**Table of Contents:**

**INTRODUCTION**

If light never interacted with matter, then nonlinear optics would not exist (as well as, of course, fiber optics, lasers, etc). This is because Maxwell’s equation in vacuum are completely linear (although it is theoretically predicted that above the Schwinger limit, the quantum vacuum electromagnetic field would also be nonlinear, but that is beyond the scope of this page and my pay-grade). They can acquire nonlinear terms, however, when they interact with matter with sufficient intensity to produce a nonlinear reaction of the material. The nonlinear reaction of the material then couples back to the light field and voila! You have nonlinear optics! Typically in most books and courses in optics, you will find that we treat the nonlinearities arising from the electric dipole moment response, or the polarization. However, there can be some nonlinear effects arising from the magnetic dipole moment response as well.

Starting with the electric flux density:

\begin{equation} D = \epsilon_0E + P \end{equation}

We know that typically in a linear medium, the polarization is proportional to the applied field, $P = \epsilon_0\chi_eE$, where $\chi$ could be a rank tensor for a medium if it is anisotropic. To make the formulation easier, however, we will assume that the $\chi$ tensors are scalars for the moment and write the polarization in terms of the nonlinear susceptibility:

\begin{equation} P = \epsilon_0[\chi^{(1)}E+\chi^{(2)}E^2+\chi^{(3)}E^3+…] \end{equation}

where the nonlinear polarization is represented by $E^2,~E^3$, etc. These nonlinear terms can arise due to motions of bound electrons, field-induced motions of atoms in their constituent materials, acoustic wave generation, thermal effects, etc. Since the nonlinearities associated with bound electrons are generally very fast, on the order of 1-2 fs, the motion of bound electrons as a result of an applied field are of particular importance in ultrafast optics. Full calculation of the susceptibility in the presence of a high intensity field would require quantum mechanics, but we can get the conceptual understanding using the simple example of an anharmonic oscillator. We discuss the harmonic oscillator in both the “Lasers” and the “Classical Optics” portions of this web page, so look there for refreshment if you don’t remember this model, or even if it’s completely new to you!

**FORCED WAVE EQUATION**

As a standard treatment in nonlinear optics, we can consider the nonlinear optical polarization as a source term in the wave equation. Assuming a uniform medium with the polarization as the only source, we can write the wave equation as:

\begin{equation} \nabla \times \nabla \times E = \nabla(\nabla\cdot{E})-\nabla^2E = -\mu_0\frac{\partial^2D}{\partial{t^2}} \end{equation}

The \textbf{D} field can then be written:

\begin{equation} D = \epsilon_0E+P_{(1)}+P_{NL} = D_{(1)}+P_{NL} \end{equation}

where $P_{NL}$ is the nonlinear polarization, $P_{(1)}$ is the first order polarization, and $D_{(1)}$ is the first-order portion of the $D$ field. Gauss’s Law then yields:

\begin{equation} \epsilon_{(1)}\nabla\cdot{E} = -\nabla\cdot{P_{NL}} \end{equation}

We can then make the simplifying assumption that the polarization is weak, so we have:

\begin{equation} \nabla\cdot{E} \approx 0 \end{equation}

The wave equation now simplifies to give:

\begin{equation} \nabla^2E – \mu_0\epsilon_{(1)}\frac{\partial^2E}{\partial{t^2}} = \mu_0\frac{\partial^2P_{NL}}{\partial{t^2}} \end{equation}