Coherence and Interference

1
Coherence and Interference
Prof. Rick Trebino Georgia Tech

• Coherence
• Temporal coherence
• Spatial coherence
• Interference
• Parallel polarizations
• interfere perpendicular
• polarizations don't.
• The Michelson Interferometer
• Fringes in delay
• Measure of Temporal Coherence
• The Fourier Transform Spectrometer
• The Misaligned Michelson Interferometer
• Fringes in position

Opals use interference between tiny structures to
yield bright colors.
2
The Temporal Coherence Time and the Spatial
Coherence Length

• The temporal coherence time is the time the
  wave-fronts remain equally spaced. That is, the
  field remains sinusoidal with one wavelength

Temporal Coherence Time, tc
The spatial coherence length is the distance over
which the beam wave-fronts remain flat
Since there are two transverse dimensions, we can
define a coherence area.
Spatial Coherence Length
3
Spatial and Temporal Coherence
Spatial and Temporal Coherence Temporal Coherenc
e Spatial Incoherence Spatial
Coherence Temporal Incoherence Spatial
and Temporal Incoherence

• Beams can be coherent or only partially coherent
  (indeed, even incoherent)in both space and time.
  

4
The coherence time is the reciprocal of the
bandwidth.

• The coherence time is given by
• where Dn is the light bandwidth (the width of the
  spectrum).
• Sunlight is temporally very incoherent because
  its bandwidth is
• very large (the entire visible spectrum).
• Lasers can have coherence times as long as about
  a second,
• which is amazing that's gt1014 cycles!

5
The spatial coherence depends on the emitter size
and its distance away.

• The van Cittert-Zernike Theorem states that the
  spatial
• coherence area Ac is given by
• where d is the diameter of the light source and D
  is the distance away.
• Basically, wave-fronts smooth
• out as they propagate away
• from the source.
• Starlight is spatially very coherent because
  stars are very far away.

6
Irradiance of a sum of two waves
Different polarizations
Same polarizations
Same colors
Different colors
Interference only occurs when the waves have the
same color and polarization.
7
The irradiance when combining a beam with a
delayed replica of itself has fringes.
Okay, the irradiance is given by
Suppose the two beams are E0 exp(iwt) and E0
expiw(t-t), that is, a beam and itself delayed
by some time t
Fringes (in delay)
I
Bright fringe
Dark fringe
t
8
Varying the delay on purpose
Simply moving a mirror can vary the delay of a
beam by many wavelengths.
Moving a mirror backward by a distance L yields a
delay of
Do not forget the factor of 2! Light must travel
the extra distance to the mirrorand back!
Since light travels 300 µm per ps, 300 µm of
mirror displacement yields a delay of 2 ps. Such
delays can come about naturally, too.
9
We can also vary the delay using a mirror pair
or corner cube.
Input beam
E(t)
Mirror pairs involve two reflections and displace
the return beam in space But out-of-plane tilt
yields a nonparallel return beam.
Mirrors
Output beam
E(tt)
Translation stage
Corner cubes involve three reflections and also
displace the return beam in space. Even better,
they always yield a parallel return beam
Hollow corner cubes avoid propagation through
glass.
10
The Michelson Interferometer
Input beam

• The Michelson Interferometer splits a beam into
  two and then recombines them at the same beam
  splitter.
• Suppose the input beam is a plane wave

L2
Output beam
Mirror
L1
Beam- splitter
Delay
Mirror
Fringes (in delay)
I
Bright fringe
Dark fringe
and DL 2(L2 L1)
DL 2(L2 L1)
11
The Michelson Interferometer
Input beam
L2
Output beam
Mirror

• The most obvious application of the Michelson
  Interferometer is to measure the wavelength of
  monochromatic light.

L1
Beam- splitter
Delay
Mirror
12
Huge Michelson Interferometers may someday detect
gravity waves.
Gravity waves (emitted by all massive objects)
ever so slightly warp space-time. Relativity
predicts them, but theyve never been
detected. Supernovae and colliding black holes
emit gravity waves that may be detectable.
Gravity waves are quadrupole waves, which
stretch space in one direction and shrink it in
another. They should cause one arm of a Michelson
interferometer to stretch and the other to shrink.
L2
Mirror
L1
Beam- splitter
L1 and L2 4 km!
Mirror
Unfortunately, the relative distance (L1-L2
10-16 cm) is less than the width of a nucleus!
So such measurements are very very difficult!
13
The LIGO project
The building containing an arm
CalTech LIGO
A small fraction of one arm of the CalTech LIGO
interferometer
Hanford LIGO
The control center
14
The LIGO folks think big
The longer the interferometer arms, the better
the sensitivity.
So put one in space, of course.
15
Interference is easy when the light wave is a
monochromatic plane wave. What if its not?
For perfect sine waves, the two beams are either
in phase or theyre not. What about a beam with a
short coherence time?
The beams could be in phase some of the time and
out of phase at other times, varying
rapidly. Remember that most optical measurements
take a long time, so these variations will get
averaged.
16
Adding a non-monochro-matic wave to a delayed
replica of itself
17
The Michelson Interferometer is a Fourier
Transform Spectrometer

