1. Linear optical properties of dielectrics
Introduction to crystal optics
Introduction to nonlinear optics
Relationship between nonlinear optics and electro-optics
Bernard Kippelen

2. Maxwell's Equations and the Constitutive Equations
Light beams are represented by electromagnetic waves propagating in space. An electromagnetic wave is described by two vector fields: the electric field E(r, t) and the magnetic field H(r, t).
In free space (i.e. in vacuum or air) they satisfy a set of coupled partial differential equations known as Maxwell's equations.
0 and 0 are called the free space electric permittivity and the free space magnetic permeability, respectively, and satisfy the condition
c2 = (1 / (0 0)), where c is the speed of light.
MKS

3. In a dielectric medium: two more field vectors
D(r,t) the electric displacement field, and B(r,t) the magnetic induction field
Constitutive equations:
 is the electric permittivity (also called dielectric function) and P = P (r, t) is the polarization vector of the medium.
MKS
 and j are the electric charge density (density of free conduction carriers) and the current density vector in the medium, respectively. For a transparent dielectric  and j =0.

4. Linear, Nondispersive, Homogeneous, and Isotropic Dielectric Media
Linear: the vector field P(r, t) is linearly related to the vector field E(r, t).
Nondispersive: its response is instantaneous, meaning that the polarization at time t depends only on the electric field at that same time t and not by prior values of E
Homogeneous: the response of the material to an electric field is
independent of r.
Isotropic: if the relation between E and P is
independent of the direction of the field vector E.
 is called the optical susceptibility

5. Wave equation
From Maxwell’s equations and by using the identity:

6. Solutions of Wave Equation Dispersion relationship
Real number
Complex number
Complex conjugate
Period T = 2/
Dispersion relationship
Time
Wavelength  = 2/k
Space

7. Which can be rewritten in the frequency domain
In real materials: polarization induced by an electric field is not instantaneous
Which can be rewritten in the frequency domain
the susceptibility is a complex number: has a real and imaginary part (absorption)
the optical properties are frequency dependent

8. Lorentz oscillator model
Electron
Nucleus
Displacement around equilibrium position due to Coulomb force exerted by electric field

9. Optics of Anisotropic Media
Optical properties (refractive index depend on the orientation of electric field vector E with respect to optical axis of material X Y Z
Need to define tensors to describe relationships between field vectors

10. Uniaxial crystals – Index ellipsoid
nX = nY ordinary index
nZ = extraordinary index
Refractive index for arbitrary direction of propagation can be derived from the index ellipsoid

11. Introduction to Nonlinear Optics
Linear term
Nonlinear corrections
Example of second-order effect: second harmonic generation (Franken 1961):
Symmetry restriction for second-order processes

12. Several electric fields are present
Nonlinear polarization
Tensorial relationship between field and polarization

13. Second-order nonlinear susceptibility tensor
Contracted notation for last two indices: xx = 1; yy = 2, zz = 3; zy or yz = 4; zx or xz = 5; xy or yx = 6
18 independent tensor elements but can be reduced by invoking group theory
Example: tensor for poled electro-optic polymers

14. Introduction to Electro-optics
John Kerr and Friedrich Pockels discovered in 1875 and 1893, respectively, that the refractive index of a material could be changed by applying a dc or low frequency electric field
In this formalism, the effect of the applied electric field was to deform the index ellipsoid
Corrections to the coefficients
Index ellipsoid equation

15. Example of tensor for electro-optic polymers
Electro-optic tensor
Relationship with second-order susceptibility tensor:
Example of tensor for electro-optic polymers
Simplification of the tensor due to group theory

16. Application of Electro-Optic Properties
Light
Applied voltage changes refractive index

17. Electro-Optic Properties of Organics
If the molecules are randomly oriented inversion symmetry
nonlinear susceptibilities 
hyperpolarizabilities

18. Convert an intensity distribution into a refractive index distribution
The Photorefractive Effect
Convert an intensity distribution into a refractive index distribution