You play a song on two speakers (not
stereo). How far back can you place the
2nd speaker and still hear interference
effects?
(Ignore 1/r intensity drop off)
A simpler problem,
clearly defined:
You play the same song on two
speakers. With a microphone you
average the intensity over the
entire song.
How does Iavg(x) depend on the
separation, and on the song?
A simpler problem,
clearly defined:
How does Iavg(x) depend on the
separation, and on the song?
Only on the power spectrum
Iavg(w) of the song! No phase
information matters.
Iavg(x) will be greatest at x=0,
will oscillate with period of
lavg, and the oscillations die
out in a distance (v/Dw)
What if I play identical
random noise on both
speakers?
Iavg(x) still comes from Iavg(w)
: will be greatest at x=0, will
oscillate with period of lavg,
and the oscillations die out in
a distance (v/Dw).
Noise might be random, but
can be interfered with itself
over a distance a distance
(v/Dw).
What if I play two
different songs?
Iavg(x) will be constant =Iavg1 +
Iavg2 …same as if I played
independent noise on each
speaker.
Temporal coherence and interference of light
How thick can a piece
of glass be to still see
interference fringes?
…it depends on the coherence length
of the light we use!
Coherence time and coherence. length
“Longitudinal” coherence time
tc, or length lc = ctc : time
(distance) interval over which
we can reasonably predict the
phase of a wave at another
time (or distance
backward/forward in the
wave), from a knowledge of
the present phase
20
15
10
5
0
-5
-10
-15
-20
0
500
1000
1500
2000
2500
OR: time (distance) shift for an amplitude-splitting
interference experiment, over which we can expect to see
sharp fringes
3000
10
8
What’s similar about
these waves?
What’s different?
6
4
2
0
-2
-4
-6
-8
-10
0
100
200
300
400
500
600
700
800
900
1000
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
0
100
200
300
400
500
600
700
800
900
1000
20
15
10
5
0
-5
-10
-15
-20
0
500
1000
1500
2000
2500
3000
Michelson Interferometer
What can we learn from I(t),
the interferogram?
Frequencies, phases?
Length of a pulse?
Beam diagnostic interferogram for light
emitted by electron beam at Brookhaven
This light has
coherence length
of 1-2 mm
Intensity measurements
Let Io be the intensity in
each arm of the
interferometer.
If t<< tc , we get typical
interference, so at a bright
fringe we should get ____ Io.
At a dark fringe we should get
____ Io.
If we move one arm so that t >> tc, there’s no
interference (no fringes), and we should measure
_____ Io. Why?
Io
Io
I
Single frequency case
Time averaged intensity in one arm
c o
I t ) 
Eo exp  iwt ) Eo exp  iwt )  I o
2
Averaged intensity combined at detector
I  t ,t )  I (t )  c o Eo exp  iwt )  exp  iw  t  t ) ) 
 exp  iwt )  exp  iw  t  t ) ) 
I t )  2 I o 1  cos wt 
Fringes keep going as t increases! So tc is infinite for
single frequency
Many-frequency case
I t ) 

 2I w ) 1  cos wt ) dw

Interferogram of gaussian
pulse.
Many-frequency case
I t ) 

 2I w ) 1  cos wt ) dw
It)

I t )  2 I o 1  Re g t ) 

g t ) 


d w I w ) e  iwt

 I w ) d w

FT 1  I (w )
Io

a dimensionless complex function to
represent the oscillations in I t )

Io 
 I w ) d w

, the intensity in one arm
gt)
Suppose we have a short
pulse, and put a thick piece of
glass in the beam before the
interferometer. The ___
a) wiggles shift
b) wiggles narrow
c) envelope shifts
d) envelope broadens
e) pattern stays the same
Why a long, dispersed pulse will have the
same I(t) as the original short one.
print transparencies
Suppose we put a thick piece
of absorbing colored glass
that absorbs the outer parts of
the spectrum The ___
a) envelope shifts
b) envelope narrows
c) envelope broadens
d) pattern stays the same
Suppose we put the thick
piece of glass in one arm of
the interferometer. What will
happen?
I  t ,t )  c o Eo  d w exp  iwt )  exp  iw  t  t )  i w ) ) 
 exp  iwt )  exp  iw  t  t )  i w ) ) 
This is a different theory from what we’re developing today
Summary
What can we learn about a beam of light from Michaelson
interferometry?
Only things related to the power spectrum! No phase
info.
I t ) 

 2 I w ) 1  cos wt ) dw  2 I

tc 
o
1  Re g t ) 

 g t )dt

1
tc 
For estimates use this!
Dw
We could also measure I w ) with a grating and
detector, and get all the info from that.
If we FT-1 E(w), we get E(t)
If we FT-1 I(w), we get …..
g t ) 
FT
1
I (w )
Io
… g (t), something that gives us the coherence
time of the beam E(t)!
E t )
I t )
FT of
I w )
g t )
I t )
Suppose with filters we
take sunlight and form
I(w) as a rectangular
function centered at wo.
The form of the wiggles
gt) of the interferogram
will be _____
a) sinc
b) gaussian
c) rectangular
I
w
If the width of the
rectangle is wo /10
The coherence time will
be about
a)
b)
c)
d)
I
10 wo
1 /10wo)
10 /wo
100 wo
How many oscillations will gt) make
before it dies down to about ½ or
so of its peak amplitude?
w
wo
20
15
10
5
0
-5
-10
-15
-20
0
500
1000
1500
2000
2500
3000
time t (fs)
E(t) is shown with time increments of femtoseconds (10-15 sec).
The approx. frequency w=2p/T of the light is ______x1012 rad/sec
a) 5
b) 15
c) 30 fs
20
15
10
5
0
-5
-10
-15
-20
0
500
1000
1500
2000
2500
3000
t (fs) t (fs)
time
How many typical periods does it take for this light to get out of phase with
previous part of the beam?
20
15
10
5
0
-5
-10
-15
-20
0
500
1000
1500
2000
2500
3000
time t (fs)
Sketch what the interferogram I(t) would look like, in femtoseconds of delay t.
Mark the coherence time and the average period of light.
Actual unnormalized interferogram shape (half of it).
We know I(w) is “boxy” because of the ringing in gt)!
250
200
150
100
50
0
0
500
1000
1500
2000
delay t (fs)
2500
3000