EDU-MINT1
EDU-MINT1/M
Michelson Interferometer Kit
User Guide
Michelson Interferometer
Table of Contents
Chapter 1
Warning Symbol Definitions ............................................1
Chapter 2
Safety .................................................................................2
Chapter 3
Brief Description ...............................................................3
Chapter 4
Kit Components ................................................................4
4.1
Basic Components of the Interferometer ........................ 4
4.2
Components to Observe the 2nd Interferometer Output .. 6
4.3
Components for the Refractive Index Measurement ....... 6
4.4
Components for the Interference with LEDs .................... 7
4.5
Components for the Thermal Expansion Setup ............... 8
Chapter 5
Setup and Adjustment....................................................10
5.1
Assembly of the Components ....................................... 10
5.2
Setting Up and Adjusting the Michelson Interferometer14
Chapter 6
Theoretical Background.................................................19
6.1
Interference using the Michelson Interferometer ......... 19
6.2
Determining the Wavelength ....................................... 23
6.3
Coherence .................................................................... 24
6.4
Interferometric Determination of the Refractive Index . 28
6.5
Determining a Thermal Expansion Coefficient .............. 29
6.6
Using the Interferometer as a Spectrometer................. 30
Chapter 7
Experiments and Examples ...........................................32
7.1
Preliminary Tests ......................................................... 32
7.2
Determining the Laser Wavelength .............................. 34
7.3
Using the Interferometer as a Spectrometer................. 35
7.4
Interference with LEDs, Coherence ............................... 36
7.5
Refractive Index Determination ................................... 39
Michelson Interferometer
7.6
Thermal Expansion Coefficient ..................................... 42
Chapter 8
Experiment Overview .....................................................44
Chapter 9
Questions ........................................................................46
Chapter 10
Ideas for Additional Experiments .................................47
Chapter 11
Modern Michelson Interferometry – LIGO ....................48
Chapter 12
Troubleshooting .............................................................49
Chapter 13
Appendix .........................................................................51
Chapter 14
Regulatory .......................................................................53
Chapter 15
Thorlabs Worldwide Contacts .......................................54
Michelson Interferometer
Chapter 1
Chapter 1: Warning Symbol Definitions
Warning Symbol Definitions
Below is a list of warning symbols you may encounter in this manual or on your device.
Symbol
Description
Direct Current
Alternating Current
Both Direct and Alternating Current
Earth Ground Terminal
Protective Conductor Terminal
Frame or Chassis Terminal
Equipotentiality
On (Supply)
Off (Supply)
In Position of a Bi-Stable Push Control
Out Position of a Bi-Stable Push Control
Caution: Risk of Electric Shock
Caution: Hot Surface
Caution: Risk of Danger
Warning: Laser Radiation
Page 1
Rev E, November 6, 2017
Michelson Interferometer
Chapter 2
Chapter 2: Safety
Safety
WARNING
The laser module is a class 2 laser, which does not require any protective eyewear.
However, to avoid injury, do not look directly into the laser beam.
DO NOT STARE INTO BEAM
CLASS 2 LASER PRODUCT
MTN006503-D02
Page 2
Michelson Interferometer
Chapter 3
Chapter 3: Brief Description
Brief Description
The objective of this experiment package is to become familiar with the interferometer as
a highly sensitive measuring instrument. Since the applications in technology and industry
are extremely diverse, this package includes various experiments for different aspects of
physics.
First a straightforward Michelson interferometer is built. It is distinguished by its ease of
adjustment, compared for example to the Mach-Zehnder interferometer. In particular, the
influence of the interferometer’s arm length on the interference pattern is discussed here.
Also, it is demonstrated how an interferometer can serve as a spectrometer.
The topic of coherence takes center stage in the next step. Initially an interference pattern
is generated with a narrow-band red LED. If the interferometer is adjusted precisely
enough, one can replace the red LED with a white one which allows white light interference
to be observed within the coherence length.
Determining the refractive index of a Plexiglas plate is the object of another experiment.
Here, a Plexiglas plate is positioned in an arm of the interferometer and rotated. The
rotation increases the optical path of the laser in this arm, so that the interference pattern
shows characteristic light-dark transitions (swelling / disappearance of rings). Then the
refractive index is determined based on their number, the plate thickness, and the rotation
angle.
The final experiment illustrates how even extremely slight expansion can be measured
with an interferometer. Here a metal rod to which an interferometer mirror has been
attached is heated so that it expands. Expansion in turn results in light-dark transitions.
The expansion coefficient can be calculated from the number of light-dark transitions and
the temperature of the rod.
This Michelson interferometer setup therefore opens up a wide range of possibilities to
become familiar with various fields of application for interferometry. In particular, a broad
spectrum of phenomena is discussed ranging from interference, coherence, and refraction
to the expansion of solids.
Page 3
Rev E, November 6, 2017
Michelson Interferometer
Chapter 4
Chapter 4: Kit Components
Kit Components
In cases where the metric and imperial kits contain parts with different item numbers,
metric part numbers and measurements are indicated by parentheses unless otherwise
noted.
Basic Components of the Interferometer
1 x B1218FX (B3045AX)
Steel Breadboard,
12" x 18" (30 cm x 45 cm)
1 x RDF1
Rubber Dampening Feet,
Set of 4
1 x LDS5(-EC)
5 VDC Regulated Power
1 x AD11NT
Supply, 2.5 mm Phono Plug, Ø1" Unthreaded Adapter for
120 VAC (230 VAC)
Ø11 mm Components
2 x PF10-03-P01
Ø1" Protected Silver Mirror
1 x SM1ZP(/M)
Z-Axis Translation Mount
MTN006503-D02
1 x CPS532-C2
Collimated Laser Diode
Module, 532 nm, Class II,
Round Beam
2 x KM100
Ø1" Mirror Mount
1 x BA2S5(/M)
Spacer,
2" x 3" (50 mm x 75 mm),
0.250" (6.25 mm) Thick
Page 4
Michelson Interferometer
2 x BA2(/M)
Base,
2" x 3" x 3/8"
(50 mm x 75 mm x 10 mm)
1 x LB1471
N-BK7 Bi-Convex Lens,
Ø1", f = 50.0 mm, Uncoated
Chapter 4: Kit Components
1 x EDU-VS1(/M)
Plastic Viewing Screen
1 x LMR1(/M)
Ø1" Lens Mount
1 x CCM1-BS013(/M)
Non-Polarizing Beamsplitter
Cube, 400 - 700 nm
4 x UPH1.5 (UPH30/M)
1.5" (30 mm) Long
Universal Post Holder
4 x TR1.5 (2 x TR30/M + 2 x TR40/M)
Ø1/2" (Ø12.7 mm) Post, 1.5" (2 x 30 mm, 2 x 40 mm)
Page 5
Rev E, November 6, 2017
Michelson Interferometer
Chapter 4: Kit Components
Components to Observe the 2nd Interferometer Output
1 x EBS1
Ø1" Economy
Beamsplitter
1 x LMR1(/M)
Ø1" Lens Mount
1 x TR1.5 (TR40/M)
Ø1/2" (Ø12.7 mm) Post,
1.5" (40 mm) Long
Components for the Refractive Index Measurement
1 x PR01(/M)
High-Precision Rotation
Mount
MTN006503-D02
1 x FP01
General Purpose Plate
Holder
Plexiglas Plates
8 mm and 12 mm Thick
Page 6
Michelson Interferometer
Chapter 4: Kit Components
Components for the Interference with LEDs
2 x LED631E
(with USB Connector)
635 nm, 4 mW,
FWHM 10 nm, 150 Ω
2 x LEDWE-15
(with USB Connector)
White light LED, Half
Viewing Angle 7.5°, 91 Ω
2 x UPH1.5 (UPH40/M)
1.5" (40 mm) Long
Universal Post Holder
2 x TR1.5 (TR40/M)
Ø1/2" (Ø12.7) mm Post,
1.5" (40 mm) Long
1 x USB-C-72
72” USB 2.0 Type-A
Extension Cable
Page 7
2 x LEDMF
LED Mount
1 x DS5
5 VDC USB Power Supply
1 x Ruler, 12" (30 cm)
Rev E, November 6, 2017
Michelson Interferometer
Chapter 4: Kit Components
Components for the Thermal Expansion Setup
1 x ME1-G01
Ø1" Aluminum Mirror
1 x MH25
Mirror Holder for Ø1"
Optics 2.5 - 6.1 mm Thick
1 x RA90(/M)
Right-Angle Post Clamp,
Fixed 90° Adapter
1 x BA1S(/M)
Base,
1" x 2.3" x 3/8"
(25 mm x 58 mm x 10 mm)
1 x HT10K
Foil Heater with 10 kΩ
Thermistor
MTN006503-D02
1 x TR2 (TR50/M)
Ø1/2" (Ø12.7 mm) Post,
2" (50 mm) Long
1 x Aluminum Post
Ø12.7 mm, 90 mm Long,
Thread for MH25
1 x PH1.5 (PH40/M)
1.5" (40 mm) Long Post
Holder
1 x Sensor
1 x Digital Thermometer
2 Cables, Crocodile Clip
to Banana Plug
1 x Electrical Tape
Page 8
Michelson Interferometer
Chapter 4: Kit Components
Imperial Kit
Type
Quantity
Type
Quantity
1/4"-20 x 1/4" Cap Screw
2
1/4"-20 Washer
15
1/4"-20 x 3/8" Cap Screw
1
8-32 x 1/4" Cap Screw
3
1/4"-20 x 1/2" Cap Screw
4
8-32 x 3/8" Cap Screw
1
1/4"-20 x 5/8" Cap Screw
9
3
1/4"-20 x 3/4" Cap Screw
4
8-32 x 5/8" Cap Screw
Counterbore Adapter
Ring for 8-32 Screws
1 x Hex Key 1/8",
1 x Hex Key 3/32",
1 x Hex Key 1/16"
4
1 x BD-3/16L
Balldriver for 1/4"-20 Screws
Metric Kit
Type
Quantity
Type
Quantity
M6 x 8 mm Cap Screw
2
M6 Washer
15
M6 x 10 mm Cap Screw
1
M4 x 6 mm Cap Screw
3
M6 x 12 mm Cap Screw
4
M4 x 10 mm Cap Screw
1
M6 x 16 mm Cap Screw
9
3
M6 x 20 mm Cap Screw
4
M4 x 16 mm Cap Screw
M6 Counterbore Adapter
Ring for M4 Screws
1 x Hex Key for M4 Screws (3 mm),
1 x Hex Key 2 mm,
1 x Hex Key 1.5 mm
Page 9
4
1 x BD-5ML
Balldriver for M6 Screws
Rev E, November 6, 2017
Michelson Interferometer
Chapter 5
Chapter 5: Setup and Adjustment
Setup and Adjustment
This chapter discusses how to assemble the various components and explains how to set
up and adjust the interferometer.
