Experiment 4 – The Michelson Interferometer 1

Experiment 4

The MichelsonInterferometer

1 Introduction

There are, in general, a number of types of optical instruments thatproduce optical interference. These instruments are grouped under thegeneric name of interferometers. The Michelson interferometer causesinterference by splitting a beam of light into two parts. Each part ismade to travel a different path and brought back together where theyinterfere according to their path length difference.

You will use the Michelson interferometer to observe the interferenceof two light sources: a HeNe laser and a sodium lamp. You will studyinterference patterns quantitatively to determine the wavelengths andsplitting of the Na D lines empirically. You will use the HeNe laserinterference spectrum to calibrate the interferometer.

2 Background - see Hecht, Chap. 9

2.1 The Michelson Interferometer

The Michelson interferometer is a device that produces interferencebetween two beams of light. A diagram of the apparatus is shown inFig. 1. The basic operation of the interferometer is as follows. Lightfrom a light source is split into two parts. One part of the light travelsa different path length than the other. After traversing these differentpath lengths, the two parts of the light are brought together to interferewith each other. The interference pattern can be seen on a screen.

Light from the source strikes the beam splitter (designated by S).The beam splitter allows 50% of the radiation to be transmitted to thetranslatable mirror M1. The other 50% of the radiation is reflected tothe fixed mirror M2. The compensator plate C is introduced along this

Experiment 4 – The Michelson Interferometer 2

Figure 1: Schematic illustration of a Michelson interferometer.

path to make each path have the same optical path length when M1

and M2 are the same distance from the beam splitter. After returningfrom M1, 50% of the light is reflected toward the frosted glass screen.Likewise, 50% of the light returning from M2 is transmitted to theglass screen. At the screen, the two beams are superposed and one canobserve the interference between them.

2.2 Interference of Waves With a Single Frequency

If two waves simultaneously propagate through the same region ofspace, the resultant electric field at any point in that region is thevector sum of the electric field of each wave. This is the principle ofsuperposition. (We assume all waves have the same polarization).

If two beams emanate from a common source, but travel over twodifferent paths to a detector, the field at the detector will be determinedby the optical path difference, which we will denote by ∆x = x2 − x1.A related quantity is the phase difference, ∆φ, given by

∆φ =2π

λ∆x = k∆x, (1)

where k is the wavenumber. Constructive interference occurs when

∆φ = 2mπ, m = 0, ±1, ±2, ±3, . . . . (2)

Destructive interference occurs when

∆φ = ± (2m + 1) π, m = 0, 1, 2, 3, . . . . (3)

Experiment 4 – The Michelson Interferometer 3

Figure 2: Beat signal from two input frequencies into a Michelson in-terferometer

2.3 Interference of Waves with Two Frequencies

We will now consider the case of two frequencies with wavenumbers k1

and k2 that together follow two different paths with a difference of ∆x.The sum of the waves with different amplitudes at point x along thex-axis is given by:

ET =(eixk1 + ei(x+∆x)k1

)E1 +

(eixk2 + ei(x+∆x)k2

)E2 (4)

If we let a = E2/E1 and define δk = (k1 − k2) /2, after a lot of algebra,we can write the intensity (E∗

T ET ) as:

2(1 + a + a2 + a cos 2δk∆x + (1 + a) (cos k1∆x + a cos k2∆x)

)(5)

Figure 2 shows the expected signal, which consists of a fast oscillationas well as a slow oscillation characteristic of δk.

Experiment 4 – The Michelson Interferometer 4

3 Experiment

In the following experiments, you will calibrate the movement of M1

with the HeNe laser and use the interferometer to accurately measurethe wavelengths of the fine structure doublet of the sodium D line, aconsequence of the spin of the electron.

3.1 Calibration with HeNe Laser Light

Inject the laser beam into the Michelson intererometer. Make sure thebeam is properly retro-reflected. Initially, you will see two bright spotson the screen. Adjust the angle of the fixed mirror until these two spotsoverlap. You can use lenses to expand the beam if necessary.

