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Figure 1 shows the traditional setting for a Michelson interferometer. A beamsplitter (a glass plate which is partially silver-coated on the front surface and angled ...

Figure 1 shows the traditional setting for a Michelson interferometer A beamsplitter a glass plate which

is partially silver coated on the front surface and angled at 45 degrees splits the laser beam into two parts

of equal amplitude One beam that was initially transmitted by the beamsplitter travels to a fixed mirror

M1 and back again One half of this amplitude is then reflected from the partially silvered surface and

directed at 90 degrees toward the observer you will use a viewing screen At the same time the second

beam reflected by the beamsplitter travels at 90 degrees toward mirror M2 and back Since this beam

never travels through the glass beamsplitter plate its optical path length is shorter than for the first beam

To compensate for that it passing twice through a clear glass plate called the compensator plate that has

the same thickness At the beamsplitter one half of this light is transmitted to an observer overlapping with

the first beam and the total amplitude of the light at the screen is a combination of amplitude of the two

Etotal E1 E2 E0 E0 cos k l 1

Here k l is a phase shift optical delay between the two wavefronts caused by the difference in optical

pathlengths for the two beams l k 2 n is the wave number is the wavelength of light in vacuum

and n is the refractive index of the optical medium in our case air

Mirror M2 is mounted on a precision traveling platform Imagine that we adjust its position by turning

the micrometer screw such that the distance traversed on both arms is exactly identical Because the

thickness of the compensator plate and the beamsplitter are the same both wavefronts pass through the

same amount of glass and air so the pathlength of the light beams in both interferometer arms will be

exactly the same Therefore two fields will arrive in phase to the observer and their amplitudes will add

up constructively producing a bright spot on the viewing screen If now you turn the micrometer to offset

the length of one arm by a half of light wavelength l 2 they will acquire a relative phase of and

total destructive interference will occur

E1 E2 0 or E1 E2

It is easy to see that constructive interference happens when the difference between pathlengths in two

interferometer arms is equal to the integer number of wavelengths l m and destructive interference

corresponds to the half integer number of wavelengths l m 1 2 here m is an integer number Since

the wavelength of light is small 600 700 nm for a red laser Michelson interferometers are able to measure

distance variation with very good precision

In a Fabry Perot configuration input light field bounces between two closely spaced partially reflecting

surfaces creating a large number of reflections Interference of these multiple beams produces sharp spikes

in the transmission for certain light frequencies Thanks to the large number of interfering rays this type

of interferometer has extremely high resolution much better than for example a Michelson interferometer

For that reason Fabry Perot interferometers are widely used in telecommunications lasers and spectroscopy

to control and measure the wavelengths of light In this experiment we will take advantage of high spectral

resolution of the Fabry Perot interferometer to resolve two very closely spaces emission lines in Na spectra

by observing changes in a overlapping interference fringes from these two lines

A Fabry Perot interferometer consists of two parallel glass plates flat to better than 1 4 of an optical

wavelength and coated on the inner surfaces with a partially transmitting metallic layer Such two mirror

arrangement is normally called an optical cavity The light in a cavity by definition bounces back and forth

many time before escaping the idea of such a cavity is crucial for the construction of a laser Any light

transmitted through such cavity is a product of interference between beams transmitted at each bounce as

diagrammed in Figure 2 When the incident ray arrives at interface point A a fraction t is transmitted and

the remaining fraction r is reflected such that t r 1 this assumes no light is lost inside the cavity

The same thing happens at each of the points A B C D E F G H splitting the initial ray into parallel

rays AB CD EF GH etc Between adjacent ray pairs say AB and CD there is a path difference of

where BK is normal to CD In a development similar to that used for the Michelson interferometer you

can show that

Figure 2 Sequence of Reflection and Transmission for a ray arriving at the treated inner surfaces P1 P2

If this path difference produces constructive interference then is some integer multiple of namely

m 2d cos 4

This applies equally to ray pairs CD and EF EF and GH etc so that all parallel rays to the right of