Assembly of the Components
First screw the four rubber feet to the four holes in the bottom of the breadboard using the
1/4"-20 x 1/2" (M6 x 12 mm) screws. Now assemble the various parts of the setup:
Screen
Lens
Mirror
Components:
EDU-VS1(/M) screen
BA2(/M) base
Components:
LB1471 Lens
LMR1(/M) Lens Mount
1.5" (40 mm) Long Post
1.5" (30 mm) Long
Universal Post Holder
Components:
PF10-03-P01 Mirror
KM100 Mirror Mount
1.5" (30 mm) Long Post
1.5" (30 mm) Long
Universal Post Holder
Screen Assembly: First, screw the screen onto the BA2(/M) base using a counterbore
adapter and the 8-32 x 3/8" (M4 x 10 mm) cap screw.
Lens and Mirror Assembly: Screw the 1.5" (40 mm) long post into the lens holder and
place it in the universal post holder. To insert the lens, first remove the retaining ring, then
insert the lens and screw the retaining ring back into place. The mirror is installed in the
KM100 mirror holder in a similar fashion. To mount the holder on the post, proceed as
shown in the photos that follow.
MTN006503-D02
Page 10
Michelson Interferometer
Chapter 5: Setup and Adjustment
The mirror is locked in place using the nylon-tipped setscrew. Now you need to assemble
the laser, beamsplitter and movable mirror as additional components of the setup.
Laser
Beamsplitter
Movable Mirror
Components:
Laser
KM100 Mount
AD11NT Adapter
1.5" (30 mm) Long Post
1.5" (30 mm) Long
Universal Post Holder
Components:
Beamsplitter Cube
1.5" (40 mm) Long Post
1.5" (30 mm) Long Universal
Post Holder
Components:
PF10-03-P01 Mirror
SM1ZP(/M) Stage
BA2(/M) Base
BA2S5(/M) Spacer
Laser: Screw the KM100 mount onto the post as shown above. Then place the laser in
the adapter and lock it using the two screws. Finally, insert the adapter into the KM100
and lock it using the nylon-tipped setscrew. Connect the laser to the LDS5(-EC) power
supply and check the bottom of the LDS5(-EC) to make sure the correct voltage is used.
Beamsplitter: The beamsplitter cube is screwed onto the 1.5" (40 mm) long post. For the
imperial kit, use the thread adapter provided with the cube.
Movable Mirror: To mount the SM1ZP(/M) stage, put it on the BA2S5(/M) spacer and the
BA2(/M) base (at the bottom). Secure the stage by using three counterbore adapters and
three 8-32 x 5/8" (M4 x 16 mm) cap screws. Secure the mirror in the stage by removing
one retaining ring, placing the mirror in the mount, and screwing the retaining ring back in
its place.
Page 11
Rev E, November 6, 2017
Michelson Interferometer
Chapter 5: Setup and Adjustment
The elements installed so far are the basic components of the interferometer. The next set
of components will be used with the interferometer in the various experiments explained
in this manual.
Ø1" Beamsplitter: To assemble the second beamsplitter, remove the retaining ring from
the LMR1(/M) mount, place the beamsplitter in the mount and attach the retaining ring.
Wear gloves while touching the beamsplitter. Screw the beamsplitter mount onto a TR1.5
(TR40/M) post.
Rotation Platform
LED Mount
Thermal Expansion
Components:
Rotation Stage
FP01 Universal Mount
Plexiglas Plate
Components:
LED Holder
LED
1.5" (40 mm) Long Post
1.5" (40 mm) Long
Universal Post Holder
DS5 power supply
USB Extension Cable
Components:
ME1-G01 Mirror
MH25 Mirror Holder
90° Adapter
Aluminum Rod
Foil Heater + Tape
2" (50 mm) Long Post
1.5" (40 mm) Long Post Holder
BA1S(/M) Base
Rotation Platform: First, attach the universal holder to the outer part of the stage using
an 8-32 x 1/4" (M4 x 6 mm) cap screw. Next, insert the thin Plexiglas plate. Make sure to
remove the protective films from both sides of both Plexiglas plates.
LED Mount: For the component with the LED, screw the LED holder to the post (as
described above for the KM100 mirror holder) using an 8-32 x 1/4" (M4 x 6 mm) cap screw
and place it in the post holder. Slide the LED with the corresponding adapter ring into the
holder and tighten the screw from above. To operate the LED, plug the USB connector
into the DS5 power supply and use the USB extension cable if needed.
MTN006503-D02
Page 12
Michelson Interferometer
Chapter 5: Setup and Adjustment
Thermal Expansion: Finally, assemble
the setup for the thermal expansion
experiment. Screw the BA1S(/M) base
onto the post holder with a 1/4"-20 x 3/8"
(M6 x 10 mm) cap screw. Place the 2"
(50 mm) post into the post holder facing
upside down. Attach the 90° adapter
and place the Aluminum rod in it. Orient
the rod so that the end with the
unthreaded hole is in the 90° adapter;
the threaded hole needs to point
away from the adapter. The end face
of the Aluminum post should be
aligned right below the screw of the RA90(/M), as shown in the photo above. Remove
the protective film from the foil heater and carefully affix it to the rod using the provided
tape. Finally, place the mirror into the mirror holder and screw it onto the Aluminum rod.
Page 13
Rev E, November 6, 2017
Michelson Interferometer
Chapter 5: Setup and Adjustment
Setting Up and Adjusting the Michelson Interferometer
In the Michelson interferometer, a laser beam is split by a 50:50 beamsplitter; the split
beams are then reflected back by mirrors and recombined at the beamsplitter. A screen
at the output of the interferometer shows an interference pattern. A lens is used to diverge
the beam in order to obtain an interference pattern consisting of light and dark rings
(constructive or destructive interference, respectively). To set up the interferometer, follow
these steps:
1. First, position the translatable mirror assembly at the edge of the breadboard using four
1/4"-20 x 3/4" (M6 x 20 mm) cap screws.
2. Position the laser assembly on the opposing side of the breadboard. Now tip and tilt the
laser so that the reflected beam falls back into the laser aperture. You may need to move
the laser in order to hit the center of the mirror. After that, secure the base to the
breadboard with a 1/4"-20 x 5/8" (M6 x 16 mm) cap screw.
WARNING
The laser module is a class 2 laser, which does not require any protective eyewear.
However, to avoid injury, do not look directly into the laser beam.
Figure 1
3.
Placing the Laser and the First Mirror
Install the beamsplitter and the screen. Ensure that the beam is split at a 90° angle.
This can be achieved by observing the secondary reflections on the screen. When
they coincide with the primary reflection, the beamsplitter is at a 90° angle.1
1
It is possible that a slight deviation occurs in vertical direction (as shown in Figure 3). This does not
affect the measurements in the experiments. The deviation means that there is a slight tilt among
the laser, beam splitter and translation stage. To get rid of it, either the translation stage or the beam
splitter would have to be mounted kinematically which in turn would compromise the stability.
MTN006503-D02
Page 14
Michelson Interferometer
Chapter 5: Setup and Adjustment
Figure 2
Figure 3
4.
Placing the Beamsplitter
Laser spots with misaligned (left) and aligned (right) beamsplitter
Next, install the second mirror. One should ensure that the distance between the
beamsplitter and the mirrors is about the same along both interferometer arms.
Page 15
Rev E, November 6, 2017
Michelson Interferometer
Chapter 5: Setup and Adjustment
Figure 4
Placement of the Second Mirror
5.
You should now see the two partial beams as bright spots on the screen. Tip and tilt
the second mirror until they overlap.
6.
