Coherence

Concept of coherence is related to stability or

predictability of phase

Spatial coherence describes the correlation between

signals at different points in space.

Temporal coherence describes the correlation between

signals at different moments of time.

Coherence is measured by the path difference between

two rays of the same source

1
The amplitude of a single frequency wave as a function of time t (red)
and a copy of the same wave delayed by τ(green). The coherence time
of the wave is infinite since it is perfectly correlated with itself for all
delays τ.

2
The amplitude of a wave whose phase drifts significantly in time τc as
a function of time t (red) and a copy of the same wave delayed by
2τc(green). At any particular time t the wave can interfere perfectly
with its delayed copy. But, since half the time the red and green waves
are in phase and half the time out of phase, when averaged over t any
interference disappears at this delay.

3
 Methods to obtain two coherent source

Wavefront splitting Involves taking one wavefront and

dividing it up into more than one
wave.
Eg: Young’s double slit interference;
Diffraction grating

Amplitude splitting Involves splitting a light beam into

two beams at a surface of two media
of different refractive index.
Eg: Michelson interferometer

4
Each wavefront represent a crest.
So, every two wavefronts are separated by λ.
Coherence area (Ac):The area of a surface perpendicular to the direction
of propagation, over which the electromagnetic wave maintains a specified degree
of coherence.

Any two points on

the same wavefront
The spatial coherence is always correlated
depends on the emitters
size and its distance

D 2 2
Ac  
d 2

2
where d is the size Lc  
and D is the distance 

A plane wave with an infinite coherence length. 5

A wave with a varying profile (wavefront) and infinite
coherence length.
6
 2
Lc 


A wave with a varying profile (wavefront) and finite coherence

length.
7
The wave with finite coherence length from last Figure is passed
through a pinhole. The emerging wave has infinite coherence area.
The Lc or c are unchanged with pin hole.
8
Waves of different frequencies (i.e. colors) interfere to form a
pulse if they are coherent.
9
Spectrally incoherent light interferes to form continuous light
with a randomly varying phase and amplitude
1
0
 Electric field distribution around the
focus of a laser beam with perfect
spatial and temporal coherence.

A laser beam with high spatial

coherence, but poor temporal
coherence.

A laser beam with poor spatial

coherence, but high temporal
coherence.
11
 Quantifying Coherence
Physically, monochromatic sources are fictitious.
Band of frequencies =

Wave train

Many sinusoidal nearby frequencies

are needed to construct the above
 Quantifying Coherence

The coherence time is the time over which a

Coherence time: propagating wave may be considered coherent.
In other words, it is the time interval within
which its phase is, on average, predictable.

: Spectral width of the

source in units of frequency.

The coherence length is the coherence time times

Coherence length: velocity of light (in vacuum), and thus also
characterizes the temporal (not spatial!) coherence
via the propagation length (and thus propagation
time) over which coherence is lost.
Red Cadmium = 6438 Å
= 10 9
Hz, 30 cm
Yellow Sodium = 5893 Å
10
= 10 Hz, 3 cm
He-Ne Laser = 6328 Å
6
= 10 Hz, 300 m
Wavefront splitting interferometers

Young’s double slit

Fresnel double mirror

Fresnel double prism

Lloyd’s mirror

1
5
Plane wave:

Plane wave:

Cylindrical wave: Cylindrical wave:

Spherical wave:

Spherical wave:

16
Young’s Double Slit Experiment

1
7
Young’s double slit

1
8
Path
difference:

For, y  1;

For a bright fringe, m: any integer

For a dark fringe,
The distance of mth bright fringe from central maxima

Fringe separation/ Fringe width,

19
20
Interference Animation If the separation between
the slits decreases, then

1. Angular spacing of the

fringes increases

2. Fringe width increases

The angular spacing of the fringes, θ, is given by:


 (where d is the separation
d between slits)

𝜆𝐷
𝑥 = 𝛽 = (Fringe width) 21
𝑑
Fresnel’s double mirror

SA  S1 A and SB  S2 B, so that SA  AP  r1 and SB  BP  r2 and   r1  r2  m  , (maxima)

L (R  d ) a
The finge width,     , (since,  R sin   R  , CP  d )
a 2 R 2
L: distance between the plane of S1 & S2 and the screen
Fresnel’s biprism

B
P


  𝜆𝐷
d 
  Fringe width 𝛽=
𝑑
Deviation angle  = (-1)
a b
D The distance between the
two virtual sources is : d=
30’ 2a

