Download Crystal Optics

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.
Birefringence
Birefringence
n2 = 1 + χ = ε
[2]
and
D =ε ⋅E
[3]
as n depends on the direction, ε is a tensor
optically isotrop crystal
(cubic symmetry)
uniaxial crystal
(e.g. quartz, calcite, MgF2)
n x = n y = nz
constant phase delay
n x = n y ≠ nz
Birefringence
extraordinary / optic axis
linear anisotropic media:
Di = ∑ ε ij E j
inverting [4] yields:
[4]
j
defining
ε ij = ε ji
principal axes coordinate system:
off-diagonal elements vanish,
D is parallel to E
Dx = ε11E x
D y = ε 22 E y
E = ε −1D
Dz = ε 33 E z
1
η=
ε
in the pricipal coordinate system η is
diagonal with principal values:
1
1
= 2
ε i ni
[5]
Birefringence
Birefringence
the
theindex
indexellipsoid
ellipsoid
a useful geometric representation is:
the index ellipsoid:
∑η x x
ij i
j
=1
[6]
ij
is in the principal coordinate system:
x12 x22 x32
+ +
=1
n12 n22 n32
[7]
uniaxial crystals (n1=n2≠n3):
1
cos2 (θ ) sin 2 (θ )
=
+
n 2 (θ )
n02
ne2
na = n0
[8]
nb = n (θ )
n (0°) = n0
n (90°) = ne
Birefringence
Birefringence
double
double refraction
refraction
refraction of a wave has to fulfill the
phase-matching condition
(modified Snell's Law):
nair ⋅ sin (θ1 ) = n (θ ) ⋅ sin (θ )
two solutions do this:
• ordinary wave:
n1 ⋅ sin(θ1 ) = n0 ⋅ sin (θ 0 )
• extraordinary wave:
n1 ⋅ sin (θ1 ) = n (θ e ) ⋅ sin (θ e )
Birefringence
Birefringence
uniaxial
uniaxial crystals
crystals and
and waveplates
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: Γ =
2π
(ne − no )L
λ
Electro-Optic
Electro-Optic
Effect
Effect
for certain materials n is a function of E,
as the variation is only slightly we can Taylor-expand n(E):
1
n (E ) = n + a1E + a2 E 2 + ...
2
Friedrich Carl Alwin
Pockels (1865 - 1913)
Ph.D. from Goettingen
University in 1888
1900 - 1913 Prof. of
theoretical physics in
Heidelberg
linear electro-optic effect
(Pockels effect, 1893):
1
n (E ) = n − r ⋅ n 3 E
2
r = −2
a1
n3
quadratic electro-optic effect
(Kerr effect, 1875):
1
n (E ) = n − s ⋅ n 3 E 2
2
s=−
a2
n3
Kerr
Kerr vs
vs Pockels
Pockels
the electric impermeability η(E):
η=
ε0 1
=
ε n2
1
−2  1
 dη 
3
3 2
2
∆η (E ) = 
 ⋅ ∆n =  3  ⋅  − r ⋅ n E − s ⋅ n E  = r ⋅ E + s ⋅ E
2

n   2
 dn 
...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 E=106 V/m : 10-6 to 10-2 (crystals)
10-10 to 10-7 (liquids)
∆n for E=106 V/m : 10-6 to 10-4 (crystals)
Electro-Optic
Electro-Optic Effect
Effect
theory
theorygalore
galore
from simple picture
η (E ) = η (0) + r ⋅ E + s ⋅ E 2
[9]
to serious theory:
ηij (E ) = ηij (0) + ∑ rijk ⋅ Ek + ∑ sijkl ⋅ Ek El
k
diagonal matrix with
elements 1/ni2
i, j, k , l , = 1,2,3
kl
rijk =
∂ηij
∂Ek
E =0
sijkl
2
1 ∂ ηij
=
2 ∂Ek ∂El
E =0
Symmetry arguments (η ij= η 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.
[10]
Pockels
Pockels Effect
Effect
doing
doingthe
themath
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:
ηij (E ) = ηij (0) + ∑ rijk Ek
k
• Write the equation for the modified index ellipsoid:
∑η ( E ) x x
ij
i
j
=1
ij
• Determine the principal axes of the new index
ellipsoid by diagonalizing the matrix ηij(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
Pockels Effect
Effect
what
whatititdoes
doesto
tolight
light
Phase retardiation Γ(E) of light after passing through a Pockels Cell of
lenght L:
Γ (E
with
this is
with
)=
2π
λ
[n a (E ) −
n b (E
)]L
[11]
1
n (E ) = n − r ⋅ n 3 E
2
Γ (E
E=
V
d
)=
2π
λ
[n a
[12]
− n b ]L −
1 2π
2 λ
[r
a
]
n a3 − r b n b3 EL
[13]
a Voltage applied between two surfaces of the crystal
the retardiation is finally:
Γ = Γ0 − π
V
Vπ
Γ0 =
2π
[na − nb ]L
λ
d
λ
Vπ =
L ra na3 − rbnb3
[14]
Pockels
Pockels Cells
Cells
building
building aa pockels
pockels cell
cell
Construction
Longitudinal Pockels Cell (d=L)
•
Vπ =
λ
r ⋅ n3
• Vπ scales linearly with λ
• large apertures possible
Transverse Pockels Cell
•
Vπ =
d λ
L r ⋅ n3
• Vπ scales linearly with λ
• aperture size restricted
from Linos Coorp.
Pockels
Pockels Cells
Cells
Dynamic
DynamicWave
WaveRetarders
Retarders//Phase
PhaseModulation
Modulation
Pockels Cell can be used as
dynamic wave retarders
Input light is vertical, linear polarized
with rising electric field (applied Voltage) the
transmitted light goes through
• elliptical polarization
• circular polarization @ Vπ/2 (U π /2)
• elliptical polarization (90°)
• linear polarization
(90°)
@ Vπ
Γ = Γ0 − π
V
Vπ
Pockels
Pockels Cells
Cells
Phase
PhaseModulation
Modulation
Phase modulation leads to
frequency modulation
definition of frequency:
2π ⋅ f (t ) ≡
dΦ(t )
=ω
dt
[15]
with a phase modulation
2π ⋅ f (t ) ≡
dΦ (t )
dφ (t )
=ω +
dt
dt
φ (t ) = m sin (Ωt )
⇒ frequency modulation at frequency Ω
with 90° phase lag and peak to peak
excursion of 2mΩ
⇒ Fourier components:
power exists only at discrete optical
frequencies ω±k Ω
Pockels
Pockels Cells
Cells
Amplitude
AmplitudeModulation
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
-> sin2 transmittance
characteristic
Pockels
Pockels Cells
Cells
the
thespecs
specs
• Half-wave Voltage
O(100 V) for transversal cells
O(1 kV) for longitudinal cells
• Extinction ratio
up to 1:1000
• 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)
• KD*P (KD2PO4)
• ADP (NH4H2PO4)
• BBO (Beta-BaB2O4)
longitudinal cells
Pockels
Pockels Cells
Cells
temperature
temperature"stabilization"
"stabilization"
an attempt to compensate thermal birefringence
Electro
Electro Optic
Optic
Devices
Devices
Liquid
Liquid Crystals
Crystals
Faraday
Faraday Effect
Effect
Optical activity
Faraday Effect
Photorefractive
Photorefractive Materials
Materials
Acousto
Acousto Optic
Optic