Physics 233 Experiment 33

The Michelson Interferometer

References

1. Jenkins and White, Fundamentals of , 3rd Edition (1957) McGraw-Hill, Chapters 13 and 14 (QC355 J4)

2. Sears, Optics, (1949) Addison-Wesley, Chapter 8, (QC355 S45)

3. Morgan, J., Introduction to Geometrical and Physical Optics, (1953), McGraw-Hill, Chapter 12. (QC355 M65)

4. Operation Instructions for the M4 Interferometer, Atomic Laboratories, Inc., Cenco Instruments Corporataion.

5. Instruction Manual, Beck Interferometer M300/6407, R. & J. Beck Ltd.

Introduction

Interference is naturally suggested by a theory. However, interference is only observable if the interacting originate from coherent sources. This restriction is understandable in terms of the properties of the emitting atoms of the source. Since the half-life of an emitting atom is short and the phase differences between the radiations of the same wavelength emitted by different atoms are random, there is no possibility of observing sustained interference between from different atoms, even of the same source. However, interference can be observed if light emitted from a single atom is divided into two or more parts, these parts made to travel different optical paths, and then superimposed at the points of observation. In this way the interfering parts will have a phase difference which is independent of time, depending only on the optical path difference.

There are two main methods of obtaining interference: 1. division of wave front, and 2. division of amplitude. In division of wave front, a point source gives rise to an extended wave front having constant phase in all directions from the source. Different portions of this wave front are then separated by mirrors, lenses or prisms. In the latter, an extended source is used. The wave is divided by partial reflection, the two resulting wave fronts having the original extent, but having reduced amplitude.

The Michelson Interferometer produces interference by means of division of amplitude. 2 Physics 233

M2 C

y

M3

A x B S

P1 P2 M1

O

Figure 1 The Michelson Interferometer

In the diagram refraction is ignored and anti-parallel rays are slightly displaced for the sake of clarity.

Consider a ray of light from the source S incident at 45° on the half-silvered rear surface M3 of the beam splitter P1 at point A. The transmitted ray x passes through the compensator P2 reflects from stationary mirror M1 and returns through the compensator to point A where it reflects to the observer O. Meanwhile, the reflected ray y traverses P1, reflects from movable mirror M2, passes through P1 and recombines with ray x as it goes to observer O. Both rays are thus viewed superimposed.

To calculate the phase difference Df between rays x and y, we need to consider the optical path difference between the two beams. The optical compensator P2 ensures that both beams travel the same distance in glass so the path difference is just

2|AB – AC| º 2d (1)

Therefore the phase difference, f o, due to the optical path difference is just

2p f o = 2d (2) l

Ray x undergoes two external reflections (at M3 and M1) whereas y undergoes only one (at M2) so an additional phase difference f r = p is introduced. The condition for constructive interference is that the total phase difference Df be an integral multiple of 2p:

2p Df= f o +f r = 2d - p where m = 0, 1, ... (3) l Physics 233 3

or 1 2d = (m + /2)l (4)

If the mirror M2 is moved a distance Dd then Dm light or dark patterns successively appear at the centre of the field of view, where

2Dd = lD m (5)

Not all the rays from the source are of the normal reflection type as discussed above. The more general case is depicted in Figure 2.

P¢ 2d P¢¢ P¢ P¢¢ q P M ¢ M 2d 1 2 cos q q d

2d

Figure 2 Reflection at arbitrary angles of incidence

The real mirror M1 has been replaced by its virtual image M1¢ formed by reflection in P1. M1¢ is parallel to M2. The extended source S, behind the observer, forms two virtual sources S1 and S2 by reflection in the mirrors M1¢ and M2. Note that these virtual sources are coherent since the phase difference between corresponding points in the two is the same at all times. When the separation of the mirrors M1¢ and M2 is d, the optical path difference between the parallel rays coming to the observer from the corresponding points P¢ and P¢¢ is 2d cos q. Taking account of the phase changes occurring upon reflection the condition for constructive interference becomes

2d cos q = ml (6)

This equation shows that the fringes produced are circular. For more details see ref. 1, p 246.

