Michelson Lab Guide Supplement
Laser Safety: The interferometer should be use with a class 2 He-Ne laser ( <1mW, visible). Looking
directly into the beam of the laser can lead to retinal burns and must be categorically avoided. The
safety instructions accompanying the laser must be strictly obeyed.
Adjust Slowly: As you align the mirrors to observe the ring pattern or interference walk the mirror left and
right and up and down slowly. If you pass through alignment quickly you will not observe the effect.
Air Cell: DO NOT exceed a pressure of 100 kPa over atmosphere.
Interference occurs when two or more coherent beams overlap. Coherent beams maintain a constant relative
phase(s). For optical interference (4 x 1014 Hz < f < 8 x 1014 Hz), the beam frequencies must match to the
inverse of the minimum observation time or about 60 Hz for direct visual observation. The part in 1012 match
means that optical interference is rarely observed except in the cases that light from a single source is divided
into several beams to yield mutually coherent beams which can propagate along separate paths and then be
overlapped to interfere. In our experiment, the light from the laser is split into two beams at the blue dot. The
beams travel along their separate paths to the right and lower mirrors and return to be co-propagating,
overlapped beams. The interference is observed as a ring pattern on the screen. If the mirrors are misaligned,
the overlapped beams have directions differing by a small angle yielding eyebrow fringes for small angles
and almost straight fringes for larger angles.
The laser is a relatively stable clock and its beams are coherent for path length differences up about 0.5 m.
(Special single mode lasers are coherent for path length differences up about 1 km.1) A atomic spectral tube
filtered down to light from a single spectral line is coherent enough to source interfering beams for path
length differences up to millimeters.
Find the blue dot on the beam splitting surface.
The beams travel as one to that point.
Differences accrue after the beams split.
Note that beam one passes through the glass
substrate of the beam splitter two more times
after separating from beam two. If mirror one is
moved downward a distance d, the distance that
the light must travel along path one increases by
2d. Therefore, the interference pattern will shift
a full fringe for a /2 movement of the mirror.
Due to phase shifts associated reflections at the
beam splitter, the beams interfere destructively
when path one and two have equal optical path
lengths.
NOTE: The beam splitting surface is on the
side toward the angular scale for the apparatus
that we are using. Manual page 3 (numbered as 9).
1
The proposed LISA lasers are to be coherent for 1010 m path length differences.
http://en.wikipedia.org/wiki/Laser_Interferometer_Space_Antenna
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The Michelson interferometer forms two virtual images of each source point in the input
plane. When properly aligned, both images lie on a line perpendicular to the lower mirror and
the beams generate an interference pattern of concentric rings.
Below, we have an idealized case in which virtual image is viewed at a distance L and the
other at a distance L + m when viewed in the forward direction. The result would be
constructive interference in the forward direction2. (The circular lines below are cross
sections of spherical wavefronts. When aligned, the observed pattern has rotational symmetry
about the line of sight.)
The condition for constructive interference is also met at an angle k with the light from the
more distant virtual image traveling an integer multiple k of  shorter distance than in the
forward direction. The pattern is symmetric about the line of sight so the interference pattern
appears as rings concentric with the line of sight. Note that the drawing is for the k = 1 case.
2
The situation actually yields nearly destructive interference due to phase shifts of the light associated with the beam splitter.
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Imagine what happens as the more distant yellow3 object moves forward. The constructive
interference points move inward and the two waves no longer interfere constructively at the
center. That is to say that the rings shrink toward their centers. Once the path lengths are
equal, continued mirror motion in the same sense means that the distance from the more
distant source is increasing so the rings are growing out of the center rather than collapsing
into it. This change is one indication that you have passed through the equal path condition.
Using the law of cosines: L2 + (m)2 – 2 mL cosk = (L + (m – k) from which it follows
that k = [2k/m]½ once corrections of order /L have been ignored.
Exercise: Complete the derivation of k = [k/2m]½. Estimate /L for the Michelson in the lab.
Use small angle approximations.
The relation k = [2k/m]½ indicates that the rings are fat near the equal path condition and
thinner the further that you are from equal paths. For equal paths, you would have m = 0 and
huge angles. Take care! The results cannot be applied literally if the small angle
approximations are not valid.
We have a second indication that a mirror adjustment has taken the device through the equal
paths condition. The rings grow thicker and thicker as the condition is approached and then
thin again once it is past.
Movie: passing through equal path lengths:
http://usna.edu/Users/physics/tank/TankMovies/Michelson/MichelsonGreen1.gif
Phase Shifts: Consider two co-linear beams of equal amplitude interfering
E(x,y) = Eo cos(1 t) + Eo cos(2 t + ) = Eo cos(1 t) + Eo cos(2 t + )
E(t )  2Eo cos(½[t   ])cos(t  ½ );
  2  1;   ½[2  1]
The effective amplitude is 2Eo cos(½  t). The human eye has an integration time of about
30 ms so, at frequency differences greater than say 60 Hz, the interference pattern would not
be observable by eye. Again, as optical frequencies are over 1014 Hz, we can only expect to
3
The light from the two sources is the same color. The color coding in the figure above indicates the source, not the color of the
light.
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observe interference patterns with both beams originate from the same source.4 We square the
amplitude to get the intensity pattern:
I (t )  4
 2c E   1cos(2 x )  2 cos ( x  ½ ) 
o
2
2
o
We observe that the highest intensity is four times I1 that due to a single beam. The light and
dark regions are of equal width. The average intensity is ½ (4 I1) + ½ (0) = 2 I1. The average
intensity for the two beams is just 2
 2c E  , twice the intensity of one beam, a result
o
2
o
expected in view of energy conservation.
The beam splitter is a glass plate with a thin film on one surface that partially reflects and
partially transmits light. For definiteness, we will assume that it is a few-nanometer-thick
aluminum film. In Physics II, you studied thin films and learned that light was phase shifted
by  when reflected form a higher index of refraction material and by 0 when reflected from
a lower index material. Examine the first figure. Beam 1 is in glass and reflected from air for
a 0 shift while beam two (as it returns from the mirror) is in air and reflected off glass for a 
shift. The action of the thin aluminum film may be discussed in your optics course.
The approximate condition for constructive interference becomes:
2 - 1 = m (2or d2 – (d1 + ½ m
Where  ½ or shift is due to the beam splitter
Recall that moving a mirror back a distance s increases the path length for the corresponding
beam by 2s. A travel distance of s in glass represents an increase in the optical path length (c
times travel time) of ng s where ng is the index of refraction of the glass. For our
interferometer, one beam travels through the glass substrate of the splitter two more times
than the other beam for a optical path additional contribution of 2 (ng – 1) D secR, where R
is the angle that the refracted beam makes with respect to the normal. We can move the
mirror to make the optical path lengths for the two beams nearly equal. 2 s = 2 (ng – nave)D
secR Unfortunately the dispersion of the glass prevents equalizing the optical path lengths
for several wavelengths at once. Michelson addressed the issue by adding a compensator
plate to his interferometer to balance the path length in glass for the two beams. I wish our
supplier had done so.
4
If two lasers are carefully designed and stabilized, the light form the two lasers can interfere to yield an interference pattern that
persists long enough to be viewed by the unaided eye.
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
Exercise: What change might be made to the orientation the compensator plate that would
still equalize the optical path for all wavelengths? Would there be any advantage to choosing
the alternate orientation?

