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Chapter 1 The Michelson Interferometer 1.1 Prelab In this lab you have to nd the position of mirror M1 (see Fig. 1.1) such that the optical path length from the beam ...

2 CHAPTER 1 THE MICHELSON INTERFEROMETER

given in both csv format and as a saved MATLAB workspace

1 2 Introduction

A well known interferometer operating on the principle of division of amplitude

is the Michelson interferometer The incident light beam is divided into two

parts by means of a beam splitter The divided beams traverse different paths

and then are recombined Depending on the difference in optical path lengths

of the two beams the recombined beams may be 180 out of phase produc

ing destructive interference or in phase producing constructive interference

Observation of these interference fringes with changing optical path difference

allows one to determine wavelengths wavelength differences and or small dif

ferences between the two optical paths

Use of this interferometer as a spectrometer to measure the power spectrum

of an emitting source became feasible with the advent of high speed large

storage capacity computers which could calculate the Fourier transform of the

output irradiance of the interferometer You will use this instrument to realize

such a Fourier transform spectrometer

1 3 The Michelson Interferometer

Fig 1 1 illustrates the physical arrangement of the Michelson interferometer

The beam splitter oriented at 45 to the incoming light is partially metallized

on one surface reflecting half of the beam to mirror M1 and transmitting the

other half of the beam to mirror M2 After reflection from these plane mirrors

the two beams are recombined at the beam splitter and part of this light passes

out of the interferometer towards the CCD camera Mirror M1 is mounted on

a computer controlled stage which can be moved via software or the jog toggle

buttons on the DC servo controller You will notice there is a second piece

of glass in one arm of the interferometer What is the purpose of this

1 4 Finding the Fringes

Be sure that the vacuum cell is not in the interferometer for this part Place

the sodium lamp at the input of the interferometer and make sure the diffusing

screen is in front of it see Fig 1 1 Note that the lamp will take a few minutes

to warm up The output of the interferometer will be examined using the CCD

camera In this case use the CCD camera with its lens assembly installed

Also you should reset the translation stage zero so that your measurements are

properly referenced Open up the APTUser program on the desktop Using

the Settings button on the GUI make sure the max velocity is 0 5 mm s for both

the move and jog Next push the Home Zero button to zero the stage The

stage will move to the end of its travel and the computer will note this as the

home or zero position The interferometer has been set so that the position of

zero path difference falls near the centre of the range of motion of the computer

controlled stage that M1 sits on you should ask the TA about where it might

be since the exact position depends on the setup and changes from week to week

1 4 FINDING THE FRINGES 3

Servo Stage Controller

HeNe Laser

White light lamp

Diffusing Orange Filter

Splitter Mirror M1 Stage

Lens Assembly

Figure 1 1 The Michelson Interferometer Note the sodium lamp is not pic

due to misalignments Position the stage near the center of the range around

10 to 12 mm Then align the mirrors by adjusting the two knobs of M2 and

making the two images of the focussing pin coincide Fringes should appear on

the monitor You will need to make sure the CCD camera is not saturated by

adjusting the aperture on the lens assembly Fringes of different shapes may be

produced by appropriate adjustment of the mirror M2

1 4 1 Circular Fringes

When the mirrors are effectively parallel to each other which means they are

actually perpendicular since the arms of the interferometer are perpendicular

see Fig 1 2 circular fringes are produced see Fig 1 3 These fringes normally

formed at infinity may be focussed on a screen or onto the CCD camera chip

with the use of a lens as shown in Fig 1 2

4 CHAPTER 1 THE MICHELSON INTERFEROMETER

Figure 1 2 Formation of circular fringes M20 indicates the effective position

of the mirror M2 When both mirrors are effectively parallel to each other but

at slightly different positions the optical path lengths are different and circular

fringes occur These fringes are focused with a lens on to the CCD camera chip

1 4 2 Straight Fringes

If the mirrors are slowly angled so that they are not effectively parallel to each

