Non Linear Optics

According to the classical description, an oscillating electric charge produces an electromagnetic wave. Equivalently, an oscillating electromagnetic wave can displace or polarize an electric charge into oscillation. Considering the second situation, the induced polarization contains more than a linear response to the applied field. Given a nonlinear susceptibility c , the charge separation or polarization P can be expanded as a power series of the applied electric field

(5.1)

Now consider what happens when two collinear electromagnetic waves with different frequencies induce a polarization inside a medium. The applied electric field inside the medium is

Ē = E_j(1) cos(w₁t) + E_j(2) cos(w₂t),

and the second order (quadratic term) of the polarization is:

Expanding the squared term yields

The second order induced polarization has one term at the sum of the two frequencies and other additional terms. Contributions of the other terms include doubling of each frequency, difference of frequencies, and a DC polarization. In principle, the induced polarization can emit light at any of these frequencies. A condition called phase matching determines if a macroscopic beam is emitted at a specific frequency.