Holography and temporal coherence
Coherence lengths
required?
Two-slit interference
Two-slit interference pattern from a plane wave
from narrow slits, incoming light at q = 0
d1
d2
h
D
Conditions for 2-slit interference peaks:
q peak 
y peak 
k d 
Find the intensity pattern I(q)
y,q
h
E (q )  E1  E1e

I (q   E1  E1e
ikhq
ikh sin q
 E1  E1e
ikhq
 E  E e   E
 ikhq
1
1
I (q   2 I one  slit 1  2 cos khq 
1
2
 E1  2 E1E1 cos khq
2
What is different for a point source off axis by q’?
E (q )  E1  E1ei ???
I screen (q   2 I one  slit 1  cos kh q  q '  
What happens if I have two uncorrelated point sources
at different q’1 and q’2?
I screen (q  
What happens if my source is a star or the sun?
I screen (q  
Many two-slit patterns are superposed
Contrast (visibility) is less as width of source is increased
So we can use
2-slit fringe
visibility to
measure the
width of a star!
Narrabri , Australia (optical), 640 m
Single mirror resolution (it’s diffraction limited)
q res  1.22

D
Two interfering mirrors separated by h have same angular
resolution as one mirror of diameter h
q res 

h
Note: modern stellar interferometers use several different
interference techniques, but resolution principle is the same.
Spatial coherence can be measured by 2 slits
Real light source with many angles at y’, q’

 h 


dq ' I q ' e  ikhq q '

 I q ' dq '

e  ikhq



dq ' I q '  e ikhq '


 I q ' dq '

The spatial coherence (2-slit fringe) function  ( h)
The e  ikhq
factor in front
doesn’t
affect the
visibility
V ( h)    h 
comes from the spatial FT-1 of the source intensity I (q ')
Fringe visibility or contrast
V ( h)    h 

hc 


I max (h)  I min (h)
V ( h) 
I max (h)  I min (h)

 (h) dh  2  V (h) 2 dh
2
0
Temporal or “longitudinal” coherence
length lc or time tc..
1
c 


Spatial or “transverse” coherence length hc hc 
q source
of a beam of light from a star, distant
streetlamp.
Hey, we can get the diameter of a star if we
know hc!
Spatial coherence can be measured by 2 slits
Real light source with many angles at y’, q’

 h 


dq ' I q ' e  ikhq q '

 I q ' dq '

e  ikhq



dq ' I q '  e ikhq '


 I q ' dq '

The spatial coherence (2-slit fringe) function  ( h)
The e  ikhq
factor in front
doesn’t
affect the
visibility
V ( h)    h 
comes from the spatial FT-1 of the source intensity I (q ')

hc 




0
0
 (h) dh   V (h) 2 dh  2 V (h) 2 dh
2
fringe contrast or visibility
To test the spatial coherence we can
vary slit spacing h and look at the
fringe contrast.
Spatial coherence and interference of light
Can we see 2-slit
interference from
sunlight?
…yes, if we use only a
small spot from the
beam to illuminate the
slits.
…how small? Less
than the transverse
coherence length
To test the spatial coherence we can
vary slit spacing h and look at the
fringe contrast. When they are
mostly washed out, we’re past hc

Or use:
hc 
  ( h)


2
dh  2  V (h) dh
2
0
Sketch visibility V(h) for the sources far from the slits:
1. A point of light on the axis : I q '   I1 q ' 
2. A uniform bar of light perpendicular to the axis
3. A light source that has brightness like a gaussian
I q '  I1 q ' q '2 
fading away from the axis.
4. A point of light moved off the axis:
a) I got it mostly right
b) I got it mostly wrong, but tried
Connection with text’s notation
 h 
e
i
khy 
D



I  y  e
i
khy 
R
 I  y dy

e  ikhq
dy


 ikhq '
Angular
d
q
'
I
q
'
e



version is

much
I
q
'
d
q
'
  
simpler!
The phase factors in front don’t affect what we measure, the
fringe visibility, V (h)    h  , so we ignore them.