Michelson Interferometer:
The Michelson interferometer was
invented in the year 1893 by Albert Michelson, to measure a standard meter in units of
the wavelength of the red line of the cadmium spectrum. With an optical
interferometer, one can measure distances directly in terms of wavelength of
light used, by counting the interference fringes that move when one or the
other of two mirrors are moved. In the Michelson interferometer, coherent beams
are obtained by splitting a beam of light that originates from a single source
with a partially reflecting mirror called a beam splitter. The resulting
reflected and transmitted waves are then re-directed by ordinary mirrors to a
screen where they superimpose to create fringes. This is known as interference
by division of amplitude. This interferometer, used in 1817 in the famous
Michelson- Morley experiment, demonstrated the non-existence of an
electromagnetic-wave-carrying ether, thus paving the way for the Special theory
of Relativity.
A simplified diagram of a Michelson
interferometer is shown in the fig: 1.
Light from a monochromatic source S is
divided by a beam splitter (BS), which is oriented at an angle 45° to the
beam, producing two beams of equal intensity. The transmitted beam (T) travels
to mirror M2 and it is reflected back to G1. 50% of
the returning beam is then reflected by the beam splitter and strikes the
screen, E. The reflected beam (R) travels to mirror M1, where it is
reflected. 50% of this beam passes straight through beam splitter and reaches
the screen.
Since the reflecting surface of the
beam splitter G1 is the surface on the lower right, the light ray starting from
the source S and undergoing reflection at the mirror M1 passes
through the beam splitter twice. The optical path length through the glass plate depends
on its index of refraction, which causes an optical path difference between the
two beams. To compensate for this, a glass plate G2 of the same thickness and
index of refraction as that of BS is introduced between M2 and G1. The recombined beams interfere and produce fringes at the screen E.
The relative phase of the two beams determines whether the interference will be
constructive or destructive. By adjusting the inclination of M1 and
M2, one can produce circular fringes, straight-line fringes, or
curved fringes. This lab uses circular fringes, shown in Fig. 2.
From the screen, an observer sees M1 directly
and the virtual image M2' of the mirror M2, formed
by reflection in the beam splitter, as shown in Fig. 3. This means that one of
the interfering beams comes from M1 and the other beam appears
to come from the virtual image M2'. If the two arms of the
interferometer are equal in length, M2' coincides with M1.
If they do not coincide, let the distance between them be d, and consider a
light ray from a point S. It will be reflected by both M2' and
M1, and the observer will see two virtual images, S1 due
to reflection at M2', and S2 due to reflection
at M1. These virtual images will be separated by a distance 2d.
If θ is the angle with which the observer
looks into the system, the path difference between the two beams is 2dcosθ.
When the light that comes from M2 undergoes reflection
at G1, a phase change of π occurs, which
corresponds to a path difference of λ/2.
Fig. 3
Therefore, the total path difference between the two beams
is,
The condition for constructive
interference is then,
For a given mirror separation d, a given wavelength λ, and order m, the angle of
inclination θ is a constant,
and the fringes are circular. They are called fringes of equal
inclination,
or Haidinger fringes. If M2' coincides with
M1, d = 0, and the path
difference between the interfering beams will be λ/2. This corresponds
to destructive interference, so the center of the field will be dark.
If one of the mirrors is moved through
a distance λ/4, the path difference changes by λ/2 and a maximum is
obtained. If the mirror is moved through another λ/4, a minimum is
obtained; moving it by another λ/4, again a maximum is obtained and so
on. Because d is multiplied by cosθ,
as d increases, new rings appear in the
center faster than the rings already present at the periphery disappear, and
the field becomes more crowded with thinner rings toward the outside. If d decreases, the rings
contract, become wider and more sparsely distributed, and disappear at the
center.
For destructive interference, the total
path difference must be an integer number of wavelengths plus a half
wavelength,
If the images S1 and S2 from the two mirrors
are exactly the same distance away, d=0 and there is no
dependance on θ. This means that only
one fringe is visible, the zero order destructive interfrence fringe, where
and the observer sees a single, large,
central dark spot with no surrounding rings.
Measurement of wavelength:
Using the Michelson interferometer, the
wavelength of light from a monochromatic source can be determined. If M1 is moved forward or
backward, circular fringes appear or disappear at the centre. The mirror is
moved through a known distance d and the number N of fringes appearing
or disappearing at the centre is counted. For one fringe to appear or
disappear, the mirror must be moved through a distance of λ/2. Knowing this, we
can write,
so that the wavelength is,
Applications
1. The Michelson - Morley experiment is
the best known application of Michelson Interferometer.
2. They are used for the detection of
gravitational waves.
3. Michelson Interferometers are widely
used in astronomical Interferometry.
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