Light and Matter Tim Freegarde School of Physics & Astronomy University of Southampton The tensor nature of susceptibility

2 Birefringence asymmetry in crystal structure causes polarization dependent refractive index ray splits into orthogonally polarized components, which follow different paths through crystal note that polarization axes are not related to plane of incidence

3 Anisotropic media difference in refractive index (birefringence) or absorption coefficient (dichroism) depending upon polarization recall that where susceptibility cannot be a simple scalar difference in or implies difference in

4 Linear dichroism fields normal to the conducting wires are transmitted current flow is not parallel to field WIRE GRID POLARIZER susceptibility not a simple scalar fields parallel to the conducting wires are attenuated

5 Birefringence – mechanical model springs attach electron cloud to fixed ion different spring constants for x, y, z axes polarization easiest along axis of weakest spring polarization therefore not parallel to field susceptibility not a simple scalar

6 Susceptibility tensor the Jones matrix can map similarly, a tensor can describe the susceptibility of anisotropic media JONES MATRIX SUSCEPTIBILITY TENSOR e.g.

7 Diagonizing the susceptibility tensor if the polarization axes are aligned with the principal axes of birefringent crystals, rays propagate as single beams the susceptibility tensor is then diagonal DIAGONAL SUSCEPTIBILITY TENSOR a matrix may be diagonalized if symmetrical: the optical activity tensor is not symmetrical; it cannot be diagonalized to reveal principal axes

8 The Fresnel ellipsoid surface mapped out by electric field vector for a given energy density FRESNEL ELLIPSOID symmetry axes x’,y’,z’ are principal axes semi-axes are

9 The Fresnel ellipsoid allows fast and slow axes to be determined: FRESNEL ELLIPSOID establish ray direction through Poynting vector electric field must lie in normal plane fast and slow axes are axes of elliptical cross-section axis lengths are allows fast and slow axes to be determined:

10 The optic axis if the cross-section is circular, the refractive index is independent of polarization FRESNEL ELLIPSOID the Poynting vector then defines an optic axis if, the single optic axis lies along (uniaxial crystals) if, there are two, inclined, optic axes (biaxial crystals)

11 Uniaxial crystals FRESNEL ELLIPSOID the single optic axis lies along optic axis one polarization is inevitably perpendicular to the optic axis (ordinary polarization) the second polarization will be orthogonal to both the ordinary polarization and the Poynting vector (extraordinary polarization) positive uniaxial: negative uniaxial:

12 Poynting vector walk-off Poynting vector obeys wavevector vector obeys in anisotropic media, and are not necessarily parallel the Poynting vector and wavevector may therefore diverge wavevector Poynting vector Fresnel ellipsoid

13 How light interacts with matter atoms are polarized by applied fields Lorentz model: harmonically bound classical particles

14 Quantum description of atomic polarization energy 0 harmonic oscillator two-level atom electron density depends upon relative phase of superposition components weak electric field

15 Quantum description of atomic polarization x/a 0 electron density depends upon relative phase of superposition components energy 0 harmonic oscillator two-level atom weak electric field

16 Optical nonlinearity potential is anharmonic for large displacements polarization consequently varies nonlinearly with field in quantum description,uneven level spacing distortion of eigenfunctions higher terms in perturbation

17 Optical nonlinearity potential is anharmonic for large displacements polarization consequently varies nonlinearly with field in quantum description,uneven level spacing distortion of eigenfunctions higher terms in perturbation

18 Electro-optic effect nonlinearity mixes static and oscillatory fields susceptibility at hence controlled by exploit the nonlinear susceptibility Pockels effect; Kerr effect

19 Second harmonic generation again exploit the nonlinear susceptibility distortion introduces overtones (harmonics) where

20 Nonlinear tensor susceptibilities nonlinear contributions to the polarization depend upon products of electric field components each product corresponds to a different susceptibility coefficient e.g. terms in the susceptibility expansion are therefore tensors of increasing rank the induced polarization has three components (i =x,y,z):

21 Nonlinear tensor susceptibilities the susceptibility depends upon the frequencies of the field and polarization components e.g. if, any susceptibility unless

22 Symmetry in susceptibility the susceptibility tensor may be invariant under certain symmetry operations e.g. rotation reflection the symmetries of the susceptibility must include – but are not limited to – those of the crystal point group optically active materials fall outside the point group description (nonlocality) materials showing inversion symmetry have identically zero terms of even rank inversion

23 Properties of susceptibility depends upon frequency dispersion and absorption in material response depends upon field orientation anisotropy in crystal and molecular structure depends upon field strength anharmonicity of binding potential hence nonlinearity tensor nature of susceptibility series expansion of susceptibility

24 Pockels (linear electro-optic) effect nonlinearity mixes static and oscillatory fields applying intrinsic permutation symmetry, in non-centrosymmetric materials, dominates

25 Kerr (quadratic electro-optic) effect nonlinearity mixes static and oscillatory fields applying intrinsic permutation symmetry, in centrosymmetric materials,

26 Pockels cell polarizer modulation voltage voltage applied to crystal controls birefringence and hence retardance mounted between crossed linear polarizers longitudinal and transverse geometries for modulation field allows fast intensity modulation and beam switching