Finally, place the lens between the laser and the beamsplitter. You may already see
interference rings. If not, turn the screws on the adjustment mirror and try to create
interference. It may help to move the screen away from the breadboard.
MTN006503-D02
Page 16
Michelson Interferometer
Figure 5
7.
Chapter 5: Setup and Adjustment
Placement of the Lens to Observe Concentric Circles
Depending on the orientation of the lens, stripes are visible instead of rings, as shown
in Figure 7. A stripe pattern is produced by placing the lens behind the beamsplitter
and moving the screen away from the breadboard, see Figure 6. Whether you choose
to count rings or stripes is a matter of personal preference.
Page 17
Rev E, November 6, 2017
Michelson Interferometer
Figure 6
Figure 7
MTN006503-D02
Chapter 5: Setup and Adjustment
Alternative Placement of the Lens to Observe Lines
Interference Pattern Produced by the Lens Position shown in Figure 6
Page 18
Michelson Interferometer
Chapter 6
Chapter 6: Theoretical Background
Theoretical Background
This chapter discusses the essential theoretical foundations that apply to the experiments
which follow. The chapter begins with a brief discussion of interference in general and an
explanation of the form of the interference pattern. Coherence is discussed next since one
experiment involves interference with LEDs. This chapter also explains the theoretical
basis for an experiment described later in the manual where the refractive index of a
Plexiglas plate is determined. Thermal expansion and the thermal expansion coefficient
are addressed at the chapter’s conclusion.
Interference using the Michelson Interferometer
Before building the interferometer, we are first going to examine the underlying theory2.
Initially we represent the situation schematically in Figure 8.
4
3
2
1
s2
s1
3
Figure 8 Sketch of a Michelson interferometer. The laser (1) is aimed at the beamsplitter (2)
which divides the beam into two partial beams. These are reflected by the respective
mirrors (3). An interference pattern can be observed on the screen (4).
The laser is divided by the beamsplitter, and the partial beams reflected by the mirrors
overlap again at the beamsplitter. Naturally half the light travels back in the direction of the
laser here. We can mathematically describe how the light intensity on the screen depends
on the path length difference Δ between the two paths and . Limiting ourselves to
examining an incident plane wave along the optical axis:
cos
(1)
Here is the angular frequency, the time, k the wave number (i.e., 2 / ) and the local
variable. In the following example, we represent the transmission capacity of the
beamsplitter with and the reflection capacity with . Now let us examine the amplitude
of the partial wave of one interferometer arm at the location of the screen:
|
|
√ ⋅
⋅
⋅ cos
(2)
2
Illustrations on this topic are found in numerous physics textbooks. In the following, we are guided
by Demtröder: Experimentalphysik 2, 5th edition (2008).
Page 19
Rev E, November 6, 2017
Michelson Interferometer
Chapter 6: Theoretical Background
Here
is the phase, the value of which is established by the actual optical path. The
factor √ ⋅ is therefore explained because the beam in path 1 is first transmitted and
then reflected. The description of the beam in path 2 is similar, but the beam is first
reflected and then transmitted. This results in the same factor and the amplitude of the
partial wave of the second interferometer arm is given on the screen by3
|
|
√ ⋅
⋅
(3)
⋅ cos
where
is the corresponding phase for the second path. The intensity on the screen is
then determined by
|
|
cos
cos
(4)
.
Naturally we only perceive the temporal averaging of the light field oscillation on the
screen, so that only the averaging
1
2
cos
cos
1
(5)
cos
is incorporated in the observed intensity. Furthermore, we are going to assume that both
the transmission and the reflection capability have the value of 0.5, which is a good
approximation for the beamsplitter being used. As the average intensity, we therefore
find ̅
̅
1
cos Δ
,
(6)
where the phase difference of the two partial waves translates directly into the path
length difference Δ between them:
Δ
2
(7)
Δ
Therefore the intensity dependence on the path length difference between the two
interferometer arms is described by a cosine function; see Figure 9.
A fundamental question arises though: when the intensity at the screen drops to zero,
where does the light and the energy go? This simple question often tricks students. To
answer this it’s important to recall that there are two outputs of the interferometer: in the
direction of screen and in the direction of the laser! So when the intensity at the screen
drops to zero, there’s constructive interference in the other output path. This means that
all energy is propagating in the direction of the laser.
3
Here an interesting aside is that the factor
does not apply to the radiation falling back into
the laser. The factors of the partial paths are given by
MTN006503-D02
and
in this case.
Page 20
Michelson Interferometer
Figure 9
Chapter 6: Theoretical Background
Normalized intensity distribution on the screen depending on the path length
difference
To understand this, one has to look at
AR
Cement
the phase shifts on a beamsplitter.
Coating
The cube beamsplitter is composed
of two prisms cemented together with
a beamsplitter coating applied to the
hypotenuse of one of the prisms, as
shown in the sketch to the right.
AR
When the light travels through the
Coating
glass and is reflected by the
beamsplitter coating, no phase shift
occurs. When the light comes from
the other direction (with a plate
BS
AR
beamsplitter that would correspond to Entrance
Coating
Coating
Faces
the “air
coating” interface) the
phase shift is (or 180°). Figure 10 shows that the phase shifts between the two
interferometer arms differs for each output. Regardless of the mirror position, the two
outputs always differ by which results in complementary patterns.
Figure 10 Phase shifts at the beamsplitter coating. Along each optical path, the total phase
shift, Σ, is calculated. The two output ports of the interferometer show a relative phase
shift that differs by . This means that the interference patterns are complementary.
Page 21
Rev E, November 6, 2017
Michelson Interferometer
Chapter 6: Theoretical Background
Size and Shape of the Pattern
We have now clarified what the interference pattern of a plane wave and/or at the central
point looks like. Naturally the real interference pattern appears different than that of a plane
wave, since the laser diverges on its way to the screen. This results in a characteristic ring
pattern with a size that largely depends on the path length difference Δ . The following
brief explanation examines why this is the case and why a ring-shaped pattern forms at
all.
When both interferometer arms are not of equal length (which is always the case since it’s
practically impossible to adjust the interferometer with nanometer precision), then there
exist two (virtual) light sources as seen by the screen which correspond to the different
light paths through the interferometer. If the path is stretched out in one dimension, one
source is behind the other due to the different lengths of the interferometer arms.
As with all interference patterns (such as for the double slit), one can now determine the
difference in path length between the path from light source A to point X and the path
from light source B to point X (as shown in Figure 11) which then translates to, e.g.,
constructive or destructive interference (see Figure 9).
A
B
X
Figure 11 Explanation of a Circular Interference Pattern
If the arms of the interferometer have very different lengths, the two virtual light sources
are far apart. In this case, a small position change on the screen corresponds to a large
change in the path length difference, which again translates into a smaller spacing
between the fringes. This explains why the interference pattern gets smaller when the
interferometer arms have very different lengths.
This line of argument is the same for all points on the screen. Since the lens diverges the
beam symmetrically around the optical axis, the interference pattern needs to be
symmetric, i.e., concentric, as well.
When the two interferometer arms are nearly identical in length, the interference pattern
will show very large rings (with the inverse argument from before). This can be used to
adjust the interferometer. In order to find a configuration with nearly identical arm lengths,
the central maximum needs to be as large as possible.
MTN006503-D02
Page 22
Michelson Interferometer
Chapter 6: Theoretical Background
A nice way to visualize the ring pattern
is to overlap two wavefronts. In practice,
we do not have a plane wave but (due
to the laser’s divergence) a spherical
wave. Since the beam is split in two, we
have two spherical waves from two
virtual sources overlapping. The result
is shown in the left sketch. When the
centers of the spherical waves move
closer together (i.e., the interferometer
arms have nearly equal length, right
sketch), then the central fringe becomes larger.
When the one of the interferometer’s mirrors is tilted, the wavefront from this mirror
changes and, thus, alters the interference pattern. The series of photos below shows the
pattern when the mirror is rotated with respect to the post axis. On first sight, it looks like
the pattern is “moved” to the left. However, this is not quite true, observe the central fringe
in the first and second image (dark and bright). So while tilting the mirror, the central fringe
changes from bright to dark and bright again several times. Whether a clockwise rotation
of the mirror results in a movement of the pattern to the left or to the right depends on
which arm of the interferometer is longer.
Determining the Wavelength
Equation (6) and Figure 9 offer an elegant way to carry out a wavelength measurement.
This is most easily realized when the light source emits only one wavelength, such as with
a laser. If only one of the mirrors is shifted, the path length difference between the
interferometer arms changes and light-dark-light transitions occur in the center, as
summarized by the intensity plot in Figure 9.
One therefore shifts the mirror by a defined path Δ and counts the number of the
maximums (or minimums). According to Figure 9, a light-dark-light transition corresponds
to a path length difference of . However, the light passes along the adjustment path of
the mirror twice (on the way there and back), so that the following applies:
⋅
Page 23
2⋅Δ
⇒
λ
.
(8)
Rev E, November 6, 2017
Michelson Interferometer
Chapter 6: Theoretical Background
Coherence
The topic of coherence is highly complex. Terms such as the contrast of an interference
pattern, correlation functions and the Wiener-Chintschin theorem are essential for a
deeper understanding. Therefore only a brief abstract can be provided here, limited to the
descriptive values of coherence time and coherence length4.