179° d= 2a(-1)

30’
Lloyd’s mirror

2
4
Billet’s split lens

2
5
 Uses of Coherence
Michelson interferometer [spatial; high resolution astronomical
imaging at optical frequencies]

Radio telescopes, e.g. the Very Large Array (VLA) [spatial;

astronomical imaging at RF frequencies]
Optical Coherence Tomography (OCT) [temporal; bioimaging with
optical sectioning, chromatic aberration is typical evidence of
temporal incoherence]
3-dimensional imaging technique with high spatial resolution and
large penetration depth even in highly scattering media
Based on measurements of the reflected light from tissue discontinuities
e.g. the epidermis-dermis junction
Based on interferometry : interference between the reflected light and
the reference beam is used as a coherence gate to isolate light from
specific depth
26
 Uses of Coherence

Multipole illumination in optical lithography [spatial; sub-µ feature

patterning]

measurement of line width of source

Hologram (every path-length should be similar to Lc)

Size of stars: Using Michelson Stellar Interferometer: The

interference fringes will disappear if the distance between the
pinholes is given by d = 1.22λ/, where  is the angle
subtended by circular source

27
 Coherence and Polarization
There are additional conditions to be satisfied in order to
observe interference effects with polarized light.

Two waves linearly polarized in the same plane can interfere.

Two waves linearly polarized with perpendicular polarizations

cannot interfere.

Two waves linearly polarized with perpendicular polarizations, if

derived from the perpendicular components of unpolarized light
and subsequently brought into the same plane, cannot interfere.

Two waves linearly polarized with perpendicular polarizations, if

derived from the same linearly polarized wave and subsequently
brought into the same plane, can interfere. 28
 Shape of interference pattern (YDSE)
O (0,0,0)
P ( x, y,0)
d
S1 (0, , L)
2
d
S2 (0, , L)
2

S 2 P  S1 P  [ x 2  ( y  d / 2) 2  L2 ]1/ 2  [ x 2  ( y  d / 2) 2  L2 ]1/ 2  (say)

x 2  ( y  d / 2) 2  L2  2  x 2  ( y  d / 2) 2  L2  2 x 2  ( y  d / 2) 2  L2
 ( y d  2 / 2) 2  2 ( x 2  ( y  d / 2) 2  L2 )
1 2
  x  (d   ) y   [ L  (  d 2 )]
2 2 2 2 2 2 2

4
x2 (d 2  2 ) y 2 x2 y2
 1 2  2 Equation of hyperbola
1 1 a b
[ L2  (2  d 2 )] 2 [ L2  (2  d 2 )]
4 4
  1 
2   L2  ( 2  d 2 ) 
where, 1  4 
a 2   L2  ( 2  d 2 ); b 2 
4 ( d 2  2 )
1/ 2
 2 
1/ 2
 2 1 2 
On rearranging we get, y   2 
  x  L2
 ( d 2
  )
 d  2
  4 

since, x 2  L2 , neglect x 2

y  f (  )  Constant for a given delta

The loci are straight lines parallel to the x-axis.
If we use slit instead of hole, each pair of points would have
produced the same straight line fringes which would have
overlapped with each other-thus we would again obtain straight
line fringes
Photograph of real fringe pattern for Young’s double slit

31
 Intensity of the fringe

Irradiance,

Interference term,
32
The interference term

33
34
For maximum irradiance cos   1
 I max  I1  I 2  2 I1 I 2

Total constructive interference

For minimum irradiance cos   1

I min  I1  I 2  2 I1 I 2
Total destructive interference

Components out of phase Components 90o out of phase

if if

35
 For I0=I1=I2

For the spherical wave emitted by two sources, in-phase

at the emitter   k (r1  r2 )  2    2  x d
  D

36
 For two beams of equal irradiance (I0)

Visibility of the fringes (V)

Maximum and adjacent minimum of the fringe system

3
7
Interference by division of Amplitude

Michelson Interferometer

Newton’s ring

Thin film

Fabry-Perot Interferometer

Color of thin films

 Michelson Interferometer
Experimental set up

Albert Abraham Michelson

(1852-1931)
This instrument can produce both types of interference
fringes i.e., circular fringes of equal inclination at infinity and
localized fringes of equal thickness
40
 Michelson
Interferometer
Mirror 1

