As discussed in the last section, since $\chi^{(3)}$ can occur in materials of any symmetry, there a lot of materials that support third order nonlinearity that are isotropic, and therefore cannot support birefringent phase-matching. Anomalous dispersion is then used to provide phase-matching, but there is also then in this case an added phase shift known as the “Gouy Phase Shift” (discussed in the “Classical Optics section) that affects the phase matching. In this section, we review some of the concepts talked about in the “Classical Optics” section for beams of finite width and then analyze how we can use our understanding of these beams to properly phase match.

**Table of Contents:**

**PARAXIAL WAVE EQUATIONS**

As is typically done in optics, we started with the assumption of interactions of monochromatic plane waves to derive the essential aspects of nonlinear polarization. However, as you may recall, plane waves are our form of living in denial, because they do not exist in real life. As such, since the nonlinearity is dependent on the intensity of the interacting light, one way we can increase the intensity is to take advantage of the fact that the light is in fact a beam and focus it. However, this creates a Gouy phase shift, which then affects the overall phase matching in the nonlinear crystal. To analyze this further, we can remind ourselves of the paraxial wave equation and the Gaussian beam solution.

Starting with the wave equation for Fourier amplitudes:

\begin{equation} \nabla^2\bar{E}^{\omega_0}+\frac{\omega_n^2}{c^2}\tensor{\epsilon}\cdot\bar{E^{\omega_0}} = -\mu_0\omega_0^2\bar{P}^{\omega_0 NL} \end{equation}

Then the paraxial wave equation is given as:

\begin{equation} 2ik_n\frac{\partial A_n}{\partial{z}} + \nabla_T^2A)n = -\mu_0\omega_n^2P^{\omega_0 NL}e^{-ik_n z} \end{equation}

**INCLUSION OF NONLINEAR TERM**

**OPTIMUM FOCUSING**

Now that we know that focusing can aid the nonlinear interaction, we may be inclined to use it willy-nilly. However, there is an optimum focal spot size that exists because too tight of focusing can lead to faster divergence which would destroy the enhanced interaction beyond a certain length. Since the confocal parameter gives us a measure of the length over which a beam can be considered collimated i.e. not divergent, we can reasonably assume that the optimum crystal length would be no longer than the confocal parameter. We then can find the optimum focusing by neglecting the effects of diffraction.

**APPLICATIONS TO THG IMAGING**

If sending a focused beam across an interface between two different $\chi^{(3)}$ materials, the Gouy phase shift does not completely cancel the interacting beams and thus THG can be used across the interface for imaging.