P 2 will constructively interfere with one another when brought together

Issues of intensity of fringes contrast between fringes and dark background are addressed in Melissinos

Experiments in Modern Physics pp 309 312

Laser Safety

Never look directly at the laser beam Align the laser so that it is not at eye level Even a weak laser

beam can be dangerous for your eyes

Alignment of Machelson interferometer

Equipment needed Pasco precision interferometry kit a laser Na lamp adjustable hight platform

To simplify the alignment of a Michelson interferometer it is convenient to work with diverging optical

beams In this case an interference pattern will look like a set of concentric bright and dark circles since the

components of the diverging beam travel at slightly different angles and therefore acquire different phase

as illustrated in Figure 3 Suppose that the actual pathlength difference between two arms is d Then the

path length difference for two off axis rays arriving at the observer is l a b where a d cos and

Recalling that cos 2 2 cos 2 1 we obtain l 2d cos The two rays interfere constructively for any

angle c for which l 2d cos m m integer at the same time two beams traveling at the angle

d interfere destructively when l 2d cos m 1 2 m integer Because of the symmetry about

the normal direction to the mirrors this will mean that interference bright and dark fringes appears in a

circular shape If fringes are not circular it means simply that the mirrors are not parallel and additional

alignment of the interferometer is required

When the path length difference l is varied by moving one of the mirrors using the micrometer the

fringes appear to move As the micrometer is turned the condition for constructive and destructive

Figure 3 Explanation of circular fringes Notice that to simplify the figure we have unfold the interfer

ometer by neglecting the reflections on the beamsplitter

interference is alternately satisfied at any given angle If we fix our eyes on one particular spot and count for

example how many bright fringes pass that spot as we move mirror M2 by known distance we can determine

the wavelength of light in the media using the condition for constructive interference l 2d cos m

For simplicity we might concentrate on the center of the fringe bull s eye at 0 The equation above

for constructive interference then reduces to 2 l m m integer If X1 is the initial position of the

mirror M2 as measured on the micrometer and X2 is the final position after a number of fringes m has

been counted we have 2 X2 X1 m Then the laser wavelength is then given as

2 X2 X1 m 6

Using Pasco kit set up the interferometer as shown in Figure 1 using the components of Pasco precision

interferometry kit A mirrors M1 2 are correspondingly a movable and an adjustable mirror from the kit

Make initial alignment of the interferometer with a non diverging laser beam Adjust the beams so that it

is impinging on the beamsplitter and on the viewing screen Try to make the beams to hit near the center

of all the optics including both mirrors the compensator plate and beam splitter The interferometer has

leveling legs which can be adjusted Align the beams such that they overlap on the viewing screen

Then insert a convex lens after the laser to spread out the beam ideally the laser beam should be pass

through the center of the lens to preserve alignment Adjust the adjustable mirror slightly until you see the

interference fringes in the screen Continue make small adjustments until you see a clear bull s eye circular

pattern A word of caution sometimes dust on a mirror or imperfections of optical surfaces may produce

similar intensity patterns True interference disappears if you block one of the beam

Note before starting the measurements make sure you understand how to read the micrometer properly

Wavelength measurements using Michelson interferometer

Calibration of the interferometer

Record the initial reading on the micrometer Focus on a the central fringe and begin turning the micrometer

You will see that the fringes move for example the central spot will change from bright to dark to bright

again that is counted as one fringe Count a total of about m 100 fringes and record the new reading

on the micrometer

Each lab partner should make at least two independent measurements starting from different initial

positions of the micrometer For each trial approximately 100 fringes should be accurately counted and

related to an initial X1 and final X2 micrometer setting Make sure that the difference X2 X1 is consistent

between all the measurements Calculate the average value of the micrometer readings X2 X1

When your measurements are done ask the instructor how to measure the wavelength of the laser using

a commercial wavemeter Using this measurement and Equation 6 calculate the true distance traveled by