other the centre of the circular fringes moves away and the fringes will appear

like a series of curved lines the edges of a set of concentric circles When the

path length is set to be very near the ZPL straight line fringes or fringes of

equal thickness occur see Fig 1 3 They are identical to the fringes formed

by an air wedge the smaller the wedge angle the broader the fringes

1 4 3 Changing from One Type of Fringe to Another

When the interferometer is set up it is unlikely that the mirrors will be aligned

or that stage will be already close to the zero path length Therefore the fringes

may be either horizontal or vertical or anything in between By adjusting the

mirrors the fringes may be changed from one form to another As the wedge

angle between the mirrors is decreased the fringes become more curved until

circular fringes theoretically appear

Find the fringes described above and practice forming straight and circular

fringes until fringes of a specific kind can be produced at will Note that perfectly

straight fringes will only occur right at the ZPL In practice of course the fringes

may also be distorted due to imperfections in the flatness of the mirrors In fact

Michelson and other amplitude division interferometers are used in industrial

fabrication plants exactly for this purpose to verify the flatness or precise

curvature of mirrors Using the WinTV2000 program on the desktop record

some images of various fringe patterns for your lab book The GUI button

SNAP will grab a frame for you Right click on the captured image to save it

1 5 THE POSITION OF ZERO PATH DIFFERENCE 5

Figure 1 3 a Straight fringes and b circular fringes which occur when the

misalignment of the mirrors is corrected so that they are effectively parallel see

1 5 The Position Of Zero Path Difference

For some applications in this lab it will be necessary to locate the position of

zero path difference The following section describes how this is done

1 5 1 Procedure

Be sure that the vacuum cell is not in the interferometer for this part

If you haven t already done so you should reset the stage zero so that your

measurements are properly referenced It will take some careful adjustment to

find the exact position for the translation stage to achieve a zero path length

difference Therefore start by setting up the sodium lamp and diffuser and

search for the location with the strongest highest contrast fringes possible

Do your best to insure the fringes are circular nicely concentric well spaced

and clearly defined during your search

Actually you will find that there is a series of stage positions where the

contrast is high then low and then high again see Fig 1 6 Take note of the

exact stage positions where the contrast is minimum or disappears near the

6 CHAPTER 1 THE MICHELSON INTERFEROMETER

stage position where the contrast appears best You will next look systematically

and precisely between each of these minima for the ZPL using the white light

Another sign that you are near the zero path length difference is that the

radius of curvature of the circular fringes is the largest Why is that At this

location you should be within a fraction of one millimeter from the zero path

length difference To look for changes of the radius of curvature as you move

the stage position you could also look at the curvature of the straight fringes

with the mirrors slightly angled

Be very careful not to bump the equipment at this stage since you can easily

knock the system out of alignment and ruin your pre alignment work

Now replace the sodium lamp with the white light source Note that the

Sodium lamp will be hot so only grab it by the post You will notice that fringes

do not appear When using the white light source they will only appear when

the arms of the interferometer have equal path length Why What does

this tell you about the coherence length of the white light source

What is the definition of the coherence length You must adjust the

stage until the path lengths are equal and fringes are observed

Since finding the white light fringes is like looking for a needle in a haystack

you should cheat by using a narrow band filter with the white light source to

find fringes with filtered light and then remove the filter to find the zero path

