and the single wing nut* tightened gently so that the protective microswitch is activated and the laser operates.

*Do not touch any other nut or bolt on laser/detector module since this will affect the relative alignment of the laser and the photodiode. If necessary, this task must be done by the Laboratory Technician ALONE.

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12. This final experiment is basically identical to the one you have just performed, except that the fringes will be counted from a graphical output of the photodiode detector and the evacuated gas cell will be refilled with one of the three unknown gases in turn The three coloured taps (red, blue and black) on the gas manifold are in pipes connected to three different gas cylinders and the same needle valve can be used to control the rate of gas flow,as shown in Figure 4 The pipework to the gas cylinders is installed under the experimental workbench and one vacuum pump serves all the interferometers in the Laboratory. In the previous measurement of the refractive index of air, the pressure obviously returns exactly to atmospheric pressure when the gas cell was opened to air. The gas cylinders used in this present experiment also serve the Speed of Sound experiment G5. They may therefore be set to a slight over pressure. If this is the case, make sure that you count only those fringes which correspond to the evacuated gas cell being refilled back to just one atmosphere of the gas.

Note: If you are at all uncertain about the correct procedure please consult a demonstrator. Take care not to leave the coloured taps open, because the gas cylinders may become empty and they are expensive to replenish.

13. Click on the Picoscope icon. The default setting, as the name suggests, is a “chart recorder” mode in which the x and y axes can be chosen to suit the observations.

Note: Make sure that in the left bottom corner of the screen there is an icon which shows either STOP (running) or START (stopped) and if this is absent, re-load Picoscope

14. The output from the photodiode is about ≈ 100mV, depending on the exact alignment, so that the y-axis multiplier × 10 or × 20 should be chosen and the y-axis brought down to the origin using the left hand slider bar.

15. The aim is to display the whole fringe pattern on the screen as the gas cell refills with the selected gas. This means that if the scale of the x-axis is set to say, 10 seconds per division, then the whole x-axis on the screen will correspond to 100 seconds. The needle valve should therefore be adjusted so that the gas cell completely refills in less than 100 seconds (this probably means opened by about one and a quarter or one and a half turns). Make a number of tests to achieve this condition, using just air to conserve the unknown gases. The process is the same as before, - evacuate the gas cell completely (as indicated by the vacuum gauge) and close the green valve to isolate the pump. Open the needle value the chosen amount and then, OPEN the main green value to the air and at the same time click on the START icon at the bottom left hand corner of the screen.

16. Watch the evolution of the fringe pattern on the screen and click on the STOP icon to stop the data acquisition when the manometer has indicated that the gas cell has been refilled with exactly one atmosphere of gas. Ideally this point will be reached before the fringe pattern moves off the left of the screen if you have adjusted the needle valve correctly. You may find that the first fringes are very close together when the whole fringe pattern is on the screen at once, as shown in Figure 5 below. This can be overcome in the next steps of the analysis.

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fringe shift air

0

20

40

60

80

100

120

140

160

180

200

15000 25000 35000 45000 55000 65000 75000 85000 95000 105000

sampling time

frin

ge in

tens

ity

Figure 5 The fringe pattern recorded with the Picoscope system is shown.

17. Click on EDIT and then COPY AS TEXT.

18. Open an Excel workbook and paste in the copied text (it will consist of a header and two columns of numbers)

19. Use the Chart Wizard facility in Excel to plot the fringe pattern. You can now for example, copy the basic chart a number of times and then by altering the scales on the x-axis of each sub-chart produce a wide graph of the fringe pattern which will be easy to measure accurately.

20. Count the number of fringes (and part fringes) and hence re-determine the refractive index of air. The wavelength of the laser is 633.0 nm. Not unaturally there is a lot of information known and tabulated about the refractive index of air. You can check your value against the accepted wavelength-dependent refractive index of air using the following formula:

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24 10315.15358.11072599.21

λλµ ×

++×=− −air ,

where the wavelength λ is in nm. Note that there are also corrections for temperature, pressure and relative humidity in the usual text books like Kaye and Laby which you may apply if you have time.

21. Once you have obtained reasonable results for air, continue the procedure for the three unknown gases. Hence identify which colour of tap supplies which gas, by reference to Table 1 below. Remember to repeat your measurements as many times as necessary to obtain accurate results.

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22. Is there any clever way that you can combine your results in Excel and find, as Michelson did, if there are fractional fringe shifts? You may also need to consider if there is any “dead volume” in the pumping lines, which might influence the experiment, or cause cross contamination of the gases.

Table 1: Refractive indices of gases

Gas µGas

Air 1.000292

Argon 1.000281

Carbon Dioxide 1.000451

Helium 1.000036

Nitrogen 1.000297

Oxygen 1.000272

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Second Year Laboratory - University of Sheffield/file/L9.pdf · difference between the mirrors ... mechanical lever connected to a the micrometer screw gauge as shown ... University