18' Coherence and Interference

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18. Coherence and Interference

• 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.
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Spatial and Temporal Coherence
Spatial and Temporal Coherence Temporal Coheren
ce 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.
  

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The Temporal Coherence Time and the Spatial
Coherence Length

• The temporal coherence time is the time over
  which the beam wave-fronts 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 wave-fronts remain flat
Since there are two transverse dimensions, we can
define a coherence area.
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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!

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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.

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Orthogonal polarizations dont interfere.

• The most general plane-wave electric field is
• where the amplitude is both complex and a vector
• The irradiance is

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Orthogonal polarizations dont interfere (contd)

• Because the irradiance is given by
• combining two waves of different polarizations is
  different from combining
• waves of the same polarization.

Different polarizations (say x and y) Same
polarizations (say x and x, so we'll omit the
x-subscripts) Therefore
Cross term!
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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
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Varying the delay on purpose
Simply moving mirror can vary the delay of a
beam by many wavelengths.
Input beam
E(t)
Mirror
Output beam
E(tt)
Translation stage
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.
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We can also vary the delay using a mirror pair
or corner cube.
Mirror pairs involve two reflections and displace
the return beam in space But out-of-plane tilt
yields a nonparallel return beam.
Corner cubes involve three reflections and also
displace the return beam in space. Even better,
they always yield a parallel return beam
Edmund Scientific
Hollow corner cubes avoid propagation through
glass.
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The Michelson Interferometer

• 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

Iout
Fringes (in delay)
DL L2 L1
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The Michelson Interferometer is a "Fourier
Transform Spectrometer"

• Suppose the input beam is not monochromatic
• (but still has constant amplitude throughout
  space)
• Þ Iout 2I c e
  ReE(tL1 /c) E(tL2 /c)
• Now, Iout will vary rapidly in time, and most
  detectors will simply
• integrate over a relatively long time, T
• Changing variables t' t L1 /c and letting
  t (L2 - L1)/c and T
• The Fourier Transform of the Field
  Autocorrelation is the spectrum!!

The Field Autocorrelation!
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Adding a wave to a delayed replica of itself

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Fourier Transform Spectrometer Data

• A Fourier Transform Spectrometer's detected light
  energy vs. delay is called an interferogram.

Fourier Transform Spectrometers find use in the
infrared where the fringes in delay are most
easily generated. As a result, they are often
called FTIR's.
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Crossed Beams
Cross term is proportional to
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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
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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

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

Crossing beams maps delay onto position.
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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