Transcript Interferometers

Michelson Interferometer
This instrument can produce both types of interference
fringes i.e., circular fringes of equal inclination at infinity
and localized fringes of equal thickness
1
INTERFEROMETER
Michelson Interferometer
Albert Abraham Michelson
(1852-1931)
Michelson-Morley Experiment
In 1878, Michelson thought the detection of motion through the ether might be
measurable.
In trying to measure the speed of the Earth through the supposed "ether", you
could depend upon one component of that velocity being known - the velocity of
the Earth around the sun, about 30 km/s. Using a wavelength of about 600 nm,
there should be a shift of about 0.04 fringes as the spectrometer was rotated 360°.
Though small, this was well within Michelson's capability.
Michelson, and everyone else, was surprised that there was no shift. Michelson's
terse description of the experiment: "The interpretation of these results is that
there is no displacement of the interference bands. ... The result of the hypothesis
of a stationary ether is thus shown to be incorrect." (A. A. Michelson, Am. J. Sci,
122, 120 (1881))
Experimental set up
Michelson
Interferometer
Michelson
Interferometer
Effective arrangement of the interferometer
Circular fringes
An observer at the detector looking into B will see M1, a
reflected image of M2(M2//) and the images S’ and S” of the
source provided by M1 and M2. This may be represented by a
linear configuration.
Longitudinal section –Circular fringes
P
rn
N
q
S
O
S
d
D
In Young’s double-hole experiment:
For small qm
SP  SP  SN  d cos q m  m
2(m  m0 ) 2n
q 

d
d
2
m
(n  m  m0 )
Radius of nth bright ring
D 2n
r Dq 
d
2
2
n
2
2
m
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.
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In Michelson interferometer
2d cos q m  m (m  0,1,2,...) : Minima
1

2d cos q m   m   (m  0,1,2,...) : Maxima
2

Order of the fringe:
When the central fringe is dark the order of the fringe is
m
2d

As d is increased new fringes appear at the centre and the existing
fringes move outwards, and finally move out of the field of view.
For any value of d, the central fringe has the largest value of m.
In Michelson interferometer
2d cos q m  m
For central dark fringe:
2d  mo 
The first dark fringe satisfies: 2d cos q  (m  1)
For small θ
 q2 
2d 1    ( m  1) 


2


 q2 
2d 1    ( m  1) 


2


Radius of nth dark ring:
dq  (mo  m)  n
2
m
D n
r Dq 
d
2
2
n
2
2
m
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Haidinger Fringe
1. Measurement of wavelength of light
2dcosq  m
2d  m0
(q  0 )
Move one of the mirrors to a new position d’ so that the order of the
fringe at the centre is changed from mo to m.
2d   m
2 d   d  m  m0   n
2 d

m
2. Measurement of wavelength separation
of a doublet (λ1 and λ1+λ)
If the two fringe patterns coincide at the centre: (Concordance)
2d1  p1  q1   
2d  m0
(q  0 )
The fringe pattern is very bright
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Concordance
2d1  p1
 q1   
2. Measurement of wavelength separation
of a doublet (λ1 and λ1+λ)
2d1  p1  q1   
2d  m0
As d is increased p and q increase by different amounts, with
q  p
When
q  p  (1 / 2)
the bright fringes of λ1 coincide with the dark fringes of λ1+λ, and
vice-versa and the fringe pattern is washed away (Discordance).
(q  0 )
Discordance
2d1  p1
= (q+1/2) 1   
2. Measurement of wavelength separation
of a doublet (λ1 and λ1+λ)
2d1  p1  q1   
2d  m0
(q  0 )
- Δ can be measured by increasing d1 to d2 so that the two sets of fringes,
initially concordant, become discordant and are finally concordant again.
- If p changes to p+n, and q changes to q+(n-1) we have concordant fringes
again.
2d2   p  n1  q  n 11   
2d2  d1   n1  n 11   
 