Lens mapping infinity to its
focal plane.
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Note that the apparatus that we have is designed to be used with normal room lighting.
However, initial alignment may require a darker environment. The interference pattern is
observed as an image on a screen rather than viewed as a virtual source at infinity.
The small image to the right shows that a lens can act on light from a virtual (or real) source
at infinity to form a real image on its focal plane. In particular, all the light rays parallel to a
ray incident on the lens center point are redirected to pass through the same point in the focal
plane of the (ideal) lens. The center ray is not deflected and travels straight to the focal plane
identifying image point for that bundle of parallel rays. The previous statements are true for
thin ideal lenses in the paraxial approximation.
SAFETY: The interferometer should be operated with a class 2 type He-Ne laser. Any viewing
directly into the beam of the laser can lead to retinal burns and must be categorically avoided. The
safety instructions accompanying the laser must be strictly obeyed.
A Class 2 laser is ‘safe’ because the blink reflex will limit the exposure to
no more than 0.25 seconds. It only applies to visible-light lasers (400–
700 nm). Class-2 lasers are limited to 1 mW continuous wave, or more if the
emission time is less than 0.25 seconds or if the light is not spatially
coherent. Intentional suppression of the blink reflex could lead to eye injury.
Take special care to see that the laser beam travels only on a path above your interferometer
and that, at the ends of the two paths terminate on the screen so that there are no stray beams.
A Class II laser is limited to 1 mW for a visible laser which rarely causes permanent damage
to the retina for exposures less than 30 ms.
http://en.wikipedia.org/wiki/Laser_safety
In all laboratories at UTSA, individuals must learn of and practice all safety procedures
relevant to their activities. Safety is a factor in laboratory grade.
PreExperiment Exercise: Study the mechanism by which the micrometer drives the
movable mirror. Include a drawing. Estimate the mechanical advantage of this mechanism.
Advantage = (distance the micrometer moves)/ (distance the mirror moves)
One mm of micrometer movement is equivalent to a mirror movement of k where  = 632.8
nm. What is k?
Required: Complete the measurements and observations in section 5.
5. Michelson Interferometer