• Suppose the input beam is not monochromatic
• (but is perfectly spatially coherent)
• Þ Iout 2I c e ReE(t2L1 /c)
  E(t2L2 /c)
• Now, Iout will vary rapidly in time, and most
  detectors will simply integrate over a relatively
  long time, T

Changing variables t' t 2L1 /c and letting
t 2(L2 - L1)/c and T
The Field Autocorrelation!
Recall that the Fourier Transform of the Field
Autocorrelation is the spectrum!!
18
Fourier Transform Spectrometer Interferogram
A Fourier Transform Spectrometer's detected light
energy vs. delay is called an interferogram.
The Michelson interferometer outputthe
interferogramFourier transforms to the spectrum.
The spectral phase plays no role! (The temporal
phase does, however.)
19
Fourier Transform Spectrometer Data
Actual interferogram from a Fourier Transform
Spectrometer
Fourier Transform Spectrometers are most commonly
used in the infrared where the fringes in delay
are most easily generated. As a result, they are
often called FTIR's.
20
Fourier Transform Spectrometers
Maximum path difference 1 m Minimum resolution
0.005 /cm Spectral range 2.2 to 18 mm Accuracy
10-3 /cm to 10-4 /cm Dynamic range 19 bits (5 x
105)
Fourier-transform spectrometers are now available
for wave-lengths even in the UV! Strangely,
theyre still called FTIRs.
A compact commercial FT spectrometer from Nicolet
21
Technical point about Michelson interferometers
the compensator plate
Input beam
Beam- splitter
Output beam
Mirror
If reflection occurs off the front surface of
beam splitter, the transmitted beam passes
through beam splitter three times the reflected
beam passes through only once.
Mirror
22
Crossed Beams
x
q
z
Cross term is proportional to
Fringe spacing
23
Irradiance vs. position for crossed beams
Fringes occur where the beams overlap in space
and time.
24
Big angle small fringes.Small angle big
fringes.
Large angle
The fringe spacing, L
As the angle decreases to zero, the fringes
become larger and larger, until finally, at q
0, the intensity pattern becomes constant.
Small angle
25
You can't see the spatial fringes unlessthe beam
angle is very small!

• The fringe spacing is
• L 0.1 mm is about the minimum fringe spacing
  you can see

26
Spatial fringes and spatial coherence
Suppose that a beam is temporally, but not
spatially, coherent.
Interference is incoherent (no fringes) far off
the axis, where very different regions of the
wave interfere.
Interference is coherent (sharp fringes) along
the center line, where same regions of the wave
interfere.
27
Fresnel's Biprism

• A prism with an apex angle of about 179 refracts
  the left half of the beam to the right and the
  right half of the beam to the left.

Fringe pattern observed by interfering two beams
created by Fresnel's biprism
28
The MichelsonInterferometerand Spatial Fringes
Fringes

• 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.
• If the input beam is a plane wave, the cross term
  becomes

Crossing beams maps delay onto position. And
moving a mirror shifts the fringes.
29
Michelson-Morley experiment
19th-century physicists thought that light was a
vibration of a medium, like sound. So they
postulated the existence of a medium whose
vibrations were light aether.
Parallel and anti-parallel propagation
Michelson and Morley realized that the earth
could not always be stationary with respect to
the aether. And light would have a different
path length and phase shift depending on whether
it propagated parallel and anti-parallel or
perpendicular to the aether.
Mirror
Perpendicular propagation
Supposed velocity of earth through the aether
30
Michelson-Morley Experiment Details
If light requires a medium, then its velocity
depends on the velocity of the medium. Velocity
vectors add.
Parallel velocities
Anti-parallel velocities
31
Michelson-Morley Experiment Details
In the other arm of the interferometer, the total
velocity must be perpendicular, so light must
propagate at an angle.
Perpendicular velocity after mirror
Perpendicular velocity to mirror
32
Michelson-Morley Experiment Details
Let c be the speed of light, and v be the
velocity of the aether.
The delays for the two arms depend differently
on the velocity of the aether! If v is the
earths velocity around the sun, 3 x 104 m/s, and
L 1 m, then
33
Michelson-Morley Experiment Results
The Michelson interferometer was (and may still
be) the most sensitive measure of distance (or
time) ever invented and shouldve revealed a
fringe shift as it was rotated with respect to
the aether velocity.
Their apparatus
Interference fringes showed no change as the
interferometer was rotated.
Michelson and Morley's results from A. A.
Michelson, Studies in Optics