Michelson Interferometer(With Laser)

ELECTROMAGNETISM & OPTICS LAB

MICHELSON INTERFEROMETE R WITH LASER INSTRUCTION MANUAL UPASANA DAS ROLL – 08PH4012 M.Sc. PHYSICS 1ST YEAR.

18th February 2009

Michelson Interferometer with Laser2

INTRODUCTION Optical Interference corresponds to the interaction of two or more light waves yielding a resultant irradiance which deviates from the sum of the component irradiances. Interferometry is the technique of diagnosing the properties of two or more waves by studying the pattern of interference created by their superposition. There are, in general, a number of types of instruments that produce optical interference. These instruments are grouped under the generic name of Interferometers. In an interferometer there are arrangements for obtaining interference effect with a large path difference between the interfering rays. Interferometry is an important investigative technique in the fields of astronomy, fibre optics, engineering metrology, optical metrology, oceanography, seismology, quantum mechanics, plasma physics and remote sensing. The Michelson Interferometer is the most common and widely used configuration for optical interferometry and was invented by Albert Abraham Michelson. It produces interference fringes by splitting a beam of monochromatic light so that one beam strikes a fixed mirror and the other a movable mirror. When the reflected beams are brought back together, an interference pattern results. The Michelson Interferometer is an ingenious arrangement which was originally designed to test the ether-hypothesis in the famous Michelson- Morley experiment. However it is now primarily used to find the wavelength of light and laser radiation, fine structure of spectral lines, to test optical components and is also used as spectrometers in the infra-red region of the spectrum.

Michelson Interferometer •

Construction and Action of the interferometer :

In Michelson Interferometer two beams obtained by Division of Amplitude, are sent in two directions perpendicular to each other and are reflected from plane mirrors, so that they recombine to form interference fringes. 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, G1 and G2, form the main optical parts. Plate G2 is kept at an angle of 45o with the incident rays. The rear side of plate G1 is lightly silvered so that light coming from the source are divided into reflected and transmitted beams of equal intensity. Light reflected normally from mirror M1 passes through G1 a third time and reaches the detector as shown. Light reflected from mirror M2 passes through G2 twice – once on the outward journey and once after reflection from M2. This second glass plate G2 is called the

Michelson Interferometer with Laser3

compensating plate, and its function is to make the path of the two rays equal in glass. The 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 wave trains.

Fig. 1. The mirror M1 is mounted on carriage, and can be moved parallel to itself by means of a micrometer screw. Since during the experiment it is moved only in multiples of wavelength of light (~ 0.5µm), these distances must be amplified by a mechanical lever connected to the micrometer screw gauge attached to M1. 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 attached to M2 (see Figure 2.).Even when the above arrangements have been made, fringes will not be seen, unless two important requirements are fulfilled: i) The light must originate from an extended source. A point source or slit

source will not produce the desired fringes in this case. ii) The light must be in general monochromatic or at least nearly so.

Michelson Interferometer with Laser4

Fig. 2.

Michelson Interferometer with Laser5

• Formation of Circular Fringes:

Circular fringes of equal inclination Fig. 3.

Circular fringes are produced with monochromatic light, when the mirrors are in exact arrangement as shown in Figure 3. The real mirror M2 has been replaced by its virtual image, M2’, formed by reflection G1. Mirror M2’ is then parallel to M1. Due to several reflections in the real interferometer, we may think of the extended source as being at L, behind the observer, which will in turn have two virtual images L1 and L2, formed by the rays reflected from the mirror M1 and the image M2’, respectively. The two virtual sources are coherent and consequently the phases of the corresponding points P’ and P” in them are exactly the same at all instants.

Michelson Interferometer with Laser6

If d is the separation M1M2’, the virtual sources will be separated by 2d (separation P’P””). When 2d is exactly an integral number of wavelength, all light reflected normally to the mirrors will be in phase : 2d = nλ

(1)

The rays of light reflected at an angle, however, will in general not be in phase. If θ be the angle of inclination of the reflected rays with the normal, then the path difference P’A between the two rays coming to the eye from corresponding points P’ and P” of the two virtual sources is given by : (Condition for maxima) where,

(2)

n = order; λ = wavelength of light.