2
1
2d 2  d1 
23
•Measurement of the coherence length of a spectral line
•Measurement of thickness of thin transparent flakes
•Measurement of refractive index of gases
LIGO - Laser Interferometer Gravitational Wave Observatory
To detect Gravitational waves, one of the predictions of Einstein’s General Theory of Relativity
When Gravitational
waves pass through the
interferometer they will
displace the mirrors!
Hanford Nuclear Reservation, Washington, Livingston, Louisiana
Arm length: 4 Km
Displacement Sensitivity: 10-16 cm
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Fabry-Perot Interferometer
26
Fabry-Perot Interferometer
θ
30o
Multiple Beam Interference
28
Optical Reversibility and
Phase Changes on Reflection
G.G. Stokes used the principle of optical
reversibility to investigate the reflection of
light at an interface between two media.
The reversibility principle states that
If there is no absorption of light, a light ray
that is reflected or refracted will retrace its
original path if its direction is reversed.
r and t are fractional amplitudes reflected and transmitted respectively
According to principle of
reversibility, the
combined effect of
reversing the reflected
and transmitted beams
should just be the incident
beam (in absence of
absorption).
r  r 
r  tt   1
2
© SPK
Thin films: multiple beam interference
0
0
0
0
0
0
0
0
0
0
0
0
© SPK
0
0
0
0
Path difference between rays 2 and 1
[(OS + SR)(in film)] – [OM( in air) ]
= [(PS + SR)(in film)] – [OM( in air)]
= [(PR)(in film)] – [OM( in air)]
= μ (PN + NR) – OM
= μ (PN) = μ (OP Cos θ)
Δ= 2μ d cos θ
CASE - I
If 2μ d cos θm = m λ
then rays 2,3,4, 5, …. are in phase
and 1 out of phase.
Amplitude of 2+3+4+5 ….
= aotr’t’(1+ r’ + r’ + r’ +…)
2
= aotr’t’(1/(1 –r’ ))
= aotr’t’(1/tt’) = aor’= - aor
2
4
6
Total reflected Amplitude: 1+(2+3+4+…)
= aor +(- aor)
=0
Amplitude of transmitted beams α, β, γ, δ …
2
4
= aott’(1+ r’ + r’ + r’ +…)
= ao
6
CASE - II
If 2μ d cos θm = (m+1/2) λ
then rays 1,2,4, 6, … are in phase
and 3,5,… are out of phase.
Rays α, γ, … in phase and rays β , δ, …
are out of phase
Optical field in reflected beam
a1R  a0 re
it
a2 R  a0tr t e
i (t  )
3
i (  t  2 )


a3 R  a0tr t e
a4 R  a0tr  t e
5
i (t 3 )
.........................
aNR  a0tr 
where
a0 e
it
(2 N 3)
t e
i[t  ( N 1) ]
: is the incident wave;
 is the phase arising from the extra optical path length.
Resultant reflected scalar wave
 i
)
it  r (1  e
aR  a0e 
2  i 
 1 r e 
where,
r  r 
tt   1  r
2
If the number of terms of the
series approaches infinity, the
series converges and the
resultant becomes
aR .a
Reflected irradiance I R 
2
2r (1  cos )
I R  I0
4
2
(1  r )  2r cos
2
*
R
2

a0 
 I0  
2

 i

r
(1

e
)
it
aR  a0e 
2  i 
 1 r e 
Optical field in transmitted beam
it

a1t  a0tt e
a2t  a0tt  e
2 i (t  )
a3 R  a0tt r e
4 i (  t  2 )
.....................
.........................
aNt  a0tr
(2 N 1)
t e
i[t  ( N 1) ]
aT  a0e
it
 tt  
1  r 2 ei 
Transmitted irradiance
I 0 (tt )
IT 
4
2
(1  r )  2r cos 
2
I 0  I R  IT
For Transmitted rays
I 0 (tt ) 2
IT 
(1  r 4 )  2r 2 cos 
 IT max  I 0
cos   1
1  r 
1  r 
cos =-1
2 2
( IT ) min  I 0
2 2
= 2mπ
Path diff.
2μd cos θm = m
= (2m+1)π
Path diff.
2μd cos θm = (2m+1)/2
2r 2 (1  cos )
I R  I0
(1  r 4 )  2r 2cos
For Reflected rays
( I R ) max  I 0
4r
2
1  r 
 I R min  0
2 2
cos   1
cos   1
Interference filter
An interference filter is designed for normal incidence of 488 nm light. The
refractive index of the spacer is 1.35. What should be the thickness of the
spacer for normal incidence of light.
2d  
d  180.74 nm
2d cos q m  m
It will pass different wavelength if the angle of incidence is not 90o.
We now introduce
Coefficient of Finesse
 2r 
F 
2 
 1 r 
2
 2r 
F 
2 
 1 r 
2
F sin  / 2 
Ir

2
I 0 1  F sin  / 2 
2
It
1

2
I 0 1  F sin  / 2 
2r 2 (1  cos )
I R  I0
(1 irt 4)  2ttr2cos
aT  a0e 
2  i
1  r e 
I 0 (tt ) 2
IT 
(1  r 4 )  2r 2 cos 