Note, take care when moving M2 as the interference is verysensitive to its alignment. As you translate mirror M1, you will seefringes appearing and disappearing on the screen. The interferometerlever arm reduction factor is 5X, so that the wavelength of the lightcan be found using

λ =1

5

(2d

m

)(6)

where d is the distance mirror M1 was moved and m is the number ofrings that disappeared (or appeared) while M1 was being moved.

Use the synchronous motor to facilitate the turning of the microm-eter. As the micrometer is turning, record the interference data withthe computer. The motor runs at 0.5 rpm and the micrometer moves5 × 10−4 m/rev. This can give you a check of things, but we will usethe HeNe data (look up the HeNe wavelength on the web) to accuratelycalibrate the speed.

3.2 Sodium Light

Now use the sodium lamp to produce an interference pattern. Sincethe spectrum of this light consists primarily of two closely spaced lines(a doublet), each wavelength will produce its own set of fringes. Yourgoal will be to empirically determine λ1 and λ2 by measuring the finelyspaced fringes and the beat pattern.

It is much more challenging to get good interference patterns withthe lamps, so take your time and play with alignment, lamp placement,and possible lens placement. You will need to increase the gain in yourdetection system.

You should observe both the finely spaced pattern as well as a mod-ulation in the contrast at the difference frequency of the two lines. Yourgoal is to as accurately as possible measure the wavelength of the twosodium lines. Be sure to carefully estimate uncertainties.

Each of the doublet lines of the sodium lamp are not monochromatic

Experiment 4 – The Michelson Interferometer 5

due to broadening from pressure effects and motion of the atoms in thelamp (Doppler effect). This means the coherence length is not thatlarge. If the path length difference is too large, you will not see anyfringes. Measure the coherence length of your lamp.

Michelson Interferometer

The Michelson interferometer has also been used to provide evidence for the special theory of relativity, to detect and measure hyperfine structure in line spectra, to measure the tidal effect of the moon on the earth and to provide a substitute standard for the meter in terms of wavelengths of light.

Assuming that the given extended source (Sodium vapor lamp) is monochromatic, use eq. (7) to

determine the wavelength of the source.

Use the instructors help to align the interferometer.

The mirrors M1 and M2 are adjusted so that circular fringes are visible in the field of view. If M1 and

M2 are equidistant from the beam splitter P1 the field of view will be perfectly dark. The mirror M2 is

kept fixed and the mirror M1 is moved with the help of the fine movement screw and the number of

fringes that cross the field of view is counted. The wavelength is determined from the fact that for one

fringe shift, the mirror moves through a distance equal to half the wavelength.

Nature of fringes: If the two mirrors M1 and M2 are not aligned precisely perpendicular to one another,

the path difference will depend on the particular region of mirror M1 (and the corresponding region of

M2) which we are observing from the position O. The field of view, then, seen by looking at mirror M1

from position O will be made up of a series of alternately bright and dark fringes, nearly straight and

parallel, similar to those produced by interference from a simple wedge. Such fringes are referred to as

fringes of equal thickness, or straight-line fringes. If the path difference is near zero, the fringes will be

broad and widely spaced in the field of view. On the other hand, if the path difference is on the order

of 40 or 50 wavelengths (p = 40 or 50), the fringes will be narrow and closely spaced, so much so that

they may be unresolvable with the naked eye.

If the two mirrors are precisely aligned such that their planes are exactly perpendicular to one another,

thus ensuring that path differences over different regions of the mirrors are constant, the fringe pattern

will be seen by the observer at 0 to consist of a series of concentric rings. Each ring will correspond to

a different angle of view measured from the perpendicular direction to the mirror M1, as illustrated in

Figure 3. The fringes shown are called fringes of equal inclination. When the mirror M1 is moved so as

to approach the condition for zero path difference, the fringe pattern will appear to collapse with all

fringes moving toward the center, then disappearing.

Fig. 3 Circular fringes (equal inclination) seen in the Michelson interferometer

But wait! The Sodium light source is not monochromatic. For the typical excitation voltages used in

the labs, there are two wavelengths close to each other. These are known as the D1 and D2 lines: the

Sodium doublet. What is the origin of these doublets? The short answer: the doublet arises due to the

electrons intrinsic spin. Some of you may know this is due to the so called spin-orbit coupling. One

requires relativistic quantum mechanics to fully understand this. You will read more about this 1 year

later in your core course, nuclear atomic and molecular physics. The splitting of the doublets is also

known as fine structure splitting.