Hence when the eye is focussed to receive parallel rays, the rays will reinforce each other to produce maxima for those angles θ which satisfy the above equation. Similarly: (Condition for minima)

(3)

Since d and λ constants, hence θ will also be a constant for a given order n. Hence the maxima will be in the form of concentric circles about the foot of the perpendicular, from the eye to the mirror, as the common centre. Fringes of this kind, where parallel beams are brought to interfere, with a phase difference determined by the angle of inclination θ, are called fringes of equal inclination. These fringes are visible over a large path difference.

• Formation of localised fringes: When the mirrors M1 and M2 are not exactly at right angles, the mirror M1 and image M2’ will not be exactly parallel. They will enclose a wedge-shaped film of air between them. The fringes in this case are almost straight because the path difference across the field of view is primarily due to the variation of the thickness of the air film between the mirrors. Such fringes are called localised fringes. Figure 4. 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.

Michelson Interferometer with Laser7

Fig. 4.

→→→→→→ Increasing path difference d →→→→→→

Now the radii of the circular (bright) fringes are given by the formula:

(4) where, r = radius of fringe; l = length of the perpendicular drawn from the eye on to the mirror M2’. From the above equation (of that of a circle), it follows that if d is increased, i.e., M1 is moved away from M2’, the radius of the bright ring increases. Thus as M1 is moved outward each ring also moves outward. If d is decreased, the rings move towards the centre. When d = 0, the path difference is zero for all values of θ and hence circular fringes disappear and uniform illumination is obtained.

A NOTE ON COHERENCE OF LIGHT SOURCES:It will be noticed that the various methods of demonstrating interference has one important feature in common – the two interfering beams are always derived from the same source of light. From experiment it is seen that it is impossible to obtain interference fringes from two separate sources, such as two lamp filaments kept side by side. This is because light from any one source is not an infinite train of waves. There are sudden changes in phase occurring in the order of 10-8 seconds. Thus although interference patterns may exist on the screen for such short intervals, they will shift their position each time there is a phase change, with the result that no fringes at all will be seen. When two

Michelson Interferometer with Laser8

beams are derived from the same source, they have a point-to-point correspondence in phase. Thus the difference in phase between any pair of points in the two sources always remains constant, and so the interference pattern is stationary. Thus sources of the same frequency, that have a pointpoint phase relation, are called coherent sources.

EXPERIMENT This experiment consists of two parts. The first experimental exercise is to measure the wavelength of the Laser light. It is necessary to know this wavelength so that we can carry out the next experiment. The second experiment involves the measurement of the refractive index of a given glass plate. The experimental set up is a modern version of the Michelson Interferometer, with a self-contained Laser source, whose output is interfaced to a counter by means of a detector.

Experiment I(a): AIM:

To calculate the wavelength of the Laser light source using Michelson Interferometer.

APPARATUS:

Optical breadboard; Diode laser; Laser mount; Beam-splitter mount; Mirror mounts (2); Screen; Detector; Counter.

Theory:

Michelson Interferometer with Laser9

Experimental setup of Michelson interferometer. M1 and M2 are mirrors; BS is a beam-splitter.

Fig. 5. M1 and M2 are two plane mirrors silvered on the front surfaces. They are mounted vertically on two translation stages placed at the sides of the optical platform. Screws are provided at the back of the holders, adjustment of which allows M1 and M2 to be tilted. M1 can also be moved horizontally by a micrometer attached to the M1 holder. BS is a 50%-50% beam-splitter, which is a planar glass plate slightly silvered on one side. It is mounted vertically and at an angle 45o to the direction of incident light. When light from laser is allowed to fall on BS, one portion, beam A, is transmitted through BS to M2 and the other, beam B is reflected by BS to M1. Beam A returning from M2 is reflected at the back of BS to reach the screen, and beam B after getting reflected from M1 passes through BS to reach the screen. Now our working formula is,

(5) where, D = change in position of M2 that occurs for ‘n’ fringes to pass; λ = wavelength of laser light.

Calibrating the micrometer attached to the movable mirror M1:The movable mirror mount is mounted on a translation stage. The micrometer shaft actuates a lever arm which pushes the translation stage carrying the mirror. To make proper measurements it is first necessary to obtain the calibration factor or reduction ratio R, which gives the correspondence between

Michelson Interferometer with Laser10

the distance d’ moved by the micrometer screw and the actual motion D of the mirror M1. Here 10 microns (= 0.01 mm) on the thimble of the micrometer (= 1 division) is equivalent to 35 microns on the translation stage, i.e., when we move one step on the micrometer, the mirror is moved 0.35 microns. Hence

Thus D in eq (5) becomes:

PROCEDURE: 1) The different components of the Michelson Interferometer are already placed

on the optical breadboard (as shown in Figure 6.). The laser is turned on and it is so aligned, such that the beam is parallel with the top of the base. The beam should strike the centre of the beam-splitter (and also that of the fixed mirror M2) and should be reflected directly back into the laser aperture. 2) The beam-splitter is already positioned on the breadboard at an angle of 45o,

so that the reflected beam hits the movable mirror at its centre. 3) Now at this point, two sets of bright dots are observed on the screen – one set comes from the movable mirror and the other from the fixed mirror. Each set of dots include a bright dot with two or more dots of lesser brightness (due to multiple reflections in the thin film of the beam splitter). The beam-splitter is adjusted slightly until the two dots are as close as possible. 4) Using the screw on the back of the movable mirror, the mirror’s tilt is adjusted until the two dots on the screen coincide.

Michelson Interferometer with Laser11

Fig. 6.

5) At first a streaky interference pattern is observed on the screen due to the non-parallel alignment of the mirrors. Now a sensitive adjustment of the screws behind both the mirrors is carried out to obtain the desired concentric form of the fringes. The circular fringes with the diode laser looks as shown in Figure 7.

Fig. 7. 6) Now a detector is placed in the path between the beam-splitter and the screen, such that the circular fringes are formed on it. The detector is connected to a counter. As the micrometer attached to the movable mirror is moved, the interference pattern changes, and the counter directly gives the count of the number of fringes that pass.

Michelson Interferometer with Laser12 7) At first the initial micrometer reading is noted. Then the screw is turned till the

counter shows that 20 fringes (n) have passed. The final micrometer reading is noted. The difference of the two readings gives the distance moved by the micrometer (d’) which on multiplying by the factor 0.035 gives the actual distance moved by the mirror (D). 8) The above procedure is repeated 5 to 6 times and the average of the readings

is computed. The readings are then substituted in eq (5) to obtain the wavelength of the laser light.

OBSERVATIONS:

Least count of the micrometer attached to the movable mirror:-

Value of 1 small division of the main scale = 0.5 mm. Pitch of the screw = P = 0.5 mm. Total number of divisions on the circular scale = N = 50. Least count of micrometer =

= 0.01 mm.

Table I - Data for the distance D moved by the movable mirror for 20 fringes to pass:-

M.S.R = main scale reading; V.R. = vernier reading. SERIAL INITIAL MICROMETER FINAL MICROMETER d’ AVG D = NO. READING READING (mm)d’ 0.035x (mm) d’ 1. M.S.R V.R. TOTAL (mm)M.S.R. V.R. TOTAL (mm) (mm) (mm) (mm) 2. 5.5 0 5.50 5.5 18 5.68 0.18 3.

5.5

18

5.68

5.5

37

5.87

0.19

4.

5.5

37

5.87

6.0

5

6.05

0.18

5.

6.0

10

6.10

6.0

29

6.29

0.19

6.

6.0

30

6.30

6.0

48

6.48

0.18

7.

6.5

0

6.50

6.5

18

6.68

0.18

0.18 6.3x10-3

Michelson Interferometer with Laser13

CALCULATION FOR WAVELENGTH λ :-

n = 20, D= 6.3x10-3 mm = 6.3x10-6 m, Hence,

COMPARISON TABLE FOR THE WAVELENGTH OF LASER DIODE: CALCULATED VALUE OF λ (nm)

STANDARD VALUE OF λ (nm)

630.0

635.0

EXPERIMENT I (b): AIM:

To calculate the Refractive Index of glass using the Michelson

Interferometer.

APPARATUS:

Optical breadboard; Diode laser; Laser mount; Beam-splitter mount; Mirror mounts (2); Screen; Detector; Counter; Rotation stage mounted with glass plate.

THEORY:

Michelson Interferometer with Laser14

Fig. 8. In principle the method for

calculating the refractive index is relatively simple. The optical path lengths of one of the light paths will change if a glass plate is inserted into it. As the glass plate is rotated, the length of glass in the path will increase and therefore the number of wavelengths in that path will increase. This will change the interference pattern. The refractive index of the glass plate can be calculated from the number of interference fringes shifted during the rotation of the glass plate through some angle θ. The following equation was used to calculate the refractive index of the glass plate:

Where, N = number of shifted fringes; θ = the angle of rotation; λ = the wavelength of light used; t = thickness of the glass plate.

