32. Interference and Coherence  Brown University .32. Interference and Coherence Interference...
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32. Interference and CoherenceInterference Only parallel polarizations interfere Interference of a wave with itself
The Michelson Interferometer Fringes in delay Measure of temporal coherence Interference of crossed beams
Coherence Temporal coherence Spatial coherence
Albert Michelson18521931
Orthogonal polarizations dont interfere.
The most general planewave electric field is:
where the amplitude is both complex and a vector:
0, Re exp ( )
E r t E j k r t
0 0 0 0, ,
x y zE E E E
dont forget the complex conjugation!The irradiance is:
* * * *0 0 0 0 0 0 0 02 2
x x y y z zc cI E E E E E E E E
Orthogonal polarizations dont interfere (contd)Because the irradiance is given by:
combining two waves of different polarizations is different from combiningwaves of the same polarization.
* * * *0 0 0 0 0 0 0 02 2
x x y y z zc cI E E E E E E E E
Different polarizations (e.g., x and y):
* *1 1 2 2 1 22
x x y ycI E E E E I I
Same polarizations (e.g. both have x polarization):
**, , 1, 2, 1, 2,2 2 total x total x x x x x
c cI E E E E E E
*1 21 2 Rec EI I I E Therefore: This is what iscalled a cross term.
This cross term can give rise to very dramatic effects.
The irradiance when combining a beam with a delayed replica of itself has fringes.
Suppose the two beams are E0 exp(jt) and E0 exp[jt)]. That is, a monochromatic wave and itself delayed by some time :
02 1 cos[ ] I I
Fringes (as a function of delay)

I
*1 1 2 2Re I I c E E I
*0 0 02 Re exp[ ] exp[ ( )] I I c E j t E j t 20 02 Re exp[ ] I c E j
20 02 cos[ ] I c E
How do we vary the delay of a beam?
Varying the delay on purposeSimply moving a mirror can vary the delay of a beam by many wavelengths.
Since light travels 300 m per ps in air, 300 m of mirror displacement yields a delay of 2 ps. Delays of less than 0.1 fsec (1016 sec) can be generated using this technique.
Moving a mirror backward by a distance L yields a delay of:
2 L /cNote the factor of 2.Light must travel the extra distance to the mirrorand back!
E(t)Output beam
Translation stage
Input beam E(t)Mirror
We can also vary the delay using a mirror pair or a corner cube.
Mirror pairs involve tworeflections and displace the return beam in space:But outofplane tilt yieldsa nonparallel return beam. Translation stage
Inputbeam
E(t)
E(t)
MirrorsOutput beam
Corner cubes involve three reflections and also displace the return beam in space. Even better, they always yield a parallel return beam:
[EdmundScientific]
output
The Michelson Interferometer
beamsplitter
inputbeam
delay
mirror
mirror
*1 2 0 1 0 2Re exp ( ) exp ( ) outI I I c E j t kz kL E j t kz kL
The Michelson Interferometer splits a beam into two and then recombines them at the same beam splitter.
Fringes (as a function of delay):
L = L2 L1
Iout
Suppose the input beam is a plane wave. Then the intensity measured at the output is:
20 0 0 2 1 0 1 2 0 0 2 Re exp ( ) ( /2)since I I I jk L L I I I c E 0 2 1 cos( ) I k L
The Michelson Interferometer  clarification
0 2 1 cos( ) outI I k LIf the path length difference is zero, this becomes: 0 4outI I
Are we getting more photons at the output than we put in? No!
But both beams that reach this point have passed through the 50/50 beam splitter twice, thus reducing their intensity (relative to the intensity of the input beam) by a factor of 4.
Thus: 0 4 inI I
In this expression, the symbol I0 refers to the intensity in either one of the two beams at this location.
beamsplitter
inputbeam
delay
mirror
mirror
output
The MichelsonMorley experiment to measure the aether
Albert Michelson18521931
The most famous failed experiment of all time
The Aether Wind
Nobel Prize, 1907 (first American to win one of the science prizes)
Michelsons laboratory, CaseWestern University, 1887
Interference of crossed beams
k
k
z
xcos sink k z k x
cos sink k z k x
cos sink r k z k x
cos sink r k z k x
*0 0 02 Re exp[ ( )] exp[ ( )] I I c E j t k r E j t k r
*0 0Re exp cos sin exp cos sin E j t kz kx E j t kz kx Cross term is proportional to:
Fringes (as a function of x position)
x
Iout(x) Re exp 2 sin
cos(2 sin )
jkx
kx
Big angle: small fringes.Small angle: big fringes.