The answer you got earlier is then the mean wavelength! Adjust one of the mirrors to see two distinct

set of circular fringes due to the two wavelengths present. The instructor will show you how to do this

if necessary.

The resulting interference pattern is the superposition of the two separate fringe systems for each

wavelength. Over certain distances of the mirror position the two fringe systems can either

constructively or destructively interference with each other when the fringe patterns overlap. When the

two maxima of the patterns are at the same position, the fringes are bright and contrast is maximum.

The fringe contrast will disappear when the bright fringe of one pattern coincides with the dark fringe

of the other. If d12 is the mirror distance between the two positions corresponding to maximum or

minimum fringe contrast,

12 1 22 1d m m

or 1 212

1 2

2d

This relation allows us to determine the fine structure splitting.

Coherence can be also defined as the degree of stability of phase of a wave (light wave in

our case) both in space and in time. By stability of phase in space we mean simply a fixed

phase relationship between two separate points on the wavefront, while by stability in time

we mean the constancy of phase within a single wave.

If a source has a spectral width , (even a laser would have a (small) spectral width. In fact no

monochromatic source can have zero spectral width. Why?), we can define a coherence length,

c

cL

where c is the speed of light in vacuum.

The coherence length of a light source can be thought of simply as the length of an

uninterrupted wave train of light. Thus, a highly coherent source that gives off light with

no phase change will produce a long wave train and will be characterized by a long

coherence length. A source that gives off short bursts of light, constantly interrupted by

arbitrary and random phase changes, will produce short wave trains of light and,

consequently, will be characterized as having a short coherence length. Lasers have

coherence lengths on the order of tens of meters and longer, while white-light sources have

a coherence length as short as one millionth (10–6) of a meter! A pure spectral line source

such as the cadmium red line can have a coherence length as long as 40 centimeters,

millions of times longer than the coherence length of white light, but considerably shorter

than the coherence length of a stable laser.

The Michelson interferometer measures coherence length of light sources directly. Since

interference can never occur between overlapping wave trains whose phases have not

remained constant and related—thus preventing the formation of fringes with an instrument

such as the Michelson interferometer—we can get a rough estimate of the coherence length

of light sources by measuring the largest difference in path length of the two mirrors over

which we still observe fringe formation. This the coherence length obtained with the

Michelson interferometer allows an estimate of the spectral width of the source used. In

other words the degree of monochromaticity can be estimated. Clearly, the wider the

spectral line, the shorter the coherence length and the less coherent the light source.

Likewise, the narrower the spectral line, the longer the coherence length and, thus, the more

coherent the light source.

Sources: www.physics.usyd.edu.au/pdfs/current/sphys/3yr_lab/Expt_24.pdf

www.sfu.ca/phys/233/labscripts/Michelson.pdf

http://agamemnon.cord.org/cm/leot/course10_Mod04/Module10-4.htm

physics.nyu.edu/~physlab/Modern_1/MInterferometer.pdf

MICHELSON INTERFEROMETER

Aim: 1. Set-up a Michelson interferometer. 2. Determination of wavelength of light / Calibration of sub-micrometer movement . 3. Study the contrast of fringes formed by a diode laser as a function of path difference. 4. Observe the effect of plane of polarization of the interfering waves on contrast of the fringes .

