The Michelson Interferometer

Experiment 4 – The Michelson Interferometer
1
Experiment 4
The Michelson
Interferometer
1 Introduction
There are, in general, a number of types of optical instruments that
produce optical interference. These instruments are grouped under the
generic name of interferometers. The Michelson interferometer causes
interference by splitting a beam of light into two parts. Each part is
made to travel a different path and brought back together where they
interfere according to their path length difference.
You will use the Michelson interferometer to observe the interference
of two light sources: a HeNe laser and a sodium lamp. You will study
interference patterns quantitatively to determine the wavelengths and
splitting of the Na D lines empirically. You will use the HeNe laser
interference spectrum to calibrate the interferometer.
2 Background - see Hecht, Chap. 9
2.1
The Michelson Interferometer
The Michelson interferometer is a device that produces interference
between two beams of light. A diagram of the apparatus is shown in
Fig. 1. The basic operation of the interferometer is as follows. Light
from a light source is split into two parts. One part of the light travels
a different path length than the other. After traversing these different
path lengths, the two parts of the light are brought together to interfere
with 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 the
translatable mirror M1 . The other 50% of the radiation is reflected to
the fixed mirror M2 . The compensator plate C is introduced along this
Experiment 4 – The Michelson Interferometer
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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 returning
from M1 , 50% of the light is reflected toward the frosted glass screen.
Likewise, 50% of the light returning from M2 is transmitted to the
glass screen. At the screen, the two beams are superposed and one can
observe the interference between them.
2.2
Interference of Waves With a Single Frequency
If two waves simultaneously propagate through the same region of
space, the resultant electric field at any point in that region is the
vector sum of the electric field of each wave. This is the principle of
superposition. (We assume all waves have the same polarization).
If two beams emanate from a common source, but travel over two
different paths to a detector, the field at the detector will be determined
by the optical path difference, which we will denote by ∆x = x2 − x1 .
A related quantity is the phase difference, ∆φ, given by
∆φ =
2π
∆x = k∆x,
λ
(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
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Figure 2: Beat signal from two input frequencies into a Michelson interferometer
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 the
x-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 (ET∗ 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 oscillation
as well as a slow oscillation characteristic of δk.
Experiment 4 – The Michelson Interferometer
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3 Experiment
In the following experiments, you will calibrate the movement of M1
with the HeNe laser and use the interferometer to accurately measure
the wavelengths of the fine structure doublet of the sodium D line, a
consequence of the spin of the electron.
3.1
Calibration with HeNe Laser Light
Inject the laser beam into the Michelson intererometer. Make sure the
beam is properly retro-reflected. Initially, you will see two bright spots
on the screen. Adjust the angle of the fixed mirror until these two spots
overlap. You can use lenses to expand the beam if necessary.
Note, take care when moving M2 as the interference is very
sensitive to its alignment. As you translate mirror M1 , you will see
fringes appearing and disappearing on the screen. The interferometer
lever arm reduction factor is 5X, so that the wavelength of the light
can be found using
1 2d
(6)
λ=
5 m
where d is the distance mirror M1 was moved and m is the number of
rings that disappeared (or appeared) while M1 was being moved.
Use the synchronous motor to facilitate the turning of the micrometer. As the micrometer is turning, record the interference data with
the computer. The motor runs at 0.5 rpm and the micrometer moves
5 × 10−4 m/rev. This can give you a check of things, but we will use
the HeNe data (look up the HeNe wavelength on the web) to accurately
calibrate the speed.
3.2
Sodium Light
Now use the sodium lamp to produce an interference pattern. Since
the spectrum of this light consists primarily of two closely spaced lines
(a doublet), each wavelength will produce its own set of fringes. Your
goal will be to empirically determine λ1 and λ2 by measuring the finely
spaced fringes and the beat pattern.
It is much more challenging to get good interference patterns with
the lamps, so take your time and play with alignment, lamp placement,
and possible lens placement. You will need to increase the gain in your
detection system.
You should observe both the finely spaced pattern as well as a modulation in the contrast at the difference frequency of the two lines. Your
goal is to as accurately as possible measure the wavelength of the two
sodium 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 the
lamp (Doppler effect). This means the coherence length is not that
large. If the path length difference is too large, you will not see any
fringes. Measure the coherence length of your lamp.
Questions related to Michelson Interferometry
1. What is (i) a plane wave? (ii) a spherical wave?
2. How are plane waves and spherical waves produced?
3. How to detect planarity or sphericity of a wavefront?
4. What is spatial coherence?
5. How do we measure spatial coherence?
6. How does a Sodium vapour lamp differ from a Helium-Neon Laser?
Lab activity
You are given a Michelson Interferometer that has two tiltable mirrors, one of which is
also a movable mirror, and a finely threaded screw to control the movement of the
mirror. Generate straight line fringes, inclined fringes, elliptical fringes and circular
fringes. At each stage, move one mirror and record how the fringes change. Align the
circular fringes such that the centre is at the centre of the field of view.
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 M 1 and
M2 are equidistant from the beam splitter P 1 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,
2d12  m1   m  1 2
or 2d12 
 12
1  2
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
Lc 

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
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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.