2  
cos


1

2sin


2
F sin  / 2 
Ir

I 0 1  F sin 2  / 2 
2
It
1

I 0 1  F sin 2  / 2 
Airy function
A (q ) 
1
1  F sin
2

2
Airy function represents the transmitted flux-density distribution.
Note: q is related to path difference .
The complementary [1 - A(q)] represents the reflected flux-density
distribution.
I0
d or 
Multiple beam interference has resulted in redistribution of energy
density in comparison to sinusoidal two-beam patter.
IR/I
IT/I
d or 
Variation of intensities with phase
d or 
Bright fringes Transmitted rays
Dark fringes Reflected rays
Dark fringes Transmitted rays
Bright fringes Reflected rays
Fabry-Perot Interferometer
53
Fabry-Perot Interferometer
θ
30o
The conditions of interference are precisely those discussed earlier.
With =1, the bright fringes in transmission are given by:
2d cosqm= m
The radii of the rings are therefore given by the formula obtained in
Michelson interferometer i.e.,
Rn ≈ D2qm2 = D2n/d
However, there is an essential difference between M.I. and F.P.: One
uses a two beam interference while the other uses multiple beam
interference. Hence the formula for the intensities and the
sharpness of the fringes are quite different.
The intensity is given by:
I o (tt / ) 2
Io
It 

4
2
1  r  2r cos  1  F sin 2 
2
Where F is Coefficient of finesse of the mirror system.
F = (2r/(1-r2))2
and we also know that, for bright fringe : 2d cosqm= m
What we can conclude from these equations:
a)The intensity falls on either side of the maximum.
b)The fall in intensity is dictated by the value of the Coefficient of
finesse F.
c)The Coefficient of finesse is larger for values of the reflection
coefficient r approaching unity. Thus very sharp rings are obtained
by increasing the polish of the mirrors.
I0
Transmitted intensity
Full width at half maximum
=IT/Io
φm
wikipedia
When two mirrors are held fixed and adjusted for parallelism by
screwing some sort of spacer, it is said to be an Etalon.
A quartz plate polished and metal-coated will also serve as an Etalon
(with   1).
Chromatic resolving power
- The ability of the spectroscope or the interferometer to separate the
components of multiplets is known as chromatic resolving power (CRP).
- In a two beam interferometer, like Michelson interferometer and Young’s
double slit set-up, the bright fringes are as broad as the dark fringes. The
fringes are not sharp.
- For good resolution, the bright fringes must be as sharp as possible.
Fabry-Perot
fringes
Michelson
fringes
Doublet separation in
Fabry-Perot interferometer
Resolved wavelengths
s: separation
w: width
Unresolved wavelengths
Barely resolved
Chromatic resolving power of
Fabry Perot interferometer
- Where, λ is the minimum wavelength interval of a doublet that the instrument is
capable of barely resolving.
- The criterion for bare resolution is called the Rayleigh criterion.
- The smaller the value of λ, the higher is the resolving power of the instrument.
Barely resolved
Using: 2d cos θm= mλ ; ( Pabry-Perot - bright fringe in transmission )
I o (tt ' ) 2
IT 
(1  r 4 )  2r 2 cos 
Io

2 
1  F sin
2
FWHM: Angular distance at which the intensity falls to half the peak intensity
Io

2
F
Io
( ) w 
2 1 
1  F sin
m 

2
2 
1/ 2
m ( ) w 
sin  
 1

4 
2
F
1/ 2
m ( ) w 
sin  
 1

4 
2
;
&
 ( ) w  ( ) w
sin 


4
 4 
4
(  ) w  1 / 2
F
Using
m 
(q ) w 
4

d cos q m

F d sin q m
1/ 2
sin(a+b)
=sin a cos b+ cos a sin b
Using
(q ) w 

F 1/ 2d sin q m
F1/2
Sodium doublet
λ1= 589.0 nm
λ2= 589.6 nm
Δλ= 0.6 nm
λ/Δλ~1000
CRP<1000
Sodium doublet
λ1= 589.0 nm
λ2= 589.6 nm
Δλ= 0.6 nm
λ/Δλ~1000
CRP ~ 1000
Sodium doublet
λ1= 589.0 nm
λ2= 589.6 nm
Δλ= 0.6 nm
λ/Δλ~1000
CRP >1000
Sodium doublet
λ1= 589.0 nm
λ2= 589.6 nm
Δλ= 0.6 nm
λ/Δλ~1000
CRP >> 1000
Sodium doublet
λ1= 589.0 nm
λ2= 589.6 nm
Δλ= 0.6 nm
λ/Δλ~1000
CRP>>>1000
Types of fringes
Interference fringes
Real
Virtual
Localized
Non-localized
Real fringe
- Can be intercepted on a screen placed anywhere in
the vicinity of the interferometer without a
condensing lens system.
Virtual fringe
- Cannot be projected onto a screen without a
condensing focusing system. In this case, rays do not
converge.
Non-localized fringe
- Exists everywhere
- Result of point/line source
Localized fringe
- Observed over particular surface
- Result of extended source
POHL’S INTERFEROMETER
Real
Non-localized
Virtual
Localized
Newton’s Ring
U<<R & U>>d