length difference with the white light Why would this filter help With the

narrow band filter in place slowly scan the stage with the jog buttons to the

positions precisely between the minima you found before When you finally try

the correct maxima the stage will be near the ZPL and fringes will be observed

If you are spending more than 15 minutes without finding the white light fringes

ask a TA to help check you are doing everything correctly At the point where

the fringes appear with the unfiltered white light source the stage is set for

zero path length difference Write down the stage position for future reference

and note the direction you approached the zero path difference The latter is

important for mechanical backlash in the micrometer screw Since you will be

using a MATLAB script to take data and this script moves the stage forward

i e to higher values on the indexed stage then backwards then forward again

back to the starting point you should always approach from below your position

of zero path length difference The fringes will appear and disappear within 2

m so you will probably have to adjust the max velocity and step distance of

the stage so that you don t go too fast and miss the appearance of the fringes

Also use the actuator toggle with care since rapid changes in motion of the

stage might also misalign the interferometer

1 6 Calibration Of The Interferometer

Before the interferometer can be used to measure wavelengths it must first

be calibrated By calibrated we mean that the distance which the stage

actually moves needs to be correlated to the approximate reading provided by

the computer

1 6 CALIBRATION OF THE INTERFEROMETER 7

1 6 1 Procedure

The distance by which the stage travels is found by counting the number of

fringes which pass given a certain mirror movement and given a light source

whose wavelength is well known The Helium Neon Laser 632 8nm is ideal

for this purpose Slide the lamp up and rotate the diffusing screen and orange

filter out of beam path of the laser When using the laser place an imaging

lens L1 in the post just after the output of the laser You should remove

the lens assembly on the CCD camera By adjusting the height and tilt of the

imaging lens in the post holder locate the the main spot of the HeNe laser on

the backplane of the CCD camera Temporarily put a piece of paper just in

front of the CCD and look at the pattern cast on this paper Align the mirror

M2 to see fringes Now take out the paper and use the monitor to view the

CCD output and further adjust mirror M2 so that about five vertical straight

fringes fill the image on the CCD camera see Fig 1 3

Using the monitor to view the CCD output adjust mirror M2 so that about

five vertical straight fringes fill the image on the CCD camera see Fig 1 3

You should adjust the camera aperture and attach some neutral density filters

to the CCD camera so that it isn t saturated

The overall plan is to move the stage a certain distance and count the number

of fringes that pass You should therefore take the data plot it and count the

fringes on the plot The mirror moves a distance of 2 as one fringe passes

a reference point on the CCD Camera Thus if N fringes pass through as the

mirror moves a distance d we have

If the servo controller readings change by D from D1 to D2 then the

mirror travel is linearly related to the stage displacement through the calibration

constant K

d K D K D2 D1 1 2

To analyze the fringe movement i e to count the fringes you will use MAT

LAB to control the translation stage i e mirror position AND to detect the

variation of the intensity on a given pixel in the fringe pattern as the fringe pat

tern moves Several helpful MATLAB files are located in the desktop folder

Michelson files At this point you should NOT be running the WinTV2000

program as it will not allow MATLAB to access the video card By reading

the comments of the M files you will be able to tell what each of their functions

