Michelson Interferometer

Experimental set up

Michelson
Mirror Interferometer
1

Monochro
matic light

Mirror
source Compensa

2
tion plate

Beam
splitter

Albert Abraham Michelson

(1852-1931)

Fringe
Pattern
This instrument can produce both types of interference
fringes i.e., circular fringes of equal inclination at infinity and
localized fringes of equal thickness
2
 The transmitted wave gets
reflected by M2 and gets
The reflected wave (partially) reflected by beam
undergoes a further splitter and results in the
reflection at M1 and this interference of these waves.
reflected waves gets
(partially) transmitted
through BS

If x1 and x2 is the distances of

the mirrors M1 and M2
from the beam splitter Three types of fringes
1. When M1 and M’2 are parallel: Circular
2. When M1 and M’2 are inclined: Curve
3. When M1 and M’2 cross: Straight line
S will appear to get reflected by
two parallel mirrors M1 and M’2 2d cos θ m = mλ (m = 0,1,2,...) : Minima
separated by a distance [d=(x1-x2)].
 1
2d cos θ m =  m + λ (m = 0,1,2,...) : Maxima
 2
Michelson interferometer Michelson interferometer
with compensator without compensator

M1 M1

S S

M2 M2

Condition for central dark spot Condition for central bright spot
2d = m0 λ (θ = 0 ) 2d = m0 λ (θ = 0 )

4
 Fringes of constant inclination
2 d cos θ m = m λ ,
If d is a constant in the region then
θ m = constant,
The fringes of constant inclination
and circle localized at infinity

mth dark ring

Radius of mth dark ring:

D 2
mλ
rm =
2
θm
d D
1. Measurement of wavelength of light
If we reduce the value of d, the fringes will appear to collapse at the
centre and the fringes become less closely spaced.

If d increased, the fringe pattern will expand

If N fringes collapse to the centre as the mirror M1 moves by a

distance d0, then we must have

2d = mλ , & 2(d − d 0 ) = (m − N )λ

2 d0
Thus, λ=
N
2. Measurement of wavelength separation of a doublet
Step 1. the interferometer is first set corresponding to the zero path
difference. Near d = 0 , both the fringe patterns from doublet will overlap.
Step 2. If the mirror M1 is moved away or towards the beam splitter
through a distance d, then the maxima corresponding to the λ1 will not
occur at the same angle as λ2
2 d cos θ m = m λ1 & 2 d cos θ m = (m + 1 / 2) λ2
If the distance d is such that
2d 2d 1
− = , (Fringes will disappear)
λ1 λ2 2
2d 2d
− = 1, (Fringes will reappear)
λ1 λ2
2d 2d 1
− ≥ , (Fringes will disappear).
λ1 λ1 + ∆λ / 2 2
λ2
∆λ ≈
2d
Internal reflection implies that the reflection is from an interface to a
medium of lesser index of refraction.

External reflection implies that the reflection is from an interface to a

medium of higher index of refraction.

8
 Phase Changes Due To Reflection
An electromagnetic wave undergoes a phase change of 180° upon
reflection from a medium of higher index of refraction than the one
in which it was traveling
– Analogous to a reflected pulse on a string

µ1 < µ 2

Phase shift δ = k0 Λ ± π
 Thin Film Interference π 
 − θi 
2 
n1 nf n1 n1 n1 C
nnf f

D θt
O θt B
C
θt
θi A
D
B d
θi θt
A
AD
d cos(π / 2 − θ i ) = sin θ i =
AC
=Λ n f [AB + BC] − n1 (AD)
AC = AO + OC = 2 d tan θ t
= BC
AB = d /cosθt
n1 sin θ i = n f sin θ t
nf
= =
AD ACsin θi 2d tan θt sin θt
n1
d nf
Λ = n f (2 ) − n1 (2 d tan θ t sin θ t ) = 2 n f d cos θ t
cos θ t n1
Path Difference ni nf
ni < n f ⇒ π phase shift
Λ =2dn f cos θt ni > n f ⇒ 0 phase shift

Phase shift (in the case of external reflection)

δ = k0 Λ ± π
4πn f
δ= d cos θ t ± π
λo
For n1 > nf > n2, or n1 < nf < n2,
the ±π phase shift will not be present
 4πn f
Phase shift δ= d cos θ t ± π
λo
 λ0 
Condition for maxima (δ = 2mπ) λ =
 f n 
 f 

λf
d cos θt =
(2m + 1) m=
0, 1, 2,...
4
Condition for minima (δ = (2m+1)π)
λf
d cos θt 2=
m m 0, 1, 2,...
4
Note: Odd and even multiple of (λf /4)
Fizeau/Wedge Fringes The optiocal path difference, Λ = 2 n f d cos α
Here the thickness is variable, d = x α
Condition for constructive interference (α is small),
2 n f d m = (m + 1 / 2)λ ⇒ 2 n f xm α = (m + 1 / 2)λ ; where, m = 0,1,2
λ
xm = (m + 1 / 2)
2α n f
λ
⇒ Fringe width, β =
2α n f

The larger α, the finer are the fringes

 Fringes of constant thickness
When the two surfaces coincide in position but are not parallel, the plane
of localization is near the site of the surface.
Eg: Newton’s ring, Wedge fringes, white light fringes in Michelson expt

2 d cos θ m = m λ ,
If d is small
but vary rapidly in small region , Beam splitter
d = constant,
fringes of constant thickness
and
fringes are localized Extended source n
at the site of the surface
n2
d = xα
x n
Haidinger’s Bands:
Fringes of equal
inclination

d
n1
Beam splitter n2

Extended
Focal
source
plane Dielectric
PI P2
slab