Innovation and Development of Study Field. nano.tul.cz

Transcription

1 Innovation and Development of Study Field Nanomaterials at the Technical University of Liberec nano.tul.cz These materials have been developed within the ESF project: Innovation and development of study field Nanomaterials at the Technical University of Liberec

2 Miroslav Šulc

3 Electro-optical effects Applied electric field E can changes impermitivity tensor η and index ellipsoid of investigated materials. The relation between impermitivity and refractive index is described for isotropic materials by equation If the intensity of electric field is small, it is possible to express this tensor η by the first members of Taylor expansion Linear Pockels coefficients r ijk are the first derivation of impermitivity tensor for zero electric field 3

4 Quadratic Kerr coefficients s ijkl the second derivation of impermitivity tensor for zero electric field The changes of refractive index due the applied electric field E on electro-optical crystal is from this concludes 4

5 Pockels Electro-Optic Effect Friedrich Carl Alwin Pockels ( ) Ph.D. from Goettingen University in Prof. of theoretical physics in Heidelberg linear electro-optic effect (Pockels effect, 1893): n E 1 3 a n r n E r 1 3 n quadratic electro-optic effect (Kerr effect, 1875): 1 3 ne n s n E a s n 3 5

6 Kerr vs Pockels the electric impermeability (E): 0 1 n d dn 3 n E n r n E s n E r E s E 1...explains the choice of r and s. Kerr effect: typical values for s: to m /V n for E=10 6 V/m : 10-6 to 10 - (crystals) to 10-7 (liquids) Pockels effect: typical values for r: 10-1 to m/v n for E=10 6 V/m : 10-6 to 10-4 (crystals) 6

7 The Pockels effect can be observed only for materials without center of symmetry. Most common crystals used as Pockels cells are LiNbO 3, LiTaO 3, ADP (NH 4 H PO 4 ), KDP (KH PO 4 ). Typical value of Pockel coefficients are /V. This means, for E=10 6 V/m, the typical refractive index change is very small and it is of order from 10-6 to For materials with center of symmetry, as gases, liquids, and certain crystals, n(e) must be an even function and Pockels coefficient must be equal to zero. All materials display the Kerr effect, with varying magnitudes, but it is generally much weaker than the Pockels effect, so only for centrosymmetric materials quadratic effect is not negligible. Typical values of Kerr coefficients are m /V in crystals; m /V in liquids. So for E=10 6 V/m index change is very small about in crystals ; ~ in liquids. 7

8 ELECTRO-OPTICAL COEFFICIENTS MEASUREMENT Measurement of phase change in Mach-Zehnder interferometer arrangment Light polarization and direction of electric field determine measured coefficients Important to separate Pockels and inverse piezoelectic effect r 1 y U out 3 U U n L A p p 8

9 Compensation of piezoelectric induced displacement The second crystal, made from the same material as sample crystal, but with another length is used with mirror placed on the top of this crystal If there is applied the same electric field on investigated sample (light is passing through it) and on compensating crystal (light is reflected from this one), we can fully compensate piezoelectric effect This compensation can be made both in Michelson and Mach- Zehnder interferometer arrangement r i r n 1 n 3 d i 9

10 EO Coefficients [pm/v] elektro-optické koeficienty [pm/v] Electro-optical effects Measurement of electro-optical coefficients of crystal LiNbO 3 in wide temperature range Undoped crystal LiNbO 3, of congruent composition (48,5%Li, 51,5% Nb) Bulk shape 36x3x mm 3 This crystal was investigated in transversal configuration. Applied electric field E=(0,0,E) was along optical axis z Point grup, symmetry 3m. Only coefficient r 13, r 33, r, r 15 Correction for piezoelectric effect (d 31 = 0, C/N) was take in account ( 0, pm/v). Resulting values are r 13 = 9,7±0, pm/v and r 33 = 30,4±0,4 pm/v r , r 13 frekvence [khz] 10

11 r 13 [pm/v] r 33 [pm/v] teplota [K] temperature [K] teplota [K] temperature [K] The temperature characteristic seems to be constant for both coefficients. It seems that electro-optical coefficient r 33 became lower with decrease of the temperature, coefficients r 13 is constant 11

12 Electro-Optic Effect mathematics description from simple picture E 0 r E s E to serious theory: ij ij 0 rijk Ek E s E E i, j, k, l, 1,, 3 diagonal matrix with elements 1/n i k r ijk kl [9] ijkl ij E k k E0 l s ijkl 1 E k ij E l E0 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. 1

13 How to find the new refractive indices: Pockels Effect doing the math Find the principal axes and principal refractive indices for E=0 Find the r ijk from the crystal structure Determine the impermeability tensor using: ij E ij0 k r Write the equation for the modified index ellipsoid: ij ( E) x x 1 ij i j 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) ijk E k 13

14 Pockels Effect what it does to light Phase retardiation (E) of light after passing through a Pockels Cell of lenght L: E n ae nbel [11] with this is n E 1 n r n 3 E 3 3 E n n L r n r n EL a b 1 a a b b [1] [13] with V E d the retardiation is finally: a Voltage applied between two surfaces of the crystal V 0 V V 0 n d L r n a a 3 a n b b r n L 3 b [14] 14

15 Pockels Cells building a pockels cell Construction Longitudinal Pockels Cell (d=l) V 3 r n V scales linearly with large apertures possible Transverse Pockels Cell V d L r n 3 V scales linearly with aperture size restricted 15 from Linos Coorp.

16 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 V / (U / ) elliptical polarization (90 ) linear polarization (90 V V 0 V 16

17 Pockels Cells Phase Modulation Phase modulation leads to frequency modulation definition of frequency: t d f t dt with a phase modulation f t d dt t t msint d t dt [15] frequency modulation at frequency with 90 phase lag and peak to peak excursion of m Fourier components: power exists only at discrete optical frequencies k 17

18 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 transmittance characteristic 18

19 Pockels Cells the 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: LiNbO 3 LiTaO 3 KDP (KH PO 4 ) KD*P (KD PO 4 ) ADP (NH 4 H PO 4 ) BBO (Beta-BaB O 4 ) longitudinal cells 19

20 Pockels Cells temperature "stabilization" an attempt to compensate thermal birefringence 0

21 Electro Optic Devices 1

22 Liquid Crystals

23 References - Further reading SALEH, B. E. A., TEICH, M.C. Fundamentals of Photonics, John Wiley & Sons, 1991 Šulc M., chapter Interferometer Based Methods for Research of Piezoelectric Materials in book Interferometry, Principles and Applications, Nova Publishers, ISBN: On the Web 3

24 THANK YOU FOR YOUR ATTENTION 4