Coherence in the largest sense describes the capacity of light to create interference. The
maximum time span Δ , during which the phase differences of random partial waves at a
point change by less than 2 , is called the coherence time. If the change falls below 2 ,
one says that the partial waves are temporally coherent.
What this means is best described with Figure 12.
a)
b)
Figure 12 (a) Pulse with spectral width ∆ (FWHM) and (b) the partial waves with frequencies
and after the coherence time ∆ have a phase offset of
.
We imagine a light source with a spectrum as shown in Figure 12(a); its spectral width is
denoted by Δ . The source therefore emits light that we can view as the overlapping of
various partial waves with frequencies in the interval
0.5Δ ,
0.5Δ . If
their phase difference at time
0 is zero, the maximum phase offset of two partial waves
is given by:
Δ
2 ⋅
⋅
(9)
If the time span has increased to 1/Δ , the phase offset is 2 . This results in the coherence
time as:
Δ
1
Δ
(10)
The phase offset of the two partial waves with the frequencies and
is shown in Figure
12(b). After the coherence time Δ , the offset has increased to 2 . For all other frequency
components in the interval
, , the phase difference is then smaller.
4
Once again we are following Demtröder: Experimentalphysik 2, 5th edition (2008).
MTN006503-D02
Page 24
Michelson Interferometer
Chapter 6: Theoretical Background
Linked to the coherence time, the coherence length is Δ . This is the path the light can
travel within the coherence time, that is:
⋅Δ
Δ
(11)
Here the propagation in air was assumed with the refractive index 1. While coherence is a
complex topic, the coherence length in case of an interferometer is more straightforward
– it corresponds to the path length difference by which the interferometer arms can differ
in order to observe interference.
For the practical calculation of the coherence time and/or length, the following
approximation is helpful (denotation of the wavelengths similar to Figure 12(a):
Δ
Δ
≅
Δ
(12)
It allows the coherence length of a given spectrum to be estimated approximately by:
Δ
≅
Δ
(13)
For a quantitative evaluation of the coherence, one would have to measure the contrast
function of the interference pattern5. In this experiment package, the coherence length is
estimated by shifting one of the mirrors in the interferometer and observing the
disappearance of the interference pattern (or the pronounced decrease in contrast).
Selecting the Beamsplitter
When constructing an interferometer, one can in principle choose different types of
beamsplitters. The most common types are the plate beamsplitter and beamsplitter cube.
Plate beamsplitters typically consist of a
reflective layer applied to a glass substrate,
while beamsplitter cubes consist of two prisms
cemented together with a layer of beamsplitter
coating applied to the hypotenuse of one of the
prisms.
Figure 13 Plate and Cube Beamsplitters
For a normal interferometer operated with a
laser, it often does not matter which version is chosen. The prices of beamsplitter cubes
exceed those of plate beamsplitters and they also differ in their transmission factors. But
when one examines white light interference, a problem arises: with a plate beamsplitter,
the light traveling through one arm of the interferometer passes through the glass substrate
fewer times than the light traveling through the other arm. A method for compensating for
this difference is illustrated in Figure 14.
5
Note that there are different definitions of coherence. Depending on the definition, the contrast
function has to drop to a different value (e.g., 1/ ).
Page 25
Rev E, November 6, 2017
Michelson Interferometer
(a)
Chapter 6: Theoretical Background
(b)
2
1
Figure 14 Interferometer with (a) plate beamsplitter and (b) compensator plate (refraction on
the surface was disregarded in this sketch)
First let us examine Figure 14(a). Here the laser is reflected off of the beamsplitter when it
is inside of the glass substrate, then reflected by the mirror, and finally transmitted through
the beamsplitter, therefore propagating through the glass substrate thrice on its way to the
screen. In Figure 14(b) it turns out that, without the compensation plate (2), the beam only
passes through the glass once since it is reflected on the outside of the beamsplitter on
the way back from the mirror to the screen.
This means there is an optical path length difference without the plate (2). If this path length
difference is now greater than the coherence length of the light being used, one cannot
observe any interference. However, one can counteract this effect with the compensator
plate (2) – it consists of the same material as the beamsplitter, only the metal layer is
missing. If one now counts how often the partial beams pass through the glass, one notes
that both paths pass through the glass substrate the same number of times (thrice).
For a beamsplitter cube, the beam in each arm travels through the cube twice.
Therefore, no compensator plate is needed, see Figure 15.
(a)
(b)
Figure 15 The beam travels through the whole beamsplitter twice for each arm of the
interferometer. Thus, no compensation plate is needed with a cube beamsplitter.
Furthermore, one might think that the compensator plate is not needed at all if one shifts
one mirror by the correct distance – however this is not the case, since every wavelength
“sees” a different optical path due to the dispersion of the glass. This means that shifting
the mirror could only compensate for a single wavelength.
A beamsplitter cube is used in this setup instead of the combination of plate beamsplitter
and compensator plate, since there is little price difference between the two versions but
adjustment with the cube is significantly simpler and more reliable.
MTN006503-D02
Page 26
Michelson Interferometer
Chapter 6: Theoretical Background
Selecting the Mirrors
The surface flatness of the mirrors used in this kit are specified as better than /10,
compared to economy mirrors which have a surface flatness of 5 . It is not absolutely
necessary to use this quality of mirror but it provides some advantages.
The difference between the two types of mirrors can be observed when using the white
light LED: the quality of the mirrors allows for the white light interference to be observed
over a wider range of relative mirror positions. The interference pattern also shows fewer
distortions.
White Light Interference Pattern
At first glance, one might expect that the interference pattern comprises concentric circles
with the colors of the rainbow. However, this is not the case.
In reality, each point on the screen is illuminated with white light. For most wavelengths,
the path difference is neither nor /2. However, for a small range of wavelengths the
condition for destructive interference is (nearly) met and the intensity on the screen (for
these particular wavelengths) is close to zero. Thus, the pattern on the screen is the white
light minus these few wavelengths. The pattern is, therefore, not colored like a rainbow but
made up of subtractive colors.
Also, concentric circles aren’t observable. The reason lies in the low coherence length of
white light, which requires the difference between the two interferometer arms to be very
small. This, in turn, results in a large central maximum (as discussed in Section 6.1:
Interference using the Michelson Interferometer). To observe the white light interference
pattern, one can adjust the interferometer either (a) so that the central maximum is showing
which basically results in a colored maximum or (b) tilt the mirror slightly such that we
move away from the central maximum in which case you see colored lines.6
6
When a gas discharge lamp is used (e.g., mercury), the coherence length is a lot higher and the
arm difference can be increased. Then, colored, concentric circles can be observed.
Page 27
Rev E, November 6, 2017
Michelson Interferometer
Chapter 6: Theoretical Background
Interferometric Determination of the Refractive Index
Determining the refractive index of a solid (transparent if possible) is one application where
a Michelson interferometer can be used as a sensitive measuring instrument. Here the
solid is first placed in one arm of the interferometer. Then it is slowly rotated so that the
optical path in this interferometer arm changes. This change in the optical path in turn
results in a change of the interference pattern. Now the number of light-dark-light
transitions, the plate thickness, and the angle of rotation can be used to derive the change
in the optical path length, which ultimately allows the refractive index to be determined.
First let us examine the situation when the solid is brought into the light path; see Figure
16(a).
(a)
(b)
Screen
Screen
α
L1
ΔL1
w
β
t
α
ΔL2
L2
Mirror
Mirror
Figure 16 (a) Placing the plate in an interferometer arm (b) Rotation and change in the
(optical) path
The length of the physical light path is:
Physical path length (not rotated):
1
2
(14)
On the other hand, the optical path length is given by:
Optical path length (not rotated):
1
⋅
2
(15)
Here is the refractive index of the material being examined. The refractive index of the
surrounding air is assumed as 1. When the plate is now rotated in the beam, both the
physical and the optical paths change; see Figure 16(b). These paths can be described
as:
Physical path length (rotated):
1
Δ 1
Optical path length (rotated):
1
Δ 1
Δ 2
⋅
Δ 2
2
(16)
2
(17)
Rotation by the angle therefore results in the optical path length difference derived from
the difference between (17) and (15).
MTN006503-D02
Page 28
Michelson Interferometer
Chapter 6: Theoretical Background
ΔOptical Path
2⋅
Δ 1
⋅
Δ 2
⋅
(18)
,
The factor of 2 appears in the equation because the light passes along the path twice. The
path length difference is derived from the light-dark transitions via
⋅
where
ΔOptical Path
(19)
denotes the number of transitions.
Now the values Δ 1, Δ 2 and have to be measured so the refractive index can be
determined. This can be studied in detail in Chapter 13; only the resulting equation is given
here:
cos
2
2
1
sin
cos
2
(20)
1
Determining a Thermal Expansion Coefficient
When one increases the temperature of a solid, it will expand in most situations (and
depending on the environment). This expansion is described by the thermal expansion
coefficient , which represents the proportionality constant between a relative linear
expansion and a corresponding temperature change:
1 d
d
(21)
The solution to this equation describes an exponential process, where the length of the
solid is given as
⋅ exp
⋅Δ
(22)
Here Δ is the temperature change of the solid, causing its original length of
to expand
to the length . In first approximation, one can then determine the expansion coefficient
with
Δ
⋅
⋅Δ
(23)
where Δ is the expansion.
