BirefringenceBirefringence

Halite (cubic sodium chloride crystal, optically isotropic)

Calcite (optically anisotropic)

Calcite crystal with two polarizersat 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:

εχ =+=12n [2] ED ⋅= ε [3]and

as n depends on the direction, ε is a tensor

∑=j

jiji ED ε

jiij εε =

principal axes coordinate system:

off-diagonal elements vanish,D is parallel to E

xx ED 11ε= yy ED 22ε= zz ED 33ε=

[4]inverting [4] yields:

DE 1−= εdefining

εη

1=

in the pricipal coordinate system η isdiagonal with principal values:

2

11

ii n=

ε [5]

BirefringenceBirefringence

optically isotrop crystal(cubic symmetry) zyx nnn == constant phase delay

uniaxial crystal(e.g. quartz, calcite, MgF2)

zyx nnn ≠= Birefringence

extraordinary / optic axis

the index ellipsoid:

∑ =ij

jiij xx 1η

is in the principal coordinate system:

a useful geometric representation is:

[6]

[7]123

23

22

22

21

21 =++

nx

nx

nx

uniaxial crystals (n1=n2≠n3):

( )( ) ( )

2

2

20

2

2

sincos1

ennnθθ

θ+= [8]

0nna = ( )θnnb =

( ) 00 nn =° ( ) enn =°90

Birefringencethe index ellipsoid

Birefringencethe index ellipsoid

refraction of a wave has to fulfill the phase-matching condition(modified Snell's Law):

( ) ( ) ( )θθθ sinsin 1 ⋅=⋅ nnair

two solutions do this:

• ordinary wave:( ) ( )0011 sinsin θθ ⋅=⋅ nn

• extraordinary wave:( ) ( ) ( )eenn θθθ 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 −=Γλπ2

Birefringenceuniaxial crystals and waveplates

Birefringenceuniaxial crystals and waveplates

Friedrich Carl Alwin Pockels (1865 - 1913)

Ph.D. from GoettingenUniversity 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):

( ) ...21 2

21 +++= EaEanEn

linear electro-optic effect (Pockels effect, 1893):

( ) EnrnEn 3

21

⋅−= 312

na

r −=

quadratic electro-optic effect (Kerr effect, 1875):

( ) 23

21

EnsnEn ⋅−= 32

na

s −=

Electro-Optic Effect

Electro-Optic Effect

the electric impermeability η(E):

20 1

n==

εε

η

( ) 22333 2

1212

EsErEnsEnrn

ndnd

E ⋅+⋅=

⋅−⋅−⋅

−

=∆⋅

=∆

ηη

...explains the choice of r and s.

Kerr effect:

typical values for s: 10-18 to 10-14 m2/V2

∆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 ⋅+⋅+= ηη

from simple picture

[9]

to serious theory:

( ) ( ) ∑∑ =⋅+⋅+=kl

lkijklk

kijkijij lkjiEEsErE 3,2,1,,,,0ηη [10]

0=∂∂

= Ek

ijijk E

rη

0

2

21

=∂∂∂

= Elk

ijijkl EE

sη

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.

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 0ηη

• Write the equation for the modified index ellipsoid:

∑ =ij

jiij xxE 1)(η

• 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 thenormal 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 Γ(E) of light after passing through a Pockels Cell of lenght L:

[11]( ) ( ) ( )[ ]LEnEnE ba −=Γλπ2

( ) EnrnEn 3

21

⋅−= [12]with

this is( ) [ ] [ ]ELnrnrLnnE bbaaba

332212

−−−=Γλπ

λπ

[13]

withdV

E =

the retardiation is finally:π

πVV

−Γ=Γ 0

[ ]Lnn ba −=Γλπ2

0

33bbaa nrnrL

dV

−=

λπ

a Voltage applied between two surfaces of the crystal

[14]

Longitudinal Pockels Cell (d=L)

•

• Vπ scales linearly with λ

• large apertures possible

Transverse Pockels Cell

•

• Vπ scales linearly with λ

• aperture size restricted

Pockels Cellsbuilding a pockels cellPockels Cells

building a pockels cellConstruction

from Linos Coorp.

3nrV

⋅=

λπ

3nrLd

V⋅

=λ

π

Pockels CellsDynamic Wave Retarders / Phase Modulation

Pockels CellsDynamic Wave Retarders / Phase Modulation

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 @ Vπ/2 (U π /2)• elliptical polarization (90°)• linear polarization (90°) @ Vπ

π

πVV

−Γ=Γ 0

Pockels CellsPhase Modulation

Pockels CellsPhase Modulation

Phase modulation leads to frequency modulation

( ) ( )ωπ =

Φ≡⋅

dttd

tf2

definition of frequency:

[15]

with a phase modulation

( ) ( )tmt Ω= sinφ

⇒ frequency modulation at frequency Ωwith 90° phase lag and peak to peak excursion of 2mΩ

⇒ Fourier components: power exists only at discrete optical frequencies ω±k Ω

( ) ( ) ( )dt

tddt

tdtf

φωπ +=

Φ≡⋅2

Pockels CellsAmplitude Modulation

Pockels CellsAmplitude Modulation

• Polarizerguarantees, that incident beam is polarizd at 45° to the pricipal axes

• Electro-Optic Crystalacts 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

• Extinction ratioup to 1:1000

• Transmission90 to 98 %

• CapacityO(100 pF)

• switching timesO(1 µs) (can be as low as 15ns)

Pockels Cellstemperature "stabilization"

Pockels Cellstemperature "stabilization"

an attempt to compensate thermal birefringence

Electro Optic Devices

Electro Optic Devices

Liquid CrystalsLiquid Crystals

Optical activity

Faraday Effect

Faraday EffectFaraday Effect

Photorefractive MaterialsPhotorefractive Materials

Acousto OpticAcousto Optic