# Light-Matter Nonlinear Interactions

When introducing nonlinear optics, it is useful to first define what the “nonlinear” part of nonlinear optics means. Given a sufficiently small input light intensity, the response of a material is linearly dependent on the input intensity. However, when the intensity of an input light field is strong enough. the response of the material becomes nonlinearly dependent on the intensity of the input field. Boyd gives the example of second harmonic generation: in second harmonic generation, the intensity of the second harmonic light generated increases as the square of the input light intensity, which is obviously no longer a linear relation.

NONLINEAR POLARIZATION

To start, let’s look at an equation we know:

$$P(t) = \epsilon_0\chi^{(1)}E(t)$$

where “$\epsilon_0$” is the permittivity of free space and $\chi^{(1)}$ is the linear susceptibility. In this case, the induced polarization is obviously linearly dependent on E, which makes this a linear material response. By contrast, in nonlinear optics, we typically rewrite this equation as:

$$P(t) = \epsilon_0(\chi^{(1)}E(t)+\chi^{(2)}E^2(t)+\chi^{(3)}E^3(t)+…)$$

where $\chi^{(2)}$ is the second order nonlinear susceptibility responsible for phenomena such as second harmonic generation, and $\chi^{(3)}$ is the third order nonlinear susceptibility, responsible for phenomena such as third harmonic generation and the nonlinear index of refraction. It is important to note that second order nonlinear phenomena can only occur in noncentrosymmetric materials, while third harmonic generation can occur in any material.

SOURCES OF NONLINEARITY

The following analysis will come primarily from [1], although with some additional supplementary sources that will be cited along the way.

SHORT INTRODUCTION TO NONLINEAR OPTICAL PROCESSES

As a partial outline of the rest of the section, I have included here some nonlinear optical processes along with the who, what, when, why, and how they occur. This should be a good rough outline as to when these processes could be occurring in your experiments!

References