Components and Equipments : Bread board to assemble optical components, Diode Laser with power supply, Laser mount, Beam splitters with mount, Mirrors with mounts, Screen, Detector to measure intensity. Warning : The interferometer beam splitters and mirrors have very sensitive surfaces. Please do not touch the surfaces. Introduction: The Michelson Interferometer, first developed by Albert Michelson in 1881, has proved of vital importance in the development of modern physics. This versatile instrument was used to establish experimental evidence for the development for the validity of the special theory of relativity, to detect and measure hyperfine structure in line spectra and to provide a substitute standard for the meter in terms of wavelengths of light. Michelson himself pioneered much of this work. Setting Up : Set up the interferometer on the bread board as shown in Fig.1. Following components are fixed on the base plate: A diode laser on an adjustable mount, mirror (M1) whose tilt is adjustable and can be translated with a screw, mirror (M2), beam splitter on tilt adjustable mounts. The Mirror M2 is fitted with a micrometer screw for precise translation of the mirror with sub-micrometer precision via a lever. 1.Align the laser so that the beam is parallel with the top of the base. The beam should strike the centers of the mirrors. 2. The distances of the two mirrors from the beam splitter is made equal as closely as possible. A translational screw behind the mirror is used for this. 3. Adjust the angle of the beam splitter as needed so that the reflected beam hits the fixed mirror M1 near its center. 4. There should now be two beams of light reaching the screen via two paths; one comes from the mirror M1 and the other from the mirror M2. Adjust the angle of the mirrors M1 or M2. such that the two beams overlap on the screen. It is relatively easy to obtain fringes with some patience. (As against Mach Zehnder in which to many parameters need to be adjusted) 5. The fringes initially will be very close, too many and possibly curved, oriented in arbitrary direction. The tilt screws behind the mirrors can be used to reduce the number and get circular fringes. The number of circular fringes can be increased by increasing the path difference between the two arms. 6. The translation screw on the back of the mirror is used to vary the path difference between the two beams. An increase or decrease in the number of fringes can be seen as

Fig.1 : Schematic arrangement of a Michelson Interferometer(sourced from the internet) you adjust the screw. As the number of fringes increase or decrease, the fringes will be seen as emerging from the centre or collapsing at the centre. The fringe pattern is governed by the equation: 2dcosθ = mλ, where d is difference in the distance of the two mirrors from the beam splitting surface and θ is the angle of incidence on the mirrors. Note that we use a diverging light beam from laser. Satisfy your self that this equation will produce circular fringes for a given value of d. What happens when d=0 ? Measurement of wavelength of laser source / Calibration of the translational stage in sub micrometer range: The wavelength of laser light source can be determined if the mirror M1 is moved in a controlled manner by the micrometer screw of the translational stage which provides sub-micrometer displacement and the number of fringes moved counted. The micrometer screw shifts the mirror M1, in the direction of the normal to its surface, through a lever mechanism, which de-magnifies the micrometer movement. The fringe count is given by L= Nλ =2dΔ, where L is the path difference that occurs when ‘N’ fringes pass as the micrometer is moved through a distance of ‘d’ and Δ is the calibration constant that depends on the demagnification factor M of the lever. Alternatively the calibration constant of the mirror movement, can be determined if λ is known. Procedure: Determine the micrometer reading every time the fringe count changes by 10 and plot 2d vs N. The slope of this graph can be used to determine Δ.

Fringe Contrast: The fringe in a fringe pattern is defined as: C= (Imax – Imin)/ (Imax + Imin) where Imax and Imin are the intensities at the maximum and minimum respectively. The contrast is unity when he intensities of the interfering beams are equal. For quasi monochromatic light sources, such as Sodium discharge lamp, the contrast decreases rapidly as the path difference is increased , usually within a few mm. If white light is used, contrast becomes zero within a few micrometers. With laser sources larger path difference can be introduced. The maximum path difference that can be introduced in an interferometer, before the contrast becomes zero is known as the coherence length of the light source. In order to verify that the diode laser you have used has a long coherence length, or otherwise, measure the contrast of the fringes when d is close to zero, and again when the path difference is as large as you possibly can introduce with the setup. Measure the max and min intensities of 10 consecutive fringes at both locations. Plot ( manually fit) a cos2

distribution to the measurements and calculate the contrast. Plane of polarization of interfering beams and the Fringe contrast: Cross two polarizers and place them in the two arm of the interferometer. Rotate the laser in its mount to adjust equal intensity of light after the polarizers in the two arms. This will ensure that two interfering beams reaching the screen have equal intensity, which in principle should give maximum contrast. No fringes are observed. Rotate one of the polarizers, fringes should start appearing. The contrast increases with further rotation of the polarizer. Why? Alternately, one more polarizer is placed in front of the screen and rotated. Fringes with varying contrast will be seen. Measure the contrast for different orientations of the rotating polarizer.