This expansion can be examined experimentally by attaching one mirror of the Michelson
interferometer to the expanding solid. When the solid expands, the path length difference
between the interferometer arms changes, causing the interference pattern to change as
well. Then the light-dark-light transitions are measured as with the wave length
measurement. When the solid is heated by Δ and
transitions are measured, the
coefficient is derived by
1
2
Page 29
Δ
⇒
α
(24)
2
Δ
Rev E, November 6, 2017
Michelson Interferometer
Chapter 6: Theoretical Background
where the factor ½ is once again due to the path of the laser to and from the mirror that
was shifted.
Using the Interferometer as a Spectrometer
A very elegant application of an interferometer is to use it as a spectrometer. Assume the
light source emits two different wavelengths. Naturally, each wavelength will give rise to
its own interference pattern. When these interference patterns overlap well, the contrast
of the resulting pattern on the screen is high. This means that the minimum intensity and
the maximum intensity of the fringe pattern differ greatly.
However, there exist mirror positions (or path length differences) where the bright fringes
of one wavelength overlap with the dark fringes of the other wavelength. This automatically
results in a poor contrast. Essentially, the two interference patterns create an effect called
beating, meaning that the contrast changes from good to poor and back as a function of
the path length difference. The effect can be used to measure the wavelength distance
between adjacent spectral lines such as the yellow sodium doublet.
In the following section, we discuss how to determine the wavelength split by observing
the contrast as a function of the path difference. We start by recalling Equation (6)
̅ ∝
1
cos
2
(25)
Δ
which states the intensity at the central point of the interference fringes, i.e., on the optical
axis. Δ is the path difference between the two interferometer arms (and not the mirror
displacement!).
When we have two wavelengths, the total intensity is then given by
̅
∝
1
cos
2
Δ
1
cos
2
Δ
(26)
assuming that the intensities of both spectral components are identical. Equation (26) can
be written7 as
̅
∝
2
2 ⋅ cos
⋅ Δ
⋅ cos
⋅ Δ
(27)
The interference term vanishes for half-integer multiples of
⋅ Δ
7
Since cos
cos
MTN006503-D02
2 ⋅ cos /2
2
/2 ⋅ cos /2
1
2
,
∈
(28)
/2
Page 30
Michelson Interferometer
Chapter 6: Theoretical Background
which means the contrast becomes zero8. Dividing by
Δ transforms this statement to
2Δ
The term Δ
2
⋅
1
2
2
1
Δ
2Δ
1
is neglectably small in comparison to
Δs
and expressing
and
as
(29)
which results in the statement
(30)
4 Δλ
Equation (30) states that the contrast of the interference pattern will be worst when the two
interferometer arms differ by 2
1
/ 4 Δλ .
Now the question is what the optical path difference Δ Δs from one contrast
to the
1
order). This is given by
disappearance to the next is (i.e., from the
Δ Δs
2
3
4 Δλ
2
1
4 Δλ
2 Δλ
(31)
Thus, we can determine the difference of the two wavelengths in the following way:
First, search for the mirror position where the interference pattern shows a minimum
contrast. Next, move the mirror until you see the next minimum in the contrast. The mirror
movement corresponds to Δ Δs /2. The difference in the wavelength Δ can then be
determined by
Δ
2 Δ Δs
(32)
In this kit, the laser features several spectral lines. By observing the fringe contrast, their
spectral distance Δ can be measured, see Chapter 7.
8
Note: the contrast drops to zero which means that no fringes are visible, only a homogeneous
intensity distribution on the screen. So naturally, the intensity itself doesn’t drop to zero.
Page 31
Rev E, November 6, 2017
Michelson Interferometer
Chapter 7
Chapter 7: Experiments and Examples
Experiments and Examples
This chapter discusses the various experiments that can be performed with this experiment
package. Numerical examples that can be expected under realistic conditions are provided
as well.
Preliminary Tests
Assemble the Michelson interferometer if you have not already done so.
Experiment 1: Change the length of an interferometer arm by moving a mirror (the one in
the kinematic holder). What is the effect on the interference pattern?
Sample result:
Figure 17 When the interferometer arms have about equal length, the fringes are large. The
pattern becomes smaller as the difference in length between the arms increases. See
theory section for an explanation.
Experiment 2: Light a match or lighter and put it right below the laser beam in one arm.
What do you observe?
Execution: The hot air has a different refractive index than the air at room temperature.
Thus, the optical path changes in one arm of the interferometer, causing the interference
pattern to shift. Surprisingly, this does not result in a blurred pattern. Instead, the pattern
is compressed or stretched and/or moved.
Experiment 3: Verify that a Michelson interferometer has
two outputs. Compare both patterns.
Execution: It is important to recognize that the output
pointing towards the screen is not the only output of the
interferometer. In fact, the second output points towards the
laser. It can be observed by looking at the face of the
beamsplitter oriented towards the laser, where you can
observe an interference pattern as well. Alternatively, put a
piece of paper between lens and beamsplitter (without
blocking the incident laser beam).
MTN006503-D02
Page 32
Michelson Interferometer
Chapter 7: Experiments and Examples
Note: Observing the pattern on the beamsplitter requires a dark room with very little stray
light.
Finally, place the second beamsplitter (use one of the post holders from the LED assembly)
between the cube beamsplitter and the lens.
This allows the two outputs to
be observed on the same
screen. An example is shown to
the right. The light coming from
the second beamsplitter has a
lower intensity since only 50%
is reflected to the screen, the
other 50% is transmitted
towards the laser.
The key aspect to notice here is
that the two patterns are
complementary, meaning that
the left shows constructive interference where the right shows destructive interference and
vice versa. This also clearly shows that the term “destructive interference” does not imply
a loss of energy. The light is simply found in the other output.
Page 33
Rev E, November 6, 2017
Michelson Interferometer
Chapter 7: Experiments and Examples
Determining the Laser Wavelength
One typical application of an interferometer is to determine the wavelength of the incident
light. This measurement is performed through controlled shifting of one mirror, which
causes a change in the interference pattern. Depending on the direction the mirror is
moved relative to the second mirror, the concentric circles either expand out from the
center (with new ones constantly appearing in the center) or they shrink into the center
(where they disappear).
Imagine that there is constructive interference in the center, so that the path length
difference between the interferometer arms on the optical axis is a multiple of the
wavelength. When the mirror is shifted forward by half the wavelength, the optical path
length difference of both arms has changed by a full wavelength since the light passes
through each arm on the way there and on the way back. Accordingly, constructive
interference is again visible in the center. This means that shifting the mirror by half the
wavelength generates a light-dark-light transition.
Experiment 4: Determine the wavelength of the laser through translation of the mirror.
Execution: The interferometer is adjusted so that the interference pattern is neither too
large nor too small to count the light-dark-light transitions easily. Then the micrometer
screw on the mirror positioning stage is turned, which changes the interference pattern.
The wavelength is derived from the start and end values of the micrometer screw and the
number of dark-light-dark transitions. One scale division of the micrometer screw
corresponds to one micrometer. Observing numerous transitions is recommended in
order to minimize the error.
Tip: This experiment is very good for demonstrating that an interferometer is a very
accurate and sensitive measuring instrument. Even a slight touch by hand potentially
results in a fluctuation of the pattern. After performing the measurement by hand, insert an
Allen key into the screw of the mirror translator and only turn the Allen key.
Tip: While taking measurements, the mirror’s direction of translation should not be
reversed due to possible backlash of the stage. Do not reverse the direction of rotation of
the knob while measuring.
Example: The following values were determined by students in practical tests. The
reference value of the laser wavelength is 532 nm.9
Number of
Transitions
40
60
80
9
Mirror displacement
[µm]
Calculated
Wavelength
Error
[%]
11.2
17
22
560 nm
567 nm
550 nm
5.3
6.6
3.4
See also the individual spec sheet of the laser.
MTN006503-D02
Page 34
Michelson Interferometer
Chapter 7: Experiments and Examples
The average error in our testing was about 5%. This is due to the fact that counting errors
occur, that there is a certain error in reading the micrometer and that the stage’s accuracy
is limited to a few percent.10
Using the Interferometer as a Spectrometer
Experiment 5: Adjust the interferometer such that the fringe contrast is very low. Next,
move the mirror to find the next minimum in the fringe contrast. Use the measured mirror
displacement to estimate the difference between peak wavelengths in the emission profile
of the laser.
Execution: First, screw in the micrometer screw of the SM1ZP(/M) as far as possible.
Then, place the mirrors such that the contrast of the interference fringes is low. Rotate the
knob of the SM1ZP(/M) until the contrast has reached a minimum. Then, turn the knob
until you’ve reached the next minimum. The difference between good contrast and poor
contrast is shown in the following photos:
Figure 18 Interference fringes with good contrast (left) and poor contrast (right).
In our test measurements, the transition from good to poor contrast corresponded to a
mirror travel range of about 1 mm (equal to 20 full rotations of the knob). This means that
the path length difference from one contrast minimum to the next is about Δ Δ
2
,
see Section 6.6: Using the Interferometer as a Spectrometer. According to Equation (32)
and a wavelength
532 nm, this results in a wavelength splitting of 0.07 nm.