In order to take data for this section you will be using the MATLAB script

called michelson timestream m You will need to quit the APT user program

when you use the michelson timestream file since MATLAB will open APT

user again with the correct parameters When this script is executed it will

prompt you to click on the image the location where the fringe pattern is the

most visible The script will then move the stage back by a distance span

specified in mm then it will move the stage forward by twice that distance

2 x span at a slow velocity speed specified in mm s and then it will move

the stage back to the starting point To use this script your code will look like

8 CHAPTER 1 THE MICHELSON INTERFEROMETER

data michelson timestream span speed

where span and speed are to be chosen by you and the data will be returned

into the array data As an example consider this code

data michelson timestream 0 01 0 001

This will move the stage back by 10 m and then forward by 20 m at a speed of

1 m s which will take 20 seconds and then back 20 m to the starting point

The data will be returned into the array data which will have the intensity on

the pixel you selected for each frame of the video Since the video frame rate is

30 frames per second this means the stage will have moved 301frames s

For a given amount of distance moved by the stage determine the number

of fringes that passed through the reference point region of interest ROI and

then calculate the calibration constant K Repeat this measurement as many

times as necessary to obtain consistent results for K and estimate its accuracy

1 7 Measuring the index of refraction of air

Now that the constant K has been determined you can use this interferometer

to characterize extremely small changes in the optical path length differences

between the two arms You can use this sensitivity to even measure the phase

shift induced by the propagation of light through air You will measure the

phase change by counting fringes as air is slowly introduced into one of the

interferometer arms Using the vacuum pump and glass vacuum cell provided

insert the glass vacuum cell between the beam splitter and M2 and measure the

index of refraction of air at 632 8 nm i e with the HeNe laser

For your measurements you can use the vid capture MATLAB script to

capture the intensity on a small area in the CCD image as a function of time

Your code will look like this d vid capture time where you choose

time appropriately try 10 seconds

To do your measurement you should first prepare a vacuum in the cell

close the leak valve and open the pump valve and turn on the vacuum

pump After the pressure in the cell as read by the meter has dropped to the

minimum on the scale close the pump valve and turn off the pump Start

the data acquisition and slowly open the leak valve to let in air while you

count fringes You may need some practice introducing the air fast enough that

the pressure rises to atmospheric pressure during the time you have set while

not rising too quickly so that you miss the passage of fringes due to the finite

sampling time of the video capture Be careful to insert the glass vacuum cell

without hitting or misaligning the mirrors see Fig 1 4

Would you have counted the same number of fringes if you had

used the Na lamp instead of the HeNe laser for this measurement

Why or why not What is your value for the index of refraction of

air for 632 8 nm light Does this agree with the accepted value

What are your sources of error Could you use this apparatus as a

barometer or thermometer

1 8 DETERMINATION OF WAVELENGTHS 9

Figure 1 4 Vacuum vessel inserted into M2 arm of interferometer

1 8 Determination Of Wavelengths

Be sure that the vacuum cell is not in the interferometer for this part

Also knowing the calibration constant K an unknown wavelength may be

measured by the same procedure i e by noting the stage readings for the pas

sage of a certain number of fringes Carry out this measurement for the sodium

source to get the average wavelength for the two Na D1 and D2 lines You

should use the same span and speed as the previous section The fringes for

these two lines coincide at the position of zero path difference and the measure

ment should be made near this position On either side the fringes become

indistinct as they start to overlap This feature will be studied in the next

1 9 Wavelength Differences

In 1887 Michelson and Morley while studying the visibility of fringes in

Michelson s interferometer observed that the H2 line n 3 n 2 is

not one line but actually a doublet with about 0 33 cm 1 separation The

wavenumber is given in units of inverse centimeters and is the spatial analog

of frequency One can convert to frequency by multiplication with the speed

of light f c The wavenumber of electromagnetic radiation is defined as

1 where is the wavelength in vacuum

In this part of the lab you will determine the separation between the D1

and D2 lines in sodium using the Na lamp The fringes from any two lines

10 CHAPTER 1 THE MICHELSON INTERFEROMETER

leak valve Vacuum pump valve pump on off

Gauge switch

Figure 1 5 Vacuum pump valves and vacuum gauge for vacuum cell

wavelengths 1 and 2 such as the Na D1 and D2 lines always overlap at the

position of zero path difference If the stage is moved away from this position

either forwards or backwards then a position will be reached at a distance d

where the bright fringes of one line fall on the dark fringes of the other The

resulting fringe pattern is then very indistinct the fringes are said to have low

visibility This particular distance is

2d N 1 1 N 2 1 2 1 3

where N is an integer The wavelength difference is therefore

2 1 2d 1 4

If 1 and 2 differ by very little it is sufficient to replace their product by

2 where is the average of the two The expected value of 0 6nm

1 9 1 Procedure

Be sure that the vacuum cell is not in the interferometer for this part

Set up a white light source and locate the white light fringes as you did

previously This will locate the stage at zero path difference Recall that due to

backlash the stage position will differ if you approach this position from above

1 10 FOURIER TRANSFORM SPECTROSCOPY 11

versus from below Since you will eventually be using a MATLAB script to

take data and this script moves the stage forward i e to higher values on

the indexed stage then backwards then forward again back to the starting

point you should always approach from below your position of zero path length

difference

Replace the white lamp with the sodium lamp Now move the stage using the

toggle lever on the controller move the stage over a distance of at least 2 mm not

