PHYS2055 - Intro to Fields in Physics
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PHYS2055 - Intro to Fields in Physics
Laboratory – Michelson Interferometer
2019
1 Introduction
The Michelson Interferometer is one of the most commonly used amplitude splitting interferometers. Michelson and Morley used a bench-top interferometer in an attempt to detect aether - the proposed medium for light propagation, while today, large kilometre-sized interferometers are in operation to detect gravitational waves. In a Michelson interferometer, light from an extended source is divided into two beams by a beam splitter. One beam travels towards a fixed mirror and is reflected back to the beamsplitter. The other beam travels to a moveable mirror, is reflected and also travels back to the beamsplitter. The two beams recombine (electric field vectors sum) and are then detected showing interference effects.
2 Theory
The layout of an interferometer is shown in Fig. 1. In this arrangement, one mirror is fixed and the other mirror can be tilted (about any axis perpendicular to the beam) and translated (along the beam direction). Both of these processes change the physical path difference between the two arms, sown as d in the figure. For a beam travelling parallel with the axis of the interferometer, light travels an extra distance of 2d in one arm compared to the other. A beam entering the system at an angle to the axis, θ, will travel an extra distance given by 2d cos θ . Furthermore, a further phase difference of π is introduced as one beam experiences external reflection at the beam splitter while the other undergoes internal reflection. If the two beams have a phase difference of π at the exit plane then they will interfere destructively and the intensity will be zero. More generally, for destructive interference between the two beams, it follows that
2d cos θ = mλ (1)
where m, an integer, is the fringe order and λ is the wavelength of the illumi- nating light. Equation (1) can be generalised to describe the fringe order by considering m as any real number, with integer values giving destructive inter- ference and half integer values giving constructive interference. Other values describe an intermediate phase difference.
Now consider the intensity observed at the centre of the mirror, looking directly along the centre axis. For a given d, the order of the central fringe, m0 , satisfies the equation
2d = m0 λ (2)
The order of the central fringe is generally non-zero and it is not necessarily a light or dark fringe, i.e. m0 can take on any real value. As we now look outwards from the centre and hence look through the interferometer at an angle θ, we will see a variation in intensity as the phase difference varies - see equation(1). Defining the angle θ 1 as the angle of observation of the point exactly one fringe from the centre, it follows that
2d cos θ1 = (m0 · 1)λ (3)
Defining the angle θp as the angle of observation of the point exactly p fringes from the centre, we can write
2d cos θp = (m0 · p)λ (4)
For small values of θ, equations (2) and (4) can be used to show that the radius of the pth ring is
θp = / 、1/2 (5)
Thus for a perfectly aligned interferometer, circular fringes are observed. A simulation is available for demonstrating some of these concepts [2].
An alternative mode of use is when one of the mirrors is tilted with respect to the other. Consider a tilt about an axis out of the page in Fig. 1. This generates parallel or Fizeau fringes which, for this axis, we would observe as vertical fringes - the phase difference varies with horizontal distance, x. If the mirror is tilted by an angle, α, then d varies with x according to
d(x) = αx (6)
Combining equations (1) and (6) and, assuming observation for small angles, θ (cos θ ≈ 1), gives
2αx = m0 λ (7)
The distance between consecutive fringes is found by setting ∆m equal to one giving
∆x = λ/2α (8)
Thus instead of circular fringes, straight parallel fringes are observed with separations determined by the angle of tilt of the mirrors.
The interference fringes can be used to make wavelength measurements. By counting the number of fringes that pass a given observation point for a given distance of mirror travel (i.e. observing how m changes as d is changed), equation (2) can be used to determine λ .
If two wavelengths λ 1 and λ2 are present in the interferometer, two sets of fringes will be present, and one observes a pattern which results from the sum of their intensities. If d is adjusted so that dark fringes of both wavelengths overlap, then good contrast fringes are observed. Upon changing d, a plate separation can be reached where the dark fringes of one wavelength overlap the bright fringes of the other wavelength. This results in low contrast fringes. The location of the positions of high and low contrast (i.e. values of d) can be used to determine the difference between the two wavelengths if the average wavelength
Fixed
mirror
M1
Image of
mirror
M 1
Compensator
Adjustable
back mirror
M2
Exit plane
Figure 1: Schematic diagram of a Michelson Interferometer.
is know. Use equation (2) to derive an expression for how far d must be changed to go from a position with high contrast fringes to the next position with low contrast fringes.
You will also need to include the following in the theory section of your report:
(a) The derivation of relationship based on equation (2) that allows you to determine the wavelength of light while varying d.
(b) The derivation of a relationship based on equation (2) that allows you to determine the difference between two wavelengths that are used to illumi- nate the interferometer. [Hint: write general forms of equation (2) for each wavelength at low and high contrast (i.e. four equations) and then solve for what is required.]
(c) An explanation of why a compensator plate must be present when observing white light fringes. Remember that such a plate is not required to observe fringes using monochromatic light.