Figure 4 Micrometer readings The course scale is in mm and smallest division on the rotary dial is 1 m

same as 1 micron The final measurements is the sum of two

the mirror l and calibrate the micrometer i e figure out precisely what displacement corresponds to one

division of the micrometer screw dial

Experimental tips

1 Avoid touching the face of the front surface mirrors the beamsplitter and any other optical elements

2 Engage the micrometer with both hands as you turn maintaining positive torque

3 The person turning the micrometer should also do the counting of fringes It can be easier to count

them in bunches of 5 or 10 i e 100 fringes 10 bunches of 10 fringes

4 Before the initial position X1 is read make sure that the micrometer has engaged the drive screw There

can be a problem with backlash

5 Avoid hitting the table which can cause a sudden jump in the number of fringes

Measurement of the Ne lamp wavelength

A calibrated Machelson interferometer that you have just constructed can be used as a wavemeter to

determine the wavelength of different light sources In this experiment you will use it to measure the

wavelength of strong yellow sodium fluorescent light produced by the discharge lamp

Without changing the alignment of the interferometer i e without touching any mirrors remove the

focusing lens and carefully place the interferometer assembly on top of an adjustable hight platform such

that it is at the same level as the output of the lamp Since the light power in this case is much weaker

than for a laser you won t be able to use the viewing screen You will have to observe the interference

looking directly to the output beam unlike laser radiation the spontaneous emission of a discharge is not

dangerous However your eyes will get tired quickly Placing a diffuser plate in front of the lamp will make

the observations easier Since the interferometer is already aligned you should see the interference picture

Make small adjustments to make sure you see the center of the bull s eye

Repeat the same measurements as in the previous part by moving the mirror and counting the number

of fringes If it is hard to count 100 fringes you may reduce the number Again each lab partner should

make at least two independent measurements recording initial and final position of the micrometer

Figure 5 The Fabry Perot Interferometer For initial alignment the laser and the convex lens are used

instead of the Na lamp

Using the calibration done in the previous section calculate the true distance traveled by the mirror

in each measurements and use Equation 6 Calculate the wavelength of the Na light for each trial Then

calculate the average value and its experimental uncertainty Compare with the expected value of 589 nm

In reality the Na discharge lamp produce a doublet two spectral lines that are very close to each other

589 nm and 589 59 nm Do you thin you Michelson interferometer can resolve this small difference Hint

the answer is no we will use a Fabry Perot interferometer for that task

Alignment of the Fabry Perot interferometer

Disassemble the Michelson Interferometer and assemble the Fabry Perot interferometer as shown in Figure 5

First place the viewing screen behind the two partially reflecting mirrors P 1 and P 2 and adjust the mirrors

such that the multiple reflections on the screen overlap Then place a convex lens after the laser to spread out

the beam and make small adjustments until you see the concentric circles Is there any difference between

the thickness of the bright lines for two different interferometers Why

Loosed the screw that mounts the movable mirror and change the distance between the mirrors Realign

the interferometer again and comment on the difference in the interference picture Can you explain it

Align interferometer one more time such that the distance between two mirrors is 1 0 1 5 mm but make

sure the mirrors do not touch

Sodium doublet measurements

1 Turn off the laser remove the viewing screen and the lens and place the interferometer to the

adjustable hight platform With the diffuser sheet in front of the lamp check that you see the in

terference fringes when you look directly to the lamp through the interferometer If necessary adjust

the knobs on the adjustable mirror to get the best fringe pattern

2 Because the Na emission consists of two light at two close wavelengthes the interference picture consists

of two sets of rings one corresponding to fringes of 1 the other to those for 2 Move the mirror back

and forth by rotating the micrometer to identify two sets of ring Notice that they move at slightly

different rate due to the wavelength difference

3 Seek the START condition illustrated in Fig 6 such that all bright fringes are evenly spaced Note

that alternate fringes may be of somewhat different intensities Practice going through the fringe

conditions as shown in Fig 6 by turning the micrometer and viewing the relative movement of fringes

Do not be surprised if you have to move the micrometer quite a bit to return to the original condition