ing the stage readings at several positions of minimum visibility When recording

the positions you should always move the stage in same direction to avoid back

lash errors Repeat this process several times and average the stage readings

You can do this manually or you can use the script from before Your code will

again look like this

d michelson timestream span speed

where you choose span and speed appropriately A span 1 0 and speed

0 01 are probably good starting values for this part

From the average value of D obtained and the measured calibration con

stant K calculate d from equation 1 2 and hence obtain a value for from

equation 1 4 Use the average value of the two Na D1 and D2 lines that you

determined above and find the difference between the two lines

1 10 Fourier Transform Spectroscopy

So far you have investigated the behavior of the interference fringe pattern as

a function of the optical path difference for different optical sources In general

the fringe pattern intensity versus the optical path difference or equivalently

versus the time delay between the two interfering beams is related to the power

spectrum of the light entering the amplitude division interferometer by a Fourier

transform Therefore one can easily measure the optical power spectrum using

the measurement of the intensity variation as a function of stage position

Let s see how this works 1 The electric field incident on the camera is the

sum of the electric fields E1 and E2 arriving from M1 and M2 respectively after

having passed through the beam splitter Assuming for the moment that the

input source is a purely monochromatic wave we can characterize these electric

fields with a vector amplitude encoding the strength and polarization of the

wave and a spatial and time dependent phase

E1 A1 ei k1 r t 1 1 5

E2 A2 ei k2 r t 2 1 6

The phase includes terms 1 k1 l1 and 2 k2 l2 which depend on the

optical path lengths from the beamsplitter along path l1 encountering M1 and

path l2 encountering M2 The intensity on the camera is simply the square of

the total electric field

I E 2 E E E1 E2 E 1 E 2 1 7

A1 A2 2A1 A2 cos 1 8

I1 I2 2 I1 I2 cos 1 9

1 This derivation follows that in Introduction to Modern Optics by Grant R Fowles

12 CHAPTER 1 THE MICHELSON INTERFEROMETER

where we have assumed in the last step that the waves have the same polarization

and that k1 k2 r 1 2 You might be wondering about a white

light source which is probably completely unpolarized Is this assumption still

valid in that case This assumption is justified since the two fields E1 and E2

are copies of the incident field generated by the beam splitter As long as the

beam splitter and the optics which follow preserve the polarization of the light

these two fields will arrive at the detector with exactly the same polarization

whatever it was at the input of the interferometer What could the beam

splitter and following optics do to change the polarization of the input

If the interferometer is well aligned so that the waves are co linear k1 k2

and the 50 50 beamsplitter generates two waves of the same intensity I1

I2 I 2 then the interference pattern is simply

I x I 1 cos kx 1 10

where x l1 l2 is the path length difference and k k1 2 is the

magnitude of the wavevector

Okay now in general the light illuminating the interferometer will be com

posed of many different frequencies So let s model the power spectrum of the

incident light as a continuous distribution G Then the total optical power

emitted into the interferometer in the frequency range from 0 to 0 d

where d is an infinitesimally small frequency band is simply G 0 d For

convenience we can instead use the distribution over wavelength G k where

ck Then the optical power P hitting the detector from illuminating the

interferometer with this polychromatic source will be the sum of the interfer

ence patterns from each monochromatic wave composing G k Assuming the

interferometer is ideal and has no losses we have then

P x G k 1 cos kx dk 1 11

eikx e ikx

G k dk G k dk 1 12

P 0 G k eikx dk 1 13

where P 0 is simply the power measured at zero path length difference We

can rearrange this expression and define the power function W x

W x 2P x P 0 G k eikx dk 1 14

We see then that the power function W x is the inverse Fourier transform of

the power spectral density G k Inverting this expression we have that the

power spectrum is the Fourier transform of the power function

G k W x e ikx dx 1 15

or equivalently Z

G W x e i2 x c dx 1 16

1 10 FOURIER TRANSFORM SPECTROSCOPY 13

Monochromatic

Two wavelengths

close together

Wide band pass

white light

coherence length

Narrow band

pass filter

orange filter

mirror position

Figure 1 6 Fringe patterns plotted as a function of the mirror position Notice

how the fringe visibility collapses and then revives as a function of the mirror

position when the spectrum is discrete Also note that if the mirror is moved a

