The Temporal Coherence Time and the Spatial Coherence Length

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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
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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.
  

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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).
• 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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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. We also discussed
incoherence, and thats what this lecture is
about!
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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 a 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.
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
Edmund Scientific
Hollow corner cubes avoid propagation through
glass.
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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
Bright fringe
Dark fringe
Iout
where DL 2(L2 L1)
Fringes (in delay)
DL 2(L2 L1)
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Michelson-Morley experiment
19th-century physicists thought that light was a
vibration of a medium, like sound. So they
postulated the existence of 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
phase shift depending on whether it propagated
parallel and anti-parallel or perpendicular to
the aether. They observed no phase
shift. Goodbye aether.
Mirror
Perpendicular propagation
Beam- splitter
Mirror
supposed velocity of earth through the aether
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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!
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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
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The LIGO folks think big
The longer the interferometer arms, the better
the sensitivity.
So put one in space, of course.
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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.
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Adding a non-monochro-matic wave to a delayed
replica of itself
Constructive interference for all times
(coherent) Bright fringe
Delay 0
Destructive interference for all times (coherent)
Dark fringe)
Delay ½ period (ltlt tc)
Incoherent addition No fringes.
Delay gt tc
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Crossed Beams
Cross term is proportional to
Fringe spacing
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Irradiance vs. position for crossed beams
Irradiance fringes occur where the beams overlap
in space and time.
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Big angle small fringes.Small angle big
fringes.
Large angle
The fringe spacing, L
Small angle
As the angle decreases to zero, the fringes
become larger and larger, until finally, at q
0, the intensity pattern becomes constant.
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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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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.
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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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Optical interferometry

• Optical interferometer instrument that
  generates interference fringe patterns resulting
  from optical path differences
• it divides initial beam into two or more parts
  that travel different optical paths and then
  brought together again to produce interference
  pattern
• it is divided into two main classes (depending on
  how initial beam is separated)

a. Wavefront division interferometers portions
of same wavefront of a coherent beam of light are
sampled (e.g. Youngs double slit, Lloyds
mirror, Fresnels biprism) b. Amplitude-division
interferometers uses beam splitter
(semireflecting film, prisms) to divide initial
beam into two parts (amplitude is shared) e.g.
Michelson interferometer uses interference of 2
beams, Fabry-Perot interferometer uses multiple
beams
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Michelson Interferometer

• Beam (1) from extended source S split by beam
  splitter BS (which has a thin, semitransparent
  front surface metallic or dielectric film,
  deposited on glass) ? amplitude splitting
• Reflected beam (2) and transmitted beam (3) have
  equal amplitudes, are then reflected (at normal
  incidence) by mirrors M2 and M1, respectively,
  and their directions are then reversed
• Returning to BS, beam (1) is transmitted and beam
  (3) is reflected by semitransparent film, so that
  they come together again and leave interferometer
  as beam (4)

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Michelson Interferometer

• One of the mirrors has tilting adjustment screws
  that allow the surface of M1 to be made
  perpendicular to M2
• One of the mirrors is movable along direction of
  beam so that difference between optical paths of
  beams (2) (3) can be varied
• Beam (3) traverses BS 3X, beam (2) traverses BS
  1X, thus, a plate C is inserted parallel to BS in
  path of beam (2) to compensate for this
• This will ensure optical paths of two beams can
  be made precisely equal (especially when white
  light is used)

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Michelson Interferometer
Actual interferometer has 2 optical axes
perpendicular to one another

• To derive the optical path difference, we shall
  use an equivalent optical system having single
  optical axis
• We work with virtual images of source S and
  mirror M1 via reflection in BS mirror
• Take S, M1, and beams (1) (3) to be rotated
  anti-clockwise by 90 about point of intersection
  of beams with BS mirror
• New position source plane S, virtual image M1
• Light from point Q on source plane S is
  reflected from both mirrors M2 and M1 (parallel)
  at optical path difference d

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Michelson Interferometer

• Now two reflected beams appear to come from two
  virtual images Q1 and Q2 of object Q
• Separation of images S1 and S2 2 X mirror
  separation ? distance between Q1 and Q2 2d
• Optical path difference between two beams
  emerging from interferometer is
• (angle ? inclination of beams relative to
  optical axis)
• For normal beam, ? 0? and ?p 2d
• If ? m? for constructive interference, the
  beams will interfere constructively again at
  every ?/2 translation of one of the mirrors
• This optical system is now equivalent to the case
  of interference due to a plane parallel air film
  illuminated by extended source
• Virtual fringes of equal inclination is seen by
  looking into BS along ray (4)

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Michelson Interferometer

• Assuming equal amplitudes of the two interfering
  beams, irradiance of fringe system of concentric
  circles is given by
• where phase difference is
• Net optical path difference is
• Relative phase shift ? because beam (2)
  undergoes 2 external reflection but beam (3)
  undergoes only one
• For dark fringes
• or
• If d is such that centre fringe is dark (normal
  rays), then, its order is
• (neighbouring dark fringes decrease in order
  outwards, as cos ? decreases from max value of 1)