MichelsonManualEnglish.pdf
Include your observations about alignment and use of Michelson interferometers.
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Superior Report: Include elements from sections 6 and 7.
Question: The red line entering from the left represents light from a point source on the
surface of the gray box. The beam splitter and mirrors provide two virtual images of this
source when viewed from the screen position. What are the apparent locations of these
images?
Question: There are two additional beams that finally leave the beams splitter. Add them to
your figure. How were they terminated? Were they safely terminated?
Index of Refraction of Air:
http://en.wikipedia.org/wiki/Clausius%E2%80%93Mossotti_relation
When an electric field is applied to a material, the atoms in the material polarize.
p   Eapplied where  is the atomic polarizability. The dipole moment per unit volume is then
P   Eapplied where  is the number density of the polarized atoms.
Following Griffiths chapter 4, D   E   r  o E   o E  P   o [1     o ] E .
The dielectric constant is r, and for non-magnetic materials, the index of refraction is [r]½.
We conclude that the nair  1 + ½ (/o).
The equation above is an approximation for low density materials. See Griffiths 192-5.
Use your data to estimate the average polarizability of an air molecule.
Richard Feynman on the Clausius–Mossotti equation[edit]
In his Lectures on Physics (Vol.2, Ch32), Richard Feynman has a background discussion deriving the Clausius-Mossotti Equation, in reference to
the index of refraction for dense materials. He starts with the derivation of an equation for the index of refraction for gases, and then shows how this
must be modified for dense materials, modifying it, because in dense materials, there are also electric fields produced by other nearby atoms,
creating local fields. In essence, Feynman is saying that for dense materials the polarization of a material is proportional to its electric field, but that it
has a different constant of proportionality than that for a gas. When this constant is corrected for a dense material, by taking into account the local
fields of nearby atoms, one ends up with the Clausius-Mossotti Equation.[8] Feynman states the Clausius-Mossotti equation as follows:
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,
where

is the number of particles per unit volume of the capacitor,

is the atomic polarizability,

is the refractive index.
Feynman discusses "atomic polarizability" and explains it in these terms: When there is a sinusoidal electric field acting on a material,
there is an induced dipole moment per unit volume which is proportional to the electric field - with a proportionality constant
that
depends on the frequency. This constant is a complex number, meaning that the polarization does not exactly follow the electric field, but
may be shifted in phase to some extent. At any rate, there is a polarization per unit volume whose magnitude is proportional to the
strength of the electric field.
Dielectric constant and polarizability[edit]
The polarizability
, of an atom is defined in terms of the local electric field at the atom by
where


is the dipole moment,
is the local electric field at the orbital [?]
The polarizability is an atomic property, but the dielectric constant will depend on the manner in which the atoms are
assembled to form a crystal. For a non-spherical atom,
will be a tensor.[9]
The polarization of a crystal may be expressed approximately as the product of the polarizabilities of the atoms times the local
electric field:

Michelson Morley: Find a reference that discusses the Michelson-Morley experiment. In
the case that the apparatus is moving at speed v along the direction of one of its arms, the
travel-time difference along the parallel and perpendicular arms will differ by about 2 L/c
(according to a non-relativistic analysis) where L is the length of the arms. Rotating the
apparatus by 900 would therefore shifts the relative travel time by 22 L/c. (a.) We can easily
detect half a full fringe so we set 22 L/c = ½T where T is the period of the red light. Make a
rough measurement of L and estimate the speed v that you could detect. In the MichelsonMorley experiment, the light was reflected back and forth in each rather longer arm many
times. Assume Leffective = 11 m. Superstars that they were, they could detect a relative time
shift of T/50 by observing a small fraction of a fringe shift. (b.) Estimate the speed v that they
could detect. (c.)What is the earth’s orbital speed relative to the sun?
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This figure illustrates the folded light path
used in the Michelson–Morley interferometer
that enabled a path length of 11 m. a is the
light source, an oil lamp. b is a beam splitter.
c is a compensating plate so that both the
reflected and transmitted beams travel
through the same amount of glass (important
since experiments were run with white light
which has an extremely short coherence
length requiring precise matching of optical
path lengths for fringes to be visible;
monochromatic sodium light was used only
for initial alignment[4][B 2]). d, d' and e are
mirrors. e' is a fine adjustment mirror. f is a
telescope.
http://en.wikipedia.org/wiki/File:On_the_Relative_Motion
_of_the_Earth_and_the_Luminiferous_Ether_-_Fig_4.png
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