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 Birefringence
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.
BirefringenceBirefringence
linear anisotropic media:
12n [2] ED [3]and
as n depends on the direction, is a tensor
j
jiji ED
jiij principal axes coordinate system:
off-diagonal elements vanish,D is parallel to E
xx ED 11 yy ED 22 zz ED 33
[4]
inverting [4] yields:
DE 1defining
1
in the pricipal coordinate system is diagonal with principal values:
refraction of a wave has to fulfill the phase-matching condition (modified Snell's Law):
sinsin 1 nnair
two solutions do this:
• ordinary wave:
0011 sinsin nn
• extraordinary wave:
eenn sinsin 11
Birefringencedouble refraction
Birefringencedouble refraction
How to build a waveplate:
input light with polarizations along extraordinary and ordinary axis, propagating along the third pricipal axis of the crystalandchoose thickness of crystal according to wavelenght of light
Phase delay difference: Lnn oe 2
Birefringenceuniaxial crystals and waveplates
Birefringenceuniaxial crystals and waveplates
Friedrich Carl Alwin Pockels (1865 - 1913)
Ph.D. from Goettingen University in 1888
1900 - 1913 Prof. of theoretical physics in Heidelberg
for certain materials n is a function of E,as the variation is only slightly we can Taylor-expand n(E):
...2
1 221 EaEanEn
linear electro-optic effect (Pockels effect, 1893):
n for E=106 V/m : 10-6 to 10-2 (crystals)10-10 to 10-7 (liquids)
Pockels effect:
typical values for r: 10-12 to 10-10 m/V
n for E=106 V/m : 10-6 to 10-4 (crystals)
Kerr vs PockelsKerr vs Pockels
Electro-Optic Effecttheory galore
Electro-Optic Effecttheory galore
20 EsErE from simple picture
[9]
to serious theory:
kl
lkijklk
kijkijij lkjiEEsErE 3,2,1,,,,0 [10]
0
Ek
ijijk Er
0
2
2
1
E
lk
ijijkl EEs
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.
diagonal matrix with elements 1/ni
2
Pockels Effectdoing the math
Pockels Effectdoing the math
How to find the new refractive indices:
• Find the principal axes and principal refractive indices for E=0
• Find the rijk from the crystal structure• Determine the impermeability tensor using:
k
kijkijij ErE 0
• Write the equation for the modified index ellipsoid:
ij
jiij xxE 1)(
• 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)
Pockels Effectwhat it does to light
Pockels Effectwhat it does to light
Phase retardiation (E) of light after passing through a Pockels Cell of lenght L:
[11] LEnEnE ba 2
EnrnEn 3
2
1 [12]
with
this is
ELnrnrLnnE bbaaba332
2
12
[13]
withd
VE
the retardiation is finally:
V
V 0
Lnn ba 2
0
33bbaa nrnrL
dV
a Voltage applied between two surfaces of the crystal
Pockels Cell can be used as dynamic wave retardersInput light is vertical, linear polarizedwith rising electric field (applied Voltage) the transmitted light goes through• elliptical polarization• circular polarization @ V/2 (U /2)• elliptical polarization (90°)• linear polarization (90°) @ V
V
V 0
Pockels CellsPhase Modulation
Pockels CellsPhase Modulation
Phase modulation leads to frequency modulation
dt
tdtf2
definition of frequency:
[15]
with a phase modulation
tmt sin
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
dt
td
dt
tdtf
2
Pockels CellsAmplitude Modulation
Pockels CellsAmplitude Modulation
• Polarizerguarantees, that
incident beam is polarizd at 45° to the pricipal axes• Electro-Optic Crystal
acts as a variable waveplate
• Analysertransmits only the component that has been rotated-> sin2 transmittance characteristic
Pockels Cellsthe specs
Pockels Cellsthe specs
preferred crystals:
• LiNbO3
• LiTaO3
• KDP (KH2PO4)
• KD*P (KD2PO4)
• ADP (NH4H2PO4)
• BBO (Beta-BaB2O4)
longitudinal cells
• Half-wave VoltageO(100 V) for transversal cellsO(1 kV) for longitudinal cells
