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1
Michelson I nterferometer
1
This instrument can produce both types of interferencefringes i.e., circular fringes of equal inclination at infinity
and localized fringes of equal thickness
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INTERFEROMETER
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Albert Abraham Michelson
Michelson Interferometer
(1852-1931)
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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))
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Experimental set up
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MichelsonInterferometer
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Michelson
Interferometer
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Effective arrangement of the interferometer
Circular fringesAn observer at the detector looking into B will see M1, a
reflected image of M2(M2//) and the images Sand Sof the
source provided by M1and M2. This may be represented by a
linear configuration.
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Longitudinal sectionCircular fringes
P
O
rn
S Sd
D
N
q
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2 00
cos
2( ) 2 ( )
m
m
SP SP SN d m
m m nn m m
d d
q
q
Radius of nthbrightring
For small qm
22 2 2 2
n m
D nr D
d
q
In Youngs double-hole experiment:
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11
I nternal ref lectionimplies that the reflection is from an interface to a
medium of lesser index of refraction.
External ref lectionimplies that the reflection is from an interface to a
medium of higher index of refraction.
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Maxima:0,1,2,...)(m2
1cos2
Minima:0,1,2,...)(mcos2
q
q
md
md
m
m
In Michelson interferometer
Order of the fringe:
When the central fringe is dark the order of the fringe is
dm
2
As dis 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.
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q md m cos2
In Michelson interferometer
For central dark fringe: omd2
The first dark fringe satisfies: q )1(cos2 md
q
)1(2
12
2
md
For small
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14
Radius of n
th
darkring:
dnDDr
nmmd
mn
om
q
q
2
222
2 )(
q )1(2
12
2
md
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Haidinger Fringe
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1. Measurement of wavelength of light
)(md 02 0 q
md 2
m
d
nmmdd
2
2 0
2 cosd mq
Move one of the mirrors to a new position dso that the order of thefringe at the centre is changed from moto m.
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18
2. Measurement of wavelength separation
of a doublet (1and1+)
1112 qpd )(md 02 0 qIf the two fringe patterns coincide at the centre: (Concordance)
The fringe pattern is very bright
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Concordance
112 pd
1q
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2. Measurement of wavelength separation
of a doublet (1and1+)
1112 qpd )(md 02 0 q
As dis increasedpand qincrease by different amounts, with
pq
)2/1( pqWhen
the bright fringes of1coincide with the dark fringes of1+, and
vice-versa and the fringe pattern is washed away (Discordance).
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Discordance
112 pd
= (q+1/2) 1
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23
2. Measurement of wavelength separation
of a doublet (1and1+)
1112 qpd
112 12 nqnpd
1112 12 nndd
12
2
1
2 dd
)(md 02 0 q
-can be measured by increasing d1to d2so that the two sets of fringes,
initially concordant, become discordant and are finally concordant again.
- Ifpchanges top+n, and qchanges to q+(n-1)we have concordant fringesagain.
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Measurement of the coherence length of a spectral lineMeasurement of thickness of thin transparent flakes
Measurement of refractive index of gases
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25
LIGO- Laser Interferometer Gravitational Wave ObservatoryTo detect Gravitational waves, one of the predictions of Einsteins General Theory of Relativity
Hanford Nuclear Reservation, Washington, Livingston, Louisiana
Arm length: 4 Km
Displacement Sensitivity: 10-16cm
When Gravitational
waves pass through the
interferometer they willdisplace the mirrors!
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26
Fabry-Perot I nter ferometer
26
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Fabry-Perot Interferometer
30o
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28
Multiple Beam Interference
28
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Optical Reversibility and
Phase Changes on Reflection
G.G. Stokes used the principle of optical
reversibility to investigate the reflection oflight at an interface between two media.
The reversibility principlestates 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.
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2
ttr
rr
SPK
rand tare fractional amplitudes reflected and transmitted respectively
According toprinciple of
reversibility, the
combined effect of
reversing the reflected
and transmitted beams
should just be the incidentbeam (in absence of
absorption).
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Thin films: multiple beam interference
SPK
0
0 0 0 0
0
0
0 0
0 0
0
0
0
0
0
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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)]
= (PN + NR)OM
= (PN) = (OP Cos )
= 2d cos
CASE I
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If 2d cos m= m
then rays 2,3,4, 5, . are in phaseand 1 out of phase.
Amplitude of 2+3+4+5 .
= aotrt(1+ r + r + r +)2 4 6
= aotrt(1/(1r ))2
= aotrt(1/tt) = aor= - aor
CASE - I
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Amplitude of transmitted beams , , ,
= aott(1+ r + r + r +)2 4
6
= ao
= aor +(- aor)
= 0
Total reflected Amplitude: 1+(2+3+4+)
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If 2d cos m= (m+1/2) then rays 1,2,4, 6, are in phase
and 3,5, are out of phase.
Rays, , in phase and rays , ,
are out of phase
CASE - II
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1 0( )
2 0
3 ( 2 )
3 05 ( 3 )
4 0
(2 3) [ ( 1) ]
0
.........................
i t
R
i t
R
i t
R
i t
R
N i t N
NR
a a re
a a tr t e
a a tr t e
a a tr t e
a a tr t e
Optical field in reflected beam
: is the incident wave;
is the phase arising from the extra optical path length.