This is consistent with the specifications of the laser diode. Here is an example of a typical
spectrum (taken from www.thorlabs.com).11
10
For a more precise measurement, we recommend using piezo crystals, see Chapter 10: Ideas for
Additional Experiments.
11
The laser’s spec sheet will only show one peak. This is due to the fact that the spectrum’s
purpose is just to show the diode’s center wavelength.
Page 35
Rev E, November 6, 2017
Michelson Interferometer
Chapter 7: Experiments and Examples
1
20°C
25°C
30°C
0.8
Amplitude
0.6
0.4
0.2
0
531.70
Wavelength (nm)
531.90
532.10
532.30
532.50
532.70
The temperature is the temperature of the housing which will be different from the room
temperature. Also, please note that the derivation of Equation (32) assumes two spectral
components of equal intensity. This is not the case here since we have more than two
components and differing intensities. This also explains why the interference pattern
doesn’t fully vanish.
While the final result does not provide a complete quantitative measurement of the laser’s
spectrum, the exercise demonstrates that an interferometer can in fact be used as a
spectrometer and that its sensitivity is very high.
Interference with LEDs, Coherence
In order to see interference with LEDs, one has to adjust the interferometer so that both
arms have almost the same length. If the length of the arms differs significantly so that the
coherence length of the LEDs is exceeded, no interference will be observed. A preliminary
adjustment with the laser is necessary so that the mirror positions needed to observe
interference with the LED are easier to find.
Experiment 6: Use the laser as the light source initially. Adjust the interferometer so that
the arm lengths are almost identical.
Execution: Shift one of the mirrors so that the central maximum of the interference pattern
gets as large as possible. The central maximum should cover nearly all of the illuminated
area on the screen. Only then is the preliminary adjustment good enough to proceed to
aligning the setup to observe interference with an LED. This step will require several
iterations using the mirror in the kinematic holder.
Experiment 7: Connect the red LED and use it instead of the laser. Now move the mirror
until you see an interference pattern.
Execution: The chances of seeing an interference pattern directly after installing the LED
are very low. Shift the translatable mirror to find the interference pattern. If you reach the
end of the mirror’s translation range without finding a pattern, shift it to the other end. If
you have not found a pattern along the entire translation range of the mirror, the preliminary
MTN006503-D02
Page 36
Michelson Interferometer
Chapter 7: Experiments and Examples
adjustment with the laser was not accurate enough. In this case repeat the alignment
process with the laser.
Tip: Do not turn the screw too quickly and, when
changing your grip, wait briefly to see if an
interference pattern develops (~1/2 second) –
otherwise you might miss the correct mirror
position.
Tip: Place the LED close to the beamsplitter to
avoid losing too much intensity.
Experiment 8: Measure the approximate coherence length of the red LED.
Execution: Shift the mirror until the contrast of the interference pattern has decreased
significantly but the pattern can still be observed, and make note of the mirror position.
Then shift the mirror in the other direction, also until the contrast of the interference pattern
Page 37
Rev E, November 6, 2017
Michelson Interferometer
Chapter 7: Experiments and Examples
has decreased significantly. The path length difference between the mirrors corresponds
to half the coherence length.
Sample result: The result of a test measurement was approximately 25 µm. Examining
the spectrum in the datasheet for the LED (see Figure 19) one can read the spectral width
at half the maximum intensity (FWHM) to be approximately 20 nm. The peak wavelength
is about 640 nm. With Equation (13), this results in a theoretical coherence length of 20.5
µm.
Figure 19 Example spectrum of the red LED from its datasheet.
However, it has to be noted here that the coherence length can only be determined in
terms of magnitude: first of all, determining it from the interference pattern is already
problematic since the contrast can only be estimated by eye. The reference value of the
LED is not distinct either, since the wavelength of the maximum and the breadth of the
spectral distribution differ slightly for every LED.
Experiment 9: Once you have found the correct position from experiment 8 for red
interference, you can replace the red LED with the white one in order to find the mirror
position to observe white light interference (it’s within the range for the red light
interference). Adjust the mirror angle to observe
the pattern.
Experiment 10: Measure the approximate
coherence length of the white light LED and
compare it to the red LED.
Execution: Follow the steps outlined for Exercise
8.
Sample result: The result of a test measurement
was approximately 10 µm. Due to the broader
spectrum of the white LED, the coherence length
is smaller compared to the narrower band red
LED.
MTN006503-D02
Page 38
Michelson Interferometer
Chapter 7: Experiments and Examples
Refractive Index Determination
In this experiment, we use the interferometer to determine the refractive index of Plexiglas.
The theoretical basis for this has already been discussed in Chapter 6. A rotation of the
Plexiglas plate in the beam lengthens the optical path and leads to a change in the
interference pattern. Together with the rotation angle and the plate thickness, the refractive
index can be derived using Equation (20).
Experiment 11: Set the rotation platform with the Plexiglas plate in one arm of the
interferometer, replace the LEDs with the laser again and establish an interference pattern.
Adjust the plate so that it stands perpendicular to the beam.
Execution: For starters, the interference pattern should be adjusted so that it is of medium
size and the transitions can be readily observed (with the Plexiglas plate in the beam).
Tip: First be sure you understand how the rotation platform functions. When the small set
screw at the front is tightened, the entire platform is turned by the fine adjustment screw.
If the set screw is not tightened, the fine adjustment screw does not engage and one can
rotate the platform manually by greater angles.
Page 39
Rev E, November 6, 2017
Michelson Interferometer
Chapter 7: Experiments and Examples
Figure 20 The micrometer drive (fine adjustment screw) and locking setscrew on the rotation
platform. When the setscrew is locked, the fine adjustment is engaged. The stage is then
rotated by turning the micrometer drive.
First, loosen the set screw and turn the fine adjustment screw as far in as possible. Now
the correct starting point for the measurement still has to be found, i.e. the plate must be
positioned so it is precisely perpendicular to the beam. In order to do so, first remove the
lens from the setup. You will now probably see more than one spot on the screen due to
the reflections of the laser at the air-Plexiglas boundary. Rotate the stage until the spots
lie on top of each other (or are parallel if there’s also a slight tilt). Now the plate is
perpendicular to the beam.
(a)
(b)
Figure 21 Position of spots on the screen when the Plexiglas plate (a) is not perpendicular to
the beam and (b) is perpendicular to the beam.
Experiment 12: Determine the refractive index of Plexiglas by rotating the thin plate.
Execution: After the adjustment from Experiment 11, the measurement can begin – in
order to do so, replace the lens, lock the center screw of the stage to engage the fine
adjustment screw and note the starting angle on the scale. Turn the fine adjustment screw
while counting the number of light-dark-light transitions. At the end of the measurement,
read the rotation angle and measure the thickness of the Plexiglas plate. There is a Vernier
scale on the platform to aid in precisely reading the angle.
MTN006503-D02
Page 40
Michelson Interferometer
Chapter 7: Experiments and Examples
Aligned Lines: 182° 20’
Sample result: The following values were determined by students in practical tests. The
reference value for the refractive index of the Plexiglas plate that was used is 1.49.
Number of
Transitions
30
60
30
60
Rotation
Angle
Plate Thickness
[mm]
According to
Equation (20)
Error
[%]
4° 25'
6° 15'
3° 50'
5° 15'
8
8
12
12
1.50
1.50
1.42
1.46
1.0
0.9
4.6
1.8
Experiment 13: Perform the same experiment with the thick plate. How do things change?
Execution: It’s the same as before; sample results are found in the table above. The
difference between the two scenarios is that the optical path changes more quickly with
the thicker plate. Thus, a smaller rotation angle is needed to achieve the same amount of
fringe transitions. However, this means that a difference in adjustment has a higher error.
Thus, the error for the thick plate is higher than for the thin plate.
Note: The value can be determined with a higher accuracy when more data is acquired.
For example, record the rotation angle after 2 steps of light-dark-light transitions. Then, fit
the theoretical curve to the numerical data. With this method, the thickness of the plate
can also be left as a parameter to be fit.
Page 41
Rev E, November 6, 2017
Michelson Interferometer
Chapter 7: Experiments and Examples
Thermal Expansion Coefficient
Experiment 14: Remove the rotation platform from the setup and replace the moveable
mirror with the setup to measure thermal expansion. Adjust the interference pattern again
so that you can count the transitions easily.
Experiment 15: Start a controlled heating of the post by applying an external voltage to
the heater. Count the number of transitions during expansion and make note of the
respective temperature. Derive the expansion coefficient of Aluminum.
Execution: Apply an external voltage to the two outer contacts of the heater by using the
supplied crocodile clip to banana plug cables and your power supply of choice (up to 12 V
with 2 A12). The heater will increase the rod temperature, causing the rod to expand and
shift the mirror; this causes a number of fringe transitions that can be observed on the
screen and counted. Make sure you give the system enough time to reach thermal
equilibrium before increasing the voltage. The temperature is taken by inserting the
supplied measurement head of the thermometer into the rod. When the pattern does not
change any more, proceed to the next voltage step.
The heating power is 10 W/in at 70°C. With an effective heating area of 2.23 in , this requires an
input power of about 23 Watts.
12
MTN006503-D02
Page 42
Michelson Interferometer
Chapter 7: Experiments and Examples
Sample result: The following values were determined by students in practical tests.