2 /(2 sin )/(2sin )
k
The fringe spacing, :
As the angle decreases to zero, the fringes become larger and larger, until finally, at = 0, the intensity pattern becomes constant.
posi
tion
Small angle:
Large angle:
posi
tion
You can't see the spatial fringes unlessthe beam angle is very small!
sin /(2 )
/(2sin )
= 0.1 mm is about the minimum fringe spacing you can see with the naked eye:
0.5 / 200 1/ 400 rad 0.15
m m
The fringe spacing is:
The MichelsonInterferometer(Misaligned)
Suppose we misalign the mirrors, so the beams cross at an angle when they recombine at the beam splitter. And we won't scan the delay, so the lengths are equal.
Beamsplitter
Inputbeam
Mirror
Mirror
z
x
Fringes (in position)
x
Iout(x)
*0 0Re exp cos sin exp cos sin E j t kz kx E j t kz kx If the input beam is a plane wave, the cross term becomes:
Re exp 2 sincos(2 sin )
jkx
kx
Crossing beams maps delay onto position If the path length difference changes, the fringes shift.
The Michelson Interferometer: A question
beamsplitter
inputbeam
delay
mirror
mirror
Lets go back, for now, to the wellaligned Michelson interferometer.
If we move the moveable mirror further and further back, do we continue to see fringes forever?
If not, then how far can we go before they disappear?
output
L = L2 L1= large
02 1 cos( ) outI I k L
The Temporal Coherence Time and the Spatial Coherence LengthThe temporal coherence time is the time over which the beam wavefronts remain equally spaced. Or, equivalently, over which the field remains sinusoidal with a given wavelength:
The spatial coherence length is the distance over which the beam wavefronts remain flat:
Since there are two transverse dimensions, we could talk about two different coherence lengths. Instead, we define a coherence area.
Spatial and Temporal Coherence
Beams can be coherent or only partially coherent (or, even incoherent) in both space and in time.
Spatial and Temporal Coherence:
Temporal Coherence; Spatial Incoherence
Spatial Coherence; Temporal Incoherence
Spatial and Temporal Incoherence
A nearly monochromatic light source has a large coherence time:
If we know there is a maximum here...
...and we wait a time equal to an integer number of periods...
...then we know that we will find another maximum.
time
Efield ampitude
The coherence time of monochromatic light
In the real world: highly stabilized lasers can have coherence times on the order of a few seconds. Thats amazing! More than 1015 cycles!
Thus, an ideal monochromatic light source has an infinite coherence time.
For a perfect cosine, the integer could be as large as you want (up to infinity), and this would still be true.
A polychromatic light source has a smaller coherence time.
Heres an example: an Efield composed of a superposition of several monochromatic waves, each with a slightly different frequency.
Start at this maximum.
Wait N periods, where N = 1, 2, 3, ...
What is the value of the Efield at each successive value of N? Is it still a local maximum?
1 2 3 45
6 7 8 91011121314
15
1617
181920 2122
2324
25
2627
The coherence time of polychromatic light
The coherence time for a given waveform is the average amount of time one has to wait from an arbitrary starting point before coherence is lost.
What determines the coherence time?
More spectral components = more rapid loss of coherence
sum of 3 sine waves
S()
E fi
eld
time
sum of 5 sine waves
S()
E fi
eld
time
sum of 10 sine waves
S()
E fi
eld
time
The coherence time is the reciprocal of the bandwidth.
The coherence time is given by:
where is the light bandwidth (the width of the spectrum).
1/c v
Short optical pulses also have small coherence times, roughly equal to their duration.
12
354 6
Sunlight is temporally very incoherent because its bandwidth is very large (the entire visible spectrum).
Coherence length = (c0/n)c
Why are we interested in coherence time?Because the notion is relevant to measurements that we often do. Let us suppose that we take a wave and interfere it with a copy of itself:
a wave