Monochromatic
light source

Mirror 2
Compensation
plate

Beam splitter

Fringe Pattern
Effective arrangement of the interferometer
An observer at the detector will see M1, a reflected
Circular fringes 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 (general treatement)

P
 PS 'S   PQ S  m

N
rn

m S O
Q
S
2d
D

S ' P  SP  S ' N  2d cos  m  m Condition of maxima

(without any reflection)
m2 (for small m)
cos  m  1 
2 Taylor expansion
 For small m m m
2
1 
2 2d

Central bright fringe 2d  m0  (Note: There is no reflection here)

(m0  m) n
 
2
m  ( n  m0  m)
d d
Radius of mth bright ring

D n 2
r D 
2
m
2 2
m
d
 Mirror 1

In Michelson interferometer
(when the phase change of ray 2 is considered) Ray 1

Ray 2

2d cos  m  m (m  0,1,2,...) : Minima

Mirror 2
Note: Ray 2
 1 experiences an
2d cos  m   m   (m  0,1,2,...) : Maxima additional  phase

 2 change due to
external reflection
and as a result the
conditions of
Order of the fringe: maxima and minima
are exchanged
When the central fringe is dark the order of the fringe is
2d
m

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.
Fringe shape

Central dark fringe

2d  mo 

1st dark ring

2d cos 1  (m0  1)
2nd dark ring
2d cos 2  (m0  2)

And so on.............
 In Michelson interferometer
2d cos m  m

For central dark fringe: 2d  mo 

The first dark fringe satisfies: 2d cos 1  (m0  1)

12
For small θ cos 1  1 
2
 12  1
2d 1    ( m0  1)  D
 2 

d   D 2
r D 
2
1
2 2 2
1 1
d
Radius of first dark fringe
The mth dark fringe satisfies: 2d cos  m  (m0  m)

 m2 
2d 1    ( m0  m)  d m2  m (2d  mo  )
 2 
mth dark ring
Radius of mth dark ring:

d  m
2
m m
D 2
m
rm  D  m 
2 2 2

48
Michelson interferometer Michelson interferometer
with compensator without compensator

M2 M2

S S

M1 M1

Condition for central dark spot Condition for central bright spot
2d  m0 (  0 ) 2d  m0 (  0 )

49
 1. Measurement of wavelength of light

2dcosm  m
2d  m0 (  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   m
d
2
m
2. Measurement of wavelength
separation of a doublet
Concordance Discordance

2 d1  p 1  q (1   )  1
2 d1  p 1   q   (1   )
 2
 Laser Interferometer Gravitational-wave Observatory (LIGO)
General relativity predicts the existence of
gravitational waves.
In Einstein’s theory, gravity is equivalent
to a distortion of space. These distortions
can then propagate through space.

The LIGO apparatus is designed to detect the

distortion produced by a disturbance that
passes near the Earth.
The interferometer uses laser beams with an
effective path length of several kilometers.
At the end of an arm of the interferometer, a mirror is mounted on a
massive pendulum.
When a gravitational wave passes, the pendulum moves, and the
interference pattern due to the laser beams from the two arms changes.
54
• Orthogonal arm lengths change in different ways as they interact with a
gravitational wave
• Use laser to measure relative lengths L/L by observing the changes in
interference pattern at the anti-symmetric port, for example, for L ~ 4 km
and for a hypothetical wave of h ~ 10–21 , L ~ 10-18 m !
• Power-recycled Michelson interferometer with Fabry-Perot arm cavities

55
Interferometric Detectors

VIRGO 3km
Italy

56
 Interferometric Detectors
LIGO Louisiana 4km, USA
 Interferometric Detectors
LIGO Washington 2km& 4km, USA
 E&M GW
space as medium for field Space-time itself
incoherent superpositions of atoms, coherent motions of huge masses (or
molecules energy)
wavelength small compared to wavelength ~large compared to
sources - images sources -
poor spatial resolution
absorbed, scattered, dispersed by very small interaction; no shielding
matter
106 Hz and up 103 Hz and down
measure amplitude (radio) or measure amplitude
intensity (light)
detectors have small solid angle detectors have large solid angle
acceptance acceptance
59
 Measurement of the coherence length of a spectral line

Temporal Coherence
Formation of straight white light fringes (=0)
 Measurement of thickness of thin transparent flakes

t
d  2 (   1) t  

NOTE: Measurement of refractive index of the transparent material