Voltage
[V]
5.1
7
9
11.2
Temperature Temperature
Start
End
[°C]
[°C]
24
26
30
40
26
30
40
50
[°C]
# of fringes
in the
-interval
Total # of
fringes
Calculated
length
[cm]
2
6
16
26
15
32
69
86
15
47
116
202
9.0004
9.0013
9.0031
9.0054
The last column was calculated as 9 cm ⋅ 532 nm ⋅ /2 . The length of the rod is 9 cm and
it’s made of aluminum. The factor 1/2 is due to the fact that the light travels to and from the
attached mirror. Thus, an expansion of 532 nm/2 266 nm results in one bright-darkbright transition on the screen.
Plotting the calculated length over the temperature increase of the rod yields this graph:
0.09
.
Following Equation (22) we can deduce the thermal expansion coefficient to be
2.26 ⋅ 10
which is about 2.2% away from the reference value of 2.31 ⋅ 10
.
Page 43
Rev E, November 6, 2017
Michelson Interferometer
Chapter 8
Chapter 8: Experiment Overview
Experiment Overview
Preliminary Tests and Determining the Laser Wavelength
0.
Set the Michelson interferometer up and adjust it.
1.
Change the length of an interferometer arm by moving the mirror (the one in the
kinematic holder). What is the effect on the interference pattern?
2.
Light a match or lighter and put it right below the laser beam in one arm. What do
you observe?
3.
Verify that a Michelson interferometer has two outputs. Compare both patterns.
4.
Determine the wavelength of the laser through translation of the mirror.
Using the Interferometer as a Spectrometer
5.
Adjust the interferometer so that the fringe contrast is very low. Next, move the
mirror to find the next minimum in the fringe contrast. Use the measured mirror
displacement to estimate the difference between peak wavelengths in the
emission profile of the laser.
Interference with LEDs
6.
Use the laser as the light source initially. Adjust the interferometer so that the arm
lengths are almost identical.
7.
Connect the red LED and use it instead of the laser. Now shift the mirror until you
see an interference pattern.
8.
Measure the approximate coherence length of the red LED.
9.
Once you have found the correct position from experiment 4 for red interference,
you can replace the red LED with the white one. Then find the white light
interference position, which will lie within the range for the red light interference.
10. Measure the approximate coherence length of the white light LED and compare
it to the red LED.
Refractive Index Determination
11. Set the rotation platform with the Plexiglas plate in one arm of the interferometer,
replace the LEDs with the laser again and establish an interference pattern.
Adjust the plate so that it stands perpendicular to the beam.
12. Determine the refractive index of Plexiglas by rotating the thin plate.
13. Perform the same experiment with the thick plate. How do things change?
Thermal Expansion Coefficient
MTN006503-D02
Page 44
Michelson Interferometer
Chapter 8: Experiment Overview
14. Remove the rotation platform from the setup and replace the moveable mirror
with the setup to measure thermal expansion. Adjust the interference pattern
again so that you can count the transitions easily.
15. Start a controlled heating of the post by applying an external voltage to the heater.
Count the number of transitions during expansion and make note of the
respective temperature. Derive the expansion coefficient of Aluminum.
Page 45
Rev E, November 6, 2017
Michelson Interferometer
Chapter 9
Chapter 9: Questions
Questions
This list of questions also provides a starting point for topics that can be examined in
relation to the Michelson interferometer (even beyond this experiment package).
What is the difference in the interference patterns when the arms are (a) of the
same length and (b) of different lengths?
How many outputs does a Michelson interferometer have?
When the mirror is shifted and a light-dark-light transition is observed in the
interference pattern – by what path length was the mirror shifted?
Why is it best to use a laser as the light source in an interferometer and not an
incandescent lamp?
How were interference experiments conducted before lasers were invented?
What is the most well-known experiment in which a Michelson interferometer was
used? What was it intended to demonstrate?
What is the difference between the spectrum of the red LED and the white LED?
What effect does this have on the coherence length?
Instead of a beamsplitter cube, one can also use a plate beamsplitter. However,
one needs a compensator plate in one of the interferometer arms.
o What is the purpose of the compensator plate?
o Why is it not needed when a cube beamsplitter is used?
o Can the effect of the compensator plate also be achieved by merely
shifting one of the mirrors?
The interference pattern of the red LED shows red, concentric circles. Why are
no white concentric circles seen with the white light LED?
Instead of using the rotation method, the refractive index can also be determined
approximately with the help of white light interference. So, if one does not have a
rotation platform: how does one proceed with white light interference in order to
determine the refractive index?
How does the calculation of the thermal expansion coefficient change when the
metal rod is not clamped at the end but in the middle?
When the rod is heated, the interference rings sometimes shrink into the center
and sometimes expand out of the center. Why? More precisely: how do the arm
lengths have to be adjusted so that the rings expand out of the center (or shrink
into the center)?
MTN006503-D02
Page 46
Michelson Interferometer
Chapter 10
Chapter 10: Ideas for Additional Experiments
Ideas for Additional Experiments
Beyond the phenomena described in this experiment package, a Michelson setup offers
numerous other opportunities for the exciting study of physics. Here is a collection of a few
ideas for how the setup can be expanded in case of further interest.
Mirror Translation with Piezo Crystals
In this kit, the mirror is moved manually using a screw. Using a piezo crystal
makes much finer adjustments possible. Such a crystal expands in a defined way
when an external voltage is applied. As a result, the translation of the mirror can
be controlled with great precision in increments smaller than 50 nm. Since piezo
crystals are widespread in modern nano-positioning systems, their use offers a
learning experience with broad applications.
One possible setup using Thorlabs components could comprise a kinematic
holder with support arm (KM100PM(/M), PM3(/M)), a mirror (e.g., ME1S-G01), a
Piezo stack (such as AE0203D08) and a KPZ101 Controller.
However, piezo crystals exhibit hysteresis which means that the actual
displacement varies depending on whether the voltage is increased or
decreased. To make precise measurements, a feedback mechanism is needed
to compensate for the hysteresis. A possible setup comprises a kinematically
mounted mirror (e.g., ME1S-G01 in KM100) on a stage (e.g., NFL5DP20S(/M)),
a KPZ101 piezo controller and a KSG101 strain gauge reader.
Alternatively, you can use the interferometer to measure the hysteresis curve
itself.
Refractive Index of Gases
By counting light-dark-light transitions, one can also derive the refractive index
of gases (especially depending on volume and pressure). In order to do so, one
loads a closed, evacuated cuvette into the setup and slowly fills it with the gas in
question. The change in the optical path results in a corresponding change of
the interference pattern.
Automated Evaluation
In order to simplify counting the transitions and support a larger number of cycles,
it is possible to install a suitable photodetector instead of the observation screen.
The advantage of a photodetector is that the voltage values can be recorded
using a digital oscilloscope and evaluated without additional software.
A possible setup using Thorlabs components is as follows:
1x SM05PD1A Silicon Photo Diode, 1x SM05D5 Iris, 1x SM05M10 Lens Tube,
1x SM05RC/M Lens Tube Retaining Ring, 1x CA2812 SMA on BNC Cable, 1x
T3285 BNC T-Piece, 1x FT104 Termination Resistor, 1x T1452 Adapter on
Banana Plug
Page 47
Rev E, November 6, 2017
Michelson Interferometer
Chapter 11
LIGO
Chapter 11: Modern Michelson Interferometry – LIGO
Modern
Michelson
Interferometry
–
A recent application of the Michelson interferometer that attracted a lot of international
attention is gravitational-wave detection. Gravitational waves are oscillations in spacetime
curvature produced by colliding black holes, neutron stars, and other astrophysical
processes that involve a dense concentration of mass-energy moving at relativistic
speeds. A network of laser interferometers has been constructed in several countries to
detect these waves. This includes the Laser Interferometer Gravitational-wave
Observatory (LIGO) in the United States, VIRGO in Italy, GEO600 in Germany, and
KAGRA in Japan. All of these experiments consist of a Michelson interferometer with
kilometer-scale arm lengths. The mirrors are suspended and free to swing in the plane of
the interferometer. A passing gravitational wave will shrink the mirror-beamsplitter distance
in one arm of the interferometer while stretching that distance in the other arm. The
oscillating shrinking/stretching pattern induced by the passing wave is recorded as an
oscillating signal in the photodetector. On September 14, 2015 the twin LIGO detectors (in
Washington state and Louisiana) made the first direct detection of a gravitational wave.
The signal (which was measured with high confidence in both detectors) was produced by
an orbiting pair of black holes that merged together about a billion light years away. This
signal caused the LIGO mirrors to move by about 10
meters, or nearly one-thousandth
the diameter of a proton. Michelson interferometers can thus perform some of the most
sensitive length measurements possible. LIGO and its partner observatories are vastly
more complicated than the interferometer in this kit, but the fundamental physical principle
behind their operation is Michelson interferometry.13
13
We cordially thank the LIGO collaboration, in particular Marc Favata, Nancy Aggarwal and Maggie
Tse, for this addition to the manual.
MTN006503-D02
Page 48
Michelson Interferometer
Chapter 12
Chapter 12: Troubleshooting
Troubleshooting
The laser spots superpose, but there is no interference.
