Non-linear Optics: Second-Harmonic Generation and Parametric Oscillation

Abstract

The electric field of a beam of photons propagating through a medium produces polarization by spatially deforming the charge distribution in the outer shell of electrons. The relationship between the polarization and the electric field is the susceptibility. The polarization of the charge distribution is opposed by a restoring force of each atom. The first order susceptibility represents the linear response of the medium to the electric field. When the medium lacks a center of inversion symmetry, it follows that the relationship between the electric field and the polarization is no longer linear. The non-linear optical response of materials is a field of study that is rich with phenomena and applications. Second harmonic generation converts two photons having frequency $$ = \omega $$=$ω$into one photon having frequency $$ = 2\omega $$=2$ω$. Parametric amplification involves beams of two different frequencies at the input: a stronger pump beam at $$ \omega_{3} $$$ω$3and a weaker signal beam at $$ \omega_{1} $$$ω$1. A special case of parametric amplification occurs when the signal amplitude at $$ \omega_{1} $$$ω$1is zero. Analysis by Maxwell's equations shows that no amplification occurs at the output. However experiment shows that measureable spontaneous fluorescence occurs. This fluorescence is called spontaneous parametric down conversion. Its existence demonstrates the limitation of Maxwell's equations to the classical domain and beams of light composed of many photons. Spontaneous parametric down conversion is widely used to prepare single photons and pairs of entangled photons.