11/29/2010. Propagation in Anisotropic Media 3. Introduction. Introduction. Gabriel Popescu

Transcription

1 Propagation in Anisotropic Media Gabriel Popescu University of Illinois at Urbana Champaign Beckman Institute Quantitative Light Imaging Laboratory Principles of Optical Imaging Electrical and Computer Engineering, UIUC Introduction Isotropic media easy, same index & velocity in all directions Anisotropic media 1) always 2 transverse modes 2) in general, different velocities 3) ke 0 in general What is different? isotropic anisotropic Both are charge on springs 2 orthogonal modes identical springs unequal springs Propagation in Anisotropic Media 2 Introduction x o( 11 x 12 y 13 ) y o( 21 x 22 y 23 ) o( 31 x 32 y 33 ) P E E E P E E E P E E E displacement _ E x() t ee P E Always possible to diagonalie a 3x3 matrix Means a set of coordinates that is rotated with respect to crystal axis co ordinate system Propagation in Anisotropic Media 3 1

2 Introduction in that reference frame: Px o11ex Py o 22Ey P o 33E ' Values different from ij! Axis set is called principal dielectric axes Can also discuss this in terms of dielectric (versus) susceptibility tensor (no new physics!) Dx 11Ex 12Ey 13E can again diagonalie Dy 21Ex 22Ey 23E end up with same principal axes! D E E E 31 x 32 y 33 Propagation in Anisotropic Media 4 Introduction of course have DoEPij o(1 ij) if ij are real ij ji Hermitian + symmetry if ij are complex * Hermitian ij ji Propagation in Anisotropic Media 5 j( tkr) assume plane wave Ert (,) Ee j( tkr) H(,) r t He Taking F.T. in both time and space: B E t k E H D H J kh E 0 t 2 eliminating H k ( k E) E 0 x 0 0 with 0 y Propagation in Anisotropic Media 6 2

3 xˆ yˆ ˆ kye key ke kx ky k kex kxe Ex Ey E kxey kye x xˆ yˆ ˆ k ke k k k x y ke ke ke ke ke ke Look at ˆx component, for example y y x x x y y x xˆ E y x y ky Ex k Ex E x x k ke E k k E ˆ x x x x y x E E 0 x y y x Propagation in Anisotropic Media 7 Can write the whole thing in matrix form 2 x ky k x y x Ex 2 y x y kx k y E y 2 x y E kx k y 0 For non trivial solution 0 (trivial solution is E 0 ) 2 x ky k x y x i.e. 2 k k 0 y x y x y 2 x y x y k k Propagation in Anisotropic Media 8 3 dimensional surface in k define x n1 ( nx ) y n2 ( ny ) n3 ( n ) e.g. Optic axis in k x k y plane (see next slide) Propagation in Anisotropic Media 9 3

4 k surface same speed Optic axis Propagation in Anisotropic Media 10 normal surface Crossing points define optic axis (axes) in general, can have 2 optic axes Propagation in Anisotropic Media 11 Propagation in Anisotropic Media 12 4

5 For any given direction k, there are two orthogonal fields, different phase velocities (for the D vector) Fields given by k x k x Ex k y E y k y E k k along optic axis, k is degenerate 1 phase velocity Propagation in Anisotropic Media 13 Classification of Materials Propagation in Anisotropic Media 14 Classification of Materials Propagation in Anisotropic Media 15 5

6 Classification of Materials Propagation in Anisotropic Media 16 Classification of Materials 2 refractive indices no, ne uniaxial n positive uniaxial e no n negative uniaxial o ne nx, ny, n biaxial convention n ny nx (may be totally unrelated to crystal axes!) Propagation in Anisotropic Media 17 Uniaxial Crystals nx ny no n ne 2 o 2 y x y x c 2 y x no k 2 x k y c 2 x y ne k 2 x ky n k k k k k k 0 c Propagation in Anisotropic Media 18 6

7 Uniaxial Crystals some mathematics 2 k 2 x ky k kx ky k ne no c no c 2 0 ellipsoid of revolution along axis kx, ky 0 k k no c no c 0 Surfaces touch along axis modes are degenerate sphere Propagation in Anisotropic Media 19 Double Refraction at Boundaries Always have to get light into crystal to have an interaction for tangential fields to be continuous (e.g. E ) k Preserved However k not the same for all polariations 2 different waves excited Propagation in Anisotropic Media 20 Double Refraction at Boundaries waves are not collinear! waves are not necessarily orthogonal!! ( k1 k2) kosino k1sin1 k2sin2 Snell s Law because of isotropy in x y plane, 1 wave is always ordinary wave k sin n sin o o o 1 Propagation in Anisotropic Media 21 7

8 Propagation in Anisotropic Media 22 Biaxial Crystals n ny nx optic axis is propagation direction for which there are two degenerate modes k n Ѳ n y n(ѳ) varies from n to n x n y lies between these limits optic axis lie in x plane where sin cos 1 2 n n n x y k x Propagation in Anisotropic Media 23 Biaxial Crystals solve for Ѳ n ny n x tan n x n n y VNC do not touch surfaces intersect! 1/2 beam with divergence Propagation in Anisotropic Media 24 8

9 Biaxial Crystals In other planes, surfaces do not intersect k k y Propagation in Anisotropic Media 25 Biaxial Crystals again have double refraction at an interface also new phenomenon Conical Refraction back to optic axis in x plane at intersection singular point, energy can go anywhere along a cone of angles called conical diffraction no unique surface normal (group velocity property) Propagation in Anisotropic Media 26 Aplications of Birefringent Crystals Propagation in Anisotropic Media 27 9

10 Propagation in Anisotropic Media 28 Optical Activity can undo rotation by reflecting off a mirror of going back through crystal (different from Faraday rotation!) Propagation in Anisotropic Media 29 Optical Activity Ρ specific rotary power (angular rotation per unit distance) dextrarotary right handed (counter clockwise as seen by observer) levarotary left handed (clockwise as seen by observer) Propagation in Anisotropic Media 30 10

11 Faraday Rotation also rotates plane of polariation but going back through medium does not reverse the rotation occurs in all materials, need strong magnetic fields what is going on?? look at response of electros in matter Electron on springs with k 1 Equal force constants k 2 k 3 B (parallel to -axis) Also optical field present E Propagation in Anisotropic Media 31 11