An interferometer can also be used to detect the refractivity of a gas placed in one arm of the interferometer. Imagine we have an evacuated cell of length, l, in one arm of the interferometer. Light passing through the cell travels at the speed of light, c. If the cell is now filled with a gas, then the light travels at a reduced speed, v = c/n, where n is the refractive index of the gas. As the light travels more slowly, it takes longer to pass through the cell. This is exactly equivalent to changing the length of that arm of the interferometer. The change in effective length is
∆d = (n · 1)l (9)
The term n · 1 is called the refractivity of the gas. By counting fringes (observing a change in m), as the pressure in the cell is increased from a vacuum to atmospheric pressure, the value of n · 1 can be determined for the gas.
The interferometer can be used to determine the refractivity of a gas flowing through one arm of the interferometer. Equation (9) can be used in exactly the same way. However, rather than counting fringes that pass with time, the fringe shift is counted from a position in the field of view that is undisturbed by the flow, to a position within the flow. One also needs an estimate of l, the width of the flow.
3 Experiments
3.1 Single week experiment
You are provided with the Michelson interferometer, a sodium lamp, a white light source and a camera. Please exercise care not to touch any of the glass surfaces - if you do so, then please contact the tutor (any attempt to clean them will make the situation worse).
One of the two mirrors is adjustable along the axis of the beam using a micrometer (note that there is a mechanical advantage in this adjustment – the mirror moves only a fraction of the distance that the micrometer moves - see the label on your interferometer for the actual ratio). Alignment should be done using only this mirror. The two adjustments on the back of the mirror change the horizontal and vertical tilt of the mirror. Under no circumstances change the settings on the other mirror (no matter how desperate).
Set up the sodium lamp so that light passes through the interferometer and can be observed at the exit plane. The predominant output of the lamp is the sodium doublet at around 590 nm. Without the screen at the entrance to the interferometer, adjust the tilt of the mirror (two screws on the back of the mirror) so that the two brightest images of the lamp overlap precisely. You should begin to observe fringes (dark and bright stripes). If not, adjust the separation of the mirrors by several turns on the micrometer and realign. Insert the scattering plate to obtain a uniform intensity distribution. Experiment by
changing each of the three mirror settings (slowly) to see what happens. Can you explain what you observe?
The camera can be operated by logging into Windows on the computer and starting the Logitech software. In the advanced settings you will be able to adjust the intensity of the image and the zoom. First ensure that you can see good quality fringes by eye - then place the camera in front of the interferometer, put it in movie mode and adjust the settings as appropriate. The camera may be useful when counting fringes but remember that your eye is a far better detector.
In the experiment you should complete and report on the following:
(a) Measurement of mean wavelength. Use fringe counting while adjusting the mirror separation to determine the mean wavelength as described above. The fringes can be observed by eye or on the computer monitor using the camera. Design a process where linear regression can be used to extract a result.
(b) Measurement of wavelength separation. Use the fringe contrast to determine the wavelength separation by measuring the distance the mirror must be moved to change from high contrast to low contrast fringes.
3.2 Extended experiment
Students completing this experiment over two weeks should first complete the single week section and then design and execute an extension to the experiment (which includes a quantitative element). Some suggestions are given below but you may also design your own experiment - discuss your ideas with your tutor before proceeding.
Air refractivity. Place the test chamber in the holder in one of the arms of the interferometer. Use the bulb to increase the pressure in the cham- ber. Observe the fringes as you slowly let the pressure decrease through the vent. From your measurements, determine the refractivity of air by counting fringes as the pressure changes by a known amount.
Flow visualisation. Use the supplied pressure pack with the straw-like nozzle. Insert the end of the nozzle into one of the arms and then gently send a spray through the field of view (across the path of the light and take care to ensure that the spray is not aimed at one of the optical components). Use the camera to take a photo and then characterise how the fringes shift through the middle of the flow.
White light fringes. These will appear when the two arms of the inter- ferometer are set to exactly the same length. As there is no absolute scale in the adjustment, you will have to go looking for this point. Start by setting up the interferometer using the sodium lamp so that you have clear vertical fringes - perhaps about 10 across the field of view. From this point only adjust the mirror separation, not the tilt of the mirror. White light fringes will occur at a high contrast position for the sodium lamp. (Why?). Note where these high contrast positions are using the sodium lamp and then observe the white light while slowly adjusting the mirror separation across the high contrast points. The white light fringes will be clearly visible when you find them. Describe what you see including ex- plaining the colours of light that you observe. It may be difficult to obtain quantitative measurements - you may be able to draw some conclusions from measuring the fringes that you can see with white light.
References
[1] Hecht, E. (2002) ‘Optics’, 4th Edition, Addison Wesley.
[2] McIntyre, T.J. (2014) Michelson simulation. Accessible through the 5 Minute Physics PHYS2055 Module on Interference -
http://teaching.smp.uq.edu.au/fiveminutephysics/
2022-08-12