Prelab Questions

1. Show how equation 6 implies circular fringes.

2. When viewing the white light fringes which colour should appear on the outside of the circular fringes and which colour on the inside? (The colours in the figure may be inaccurate.) 4 Physics 233

Apparatus

Michelson Interferometer, sodium, mercury and incandescent light sources, glass plates, vacuum cell, gauge, pump, air, other gasses for optional part.

Two points of caution are essential:

a) The optical surfaces of the interferometer must never be touched with fingers or any other object. (With the possible exception of camel's or squirrel's hair brushes.)

b) Damage to the eye will result from direct exposure to the UV emission of the mercury vapour lamp. Always use a glass shield or filter and keep the metal housing around the bulb whenever it is on.

M2 of the Michelson optics is mounted on a carriage which is driven by a micrometer. This controls the length of the AC arm of the interferometer and hence the path difference d. The interferometer must be calibrated, i.e., the ratio K where

Change in micrometer reading K = Distance of carriage movement must be found. An ordinary lens may be used to magnify the fringe pattern. Always be careful to eliminate the effects of backlash in the micrometer screw.

Experiment

Part 1.

1. Obtain an interference pattern using sodium light which consists mainly of two yellow wavelengths. Make the distance AM1 and AM2 roughly equal and use a sighting pin during the initial mirror tilt adjustments. The fringes should be circular and centred, rather thick and of good contrast. (See ref. 4.)

2. Assuming the wavelength of green mercury light to be 546.07 nm calibrate the carriage movement.

3. Using sodium light, determine the mean wavelength and the separation of the sodium doublet.

4. White light fringes can be observed with the Michelson interferometer when the optical path difference of the interfering beams is nearly zero. Observe the striking colour changes in the pattern as the path difference is slowly varied from zero. (Ref. 4, and Appendix to this handout.)

Part 2.

1. With either light source, determine for the known wavelength the optical path length of the transparent plate provided. Physics 233 5

2. With the auxiliary vacuum apparatus provided, determine the of air.

Optional Experiment

1. Measure the indices of reflection of gases other than air (e.g., He, CO2). Does your answer depend on the wavelength of the light?

2. Measure the index of refraction of a thin glass plate (ref. 4. page 9.)

Appendix A procedure for obtaining zero path length difference and white light fringes

1. White light contains a whole mess a wavelengths. Sodium yellow light contains two wavelengths. If white light fringes are to be observed it must be possible, at least, to see high contrast sodium-yellow fringes.

2. The distance between neighbouring fringes of a given wavelength increases as the modulus of the optical path difference produced by the interferometer decreases.

The above facts are used in the following procedure for obtaining white-light fringes.

Not to scale Movable Mirror

Sodium Lamp Black Card Tungsten Lamp Diffusing Screen

What you should see (sort of)

White-light Yellow Fringes Fringes i. Set up the arrangement shown in the above diagram so that the white card obscures some of the original sodium fringes.

ii. Make sure that the sodium fringes which remain visible have good contrast and are widely spaced. 6 Physics 233

iii. Search for white light fringes by turning the micrometer screw. When you alter the path length do so at such a rate that you can always see the yellow fringes traversing the field of view. In this way you will be able to approach zero path difference at maximum speed but will not miss the appearance of the white light fringes. It is very easy in the absence of the sodium light to pass through the zero path position and not to observe the white light fringes.

It should be remembered that there are several positions of the moving mirror where the contrast of the sodium fringes is high and where the fringes are widely spaced. Only one of these will include the zero path situation. If your first attempt to obtain white light fringes is unsuccessful, move the mirror until the fringes are again sharp and widely spaced and try again. It should not be necessary to repeat the procedure more than once or twice because as one leaves the position of zero path difference the sodium fringes become more closely spaced, an effect which is easily seen.

The above method is not the only one which may be used to find the white light fringes. See standard optics reference books for other methods.

UG2/2000