Birefringence

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Birefringence
Birefringence
Halite (cubic sodium chloride crystal, optically
isotropic)
Calcite (optically anisotropic)
Calcite crystal with two polarizers at right
angle to one another
Birefringence was first observed in the 17th
century when sailors visiting Iceland brought
back to Europe calcite cristals that showed
double images of objects that were viewed through
them. This effect was explained by Christiaan
Huygens (1629 - 1695, Dutch physicist), as double
refraction of what he called an ordinary and an
extraordinary wave. With the help of a polarizer
we can easily see what these ordinary and
extraordinary beams are. Obviously these beams
have orthogonal polarization, with one
polarization (ordinary beam) passing undeflected
throught the crystal and the other (extraordinary
beam) being twice refracted.
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Birefringence
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and
as n depends on the direction, ? is a tensor
optically isotrop crystal(cubic symmetry)
constant phase delay
uniaxial crystal(e.g. quartz, calcite, MgF2)
Birefringence
extraordinary / optic axis
linear anisotropic media
inverting 4 yields
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defining
principal axes coordinate system
in the pricipal coordinate system ? is diagonal
with principal values
off-diagonal elements vanish,D is parallel to E
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Birefringencethe index ellipsoid
a useful geometric representation is
the index ellipsoid
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is in the principal coordinate system
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uniaxial crystals (n1n2?n3)
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Birefringencedouble refraction
refraction of a wave has to fulfill the
phase-matching condition (modified Snell's Law)
two solutions do this

• ordinary wave

• extraordinary wave

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Birefringenceuniaxial crystals and waveplates
How to build a waveplate
input light with polarizations along
extraordinary and ordinary axis, propagating
along the third pricipal axis of the
crystal and choose thickness of crystal according
to wavelenght of light
Phase delay difference
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Electro-Optic Effect
for certain materials n is a function of E, as
the variation is only slightly we can
Taylor-expand n(E)
linear electro-optic effect (Pockels effect,
1893)
quadratic electro-optic effect (Kerr effect,
1875)
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Kerr vs Pockels
the electric impermeability ?(E)
...explains the choice of r and s.
Kerr effect
Pockels effect
typical values for s 10-18 to 10-14 m2/V2
typical values for r 10-12 to 10-10 m/V
?n for E106 V/m 10-6 to 10-2
(crystals) 10-10 to 10-7 (liquids)
?n for E106 V/m 10-6 to 10-4 (crystals)
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Electro-Optic Effecttheory galore
from simple picture
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to serious theory
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diagonal matrix with elements 1/ni2
Symmetry arguments (? ij ? ji and invariance to
order of differentiation) reduce the number of
independet electro-optic coefficents to
6x3 for rijk
6x6 for sijkl
a renaming scheme allows to reduce the number of
indices to two (see Saleh, Teich "Fundamentals
of Photonics")and crystal symmetry further
reduces the number of independent elements.
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Pockels Effectdoing the math

• How to find the new refractive indices
• Find the principal axes and principal refractive
  indices for E0
• Find the rijk from the crystal structure
• Determine the impermeability tensor using

• Write the equation for the modified index
  ellipsoid

• Determine the principal axes of the new index
  ellipsoid by diagonalizing the matrix ?ij(E) and
  find the corresponding refractive indices ni(E)
• Given the direction of light propagation, find
  the normal modes and their associated refractive
  indices by using the index ellipsoid (as we have
  done before)

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Pockels Effectwhat it does to light
Phase retardiation ?(E) of light after passing
through a Pockels Cell of lenght L
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with
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this is
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with
a Voltage applied between two surfaces of the
crystal
the retardiation is finally
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Pockels Cellsbuilding a pockels cell
Construction

• Longitudinal Pockels Cell (dL)
• V? scales linearly with ?
• large apertures possible

• Transverse Pockels Cell
• V? scales linearly with ?
• aperture size restricted

from Linos Coorp.
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Pockels CellsDynamic Wave Retarders / Phase
Modulation

• Pockels Cell can be used as
• dynamic wave retardersInput light is vertical,
  linear polarized
• with rising electric field (applied Voltage) the
  transmitted light goes through
• elliptical polarization
• circular polarization _at_ V?/2 (U ? /2)
• elliptical polarization (90)
• linear polarization (90) _at_ V?

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Pockels CellsPhase Modulation
Phase modulation leads to frequency modulation
definition of frequency
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with a phase modulation
? frequency modulation at frequency ? with 90
phase lag and peak to peak excursion of 2m?
? Fourier components power exists only at
discrete optical frequencies ??k ?
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Pockels CellsAmplitude Modulation

• Polarizer guarantees, that incident beam is
  polarizd at 45 to the pricipal axes
• Electro-Optic Crystal acts as a variable
  waveplate
• Analyser transmits only the component that has
  been rotated -gt sin2 transmittance
  characteristic

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Pockels Cellsthe specs

• Half-wave Voltage O(100 V) for transversal
  cells O(1 kV) for longitudinal cells
• Extinction ratio up to 11000
• Transmission 90 to 98
• Capacity O(100 pF)
• switching times O(1 µs) (can be as low as
  15ns)

• preferred crystals
• LiNbO3
• LiTaO3
• KDP (KH2PO4)
• KDP (KD2PO4)
• ADP (NH4H2PO4)
• BBO (Beta-BaB2O4)

longitudinal cells
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Pockels Cellstemperature "stabilization"
an attempt to compensate thermal birefringence
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Electro Optic Devices
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Liquid Crystals
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Faraday Effect
Optical activity
Faraday Effect
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Photorefractive Materials
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Acousto Optic