(12-1)
(12-2)
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Michelson Interferometer

• In order to express order such that it increases
  in number outwards instead, we introduce p where
• Putting (12-3) into (12-1), we get
• where central fringe is now of order p 0 and
  neighbouring fringes increase in order, outward

(12-3)
(12-4)

• Eqns (12-1) (12-4) indicate
• as d varies, a particular point in fringe
  pattern (? constant) corresponds to gradually
  changing values of order of m or p

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Michelson Interferometer

• From eqn (12-1)
• (differentiate)
• This means that fringes are more widely spaced
  when optical path differences (denominator) are
  small
• If mirror translates ?d, number ?m of fringes
  passing a point near or at centre of pattern is
• ? we can measure ? when ?d is known

(12-5)
E.g. Fringes observed due to monochromatic light
in Michelson interferometer. When movable mirror
translates 0.73 mm, a shift of 300 fringes is
seen. What is the wavelength of the light? What
displacement of the fringe system takes place
when a slide of glass of index 1.51 and 0.0005 mm
thick is inserted in one arm of interferometer?
(Assume light beam to be normal to glass
surface.) Glass inserted, one arm is extended by
path difference of ?d ng t ? nair t, thus,
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Michelson Interferometer (continued)
Fringes of equal thickness
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Michelson Interferometer (continued)
Fringes of equal Inclination
Fringes of equal thickness
(Path differences increases outward from centre)
White light fringes
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Applications of Michelson Interferometer
(12-6)
(12-7)
(12-8)
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Applications of Michelson Interferometer
(determine wavelength difference)

• We see two sets of fringe patterns viewing near
  centre of circular system, i.e., cos ? ? 1
• For a fixed path difference d, m? m?
• When two fringe systems coincide (in-sync),
  resultant pattern appears sharp

• When one fringe system is midway (out-of-sync)
  with other fringe system, resultant pattern
  appears blur or wash-out
• Say at 1st coincidence, the orders (m m) of
  the two systems corresponding to ? and ? are
  related through
• where N is an interger
• if optical path difference here is d1, then from
  (12-8), we have

(12-9)
(12-10)
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Applications of Michelson Interferometer
(determine wavelength difference)

• Now at 2nd coincidence, occurring at optical path
  difference d2, we have
• or
• Subtract (12-10) from (12-11), and writing mirror
  movement as ?d d2 ? d1, we get
• Since ? and ? are very close, wavelength
  difference ?? may then be approximated by

Related to application (2)
(12-11)
(12-12)
(12-13)
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Energy levels for sodium - note Na doublets
588.99 589.59 nm related to application (3)
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Variations on the Michelson Interferometer
Twyman-Green Interferometer
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Twyman-Green interferometerTesting of flat
surfaces
The basic use of a Twyman-Green interferometer is
for measuring surface height variations. The
Twyman-Green and Fizeau give the same
interferograms for testing surface flatness the
main advantages of the Twyman-Green are more
versatility and it is a non-contact test, so
there is less chance of scratching the surface
under test, while the main disadvantage is that
more high-quality optical components are required.
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Twyman-Green interferometer(continued)Testing
spherical surface and prism
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Twyman-Green Interferometer
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Mach-Zehnder Interferometer - variation of
Michelson Interferometer
Parallel beam divided into two at beam splitter
BS Each divided beam totally reflected by mirrors
M1 M2 Then beams are brought together again by
M3 Path lengths of beam (1) and (2) around
rectangular system and through glass of BS and M3
are identical
BS M3 are half-silvered mirror
(semi-transparent) M1 M2 are mirrors
Say we want to measure the geometry of air flow
around an object in a wind tunnel (detected as
local variations of pressure and refractive
index) Put windowed test chamber into path (1)
and another identical chamber in path 2 to
maintain equality of optical paths. The model
streamline flow of air are introduced in test
chamber. Air-flow pattern will be revealed by
fringe pattern.
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Mach-Zehnder interferometerTesting transparent
medium

• Optical path SLn
• L geometrical path through medium
• Optical path difference DS2-S1
• L(n-1)
• Bright fringe L(n-1) Nl
• for N0,1,2,3
• Therefore NL(n-1)/l

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Mach-Zehnder interferometer (continued)
Example
Optical path difference
Llength of medium

• If refractive index varies in y-axis only

• Fringe separation depends on the gradient of
  refractive index n

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Mach-Zehnder interferometer (continued)Reference
fringe

• Interpreting the interferogram can be confusing
  because Df give the same fringe pattern e.g.
  figure (a). IT is ambiguous whether the fringe
  order number N increases or decreases as we
  proceed inward.
• This problem can be resolved by adding a known
  reference fringe pattern (b).
• In (c ) the fringe order N increases upwards
  while that of (d) increases downwards.

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Mach-Zehnder interferometer (continued)Assigning
fringe order number
Without reference fringe. In this case, it is
assumed that the order is known. The background
is assigned N0 and the subsequent bright fringe
N1, etc
With reference fringe. Here the order N increases
upwards. A line parallel to the reference fringe
at the position of interest is drawn. The first
intersection is assigned N1, etc,,,
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