Check whether all of the components have been positioned as precisely as
possible (Is there a 90° beam angle after reflection? Is the height of the beam
above the plate at the screen the same as it is directly at the laser?). If these
conditions exist, you may have to simply experiment a little and slightly change
one spot repeatedly without completely losing the superposition.
You have found an interference pattern, but the diameter is very small.
If this is the case, it is probable that the distance between the beamsplitter and
the mirror in one of the arms of the interferometer is much greater than in the
other arm. Therefore, move the mirror so that the distances are as equal as
possible.
The interference sometimes disappears for no apparent reason without the
setup being touched.
Temperature changes in the crystal can lead to changes in the laser modes.
Place a hand on the laser module and warm it slightly – the interference should
appear again.
The contrast of the interference pattern varies.
One reason is the temperature change mentioned in the previous point. However,
the fringe contrast/visibility also changes when there is more than one line in the
spectrum of the used laser which is the case for the CPS532-C2. So the variation
of the visibility is nothing to be concerned about and can actually be used for
precision spectroscopy, see Chapters 6.6 and 7.3. Simply move one mirror which
should increase the contrast.
Instead of the ring-shaped interference
pattern, hyperbolic-shaped interference
fringes can be seen.
These and other distortions of the
interference pattern typically occur when
the height of the beams along both arms of
the interferometer is not exactly the same.
We recommend moving the screen along
the beam to check the heights throughout
the setup.
No red light / white light interference can be found.
It is important to first adjust the interferometer with the laser so that the arm
lengths are effectively equal. Due to the low coherence of LED light, the arms
ultimately have to correspond within a few dozen micrometers. In order to
achieve this, the post-mounted mirror should be fine-tuned so that the visible
interference pattern (when using the laser!) consists almost exclusively of the
central maximum. Only then is the preliminary adjustment good enough to
proceed with fine-tuning using the red LED and the moveable mirror. When the
Page 49
Rev E, November 6, 2017
Michelson Interferometer
Chapter 12: Troubleshooting
LED is in the setup, the mirror has to be shifted until interference can be
observed. If no interference can be observed along the entire translation path of
the mirror, the preliminary adjustment has to be repeated and if possible
refined.
The thermal expansion is too fast or the calculated values for the coefficient
deviate too much from the literature values.
In these cases the voltage of the heating element should be increased more
gradually. In particular, one should give the system some time to reach a
thermal balance before reading the temperature. Also, make sure that the rod is
fixed at its very end in the 90° angle bracket. Only then the expansion will be
fully translated into a mirror movement into the right direction.
MTN006503-D02
Page 50
Michelson Interferometer
Chapter 13
Chapter 13: Appendix
Appendix
Determining the refractive index was discussed in Section 6.4. Here the calculation that
leads to equation (20) is presented in detail. The content discussed in Chapter 6 on the
physical and optical path is presumed to be known here.
(a)
Screen
α
(b)
Screen
α
ΔL1 e
α
w
t
β
α
(c)
Screen
α
e
fα g
β t
Mirror
α
ΔL2
t
Mirror
Mirror
Figure 22 Rotated plate with refractive index
ht
in the beam path
To examine the entire optical and/or physical path, it is best to segment it into three
sections as shown in Figure 22(a) through (c). The section in the plate is easy to determine.
Figure 22(a) shows that
cos
⇒
(33)
cos
applies. Similarly one sees in Figure 22(b), that
sin
Δ 1
⇒
Δ 1
(34)
sin
where we make note of the relation
(35)
⋅ tan
for subsequent steps. Determining Δ 2 is more complex. Note that
(36)
tan
applies, which for
results in
cos
⇒
⋅ cos
tan
⋅ cos
(37)
Now Δ 2 can be calculated as
Δ 2
Page 51
(38)
Rev E, November 6, 2017
Michelson Interferometer
Chapter 13: Appendix
We now have almost everything we need for equation (20). First we want to point out
Snell’s law of refraction, according to which
1 ⋅ sin
⋅ sin
(39)
applies, where we assumed the refractive index of air with a good approximation of 1. For
a subsequent calculation step, one still needs
cos arcsin
cos
sin
sin
1
(40)
where the latter equality is derived from trigonometric calculation rules.
As discussed in Chapter 6, the optical path length difference is then derived from
,
⋅
2⋅
,
,
Δ 1
⋅
Δ 2
⋅
,
2⋅
⋅
sin
cos
cos
tan
1
sin
2⋅
⋅
2⋅
⋅ tan
2⋅ ⋅
⋅
sin
⋅
cos
sin
⋅
sin
⋅ cos
cos
⋅
2⋅ ⋅
sin
2⋅ ⋅
sin
⋅ cos
tan
⋅
cos
1
n ⋅ cos
1
1
1
⋅
⋅ cos
⋅
⋅ cos
⋅
cos
cos
cos
(41)
Now equation (41) is solved to the root term and squared. We then have:
2
⇒
2
1
⇒
1
cos
cos
2 ⋅
sin
2 ⋅
2
1
cos
2
1
cos
sin
sin
2
1
cos
This immediately results in the equation (20) for the refractive index.
MTN006503-D02
Page 52
Michelson Interferometer
Chapter 14
Chapter 14: Regulatory
Regulatory
As required by the WEEE (Waste Electrical and Electronic Equipment Directive) of the
European Community and the corresponding national laws, Thorlabs offers all end users
in the EC the possibility to return “end of life” units without incurring disposal charges.
This offer is valid for Thorlabs electrical and electronic equipment:
Sold after August 13, 2005
Marked correspondingly with the crossed out “wheelie
bin” logo (see right)
Sold to a company or institute within the EC
Currently owned by a company or institute within the
EC
Still complete, not disassembled and not contaminated
As the WEEE directive applies to self contained operational
electrical and electronic products, this end of life take back
service does not refer to other Thorlabs products, such as:
Wheelie Bin Logo
Pure OEM products, that means assemblies to be built into a unit by the user
(e.g. OEM laser driver cards)
Components
Mechanics and optics
Left over parts of units disassembled by the user (PCB’s, housings etc.).
If you wish to return a Thorlabs unit for waste recovery, please contact Thorlabs or your
nearest dealer for further information.
Waste Treatment is Your Own Responsibility
If you do not return an “end of life” unit to Thorlabs, you must hand it to a company
specialized in waste recovery. Do not dispose of the unit in a litter bin or at a public waste
disposal site.
Ecological Background
It is well known that WEEE pollutes the environment by releasing toxic products during
decomposition. The aim of the European RoHS directive is to reduce the content of toxic
substances in electronic products in the future.
The intent of the WEEE directive is to enforce the recycling of WEEE. A controlled recycling
of end of life products will thereby avoid negative impacts on the environment.
Page 53
Rev E, November 6, 2017
Michelson Interferometer
Chapter 15
Chapter 15: Thorlabs Worldwide Contacts
Thorlabs Worldwide Contacts
USA, Canada, and South America
Thorlabs, Inc.
56 Sparta Avenue
Newton, NJ 07860
USA
Tel: 973-300-3000
Fax: 973-300-3600
www.thorlabs.com
www.thorlabs.us (West Coast)
Email: sales@thorlabs.com
Support: techsupport@thorlabs.com
UK and Ireland
Thorlabs Ltd.
1 Saint Thomas Place, Ely
Cambridgeshire CB7 4EX
Great Britain
Tel: +44 (0)1353-654440
Fax: +44 (0)1353-654444
www.thorlabs.com
Email: sales.uk@thorlabs.com
Support: techsupport.uk@thorlabs.com
Europe
Thorlabs GmbH
Hans-Böckler-Str. 6
85221 Dachau
Germany
Tel: +49-(0)8131-5956-0
Fax: +49-(0)8131-5956-99
www.thorlabs.de
Scandinavia
Thorlabs Sweden AB
Bergfotsgatan 7
431 35 Mölndal
Sweden
Tel: +46-31-733-30-00
Fax: +46-31-703-40-45
www.thorlabs.com
Email: europe@thorlabs.com
Email: scandinavia@thorlabs.com
France
Thorlabs SAS
109, rue des Côtes
78600 Maisons-Laffitte
France
Tel: +33 (0) 970 444 844
Fax: +33 (0) 825 744 800
www.thorlabs.com
Brazil
Thorlabs Vendas de Fotônicos Ltda.
Rua Riachuelo, 171
São Carlos, SP 13560-110
Brazil
Tel: +55-16-3413 7062
Fax: +55-16-3413 7064
www.thorlabs.com
Email: sales.fr@thorlabs.com
Email: brasil@thorlabs.com
Japan
Thorlabs Japan, Inc.
Higashi-Ikebukuro Q Building 2F
2-23-2, Higashi-Ikebukuro,
Toshima-ku, Tokyo 170-0013
Japan
Tel: +81-3-5979-8889
Fax: +81-3-5979-7285
www.thorlabs.jp
China
Thorlabs China
Room A101, No. 100
Lane 2891, South Qilianshan Road
Putuo District
Shanghai
China
Tel: +86 (0) 21-60561122
Fax: +86 (0)21-32513480
www.thorlabschina.cn
Email: sales@thorlabs.jp
Email: chinasales@thorlabs.com
MTN006503-D02
Page 54
Michelson Interferometer
Page 55
Chapter 15: Thorlabs Worldwide Contacts
Rev E, November 6, 2017
www.thorlabs.com
Purchase answer to see full
attachment