Astro 452 Fall 2004

The Michelson Interferometer

1. Introduction
This experiment consists of a set of exercises in which you use the Michelson inter-
ferometer to measure wavelengths as well as the refractive indices of air and glass. The
Michelson interferometer is an instrument of historical importance since it was used by
Michelson to demonstrate that the luminipherous ether did not exist and thus paved the
way towards the development of special relativity. This lab experiment consists of three
exercises referred to as Experiments 1, 2, and 3. In Experiment 1 you will use the
Michelson interferometer to measure the wavelength of light from a laser. You need to
know this wavelength to carry out Experiments 2 and 3 in which you will measure the
refractive indices of air and glass respectively.

2. About This Manual

Attached to this cover page is the original manual from Pasco Scientific, the manufac-
turer of the equipment that you will use in the lab. This manual describes the equipment
and two of the experiments. When reading the manual you should keep in mind that
you are interested specifically in the Michelson interferometer and you should ignore any
sections related to the Twyman-Green or Fabry-Perot interferometers. You should pay
particular attention to the following sections:
- Description of the equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 23
- Underlying theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 45
- Interferometer setup and operation . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 67
- Usage tips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . p. 910
You may need to refer back to these sections for details while you are carrying out the
experiment. Therefore, you should know them well so that you can find the information
you need right away.

3. Experiments
1. The first experimental exercise is measuring the wavelength of the light from the laser
that you will be using. It is necessary to know this wavelength so that you can carry
out the next two experiments. Orient the apparatus so that the viewing screen is in
the direction of one of the nearby walls of the lab. The reason for this is so that you
can easily project the fringe pattern onto the wall (for magnification), just by removing
the screen from its holder. Follow the setup and operation instructions on pages 67
of the Pasco manual to get an interference pattern on the screen. Then remove the
screen and project the pattern on the wall. Read the tips on fringe counting on page 9.
Use the micrometer to move the mirror and observe how the fringes travel across the
field of view. Once you are familiar with the pattern, count fringes as you move the
mirror and find how much the mirror has to be moved for 10 fringes to go by. From
Astro 452 The Michelson Interferometer Fall 2004

this information calculate the wavelength of the laser light. Repeat the experiment 4
times and take a weighted average of the results and uncertainties. Contemplate the
following questions (and write about them in your reports):
Why should you count several fringes instead of just one?
One possible source of error is miscounting the fringes as they go by (e.g., mis-
counting by 1). What error does this introduce in the final result? What are
other sources of error in this measurement?
2. The second experiment involves the measurement of the refractive index of air, and
it is described in detail on pages 1314 of the Pasco manual. Follow the procedure
described there as well as the instructions for the data analysis. At the end answer
the questions. As in Experiment 1, you may find it easier to project the fringe pattern
on the wall rather than on the small screen.
3. The second experiment involves the measurement of the refractive index of glass, and
it is described in detail on pages 1516 of the Pasco manual. Follow the procedure
described there as well as the instructions for the data analysis. As in Experiment 1,
you may find it easier to project the fringe pattern on the wall rather than on the small
screen. Consider possible sources of error and estimate their magnitude. Propagate
their effect to see how they affect the final result.

4. Lab Reports
As usual, your lab reports should include a brief description of the experiment. There
is no need to repeat any of the details given in the lab manual. However, you should
describe anything unexpected that you encountered and deviations from the manual, if
there were any. Report all of your data and make sure that you have answered all of the
questions in the lab manual.
Precision Interferometer 012-05187B

Data Analysis

1. With the above measurements, compute the index of refraction of glass from the
following equation:
2
[(m/2t) na (1 cos )] + (na sin )2
ng = , (1)
2 [(m/2t) na (1 cos )]

where t is the thickness of the glass plate, is the wavelength of the laser light, m is
the number of fringe transitions that you counted as the glass block was rotated from
being perpendicular to the beam to an angle , and na is the refractive index of air.
2. Show that the above expression can be simplified, with sensible approximations, to
the following:
(1 cos ) [1 (m/2t)]
ng = . (2)
(1 cos ) (m/2t)
What are the approximations that you need to make to accomplish this? Why are
they sensible? Compute the index of refraction of glass with equation (2) as well as
with equation (1) to illustrate that the answers are very similar.
3. Using equation (2) derive an expression for the uncertainty in ng resulting from a
small error in the measured rotation angle of the glass block, , and from a small
error in the number of fringe transitions, m. Use this expression to estimate the
uncertainty in ng .

For Extra Credit: derive equation (1).

This will earn you an additional 25% of the points of the lab report, but you should
think carefully whether it is worth investing any time in this exercise. You can go
about it as follows:
(a) Draw a figure showing the path of the beam as it travels from the beam splitter,
through the glass block at an arbitrary inclination angle , and on to the reflector.
(b) Compute the optical path length of the beam as it travels from the beam splitter,
through the glass block at an arbitrary inclination angle , on to the reflector, and
then all the way back. Pay attention to the definition of the inclination angle: it is
defined as the angle of incidence of the beam onto the face of the glass block, i.e., the
angle between the beam and the normal to the glass surface.
(c) Find the change in the optical path length as the inclination angle changes from 0 to
an arbitrary value, . If this change results in an integer number of fringe transitions,
m, then the change in the optical path length should be equal to an integer multiple
of the wavelength, m. Set the change in optical path length equal to m to get an
equation for ng .

16