0
i ta e
where
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Resultant reflected scalar wave
0 2
(1 )
1
ii t
R i
r ea a e
r e
2
1
r r
tt r
If the number of terms of the
series approaches infinity, the
series converges and the
resultant becomes
where,
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2
0 4 2
2 (1 cos )
(1 ) 2 cosR
rI I
r r
Reflected irradiance
2
00
2
aI
*.
2
R RR
a aI
0 2
(1 )
1
ii t
R i
r ea a e
r e
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1 0
2 ( )
2 0
4 ( 2 )3 0
(2 1) [ ( 1) ]
0
.....................
.........................
i tt
i t
t
i tR
N i t N
Nt
a a tt e
a a tt e
a a tt r e
a a tr t e
Optical field in transmitted beam
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0 2
2
0
4 2
1
( )
(1 ) 2 cos
i t
T i
T
tta a e
r e
I ttI
r r
Transmitted irradiance
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0 R TI I I
2( )I tt
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For Transmitted rays
0max cos 1TI I
22
min 0 22
1( ) cos =-1
1T
rI I
r
0
4 2
( )
(1 ) 2 cosT
I ttI
r r
= 2m
Path diff.2d cos m= m
= (2m+1)
Path diff.
2d cos m= (2m+1)/2
22 (1 cos )rI I
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2
max 0 22
4( ) cos 1
1R
rI I
r
For Reflected rays
min 0 cos 1RI
0 4 2
2 (1 cos )
(1 ) 2 cosR
rI I
r r
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Interference filter
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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.
nm47.180
2
d
d
It will pass different wavelength if the angle of incidence is not 90o.
q md m cos2
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We now introduce
Coeff icient of F inesse
2
2
2
1
rF
r
22
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2
2
0
2
0
sin / 2
1 sin / 2
1
1 sin / 2
r
t
FI
I F
I
I F
2cos 1 2sin
2
2
0
4 2
( )
(1 ) 2 cosT
I ttI
r r
2
0 4 2
2 (1 cos )
(1 ) 2 cosR
rI I
r r
2
2
1
rF
r
2sin / 2FI
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Airy function
2
1
( )1 sin
2F
q A
2
0
2
0
sin / 2
1 sin / 2
1
1 sin / 2
r
t
FI
I F
I
I F
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.
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d or
I0
Multiple beam interference has resulted in redistribution of energy
density in comparison to sinusoidal two-beam patter.
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IR/I IT/I
d or
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Variation of intensities with phase
d or
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Bright fringes Transmitted rays
Dark fringes Reflected rays
Dark fringes Transmitted raysBright fringes Reflected rays
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53
Fabry-Perot I nter ferometer
53
F b P t I t f t
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Fabry-Perot Interferometer
30
o
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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 D2qm
2= 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 fr inges are qui te different.
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The intensity is given by:
2sin1
cos21
)(
224
2/
F
I
rr
ttII oot
WhereFis 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 offinesse F.
c)The Coefficient of finesse is larger for values of the reflection
coefficient rapproaching unity. Thus very sharp rings are obtained
by increasing the polish of the mirrors.
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I0
T itt d i t it
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Transmitted intensity
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Full width at half maximum
m
=IT/Io
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wikipedi
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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 resol ing po er
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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 Youngs
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.
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Fabry-Perot
fringes
Michelson
fringes
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Doublet separation in
Fabry-Perot interferometer
R l d l h
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Resolved wavelengths
w: widths: separation
U l d l th
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Unresolved wavelengths
Barely resolved
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Barely resolved
Chromatic resolving power of
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Chromatic resolving power of
Fabry Perot interferometer
- Where, is the minimum wavelength interval of a doublet that the instrument is
capable of barely resolving.- The criterion for bare resolution is called theRayleigh criterion.
- The smaller the value of , the higher is the resolving power of the instrument.
Barely resolved
Using: 2d cos m= m; ( Pabry-Perot - bright fringe in transmission )
)'( 2ttII o
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2sin1
cos2)1(
2
24
F
I
rrI
o
oT
14
)(
2sin2/1
wmF
FWHM: Angular distance at which the intensity falls to half the peak intensity
2
)(
2
1sin1
2 2 wm
oo
F
II
1)(
sin2/1
wmF
sin(a+b)=sin a cos b+ cos a sin b
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142
sin
F
&
=sin a cos b+ cos a sin b
4
)(
4
)(sin
ww
2/1
4)(
Fw
;
Usingmm d q
cos
4
m
wdF q
q
sin)(
2/1
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m
wdF q
q
sin)(
2/1
F1/2
Using
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CRP
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Sodium doublet
1= 589.0 nm
2= 589.6 nm
CRP ~ 1000
= 0.6 nm
/~1000
di d bl
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CRP >1000
Sodium doublet
1= 589.0 nm
2= 589.6 nm
/~1000
= 0.6 nm
S di d bl
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CRP >> 1000
Sodium doublet
1= 589.0 nm
2= 589.6 nm
= 0.6 nm
/~1000
S di d bl
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CRP>>>1000
Sodium doublet
1= 589.0 nm
2= 589.6 nm
/~1000
= 0.6 nm
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Types of fringes
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Interference fringes
Real Virtual Localized Non-localized
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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.
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Non-localized fringe
- Exists everywhere
- Result of point/line source
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Localized fringe
- Observed over particular surface
- Result of extended source
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POHLS INTERFEROMETER
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Real
Non-localized
Virtual
Localized
N t Ri
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Newtons Ring
Ud