PROCEDURE: 1) The laser and the other components are already aligned in the Michelson

mode on the optical breadboard. The steps 1) to 6) are followed as in the case of experiment I (a). 2) The rotation stage is placed between the beam-splitter and movable mirror. 3) The glass plate is mounted on the rotation stage.

Michelson Interferometer with Laser15 4) The glass plate is positioned in such a way that the angle scale reads 0 o and

the light strikes the plate perpendicularly. 5) When the glass plate is introduced in the optical path, the fringes will be shifted and it will become blurred. To make the fringe pattern sharp again, the mirror mount is moved to and fro till a clear set of fringes is achieved on the screen. 6) Now the rotation stage is slowly rotated and the number of fringe transitions

is counted (by the counter), that occurs as the plate is rotated from 0o to 10o.

OBSERVATIONS:

Vernier constant of screw gauge used to measure thickness o f glass plate:-

Value of 1 small division of the main scale = 0.5 mm. Pitch of the screw = P = 0.5 mm. Total number of divisions on the circular scale = N = 50. Least count of screw gauge =

= 0.01 mm.

Table I - Data for the measurement of the thickness of the glass plate:-

SERIAL NO.

THICKNESS OF GLASS PLATE M.S.R V.R. TOTAL (mm) (mm)

1.

1.0

20

1.20

2.

1.0

21

1.21

AVERAGE THICKNESS t (mm)

1.20

Michelson Interferometer with Laser16

3.

1.0

19

1.19

Table II - Data for the number of shifted fringes N, when the glass plate is rotated by an angle 10o :SERIAL NO.

NO. OF FRINGES

1. 2. 3. 4. 5. 6.

15 16 17 15 15 17

AVERAGE NO. OF FRINGES N

16

CALCULATION FOR REFRACTIVE INDEX OF GLASS ng :-

Where, t = 1.2 mm ; N = 16 ; λ = 630nm [as calculated in experiment o I (a) ]; θ = 10 . Hence,

COMPARISON GLASS:-

TABLE

FOR

THE

REFRACTIVE

INDEX

OF

CALCULATED VALUE OF ng

STANDARD VALUE OF ng FOR CROWN GLASS

1.40

1.52

Michelson Interferometer with Laser17

PRECAUTIONS & DISCUSSIONS 1) The diode laser has a low power, visible red beam. Eye damage may occur if

the laser is directly shown into the eye. Hence care should be taken while working with the laser. One should not look directly into the laser beam, or even at its reflection. 2) The reflecting surfaces of the mirror and the beam-splitter should not be touched, as the finger grease eventually causes etching and permanent damage. Also the surface should not be wiped with a tissue, as it will leave a scratch, which will give rise to scattering of light. 3) The knobs attached to the mirror mounts should rotate effortlessly. No excess

force should be applied to these knobs, as that would damage the precision screw which moves the mirror or optical part. 4) To avoid backlash error in the micrometer screw, it should be rotated only in one direction, while doing repetitive experiments. 5) The micrometer must be rotated very gradually while applying a steady pressure on the thimble. 6) The detector must be positioned carefully so that the full image of the circular fringes falls on it, else the counter will not register any count as the fringes shift. 7) The entire set-up is kept on a breadboard which is itself mounted on a tire

tube. This is because the Michelson Interferometer is very sensitive to the shifts of its optical components and hence has to be well shielded from vibrations. The heavy breadboard on the tire tube acts like a low-pass frequency filter that cuts out frequencies higher than a few Hz, which may cause the mirrors to vibrate. Vibrations of the whole interferometer at lower frequencies are not a problem, since they do not change the relative position of the optical elements.

A NOTE ON LASER : A LASER is a device that emits light through a process called stimulated emission. The term laser is an acronym for “ LIGHT AMPLIFICATION BY STIMULATED EMISSION OF RADIATION “.

Michelson Interferometer with Laser18

Fig. 9.

If an atom is already in the excited state, it may be perturbed by the passage of a photon which has a frequency ν21 corresponding to the energy gap ΔE of the excited state to ground state transition. In this case, the excited atom relaxes to the ground state, and is induced to produce a second photon of frequency ν21. The original photon is not absorbed by the atom, and so the result is two photons of the same frequency. This process is known as stimulated emission. The rate at which stimulated emission occurs is proportional to the number of atoms N2 in the excited state, and the radiation density of the light. Therefore, when the numbers of atoms in the ground and excited states are equal, the rate of stimulated emission is equal to the rate of absorption for a given radiation density. The critical detail of stimulated emission is that the induced photon has the same frequency and phase as the inducing photon. In other words, the two photons are coherent. It is this property that allows optical amplification, and the production of a laser system. If the higher energy state has a greater population than the lower energy state (N1 < N2), then the emission process dominates, and light in the system undergoes a net increase in intensity. It is thus clear that to produce a faster rate of stimulated emissions than absorptions, it is required that the ratio of the populations of the two states is such that N2/N1 > 1; In other words, a population inversion is required for laser operation. (see figure 9.)

