Efim A. Khazanov

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

2. 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 Outline.

3. Introduction. Structure of polycrystalline ceramics.

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

5. y, b c, z y y y z y a b F e1 e2 Y x r x x j a b F x x, a x x y z z, c k z  r Depolarization in single crystal and ceramics. Thermo-induced birefringence. crystal/grain orientation Euler angles (a,b,F) angle of declination of eigen polarizations Y=Y(r,j, a,b,F) phases delay between eigen polarizationsd=d(r,j, a,b,F, Lg) Grain Jones matrix Ag=Ag(r,j, ag, bg, Fg,Lg)

6. Eout(r,j) Ein(r,j) ….. 1 2 3 4 ….. N Depolarization in single crystal and ceramics.Local and integrated. Jones matrix of whole element (k realization) A(r,j,k)=A1(r,j,a1,b1,F1,L1) A2(r,j,a2,b2,F2,L2)... AN(r,j,aN,bN,FN,LN) Local depolarization G(r,j ,k)=ч Eout(r,j ,k) /Ein(r,j)ч2 Average (over realizations) local depolarization <G(r,j)> Integrated depolarization: g(k) and its deviation : <g> andsg

7. Mathematical statement of the problem. Assumptions. • The number of grains, N within a beam’s path is fixed. • The orientation of crystallographic axes in a certain grain does not depend on vicinal grains. • The distribution function f(Lg,a,b,F)for a single grain is uniform with respect to the angular part and the gaussian with respect to Lg

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

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

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

12. Difference between depolarization in single crystal and ceramics. Integrated depopolarization g. Integrated depopolarization, % ceramics N=µ [111] single crystal +N=30 oN=100 р N=300 normalized heat power р

13. 900 active element active element active element Faraday mirror 2b 1b 2a 1a 1c V.Gelikonov et al. JETF lett., 13, 775, 1987 l 450 uniaxial crystal active element E.Khazanov et al. JOSA B, 19, 667, 2002 active element /2 active element active element l/4 l W.A. Clarkson. et al. Opt. Lett., 24, 820, 1999 Birefringence compensation in active elements. Typical schemes. W.Scott, M. De Wit Appl. Phys. Lett. 18, 3, 1973

14. Integrated depolarization Local depolarization 900 active element active element 1c 1a 1b а(r,) no compensation а,b,c(r,)є 0 l active element Faraday mirror N=30 Single crystal Integrated depolarization g if l<<30…100mm=<Lg>, then b(r,)є 0 uniaxial crystal active element 1 b(r,) 450 N=100 • Small scale modulation • at pN-1 <<1 • а(r,)= 2c(r,)= b(r,) (l <Lg>) N=300 c(r,) 0 ceramics Normalized heat power p p=5 Compensation of thermally induced birefringence in ceramics. Schemes1a-c. at pN-1 <<1  а(r,)=2c(r,)=b(r,) » 0.07p2 N-1 (solidlines)

15. Integrated depolarization Local depolarization 1 2b 2a (l <Lg>) 0 single crystal Integrated depolarization g 1 0 /2 active element active element active element ceramics l/4 normalized heat power p p=5 l Compensation of thermally induced birefringence in ceramics. Schemes 2а-b. • small scale modulation • weak dependence of andgonN

16. Conclusion. Main results. • Analytical expressions for mean depolarization < Г(r,j) > and <g> 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 <Lg> • Depolarization < Г(r,j) > and <g> for ceramics rod are closeto Г and gfor a single crystal [111],BUT: • Both polarized and depolarized radiation always have small-scale modulation with a characteristic size of about < Lg >. • Birefringence compensation by means of all known techniques is worse for ceramics than for a single crystal. Additional depolarization is proportional to the quantityp2N-1. • An increase in N is expedient from the viewpoint of both diminution of depth of the modulation and birefringence compensation.

17. Aknowlegements Special thanks to 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.