1. 1 Compensation of Thermally Induced Birefringence in Active Medium Made of Polycrystalline Ceramics. Efim A. Khazanov Institute of Applied Physics, Nizhny Novgorod, Russia Mikhail A. Kagan Pennsylvania State University, University Park, PA, USA

2. 2 Outline.  Introduction  Polycrystalline ceramics vs glass and a single crystal  Thermally induced birefringence in polycrystalline ceramics  Ceramics description  Depolarization in single crystal and polycrystalline ceramics  Birefringence compensation in polycrystalline ceramics  Conclusion

3. 3 Introduction. Structure of polycrystalline ceramics.

4. 4 Polycrystalline ceramics vs glass and single crystal. Properties.

5. 5 Depolarization in single crystal and ceramics. Thermo-induced birefringence. angle of declination of eigen polarizations  r,  phases delay between eigen polarizations  r,  L g  Grain Jones matrix A g =A g (r,  g  g  g, L g    r e2e2 e 1 x y z k y x c, z x z   x, a y, b x y z, c   x y x y z   r  crystal/grain orientation Euler angles ( 

6. 6 Jones matrix of whole element (k realization) A(r, ,k  A 1 (r,  1  1  1,L 1  A 2 (r,  2  2  2,L 2  A N (r,  N  N  N,L N  Local depolarization  (r, ,k)=  E out (r, ,k) /E in (r,  )   Average (over realizations) local depolarization Integrated depolarization:  k  and its deviation : and   Depolarization in single crystal and ceramics. Local and integrated. ….. 1 2 3 4 ….. N E in (r,  ) E out (r,  )

7. 7 Mathematical statement of the problem. Assumptions. Nfixed 1.The number of grains, N within a beam’s path is fixed. orientationdoes not dependvicinal grains 2.The orientation of crystallographic axes in a certain grain does not depend on vicinal grains. fL g  uniform angular partgaussianL g 3.The distribution function f(L g  for a single grain is uniform with respect to the angular part and the gaussian with respect to L g

8. 8 Ceramics description. Jones matrixes  Quaternion formalism. Media without absorption is described by a unitary matrix U, That could be presented as

9. 9 Ceramics description. Quaternion properties. (takes place for every imaginary unit) (takes place for two different imaginary units) Jones matrixes and quaternions for several typical optical elements  - angle of declination,  - phases delay between eigen polarizations

10. 10 Difference between depolarization in single crystal and ceramics. List of parameters. - crystal constant p - normalized (unitless) heat power single crystal orientation , , 

11. 11 Difference between depolarization in single crystal and ceramics. Local depopolarization Г(r,  ). J. Lu, Appl. Phys. Lett., 78, 2000 S. D. Sims, Applied Optics, 6, 1967 Analytical plot 0 1

12. 12 Difference between depolarization in single crystal and ceramics. Integrated depopolarization . Integrated depopolarization, % ceramics N=  [111] single crystal normalized heat power р + N=30 o N=100  N=300

13. 13 Birefringence compensation in active elements. Typical schemes. 90 0 active element active element 1a1a W.Scott, M. De Wit Appl. Phys. Lett. 18, 3, 1973 V.Gelikonov et al. JETF lett., 13, 775, 1987 Faraday mirror active element ll 45 0 uniaxial crystal active element 1b1b 1c1c ll active element /4 2a2a /2 active element active element 2b2b W.A. Clarkson. et al. Opt. Lett., 24, 820, 1999 E.Khazanov et al. JOSA B, 19, 667, 2002

14. 14 at pN -1 <<1  а (r,  )=2  c (r,  )=  b (r,  )  0.07p 2 N -1 (solid lines) Compensation of thermally induced birefringence in ceramics. Schemes 1a-c. 90 0 active element active element 1a1a  а (r,  ) Faraday mirror active element ll 45 0 1b1b  b (r,  ) uniaxial crystal active element 1c1c  c (r,  ) if  l, then  b (r,  )  0 N=30 N=100 N=300 no compensation Integrated depolarization Integrated depolarization  Normalized heat power p Small scale modulation at pN -1 <<1  а (r,  )= 2  c (r,  )=  b (r,  ) (  l  ) Single crystal  а,b,c (r,  )  0 Local depolarization ceramics 0 1 p=5

15. 15 small scale modulation weak dependence of  and  on N Compensation of thermally induced birefringence in ceramics. Schemes 2а-b. /2 active element active element 2b2b ll active element /4 2a2a (  l  ) Integrated depolarization Integrated depolarization  normalized heat power p Local depolarization ceramics 1 0 single crystal 1 0 p=5

16. 16 Conclusion. Main results.  Analytical expressions for mean depolarization and without compensation and with compensation by means of all known techniques  Output polarization depends on a dimensionless heat release power р, and parameter N, ratio of the rod length to mean grain length  Depolarization and for ceramics rod are close to Г and  for a single crystal [111], BUT:  Both polarized and depolarized radiation always have small-scale modulation with a characteristic size of about.  Birefringence compensation by means of all known techniques is worse for ceramics than for a single crystal. Additional depolarization is proportional to the quantity p 2 N -1.  An increase in N is expedient from the viewpoint of both diminution of depth of the modulation and birefringence compensation.

17. 17 Aknowlegements Special thanks to prof. J.Collinsprof. N.Samarth prof. J.Collins and prof. N.Samarth of Pennsylvania State University. The work of M.Kagan was supported by the Dunkan Fellowship of Physics Department of PennState University.