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## Overview of the Equations

Now, we will examine the differential equation describing the change in the electromagnetic field (Eqn. 1). The first thing to notice about this equation is that it is of the form

This means that the rate of change of the electromagnetic field is directly proportional to the electromagnetic field already present. If there is no electromagnetic field to begin with, (e = 0), then no field will ever be present in the laser. Therefore, in order for the laser described by our model to operate, an electromagnetic field must be present in the laser when it is turned on.

Also notice that, when looking at the real and imaginary parts of equations (7-8), of the three terms in the equations, one of them ( ) is always a "loss" term, that is, the electromagnetic field is decreased by this term. The second term of the equation, , is a "gain" term (increasing the field) only when n is greater than 1. Therefore, in order for the electromagnetic field to increase, e must not equal zero, n must be greater than 1, and the "gain" term must outweigh the "loss" term. The third term of the equation, , is neither a gain nor a loss term, as is used in the equation, and is used in the equation. Rather, these last terms give a dynamic coupling between the real and imaginary parts of the field.

Now, we will examine the second equation of the SRE system, the n differential equation (2):

Again, let us examine the "gain" versus the "loss" terms of this equation. The first term, J, increases n, and is called the "pump" term. It is also clear that is a loss term. The final term, however, requires a more careful examination. is negative (a "loss") when n is greater than 1, but is positive (a "gain") when n is less than one. Notice that this is the reversed situation of the term in the e equation. When the e term is a loss, the n term is a gain, and vice versa. This means that gain in one equation must be balanced by loss in the other equation.

Finally, remember that n will only increase when the positive ("gain") terms are larger than the negative ("loss") terms.

Now that we have given a cursory explanation of the terms in our model, we can turn to an examination of the behavior of solutions to these equations.

Next: Behavior of Trajectories Up: The Semiconductor Laser Rate Previous: Decomposition of Complex Field