17 interferometers

8/11/2019 17 Interferometers

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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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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



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

17 interferometers

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