# Physical Concepts

We can write down the wave equation for the laser traveling through the cavity from Maxwell's equations:

We look for a solution of the form .which describes the propagation of the electric field as a sinusoidal function. Here A represents the complex amplitude of the wave and is given by .

Due to the presence of the saturable Kerr material, A is not constant and will change as it passes through the material. Our strategy is to use the method of multiple scales, making A vary with time, but at a much slower rate than the carrier wave. We get leading terms which are explicitly in terms of and k describing the carrier wave. The second order term describes how A changes with time and is given by:

Where and . The complex factor B is the envelope of the polarization, which is assumed to be small. As the laser propagates through the material, the electric field will shift the electrons in the material, which act like dipoles. By assuming B to be small, we assume that the electrons in the material are more or less fixed.We can now make a change of variables and turn this equation into an ODE.

Now, B depends on A. If we take:

then it follows from eqn.(4) that:

So there is a small phase shift of the light as it goes through the material. If we take , we can write an expression for the amplitude coming out of the material in terms of it going in:

At the mirrors which are 100 percent reflective, we have that . But at the other mirror, we have a decrease in amplitude given by where R is the reflectivity of the mirror and T + R = 1. We can use this to write an expression for the electric field just after hitting the mirror not reflecting 100 percent:

This is our model which will describe how E changes as the laser makes the trip around the cavity. It is very sensitive to the initial conditions and solutions tend to vary a great deal when a small change is made. This is the first step in seeing the chaotic dynamics of the system. Our desire is to couple two of these systems together when they are in a chaotic regime. The goal is to be able to investigate the coupling so as to find ways of controlling the chaos.