Download The Michelson Interferometer

PHYS 233
Experiment #33
Experiment #33
The Michelson Interferometer
References
1. Your first year physics textbook.
2. Hecht, Optics, Addison Wesley - Chapter 9 in the 4th Ed. (2001).
3. Jenkins and White, Fundamentals of Optics 3rd Ed. (1957), McGraw-Hill - Chapter
13. (QC355 J4)
4. Instruction Manual, Beck Interferometer M300/6407, R.&J. Beck Ltd. (pdf available
on course website).
Introduction
The Michelson interferometer is the best known example of a class of interferometers that
are known as amplitude-splitting interferometers, that is they produce interference by means
of division of the amplitude of incident light by means of arrangements of mirrors and beamsplitters. Michelson developed this type of interferometer in order to investigate questions
about the wave nature of light, specifically whether the existence of a supporting medium,
known as the aether, was required for light propagation. The Michelson interferometer has
also been used to provide evidence for the special theory of relativity, to detect and measure
hyperfine structure in line spectra, to measure the tidal effect of the moon on the earth
and to provide a substitute standard for the meter in terms of wavelengths of light. In this
experiment, you will use the Michelson interferometer to measure the wavelengths of the two
main lines of the sodium spectrum and the refractive index of air.
The basic elements of a Michelson interferometer are shown in Fig. 1. In this diagram,
refraction is ignored and anti-parallel rays are slightly displaced for the sake of clarity. Light
from the source S is incident at 45◦ on a half-silvered rear-surface M3 of the beam splitter
P1 at point A. The transmitted ray x passes through the compensator P2 , reflects from the
stationary mirror M1 and returns through the compensator to point A where it reflects to
the observer O. Meanwhile, the reflected ray y traverses P1 , reflects from movable mirror
M2 , passes through P1 and recombines with ray x as it goes to observer O. The optical
compensator P2 ensures that both beams travel the same distance in glass.
Both rays then interfere at the observation point with a path difference d defined by the
equation
2 d = 2 |AB − AC| ,
(1)
which results in a phase difference ∆φo given by
2π
∆φo = ~k · ~r =
2d ,
λ
(2)
PHYS 233
Experiment #33
M2
C
y
M3
A
x
B
S
P1
P2
M1
O
Figure 1: The Michelson interferometer.
where ~k is the wavevector of the light. Ray x undergoes two external reflections (at M3
and M1 ) whereas y undergoes only one (at M2 ) so an additional phase difference φr = π is
introduced for a total
∆φ = ∆φo + φr
2π
2d − π
=
λ
(3)
(4)
The condition for constructive interference is that the total phase difference ∆φ be an integral
multiple of 2 π i.e.
1
2 d = m+
λ
(5)
2
where m = 0, 1, 2, ....
If the mirror M2 is moved a distance ∆d then the intensity of light detected at O will
increase and decrease as successive interference maxima and minima are experienced. Counting the number ∆m of consecutive bright or dark patterns allows the change in path length
∆d to be calculated as
2 ∆d = λ∆m .
(6)
It is important to note that the field of view will only alternate between completely bright
and completely dark if the incident light is collimated, i.e all rays are parallel. It is more
likely that you will be using an uncollimated or extended source. In this case, concentric
rings or fringes will be observed, as shown in Fig. 2. In this figure, the real mirror M1 has
′
′
been replaced by its virtual image M1 formed by reflection in P1 . M1 is parallel to M2 .
The extended source S, now behind the observer, forms two virtual sources S1 and S2 by
PHYS 233
Experiment #33
P′
2d
P′′
′
P′′
P
θ
M 1′ M 2
2d c
os θ
θ
d
2d
Figure 2: Michelson interferometer using an extended source.
′
reflection in the mirrors M1 and M2 . Note that these virtual sources are coherent since the
phase difference between corresponding points in the two is the same at all times. When the
′
separation of the mirrors M1 and M2 is d, the optical path differences between the parallel
′
′′
rays coming to the observer from the corresponding points P and P is 2 d cos θ. Taking
into account the phase changes occurring upon reflection, the condition for constructive
interferences becomes
2 d cos θ = m λ .
(7)
In this experiment, you will use a Michelson interferometer made by R.&J. Beck Ltd. In
the Beck apparatus, M2 of the Michelson optics is mounted on a carriage that is driven by
a micrometer. This controls the length of the AC arm of the interferometer and hence the
path difference d. The instrument is calibrated by measuring the relationship between d and
the change in micrometer reading
K =
Change in micrometer reading
Distance of carriage movement
(8)
Prelab Questions
1. Show schematically how an extended source implies circular fringes.
2. Show how Eq. 7 implies circular fringes.
3. It is possible to view interference fringes from a white light source if the path length
difference is very small. When viewing white light fringes, which colour should appear
on the outside of the circular fringes and which colour on the inside?
4. In Part 3 of the experiment, you are asked to determine the mean wavelength and the
separation of the sodium doublet. There are now two sets of fringes, one for each line
of the doublet. You will see fringes due to both when they constructively interfere,
PHYS 233
Experiment #33
and the fringe pattern will disappear when they destructively interfere. Once you have
determined K it should be obvious how you determine λ̄. (How will you do this?)
Determination of ∆λ is a little more subtle. Show that the mirror distance between
two positions where the fringes disappear is given by
2 (d1 − d2 ) =
λ1 λ2
λ1 − λ2
.
(9)
[Hint: consider the conditions for interference for λ1 and λ2 at the two positions.]
Caution
Two points of caution are worth noting:
• The optical surfaces of the interferometer must never be touched with fingers or any
other object. (With the possible exception of camel’s or squirrel’s hair brushes.)
• It is possible to damage your eye by looking directly at the UV emission of the mercury
vapour lamp. Always use a glass shield or filter and keep the metal housing around
the bulb whenever it is on.
Apparatus
• Beck Model No. M300/6407 Michelson interferometer
• sodium, mercury and incandescent light sources
• ground glass plate
• vacuum cell and vacuum pump
Experiment
1. Investigate the operation of the apparatus. Obtain an interference pattern using the
sodium light source, which consists mainly of two yellow wavelengths (known as the
sodium doublet). Make the distance AM1 and AM2 roughly equal and use a sighting
pin during the initial mirror tilt adjustments. The fringes should be circular and
centred, rather thick and of good contrast. (Please look at the figure in the Beck
Manual.)
2. Determine K. Using the mercury light source (green), calibrate the carriage movement.
Assume the wavelength of the green mercury light to be 546.07 nm.
3. Determine the mean wavelength of the sodium doublet and the separation of the two
lines.
PHYS 233
Experiment #33
4. Determine the refractive index of air. Measure the difference in optical pathlength that
occurs as air is removed from the vacuum cell using the vacuum pump provided.
5. (Optional) White light fringes can be observed with the Michelson interferometer when
the optical path difference of the interfering beams is nearly zero. Observe the striking
colour changes in the pattern as the path difference is slowly varied from zero. (Ref.
4, and Appendix to this handout.)
UG2/2007