1. 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

2. linear anisotropic media: [2][3]and as n depends on the direction,  is a tensor principal axes coordinate system: off-diagonal elements vanish, D is parallel to E [4] inverting [4] yields: defining in the pricipal coordinate system  is diagonal with principal values: [5] Birefringence optically isotrop crystal (cubic symmetry) constant phase delay uniaxial crystal (e.g. quartz, calcite, MgF 2 ) Birefringence extraordinary / optic axis

3. the index ellipsoid: is in the principal coordinate system: a useful geometric representation is: [6] [7] uniaxial crystals (n 1 =n 2  n 3 ) : [8] Birefringence the index ellipsoid

4. refraction of a wave has to fulfill the phase-matching condition (modified Snell's Law): two solutions do this: ordinary wave: extraordinary wave: Birefringence double refraction

5. 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: Birefringence uniaxial crystals and waveplates

6. 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): linear electro-optic effect (Pockels effect, 1893): quadratic electro-optic effect (Kerr effect, 1875): Electro-Optic Effect

7. the electric impermeability  (E):...explains the choice of r and s. Kerr effect: typical values for s:10 -18 to 10 -14 m 2 /V 2  n for E=10 6 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=10 6 V/m :10 -6 to 10 -4 (crystals) Kerr vs Pockels

8. Electro-Optic Effect theory galore from simple picture [9] to serious theory: [10] Symmetry arguments (  ij =  ji and invariance to order of differentiation) reduce the number of independet electro-optic coefficents to: 6x3 for r ijk 6x6 for s ijkl 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/n i 2

9. Pockels Effect doing the math How to find the new refractive indices: Find the principal axes and principal refractive indices for E=0 Find the r ijk 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 n i (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)

10. Pockels Effect what it does to light Phase retardiation  (E) of light after passing through a Pockels Cell of lenght L: [11] [12] with this is [13] with the retardiation is finally: a Voltage applied between two surfaces of the crystal [14]

11. Longitudinal Pockels Cell (d=L) V  scales linearly with large apertures possible Transverse Pockels Cell V  scales linearly with aperture size restricted Pockels Cells building a pockels cell Construction from Linos Coorp.

12. Pockels Cells Dynamic Wave Retarders / Phase 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 @ V  /2 (U  /2 ) elliptical polarization (90°) linear polarization (90°) @ V 

13. Pockels Cells Phase Modulation Phase modulation leads to frequency modulation definition of frequency: [15] 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 

14. Pockels Cells Amplitude 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 -> sin 2 transmittance characteristic

15. Pockels Cells the specs preferred crystals: LiNbO 3 LiTaO 3 KDP (KH 2 PO 4 ) KD*P (KD 2 PO 4 ) ADP (NH 4 H 2 PO 4 ) BBO (Beta-BaB 2 O 4 ) longitudinal cells 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)

16. Pockels Cells temperature "stabilization" an attempt to compensate thermal birefringence

17. Electro Optic Devices

18. Liquid Crystals

19. Optical activity Faraday Effect

20. Photorefractive Materials

21. Acousto Optic