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## Decomposition of Complex Field

It is important to keep in mind that e is a complex number. However, the analytic tools which we will use to examine the model require that we use only real numbers in our equations. To work around this problem, we will break the equation up into two real equations, one dealing with the real part of e, and one dealing with the imaginary part.

Therefore, Eqn. (1) becomes

These two equations will model the electromagnetic field of the laser. However, due to the complicated nature of these equations, we will use equation (1) of the SRE model when analyzing the field equation by hand. Splitting the equation into its real and imaginary component parts is very useful when applying computational tools, but is not always useful for analysis done by hand.

There is another way of decomposing Eqn. (1) which proves useful in analyzing the system. Since is an complex number, we can break it up into its amplitude and phase. By doing this, we can then easily transform the equations into differential equations describing the behavior of the electric field's amplitude - the modulus of in Eqn. (1) - and phase:

It is important to note that Eqn. (1), Eqn. (7-8), and Eqn. (9-10) are equivalent.