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

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

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.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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