Birefringence

1. 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.

2. Birefringence [2] and [3] 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: [4] defining principal axes coordinate system: in the pricipal coordinate system  is diagonal with principal values: off-diagonal elements vanish,D is parallel to E [5]

3. Birefringencethe index ellipsoid a useful geometric representation is: the index ellipsoid: [6] is in the principal coordinate system: [7] uniaxial crystals (n1=n2n3): [8]

4. 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:

5. 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:

6. Friedrich Carl Alwin Pockels (1865 - 1913) Ph.D. from Goettingen University in 1888 1900 - 1913 Prof. of theoretical physics in Heidelberg 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):

7. 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 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)

8. Electro-Optic Effecttheory galore from simple picture [9] to serious theory: [10] 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.

9. 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: • 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)

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

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

12. 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 @ V/2 (U  /2) • elliptical polarization (90°) • linear polarization (90°) @ V

13. Pockels CellsPhase 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 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

15. Pockels Cellsthe 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

16. Pockels Cellstemperature "stabilization" an attempt to compensate thermal birefringence

17. Electro Optic Devices

18. Liquid Crystals

19. Faraday Effect Optical activity Faraday Effect

20. Photorefractive Materials

21. Acousto Optic