distance d the optical path length between the two arms in the interferometer

change by 2d

1 10 1 Procedure

Be sure that the vacuum cell is not in the interferometer for this part

In this stage of the experiment you will use the interferometer to realize

14 CHAPTER 1 THE MICHELSON INTERFEROMETER

a Fourier transform spectrometer The idea is to translate the stage around

the zero path length position ZPL record the fringe pattern and use it to

determine the power function W x and then compute numerically the corre

sponding spectral density G k of various optical sources To do this you will

as before use the MATLAB script michelson timestream m which records

an array of power values for each video frame i e each stage position You will

use your calibration constant for the motion K to deduce the stage position at

each frame and therefore generate the power function W x be sure the stage

velocities are set the same or else recalibrate K for this part From this you will

numerically compute the Fourier transform to determine the power spectrum of

the light White Light

power arb units

200 300 400 500 600 700 800 900 1000

wavelength nm

Figure 1 7 Example spectrum from the incandescent bulb measured by the

Michelson Fourier transform spectrometer Your reference white light spectrum

should look something like this one Keep in mind that this measurement of

the spectrum is ultimately limited by the wavelength dependent transmission

of the optical elements in the Michelson interferometer and by the sensitivity

of the CCD detector See Fig 1 8 for a typical response curve of a silicon CCD

image sensor

As before set up a white light source find the fringes at zero path length

difference make them vertical and note the stage setting Before taking each

data run you should carefully adjust the iris on the camera lens to maximize

the fringe brightness without saturating the camera This will optimize the

signal to noise and produce much better data Start by recording the fringe

pattern and computing the spectrum of the white light source You should get

a spectrum similar to that shown in Fig 1 7 This is your reference spectrum

which you will be modifying with a filter Keep in mind that this measurement

of the white light source spectrum is ultimately limited by the transmission of

the optical elements in the Michelson interferometer and by the sensitivity of

the CCD detector See Fig 1 8 for a typical response curve of a silicon CCD

image sensor

To take the data for this section you will again be using the MATLAB

script michelson timestream m only this time you should use a very slow

speed and a very narrow span since the filtered white light has a short coher

ence length and you will need good spatial resolution which will determine the

1 10 FOURIER TRANSFORM SPECTROSCOPY 15

Figure 1 8 The spectral response of a typical silicon CCD image sensor ranges

from 400 to 1000 nm

highest frequency in your spectrum so you can see the full spectrum without

any truncation Recommended values are speed 0 001 and span 0 05

Note that this span might not be wide enough for the narrow band filter but

that it already will take around 100 seconds given the speed So first check with

a higher speed that this is the correct range for your filter and ONLY THEN

take your data at the appropriate slow speed Be aware if you choose a span

smaller than the backlash in the micrometer screw the stage will not return to

where you started You should therefore insure the span is wide enough that

the stage returns to the starting point properly

Carefully without disturbing the interferometer alignment insert the 589 nm

interference filter and record the fringe pattern generated Take your spectrum

and normalize it so that the peak of the spectrum has an amplitude of 1 Do

the same with the reference spectrum and divide the filtered spectrum by the

reference to deduce the transmission function of the filter Does the 589 nm

interference filter have a Gaussian transmission function Why or

You should now take a spectrum of the HeNe laser to check that

your calibration of the stage velocity is correct Is the peak of the

spectrum where you would expect it If not why not You should

use this information to refine and or correct the spectra you took of

the filtered white light What is the measured width of the spectral

peak of the HeNe output in nm The linewidth of the HeNe laser is

well below 100 MHz Is the measured width larger than this If so

1 10 2 Analysis with MATLAB

In order to compute the power spectra from your fringe pattern data you will

need to do some data processing First you will need to properly truncate your

data because the vector of data data may include points when the stage was

moving backwards and then sitting stationary before and after the slow sweep

through the ZPL position Therefore find the points AA and BB which are at

the beginning and end of the data and truncate the data Your code will look