Characteristics of laser light:

i) COHERENT : Coherence is one of the unique properties of laser light. It arises

from the stimulated emission process which provides the amplification. Since a common stimulus triggers the emission events which provide the amplified light, the emitted photons are in step and have a definite phase relation to each other. ii) MONOCHROMATIC : Laser light consists of essentially one wavelength, having

its origin in stimulated emission from one set of atomic energy levels. So the

Michelson Interferometer with Laser19

laser light has a single spectral colour and is almost the purest monochromatic light available. iii) COLLIMATED : Because of bouncing back between mirrored ends of a laser

cavity, those paths which sustain amplification must pass between the mirrors many times and be very nearly perpendicular to the mirrors. As a result laser beams are narrow and do not spread very much.

APPLICATIONS OF THE MICHELSON INTERFEROMETER Determination of small differences of wavelengths of light from a source: I)

If the source which produces the interference fringes emits light of a single wavelength λ, then for each displacement of the movable mirror by a distance λ/2, the position of one bright fringe will be occupied by the next, but the appearance of the interference pattern will remain unaltered. If however the source emits light of two wavelengths λ1 and λ2 (λ1 > λ2), then each wave will produce an interference pattern of its own. In this case if the movable mirror is displaced, the fringes will be alternately visible or invisible or they will be alternately in consonance and dissonance. Let the mirror be displaced by‘d’, so that the fringes pass from one consonance to the next, through the intermediate state of dissonance. This will happen if:

where, n and (n+1) represents the number of fringe shifts for lights of wavelengths λ1 and λ2 respectively. Therefore :

or, or,

Michelson Interferometer with Laser20

Thus if λ1 is known, λ2 can be found out and hence (λ1 – λ2) can be determined. The Michelson Interferometer has been also used for the detection of gravitational waves, as a tunable narrow band filter, and as the core of Fourier transform spectroscopy. There are also some interesting applications as a "nulling" instrument that is used for detecting planets around nearby stars.

II)Testing optical surfaces: Various adaptations of the Michelson interferometer enable optician engineers to test the quality of flat or spherical mirrors, of lenses, lens systems, etcetera. III)To

measure the refractive indices of gases

The Michelson Interferometer has also been used for the detection of gravitational waves, as a tunable narrow band filter, and as the core of Fourier transform spectroscopy. There are also some interesting applications as a "nulling" instrument that is used for detecting planets around nearby stars. IV)

VIVA-VOICE QUESTIONS

1) What is Interference? What are the necessary conditions for interference to occur? 2) Why is the interferometer mounted on a breadboard, itself resting on a tire tube? 3) What are the similarities and differences between the circular fringes obtained in the Newton’s Rings experiment and those of the Michelson interferometer? 4)

What are localized fringes?

5)

What do you mean by ‘fringes of equal inclination’?

How is the order number of the fringes in this case related to the radius of the rings? 6)

7) Why in this set-up of Michelson Interferometer with laser, no compensating plate is used?

Michelson Interferometer with Laser21

8) What will be your observation, if white light is used instead of the laser source? 9) How can you use the Michelson Interferometer to standardise the metre? 10) What is the working principle behind the laser? What is meant by population inversion?

References: [1] OPTICS [2] FUNDAMENTALS OF OPTICS [3] A TEXT BOOK ON LIGHT Mazumdar.

by Eugene Hecht. by Jenkins & White. by B. Ghosh and K.G.

[4] http://en.wikipedia.org/wiki/Michelson_interferometer [5] http://hyperphysics.phy-astr.gsu.edu/Hbase/phyopt/michel.html [6] http://www.colorado.edu/physics/phys5430/phys5430_sp01/PDF %20files/Michelson%20Interferometer.pdf [7] http://www.phy.davidson.edu/StuHome/cabell_f/diffractionfinal/pag es/Michelson.htm [8] http://en.wikipedia.org/wiki/Laser