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.

This leads to

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.