4 Turn the micrometer close to zero reading and then find a place on the micrometer d1 where you

have the START condition for fringes shown in Fig 6 Now advance the micrometer rapidly while

viewing the fringe pattern NO COUNTING OF FRINGES IS REQUIRED Note how the fringes

of one intensity are moving to overtake those of the other intensity in the manner of Fig 6 Keep

turning until the STOP pattern is achieved the same condition you started with Record the

micrometer reading as d2

5 Each lab partner should repeat this measurement at least one time and each group should have at

least three independent measurements Using the previous calibration convert

We chose the START condition the equally spaced two sets of rings such that for the given distance

between two mirrors d1 the bright fringes of 1 occur at the points of destructive interference for 2 Thus

the bull s eye center 0 we can write this down as

2d1 m1 1 m1 n 2 7

Here the integer n accounts for the fact that 1 2 and the 1 2 for the condition of destructive

interference for 2 at the center The STOP condition corresponds to the similar situation but the net

action of advancing by many fringes has been to increment the fringe count of 2 by one more than that of

2d2 m2 1 m2 n 2 8

Subtracting the two interference equations and solving for the distance traveled by the mirror d2 d1 we

Solving this for 1 2 and accepting as valid the approximation that 1 2 2 where is the

average of 1 and 2 589 26nm we obtain

Use this equation and your experimental measurements to calculate average value of Na doublet splitting

and its standard deviation Compare your result with the established value of N a 0 598 nm

Figure 6 The Sequence of fringe patterns encountered in the course of the measurements Note false colors

in your experiment the background is black and both sets of rings are bright yellow

Detection of Gravitational Waves

A Michelson interferometer can help to test the theory of relativity Gravity waves predicted by

the theory of relativity are ripples in the fabric of space and time produced by violent events in the distant

universe such as the collision of two black holes Gravitational waves are emitted by accelerating masses

much as electromagnetic waves are produced by accelerating charges and often travel to Earth The only

indirect evidence for these waves has been in the observation of the rotation of a binary pulsar for which

the 1993 Nobel Prize was awarded Laser Interferometry Gravitational wave Observatory LIGO sets

the ambitious goal to direct detection of gravitational wave The measuring tool in this project is a giant

Michelson interferometer Two mirrors hang 2 5 mi apart forming one arm of the interferometer and

two more mirrors make a second arm perpendicular to the first Laser light enters the arms through a beam

splitter located at the corner of the L dividing the light between the arms The light is allowed to bounce

between the mirrors repeatedly before it returns to the beam splitter If the two arms have identical lengths

then interference between the light beams returning to the beam splitter will direct all of the light back toward

the laser But if there is any difference between the lengths of the two arms some light will travel to where

it can be recorded by a photodetector

The space time ripples cause the distance measured by a light beam to change as the gravitational wave

passes by These changes are minute just 10 16 centimeters or one hundred millionth the diameter of

a hydrogen atom over the 2 5 mile length of the arm Yet they are enough to change the amount of light

falling on the photodetector which produces a signal defining how the light falling on changes over time LlGO

requires at least two widely separated detectors operated in unison to rule out false signals and confirm that

a gravitational wave has passed through the earth Three interferometers were built for LlGO two near

Figure 7 For more details see http www ligo caltech edu

Figure 8 For more details see http lisa nasa gov

Richland Washington and the other near Baton Rouge Louisiana

LIGO is the family of the largest existing Michelson interferometers but just wait for a few years until

LISA Laser Interferometer Space Antenna the first space gravitational wave detector is launched LISA

is essentially a space based Michelson interferometer three spacecrafts will be arranged in an approximately

equilateral triangle Light from the central spacecraft will be sent out to the other two spacecraft Each

spacecraft will contain freely floating test masses that will act as mirrors and reflect the light back to the

source spacecraft where it will hit a detector causing an interference pattern of alternating bright and dark

lines The spacecrafts will be positioned approximately 5 million kilometers from each other yet it will be

possible to detect any change in the distance between two test masses down to 10 picometers about 1 10th

the size of an atom