Modern Optics IVcoherence

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Modern Optics IV-coherence

• Special topics course in IAMS
• Lecture speaker Wang-Yau Cheng
• 2006/4

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Outline

• Wave properties of light
• Polarization of light
• Concept of linewidth
• Coherence of light
• Special issues on quantum optics

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• Spatial coherence and time coherence
• Interferometer and the applications
• Anti-reflection coating
• Correlation function and second order correlation

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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 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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What is the coherence of light?
Classical idea
Coherent time
Dt Dl/c
Coherent length
Phase stability is the key role! !
Linewidth should be narrow!
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Frequency-stabilized lasers is the highest
coherent light source that human being ever use

• A nice tool for detecting the subtle structures
  of our material words

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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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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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• Spatial coherence and time coherence
• Interferometer and the applications
• Anti-reflection coating
• Correlation function and second order correlation

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Mach-Zehnder Interferometer
The Mach-Zehnder interferometer is usually
operated misaligned and with something of
interest in one arm.
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Mach-Zehnder Interferogram
Nothing in either path
Plasma in one path
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The Sagnac Interferometer

• The two beams automatically take the same path
  around the
• interferometer. The paths can differ, however,
  if the device
• is rotating.

The Sagnac interferometer senses rotation.
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Sagnac Interferometer Math

• Suppose that the beam splitter moves by a
  distance, d, in the
• time, T, it takes light to circumnavigate the
  Sagnac interferometer.
• As a result, one beam will travel more, and the
  other less distance.
• If R the interferometer radius, and W its
  angular velocity
• Thus, the Sagnac Interferometer's sensitivity to
  rotation depends on its area. And it need not be
  round!

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Newton's Rings
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Newton's Rings

• Get constructive interference when an integral
  number of half wavelengths occur between the two
  surfaces (that is, when an integral number of
  full wavelengths occur between the path of the
  transmitted beam and the twice reflected beam).

This effect also causes the colors in bubbles and
oil films on puddles.
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Multiple-beam interference The Fabry-Perot
Interferometer or Etalon

• A Fabry-Perot interferometer is a pair of
  parallel surfaces that reflect beams back and
  forth. An etalon is a type of Fabry-Perot etalon,
  and is a piece of glass with parallel sides.
• The transmitted wave is an infinite series of
  multiply reflected beams.

r, t reflection, transmission coefficients from
glass to air
Transmitted wave E0t
Incident wave E0
Reflected wave E0r
n
n 1
n 1
d round-trip phase delay inside medium
Transmitted wave
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The Etalon (cont'd)

• The transmitted wave field is

The transmittance is
Dividing numerator and denominator by
where
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Etalon Transmittance vs. Thickness, Wavelength,
or Angle
Transmission maxima occur when 2pL/l 2mp or

• The transmittance varies significantly with
  thickness or wavelength.We can also vary the
  incidence angle, which also affects d.
• As the reflectance of each surface (r2)
  approaches 1, the widths of the high-transmission
  regions become very narrow.

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The Etalon Free Spectral Range
The Free Spectral Range is the wavelength range
between transmission maxima.
lFSR Free Spectral Range
lFSR
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Etalon Linewidth and Finesse

• The Linewidth dLW is a transmittance peak's
  full-width-half-max (FWHM).
• Setting d equal to dLW/2 should yield T 1/2
• For d ltlt 1, we can make the small argument
  approx
• The Finesse, F, is the ratio of the Free
  Spectral Range and the Linewidth
• Substituting we have

d 2p corresponds to one FSR
The Finesse is the number of wavelengths the
interferometer can resolve.
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Applications of Fabry-Perot interferometers and
etalons
To frequency filter a beam (this is often done
inside a laser). To measure the wavelength or
spectrum of a beam (but you must know it in
advance to within a Free Spectral Range, and you
must scan the thickness of the interferometer and
watch for the transmission vs. thickness). Money
is now coated with interferometric inks to help
foil counterfeiters.Notice the shade of
the20, which is shown from two different
angles.
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• Spatial coherence and time coherence
• Interferometer and the applications
• Anti-reflection coating
• Correlation function and second order correlation

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Anti-reflection Coating
Notice that the center of the round glass plate
looks like its missing. Its not! Theres an
anti-reflection coating there (on both the
front and back of the glass).
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Anti-reflection Coating Math

• Consider a beam incident on a piece of glass (n
  ns) with a layer of material (n nl) if
  thickness, h, on its surface.
• It can be shown that the Reflectance is

Notice that R 0 if
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Multilayer coatings

• Typical laser mirrors and camera
• lenses use many layers.

The reflectance and transmittance can be tailored
to taste!
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• Spatial coherence and time coherence
• Interferometer and the applications
• Anti-reflection coating
• Correlation function and second order correlation